Shadows Of The Truth: Metamathematics Of Elementary Mathematics Alexandre V. Borovik

Transcript

Alexandre V. Borovik Shadows of the Truth: Metamathematics of Elementary Mathematics Working Draft 0.822 November 23, 2012 American Mathematical Society To Noah and Emily Fig. 0.1. L’Evangelista Matteo e l’Angelo. Guido Reni, 1630–1640. Pinacoteca Vaticana. Source: Wikipedia Commons. Public domain. Guido Reni was one of the first artists in history of visual arts who paid attention to psychology of children. Notice how the little angel counts on his fingers the points he is sent to communicate to St. Matthew. Preface Toutes les grandes personnes ont d’abord été des enfants (Mais peu d’entre elles s’en souviennent.) Antoine de Saint-Exupéry, Le Petit Prince. This book is an attempt to look at mathematics from a new and somewhat unusual point of view. I have started to systematically record and analyze from a mathematical point of view various difficulties experiencing by children in their early learning of mathematics. I hope that my approach will eventually allow me to gain a better understanding of how we—not only children, but adults, too—do mathematics. This explains the title of the book: metamathematics is mathematics applied to study of mathematics. I chase shadows: I am trying to identify and clearly describe hidden structures of elementary mathematics which may intrigue, puzzle, and—like shadows in the night—sometimes scare an inquisitive child. The real life material in my research is limited to stories that my fellow mathematicians have chosen to tell me; they represent tiny but personally significant episodes from their childhood. I directed my inquiries to mathematicians for an obvious reason: only mathematicians possess an adequate language which allows them to describe in some depths their experiences of learning mathematics. So far my approach is justified by the warm welcome it found among my mathematician friends, and I am most grateful to them for their support. For some reason (and the reason deserves a study on its own) my colleagues know what I am talking about! The book was born from a chance conversation with my colleague Elizabeth Kimber. I analyze her story, in great detail, in Chapter 5. Little Lizzie, aged 6, could easily solve “put a number in the box” problems of the type 7 +  = 12, v vi by counting how many 1’s she had to add to 7 in order to get 12 but struggled with  + 6 = 11, because she did not know where to start. Much worse, little Lizzie was frustrated by the attitude of adults around her—they could not comprehend her difficulty, which remained with her for the rest of her life. When I heard that story, I instantly realized that I had had similar experiences myself, and that I heard stories of challenge and frustration from many my fellow mathematicians. I started to ask around—and now offer to the reader a selection of responses arranged around several mathematical themes. A few caveats are due. The stories told in the book cannot be independently corroborated or authenticated—they are memories that my colleagues have chosen to remember. I believe that the stories are of serious interest for the deeper understanding of the internal and hidden mechanisms of mathematical practice because the memories told have deeply personal meaning for mathematicians who told the stories to me. The nature of this deep emotional bond between a mathematician and his or her first mathematical experiences remains a mystery—I simply take the existence of such a bond for granted and suggest that it be used as a key to the most intimate layer of mathematical thinking. This bond with the “former child” (or the “inner child”?) is best described by Michael Gromov: I have a few recollections, but they are not structural. I remember my feeling of excitement upon hitting on some mathematical ideas such as a straight line tangent to a curve and representing infinite velocity (I was about 5, watching freely moving thrown objects). Also at this age I was fascinated by the complexity of the inside of a car with the hood lifted. Later I had a similar feeling by imagining first infinite ordinals (I was about 9 trying to figure out if 1000 elephants are stronger than 100 whales and how to be stronger than all of them in the universe). Also I recall many instances of acute feeling of frustration at my stupidity of being unable to solve very simple problems at school later on. My personal evaluation of myself is that as a child till 8–9, I was intellectually better off than at 14. At 14–15 I became interested in math. It took me about 20 years to regain my 7 year old child perceptiveness. I repeat Michael Gromov’s words: It took me about 20 years to regain my 7 year old child perceptiveness. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK vii I am confident that this sentiment is shared by many my mathematician colleagues. This is why I concentrate on the childhood of mathematicians, and this is why I expect that my notes will be useful to specialists in mathematical education and in psychology of education. But I wish to make it absolutely clear: I am not making any recommendations on mathematics teaching. Moreover, I emphasize that the primary aim of my project is to understand the nature of mainstream “research” mathematics. The emphasis on children’s experiences makes my programme akin to linguistic and cognitive science. However, when a linguist studies formation of speech in a child, he studies language, not the structure of linguistics as a scientific discipline. When I propose to study the formation of mathematical concepts in a child, I wish to get insights into the interplay of mathematical structures in mathematics. Mathematics has an astonishing power of reflection, and a self-referential study of mathematics by mathematical means plays an increasingly important role within mathematical culture. I simply suggest to take a step further (or a step aside, or a step back in life) and to take a look back in time, at one’s childhood years. A philosophically inclined reader will immediately see a parallel with Plato’s Allegory of the Cave: children in my book see shadows of the Truth and sometimes find themselves in a psychological trap because their teachers and other adults around them see neither Truth, nor its shadows. But I am not doing philosophy; I am a mathematician and I stick to a concise mathematical reconstruction of what the child had actually seen. My book is also an attempt to trigger the chain of memories in my readers: even the most minute recollection of difficulties and paradoxes of their early mathematical experiences is most welcome. Please write to me at [email protected]. B IBLIOGRAPHY. At the end of each chapter I place some bibliographic references. Here are some (very different) books most closely related to themes touched on in this introduction: Aharoni [610], Carruthers and Worthington [642, 644], Freudenthal [667], Gromov [30], and Krutetskii [826]. Alexandre Borovik Didsbury 16 July 2011 S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK Acknowledgements Fig. 0.2. Guido Reni. A fragment of Purification of the Virgin, c. 1635– 1640. Musée du Louvre. Source: Wikipedia Commons. Public domain. I am grateful to my correspondents Ron Aharoni, JA, Natasha Alechina, Tuna Altınel, RA, Nicola Arcozzi, Pierre Arnoux, Autodidact, Bernhard Baumgartner, Frances Bell, SB, Mikhail Belolipetsky, AB, Alexander Bogomolny, RB, Anna Borovik (my wife, actually), Lawrence Braden, Michael Breen, TB, BB, Dmitri Burago, LB, CB, LC, David Cariolaro, SC, Emily ˇ Jonathan Crabtree, Iain Currie, RTC, Cliff, Alex Cook, BC, VC, ix x ˘ PD, Yagmur Denizhan, Antonio Jose Di Scala, SD, DD, Ted Eisenberg, Theresia Eisenkölbl, RE, SUE, ¸ David Epstein, Gwen Fisher, Ritchie Flick, Jo French; Michael N. Fried, Swiatoslaw G., IG, Herbert Gangl, Solomon Garfunkel, Dan Garry, Olivier Gerard, John Gibbon, Anthony David Gilbert, Jakub Gismatullin, VG, Alex Grad, IGG, Rostislav Grigorchuk, Michael Gromov, IH, Leo Harrington, EH, Robin Harte, Toby Howard, RH, Jens Høyrup, Alan Hutchinson, BH, David Jefferies, Mikhail Katz, Tanya Khovanova, Hovik Khudaverdyan, Elizabeth Kimber, EMK, Jonathan Kirby, SK, Ekaterina Komendantskaya, Ulrich Kortenkamp, Charles Leedham-Green, AL, EL, RL, DMK, JM, Victor Maltcev, MM, Archie McKerrell, Jonathan McLaughlin, Alexey Muranov, Azadeh Neman, Ali Nesin, John W. Neuberger, Joachim Neubüser, Anthony O’Farrell, Alexander Olshansky one man and a dog, Teresa Patten, Karen Petrie, NP, Eckhard Pflügel, Richard Porter, Hillary Povey MP, Alison Price, Mihai Putinar, VR, Roy Stewart Roberts, FR, PR, AS, John Shackell, Simon J. Shepherd, GCS, VS, Christopher Stephenson, Jerry Swan, Johan Swanljung, BS, Tim Swift, RT, Günter Törner, Vadim Tropashko, Viktor Verbovskiy, RW, PW, JW, RW, MW, Jürgen Wolfart, CW, Maria Zaturska, WZ and Logan Zoellner for sharing with me their childhood memories and/or their educational and pedagogical experiences; to parents of DW for allowing me to write about the boy; and to my colleagues and friends for contributing their expertise on history of arithmetic and history of infinitesimals, French and Turkish languages, artificial intelligence, turbulence, dimensional analysis, subtraction, cohomology, p-adic integers, programming, pedagogy — in effect, on everything — and for sharing with me their blog posts, papers, photographs, pictures, problems, proofs, translations: Santo D’Agostino, Paul Andrews, John Baez, John Baldwin, Oleg ˘ Belegradek, Marc Bezem, Adrien Deloro, Yagmur Denizhan , Muriel Fraser, Michael N. Fried, Alexander Givental, AH, Mitchell Harris, Albrecht Heeffer, Roger Howe, Jens Høyrup, Jodie Hunter Mikael Johansson, Jean-Michel Kantor, H. Turgay Kaptanoglu, Serguei Karakozov, Mikhail Katz, Alexander Kheyfits Hovik Khudaverdyan, Eren Mehmet Kıral, David H. Kirshner, Semen Samsonovich Kutateladze, Vishal Lama, Joseph Lauri, Michael Livshits, Dennis Lomas, Dan MacKinnon, John Mason, Gábor Megyesi, Javier Moreno, Ali Nesin, Sevan Ni¸sanyan, Windell H. Oskay, David Pierce, Donald A. Preece, Thomas Riepe, Jane-Lola Seban, Ashna Sen, Alexander Shen, Aaron Sloman, Kevin Souza, Chris Stephenson, Vadim Tropashko, Sergei Utyuzhnikov, Roy Wagner, Thomas Ward, David Wells, and Dean Wyles; S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK xi and to Tony Gardiner, Yordanka Gorcheva, Dan Garry, Olivier Gerard, Stephen Gould, Mikhail Katz, Michael Livshits, Alison Pease and Frederick Ross for sending me detailed comments on, and corrections to, the on-line version of the book and /or associated papers. This text would not appear had I not received a kind invitation to give a talk at “Is Mathematics Special” conference in Vienna in May 2008, and without an invitation from Ali Nesin to give a lecture course “Elementary mathematics from the point of view of “higher” mathematics” at the Nesin Mathematics Village in Sir¸ ince, Turkey, in July 2008 and in August 2009. Section 10.1 was first published in a [106] in the proceedings volume of the Vienna conference edited by Benedikt Löwe and Thomas Müller. Parts of the text first appeared in Matematik Dünyası, a popular mathematical journal edited by Ali Nesin [627]. My work on this book was partially supported by a grant from the John Templeton Foundation, a charitable institution which describes itself as a “philanthropic catalyst for discovery in areas engaging in life’s biggest questions.” However, the opinions expressed in the book are those of the author and do not necessarily reflect the views of the John Templeton Foundation. Finally, my thanks go to the blogging community—I have picked in the blogosphere some ideas and quite a number of references— especially to the late Dima Fon-Der-Flaass and to my old friend who prefers to be known only as Owl. Alexandre Borovik Didsbury 23 November 2012 S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix 1 Dividing Apples between People . . . . . . . . . . . . . . . . . . . . 1 1.1 Sharing and dispensing . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Digression into Turkish grammar . . . . . . . . . . . . . . . . 3 1.3 Dividing apples by apples: a correct answer . . . . . . . 5 1.4 What are the numbers children are working with? . 6 1.5 The lunch bag arithmetic, or addition of heterogeneous quantities . . . . . . . . . . . . . . . . . . . . . . . . 8 1.6 Duality and pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.7 Adding fruits, or the augmentation homomorphism 10 1.8 Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2 Pedagogical Intermission: Human Languages . . . . . . . 13 3 Units of measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Fantasy units of measurement . . . . . . . . . . . . . . . . . . . 3.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 19 21 23 4 History of Dimensional Analysis . . . . . . . . . . . . . . . . . . . . 4.1 Galileo Galilei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Froude’s Law of Steamship Comparisons . . . . . . . . . . 4.2.1 Difficulty of making physical models . . . . . . . . 4.2.2 Deduction of Froude’s Law . . . . . . . . . . . . . . . . . 4.3 Kolmogorov’s “5/3” Law . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Turbulent flows: basic setup . . . . . . . . . . . . . . . 4.3.2 Subtler analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Dimension of Lagrange multipliers . . . . . . . . . . . . . . . 4.5 Length and area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 28 30 30 31 32 32 34 35 36 38 xiii xiv Contents 45 45 47 48 Adding One by One . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Adding one by one . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Dedekind-Peano axioms . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 A brief digression: is 1 a number? . . . . . . . . . . . . . . . . . 5.4 How much mathematics can a child see at the level of basic counting? . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Properties of addition . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Associativity of addition . . . . . . . . . . . . . . . . . . . 5.5.2 Commutativity of addition . . . . . . . . . . . . . . . . . 5.6 Dark clouds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Induction and recursion . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Digression into infinite descent . . . . . . . . . . . . . . . . . . . 5.9 Landau’s proof of the existence of addition . . . . . . . . 49 52 53 54 55 57 59 61 6 What is a Minus Sign Anyway? . . . . . . . . . . . . . . . . . . . . . . 6.1 Fuzziness of the rules . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 A formal treatment of subtraction . . . . . . . . . . . . . . . . 6.3 A formal treatment of negative numbers . . . . . . . . . . 6.4 Testimonies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Multivalued groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 65 67 68 71 73 7 Counting Sheep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Numbers in computer science . . . . . . . . . . . . . . . . . . . . 7.2 Counting sheep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Abstract nonsense . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Existence and uniqueness . . . . . . . . . . . . . . . . . 7.3.2 Unary algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Induction on systems other than N . . . . . . . . . . . . . . . 7.5 Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Digression: Natural numbers in Ancient Greece . . . . . . . . . . . . . . 77 77 79 81 81 81 82 82 84 8 Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Fractions as “named” numbers . . . . . . . . . . . . . . . . . . . 8.2 Inductive limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Field of fractions of an integral domain . . . . . . . . . . . 8.4 Back to commutativity of multiplication . . . . . . . . . . . 87 87 89 92 93 9 Pedagogical Intermission: Didactic Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 9.1 Didactic transformation . . . . . . . . . . . . . . . . . . . . . . . . . 97 9.2 Continuity, limit, derivatives . . . . . . . . . . . . . . . . . . . . . 100 9.3 Continuity, limit, derivatives: the Zoo of alternative approaches . . . . . . . . . . . . . . . . . 101 9.4 Some practical issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5 S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 85 c A LEXANDRE V. B OROVIK Contents xv 10 Carrying: Cinderella of Arithmetic . . . . . . . . . . . . . . . . . . 109 10.1 Palindromic decimals and palindromic polynomials 109 10.2 DW: a discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 10.3 Decimals and polynomials: an epiphany . . . . . . . . . . . 114 10.4 Carrying: Cinderella of arithmetic . . . . . . . . . . . . . . . . 115 10.4.1 Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 10.4.2 A few formal definitions . . . . . . . . . . . . . . . . . . . 117 10.4.3 Limits and series . . . . . . . . . . . . . . . . . . . . . . . . . 118 10.4.4 Euler’s sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 10.5 Unary number system . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 11 Pedagogical Intermission: Nomination and Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 125 11.1 Names . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 11.2 Nomination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 12 The Towers of Hanoi and Binary Trees . . . . . . . . . . . . . . 133 13 Mathematics of Finger-Pointing . . . . . . . . . . . . . . . . . . . . 135 13.1 John Baez: a taste of lambda calculus . . . . . . . . . . . . . 135 13.2 Here it is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 13.3 A dialogue with Peter McBride . . . . . . . . . . . . . . . . . . . 139 14 Numbers and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 14.1 Chinese Remainder Theorem . . . . . . . . . . . . . . . . . . . . 141 14.1.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 14.1.2 Simultaneous Congruences . . . . . . . . . . . . . . . . 142 14.1.3 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 14.1.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 14.2 The Lagrange Interpolation Formula . . . . . . . . . . . . . 144 14.3 Numbers as functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 15 Graph Paper and the Arithmetic of Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 15.1 Graph paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 15.2 Pizza, logarithms and graph paper . . . . . . . . . . . . . . . 151 15.3 Multiplication of squares . . . . . . . . . . . . . . . . . . . . . . . . 153 15.4 Pythagorean triples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 16 Uniqueness of Factorization . . . . . . . . . . . . . . . . . . . . . . . . 159 16.1 Uniqueness of factorization . . . . . . . . . . . . . . . . . . . . . . 159 16.2 Dialog with AL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 16.3 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 16.4 The Fermat Theorem for polynomials . . . . . . . . . . . . . 163 17 Pedagogical Intermission: Factorization . . . . . . . . . . . . 165 S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK xvi Contents 18 Being in Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 18.1 Leo Harrington: Who is in control? . . . . . . . . . . . . . . . 167 18.2 The quest for truth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 18.3 The quest for logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 18.4 The quest for understanding . . . . . . . . . . . . . . . . . . . . . 172 18.5 The quest for power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 18.6 The quest for rigour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 18.7 Suspicion of easy options . . . . . . . . . . . . . . . . . . . . . . . . 182 18.8 “Everything had to be proven” . . . . . . . . . . . . . . . . . . . . 185 18.9 Raw emotions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 18.10 David Epstein: Give students problems that interest them . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 18.11 Autodidact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 18.12 Blocking it out . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 19 Controlling Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 19.1 Fear of infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 19.2 Counting on and on . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 19.3 Controlling infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 19.4 Edge of an abyss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 20 Pattern Hunting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 21 Visual Thinking vs Formal Logical Thinking . . . . . . . . 213 21.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 21.2 EH: Visualisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 21.3 Lego . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 22 Telling Left from Right . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 22.1 Why does the mirror change left and right but does not change up and down? . . . . . . . . . . . . . . . . . . . 221 22.2 Pons Asinorum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 22.3 TB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 22.4 Maria Zaturska . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 22.5 MP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 22.6 Digression into ethnography . . . . . . . . . . . . . . . . . . . . . 226 22.7 BB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 22.8 PD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 22.9 Digression into Estonian language . . . . . . . . . . . . . . . 230 22.10 Standing arches, hanging chains . . . . . . . . . . . . . . . . . 230 22.11 Orientation of surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 231 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 1 Dividing Apples between People It is important not to separate mathematics from life. You can explain fractions even to heavy drinkers. If you ask them, ‘Which is larger, 2/3 or 3/5?’ it is likely they will not know. But if you ask, ‘Which is better, two bottles of vodka for three people, or three bottles of vodka for five people?’ they will answer you immediately. They will say two for three, of course. Israel Gelfand 1.1 Sharing and dispensing I take the liberty to tell a story from my own life1 ; I believe it is relevant for the principal theme of this book. When, as a child, I was told by my teacher that I had to be careful with “named” numbers and not to add apples and people, I remember asking her why in that case we can divide apples by people: 10 apples : 5 people = 2 apples. (1.1) Even worse: when we distribute 10 apples giving 2 apples to a person, we have 10 apples : 2 apples = 5 people (1.2) Where do “people” on the right hand side of the equation come from? Why do “people” appear and not, say, “kids”? There were no “people” on the left hand side of the operation! How do numbers on the left hand side know the name of the number on the right hand side? 1 Call me AVB; I am Russian, male, have a PhD in Mathematics, teach mathematics in a British university. 1 2 1 Dividing Apples between People Fig. 1.1. The First Law of Arithmetic: you do not add fruit and people. Giuseppe Arcimboldo, Autumn. 1573. Musée du Louvre, Paris. Source: Wikipedia Commons. Public domain. There were much deeper reasons for my discomfort. I had no bad feelings about dividing 10 apples among 5 people, but I somehow felt that the problem of deciding how many people would get apples if each was given 2 apples from the total of 10, was completely different. I tried to visualize the problem as an orderly distribution of apples to a queue of people, two apples to each person. The result was deeply disturbing: in horror I saw an endless line of poor wretches, each stretching out his hand, begging for his two apples. (I discuss these my childhood fears in more detail in Section 19.1.) Indeed, my childhood experience is confirmed by experimental studies, see Bryant and Squire [264]. To emphasize the difference between the two operations, I started to call operation (1.1) sharing and (1.2) dispensing or distribution. I discovered later that these operation were called partition and quotition in [623]. But even S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 1.2 Digression into Turkish grammar 3 sharing is not easy and may lead to mathematical discoveries! If you do not believe, read a testimony from David Cariolaro:2 When I was 3 years old I was trying to divide evenly the L EGO pieces that I had at that time with my brother—and failed in that respect and burst in tears. When I told my Mum that I could not divide evenly the pieces she recognized that I indeed discovered odd numbers, and that was my first mathematical discovery. Finally, notice that there are similar special cases of subtraction; it is worth quoting from Romulo Lins [714]: I once had a very interesting conversation with Alan Bell, at the time when he was my Ph. D. supervisor. He argued that when a store-clerk gives you the right change by ‘adding up’ he is actually doing a subtraction. For instance, I have to pay $35 and give a $100 bill to the clerk. He gives me a $5 bill and says ‘forty’, gives me a $10 bill and says ‘fifty’, and finally gives me a $50 bill and says ‘a hundred’. I argued that this and doing a subtraction were quite different things, as, unless the clerk wants to pay attention to how much he returned, he will not know, in the end, the change given (try this out in shops without those modern machines!). And how can we call ‘subtraction’ an operation that in the end leaves us without knowing ‘the result of the subtraction’? Shouldn’t we better call that a ‘change giving’ operation? The same argument applies to ‘sharing’ and ‘division’. 1.2 Digression into Turkish grammar A logical difference between the operations of sharing and dispensing is reflected in the grammar of the Turkish language by the presence of a special form of numerals, distributive numerals. What follows was told to me by David Pierce, Eren Mehmet Kıral and Sevan Ni¸sanyan. First David Pierce: Turkish has several systems of numerals, all based on the cardinals; as well as a few numerical peculiarities. The cardinals begin: bir, iki, üç, dört, be¸s, altı, . . . (one, two, three, . . . ) These answer the question Kaç? (How many?) The ordinals take the suffix -inci, adjusted for vowel harmony: 2 DC is male, Italian, has a PhD in mathematics, holds a research position. In this episode, the language of communication was his mother tongue, Italian. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 4 1 Dividing Apples between People birinci, ikinci, üçüncü, dördüncü, be¸sinci, altıncı, . . . (first, second, third, . . . ) These answer the question Kaçıncı? The distributives take the suffix -(¸s)er: birer, iki¸ser, üçr, . . . Used singly, these mean “one each, two each” and so on, as in “I want two fruits from each of these baskets”; they answer the question Kaçar? Then Eren Mehmet Kıral continues: When somebody is distributing some goods s/he might say Be¸ser be¸ser alın. (Each one of you take five) or I˙ ki¸ser elma alın. (Take two apples each) I do not know if it is a grammatical rule (or if it is important) but when the name of the object being distributed is not mentioned then the distributive numeral is repeated as in the first example. The numeral may also be used in a non distributive problem. If somebody is asking students (or soldiers) to make rows consisting of 7 people each then s/he might say Yedi¸ser yedi¸ser dizilin. (Get into rows of seven) 3 In that context, a story told to me by one of my colleagues, SUE ¸ is very interesting. His experience of arithmetic in his (Turkish) elementary school, when he was about 8 or 9 years old, had a peculiar trouble spot: he could factorize numbers up to 100 before he learnt the times table, so he could instantly say that 42 factors as 6×7, but if asked, on a different occasion, what is 6 × 7, he could not answer. Also, he could not accept the concept of division with remainder: if a teacher asked him how many 3s go into 19 (expecting an answer: 6, and 1 is left over), little SUE ¸ was very uncomfortable—he knew that 3 did not go into 19. SUE ¸ added: But I did not pay attention to 19 being prime. I had the same problem when I was asked how many 3s go into 16. It is the same thing: no 3s go in 16. Simply because 3 is not a factor of 16. This is perhaps because of distributive numerals I somehow built up an intuition of factorizing, but perhaps for the same reason (because of the intuition that distributive gave) I could not understand division with remainder. As we can see, SUE ¸ does not dismiss the suggestion that distributive numerals of his mother tongue could have made it easier for him to form concept of divisibility and prime numbers (although he did not know the term “prime number”) before he learned multiplication. 3 SUE ¸ is Turkish, male, recent mathematics graduate. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 1.3 Dividing apples by apples: a correct answer 5 SUE ¸ only made his peace with remainders during his first year at university, when the process of division with remainder was introduced as a formal technique. He is not alone in waiting years before finally being told that division with remainder is not a binary operation because it produces two outputs, not one, as a binary operation should: partial quotient and remainder. Indeed here is a story from Dan Garry4 : When I was seven, I had to take a week off because I was sick. We were studying division at the time, and during the week I missed, the concept of remainders was covered. I asked the teacher what a “remainder” was and she was rather dismissive, saying “It’s what’s left over when you divide”. This made absolutely no sense to me; I remember thinking “7 divided by 3 is 2, what exactly is there to be left over?”. Looking back on it, it occurs to me that I was thinking of division as a binary operation: 7 divided by 3 is exactly 2. As silly as it might sound, I never really figured out the relation between “division” and “remainders” of integers until I went to a lecture on the division algorithm in my first year of university, which conveniently took place a few hours after a lecture in computer science about how the J AVA programming language handles integer division. 1.3 Dividing apples by apples: a correct answer But let us return to comparing problems (1.1) and (1.2). In the first problem you have a fixed data set: 10 apples and 5 people, and you can easily visualize giving apples to the people, in rounds, one apple to a person at a time, until no apples were left. But, as I have already mentioned, an attempt to visualize the second problem in a similar way, as an orderly distribution of apples to a queue of people, two apples to each person, necessitates dealing with a potentially unlimited number of recipients. I asked my teacher for help, but did not get a satisfactory answer. Only much later did I realize that the correct naming of the numbers should be apples = 5 people. people (1.3) I was not alone in my discomfort with “named numbers” and “units”. Here is a testimony from John Gibbon5 : 10 apples : 5 people = 2 4 5 apples , people 10 apples : 2 DG is male, 21 years old, was born and raised in England. He is a final year undergraduate studying Computer Science and Mathematics in a British university. JDG is male, British, a professor of applied mathematics. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 6 1 Dividing Apples between People Fig. 1.2. Paul Cézanne. Still Life with Basket of Apples. 1890–94. The Art Institute of Chicago. Source: Wikimedia Commons. Public domain. At the age of 6 years I was asked the question “How many oranges make 5?”. I recall that I refused to answer. This indicated to her that I was unintelligent, which had been her worry. Later in life I realized why my 6 year-old mind had felt there was something wrong with the question. The issue was one of units: “How many oranges make 5 what?” was the problem turning round in my 6 year-old mind. On the one hand one cannot change oranges into something else so I rejected “How many oranges make 5 apples?” On the other hand, if the answer was “How many oranges make 5 oranges?” then we had a tautology. I did not know what a tautological argument was but I knew I felt uncomfortable with it. Therefore let us look into equations (1.3) with some attention. 1.4 What are the numbers children are working with? It is a commonplace wisdom that the development of mathematical skills in a student goes alongside the gradual expansion of the realm of numbers with which he or she works, from natural numbers to integers, then to rational, real, complex numbers: N ⊂ Z ⊂ Q ⊂ R ⊂ C. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 1.4 What are the numbers children are working with? 7 What is missing from this natural hierarchy is that already at the level of elementary school arithmetic children are working in a much more sophisticated structure, a graded ring Q[x1 , x1−1 , . . . , xn , xn−1 ]. of Laurent polynomials6 in n variables over Q, where symbols x1 , . . . , xn stand for the names of objects involved in the calculation: apples, persons, etc. This explains why educational psychologists confidently claim that the operations (1.1) and (1.2) on Page 1 have little in common [264]—indeed, operation (1.2) involves an operand “apple/people” of a much more complex nature than basic “apples” and “people” in operation (1.1): “apple/people” could appear only as a result of some previous division. This difficulty was identified already by François Viéte who in 1591 wrote in his Introduction to the Analytic Art [237] that If one magnitude is divided by another, [the quotient] is heterogeneous to the former . . . Much of the fogginess and obscurity of the old analysts is due to their not paying attention to these [rules]. The presence of grading can be felt by some children. This is what is told to me by IG7 : My story hasn’t finished yet, as the problem is still very much with me now, as it was when I was 7. The bane of my existence is the addition and multiplication of integers. Take, for example, 75. The teacher would have us believe that 75 as 5 × 5 × 3, as 15 × 5 etc. all were ’the same’ 75. For the life of me I can’t believe it, and no proof convinces me. To me, 5 × 5 × 3 is somehow 3 dimensional, and 75 is something like volume. Then, when adding numbers, I get a moment of panic as if I am trying add things of different dimension and have no way of obtaining the correct dimensions just from the volume, and so the whole thing can’t possibly be right. The only progress I made over many years is that I learned to stuff this treacherous thought away whenever it rears its ugly head. Even so, perhaps there is no need to teach Laurent polynomials to kids (or even to teachers); but we need some simple common language that addresses the subtleties without adding unnecessary 6 7 Laurent polynomials and Laurent series are named after French military engineer Pierre Alphonse Laurent (1813–1854) who was the first to introduce them. Another his major achievement was construction of the port of Le Havre. IG is female, a PhD student in a leading British university. She went to school in Russia and was educated in Russian, her mother tongue. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 8 1 Dividing Apples between People sophistication. This is why I devote Chapters 3 and 4 to discussion of dimensional analysis, that is, the use of “named” numbers in physics. To my taste, it provides a number of interesting elementary examples that may be used if not at school but then at least in teachers’ training. This need for proper language for elementary school arithmetic is emphasized by Ron Aharoni [609]: Beside the four classic operations there is a fifth one, more basic and important: that of forming a unit. Taking a part of the world and declaring it to be the “whole”. This operation is at the base of much of the mathematics of primary school. First of all, in counting: when you have another such unit you say you have “two”, and so on. The operation of multiplication is based on taking a set, declaring that this is the unit, and repeating it. The concept of a fraction starts from having a whole, from which parts are taken. At the “adult” level, “forming a unit” may be viewed as setting up an appropriate Laurent polynomial ring as an ambient structure for a particular arithmetic problem. Later we shall see that, once we set up a structure, it inevitably comes into interaction with other structures, thus leading to some (very elementary and therefore very important) category theory coming into play (see Chapter 7). 1.5 The lunch bag arithmetic, or addition of heterogeneous quantities Usually, only Laurent monomials are interpreted as having physical (or real life) meaning. But the addition of heterogeneous quantities still makes sense and is done componentwise: if you have a lunch bag with (2 apples + 1 orange), and another bag, with (1 apple + 1 orange), together they make (2 apples +1 orange)+(1 apple +1 orange) = (3 apples +2 oranges). Notice that the “lunch bag” metaphor gives a very intuitive and straightforward approach to vectors: a lunch bag is a vector (at least this is how vectors are used in econometrics and mathematical economics). 1.6 Duality and pairing The “lunch bag” approach to vectors allows a natural introduction of duality and tensors: the total cost of a purchase of amounts S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 1.6 Duality and pairing 9 g1 , g2 , g3 of some goods at prices p1 , p2 , p3 is a “scalar product”-type8 expression X g i pi . We see that the quantities gi and pi could be of completely different nature. In physics, as a rule, the dot product involves heterogeneous magnitudes. In introductory physics courses, the dot product usually makes its first appearance on the scene as work done by moving an object, which is the dot product of the force applied and the displacement of the object. The standard treatment of scalar (dot) product of vectors in undergraduate linear algebra usually conceals the fact that dot product is a manifestation of duality or pairing of vector spaces, thus creating immense difficulties in the subsequent study of tensor algebra. As the following testimony from CB9 shows, the boredom and confusion start even earlier: I remember the very first conceptual difficulty I ever had: that was the scalar product of vectors. I could not figure why an operation involving two vectors should yield a plain number, and my teachers could not explain what that number meant in relation to the two vectors. As a result I hated scalar products as all we did with them was a meaningless if easy algebraic manipulation. Indeed scalar (or dot) product as it appears in physics is a pairing of two vector spaces U and V of different nature; assuming that we are working over the real numbers R, pairing is a map U ×V → R (u, v) 7→ u · v which is bilinear, that is, (au1 + bu2 ) · v = au1 · v + bu2 · v and similarly u · (av1 + bv2 ) = au · v1 + bu · v2 , in both cases for all a, b ∈ R and all vectors u, ui ∈ U and v, vi ∈ V . If it is possible to ignore physical (or financial) meanings of the vector spaces U and V , then the two spaces become logically undistinguishable. Paradoxically, this provides another source of difficulty for those students who are sensitive to formal logical aspects of mathematical concepts. 8 9 Scalar product is also called dot product or inner product. CB is female, holds a PhD, works as an editorial director in a mathematics publishing house. Her mother tongue is French, but she was educated in English. The episode described happened at age 12. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 10 1 Dividing Apples between People Here is a testimony from BB10 : From the time I learned matrices (age 16 or so) I cannot remember which are the columns and which are the rows. Given that the arrangement of coefficients in a linear transformation can be written equally well in a matrix in two ways, it is something that always takes me 10–15 seconds to recall even now. Of course, BB has reasons to be confused: for a mathematician, a matrix is an element in the tensor product V ⊗ V ∗ of a finite dimensional vector space V and its dual V ∗ . Since the dual of the dual of a finite dimensional space is the same as the original space, that is, (V ∗ )∗ ≃ V canonically, there is no intrinsic reason to distinguish between V and V ∗ , that is, between the rows and the columns of a square matrix. PD11 touches on the same theme: Does the “transition matrix” transform the basis or the coordinates? (Actually, many books hide the appearance of the inverse of the transpose by suitably defining the transition matrix.) Given a matrix of a linear map, am I writing the map between the vector spaces or between their duals? I discuss these and other confusing issues of logical symmetry and duality—and their possible psychophysiological substrate—in Chapter 22, Telling Left from Right. 1.7 Adding fruits, or the augmentation homomorphism There is another approach to addition of heterogeneous quantities highlighted by my correspondent Alex Grad12 : I remember that I was taught too that you can’t add apples and oranges, but I “resolved” the problem saying that 3 apples + 2 oranges = 5 fruits, including both categories in one more general, I am curious if that reflects more interesting notions like in your case Laurent polynomials. 10 11 12 BB is male, Russian, has a PhD from an American university and holds a research position in a British university. PD is male, Bulgarian, has a PhD in Mathematics, teaches in an American university. AG is male, Romanian, a student of computer science. His stories can also be found on Pages 14 and 87. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 1.8 Dimensions 11 It also has a name in “adult” mathematics: it is the augmentation homomorphism Z[x, x−1 , y, y −1 ] → Z[z, z −1 ] x±1 → 7 z ±1 y ±1 → 7 z ±1 , it turns variables x and y in a variable z which might be thought to be of more general sort. 1.8 Dimensions He consider’d therefore with himself, to see if he could find any one Adjunct or Property which was common to all Bodies, both animate and inanimate; but he found nothing of that Nature, but only the Notion of Extension, and that he perceiv’d was common to all Bodies, viz. That they had all of them length, breadth, and thickness. Abu Bakr Ibn Tufail, The History of Havy Ibn Yaqzan Translated from the Arabic by Simon Ockley. Physicists love to work in the Laurent polynomial ring R[length±1 , time±1 , mass±1 ] because they love to measure all physical quantities in combinations (called “dimensions”) of the three basic units: for length, time and mass. But then even this ring becomes too small since physicists have to use fractional powers of basic units. For example, velocity has dimension length/time, while electric charge can be meaningfully treated as having dimension mass1/2 length3/2 . time Indeed, it would be natural to choose our units in such a way that the permittivity ǫ0 of free space is dimensionless, then from Coulomb’s law 1 q1 q2 F = 4πǫ0 r2 applied to two equal charges q1 = q2 = q, we see that q 2 /r2 has the dimensions of force. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 12 1 Dividing Apples between People It pays to be attentive to the dimensions of quantities involved in a physical formula: the balance of names of units (dimensions) on the left and right hand sides may suggest the shape of the formula. Such dimensional analysis quickly leads to immensely deep results, like, for example, Kolmogorov’s celebrated “5/3 Law” for the energy spectrum of turbulence, see Chapter 4.3. Meanwhile, we should not blame schoolteachers for mess with “named” numbers. Unfortunately, it is a part of a more general tradition of neglect. In 1999 NASA lost a $125 million Mars orbiter because a Lockheed Martin engineering team used English units of measurement (inches and feet) while the agency’s team used the more conventional metric system (meters and millimeters) for a key spacecraft operation. Only very recently a programming language was created, F#, which automatically keeps control of units of measurement and dimensions of quantities generated in the process of computation. Exercises Exercise 1.1 To scare the reader into acceptance of the intrinsic difficulty of division, I refer to a paper Division by three [18] by Peter Doyle and John Conway. I quote their abstract: We prove without appeal to the Axiom of Choice that for any sets A and B, if there is a one-to-one correspondence between 3 × A and 3 × B then there is a one-to-one correspondence between A and B. The first such proof, due to Lindenbaum, was announced by Lindenbaum and Tarski in 1926, and subsequently ‘lost’; Tarski published an alternative proof in 1949. Here, of course, 3 is a set of 3 elements, say, {0, 1, 2}. An exercise for the reader: prove this in a naive set theory with the Axiom of Choice. The following line is repeated in the paper [18] twice: The moral? There is more to division than repeated subtraction. Exercise 1.2 Theoretical physicists occasionally use a system of measurements based on fundamental units: • • • speed of light c = 299, 792, 458 meters per second, gravitational constant G = (6.67428 ± 0.00067) × 10−11 m3 kg−1 s−2 and Planck’s constant h = 6.62606896 × 10−34 m2 kg · s−1 . Express the more common physical units: meter, kilogram, second in terms of c, G, h—you will get what is known as Planck’s units. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 2 Pedagogical Intermission: Human Languages We have had a chance to see in Section 1.2 how children’s perception of mathematics can be affected by logical and mathematical structures (such as systems of numerals) of the language of mathematics instruction and their mother tongue. In this book, we shall encounter more stories about language. Intrinsic structures of natural languages are engaged in a delicate interplay with hidden structures of mathematics. For me personally, this is a serious practical issue. Every autumn, I teach a foundation year (that is, zero level) mathematics course to a large class of students which includes 70 foreign students from countries ranging from Afghanistan to Zambia. Students in the course come from a wide variety of socioeconomic, cultural, educational and linguistic backgrounds. But what matters in the context of the this book are invisible differences in the logical structure of my students’ mother tongues which may have huge impact on their perception of mathematics. For example, the connective “or” is strictly exclusive in Chinese: “one or another but not both”, while in English “or” is mostly inclusive: “one or another or perhaps both”. Meanwhile, in mathematics “or” is always inclusive and corresponds to the expression “and/or” of bureaucratic slang. In Croatian, there are two connectives “and”: one parallel, to link verbs for actions executed simultaneously, and another consecutive1 . But it is as soon as you approach definite and indefinite articles that you get in a real linguistic quagmire. In the words of my ˇ 2: correspondent VC [In Croatian, there are] no articles. There are many words that can “serve” as the indefinite articles (neki=some, for example), but no 1 2 Rudiments of a “consecutive and” can be found in my native Russian and traced to the same ancient Slavic origins. ˇ is male, Croatian, a lecturer in mathematical logic and computer VC science. 13 14 2 Pedagogical Intermission: Human Languages particularly suitable word to serve as definite article (except the adjective odredeni = definite, I guess). Many times when speaking ¯ mathematics, I (in desperation) used English articles to convey meaning (eg. Misliš da si našao a metodu ili the metodu za rješavanje problema tog tipa? = You mean you found a method or the method for solving problems of that type?) What is truly astonishing is that subtler linguistic aspects of mathematics can be felt by children. What follows is a story from JA3 . I was 10 or 11 years old, in the final year of primary school in London. I am a native English speaker. The lesson was about fractions, and we were working on ‘word problems’ (i.e. things like, how many is one quarter of 36 apples?). The teacher said, “When we are doing fractions, ‘of’ means ‘multiply”’, and I thought, “No it doesn’t. ‘Of’ can’t change its meaning just because we are doing fractions. We are being fooled here.” And in that moment I saw mathematics as a set of conventions for which this teacher at least did not have a coherent understanding. I needed to know why the word ‘of’ and the operation of multiplication were linked, and the teacher could not tell me. On the other hand, the realization of the linguistic nature of mathematical difficulties can come in later life. This is a testimony from Alex Grad4 : When I was about 9 years old, I first learned at school about fractions, and understood them quite well, but I had difficulties in understanding the concept of fractions that were bigger than 1, because you see we were taught that fractions are part of something, so I could understand the concept of, for example 1/3 (you a have a piece of something you divided in 3 equal pieces and you take one), but I couldn’t understand what meant 4/3 (how can you take 4 pieces when there are only 3?). Of course I got it in several days, but I remember that I was baffled at first. 3 4 JA tells about herself: “Incidentally, I went on to study mathematics at A level, and began a degree in philosophy and mathematics at university, but became very disillusioned with the way in which mathematics was taught, and simply could not keep up—but I loved philosophy, and so dropped the maths. I re-gained my love of mathematics when I began a PGCE course to become a primary school teacher, and have spent the last 25+ years as a lecturer and researcher in mathematics education.” AG is male, Romanian, a student at Computer Science faculty which belongs to Engineering School. He says about himself: “yes, my occupation is still somehow related to mathematics, but above that I keep an interest in mathematics and in the psychology of mathematics and the philosophy of it”. His stories are quoted also on Pages 10 and 87. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 2 Pedagogical Intermission: Human Languages 15 The language of my mathematical instruction was Romanian, which was also my mother tongue. The word for fraction in Romanian is fractie, and that was the terminology used by my teacher and in textbooks. The word is used frequently in common language to express a part of something bigger, like fraction is used in English (I think . . . ), so I think that it’s very likely that my difficulty could be of a linguistic nature, I can’t eliminate also the fact that actually that was how fractions were introduced to us—pupils (like parts of an object) and only later the notion was extended, and so maybe I had problems accommodating to the new notion. Or maybe both reasons . . . Tuna Altınel5 values the fact that his school (the famous Galatasaray Lisesi in Istanbul) taught him to see and feel cultural differences brought by the use of a non-native tongue as a medium of education: [. . . ] the experience and the language of teaching (French) may be relevant. Another potentially relevant detail is that this is not a French school but a Turkish school where sciences are taught in French. The reason why I am mentioning this is not a patriotic feeling about the school or my nationality but in such a school, as a general rule that may or may not apply to mathematics education, the two cultures confront each other more visibly. At least, this is what I felt as I compared my high school to other French schools or such American schools as Robert College. A lasting effect of early linguistic experiences is emphasized by Tim Swift6 : By the way, related to the issue of language, I was taught to read and write (in a Yorkshire primary school in the mid 1960s) using an ‘experimental language’ called ITA (Initial Teaching Alphabet); I don’t know if this hindered or accelerated my development of communication skills, but I do remember that I seemed to be the only person in the class who, when we finally arrived at standard English when I was six or seven, had to translate everything back to ITA before I was happy with its meaning. An echo of that ‘urge 5 6 TA is male, Turkish, holds a PhD in Mathematics from an American univerity, teaches in a French university. TS is male, English. He wrote about himself: My PhD was from an English university (namely the University of Southampton). Most of my teaching and research talks have been conducted in English, although I have, on occasions, lectured in French. My university positions have been at English and Scottish institutions. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 16 2 Pedagogical Intermission: Human Languages to translate’ remains nowadays inasmuch as when I want to understand a piece of Mathematics properly, I usually have to translate it into my own preferred notation, etc., in order that I might feel happy with its meaning. Fig. 2.1. Initial Teaching Alphabet (Sir James Pitman, 1901–1985). Source: Wikipedia, http://en.wikipedia.org/wiki/File:Initial_ Teaching_Alphabet_ITA_chart.svg. Public domain. This alphabet plays a key role in a story told by Tim Swift, Page 16. Tim Swift added further: My main mathematical interest has been that of differential geometry and its applications, and the translation issue is very relevant in this part of Mathematics. There are so many different ‘formalisms’, and everyone has their own particular favorites: vector bundles or principal bundles?; covariant derivatives or connection 1-forms?; ‘index-free notation’ or use of indices?; the framework of jet bundles?; category theory language or not? . . . There is a joke S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 2 Pedagogical Intermission: Human Languages 17 about differential geometry being the study of concepts invariant under a change of notation, and I believe that there is a lot in that! (I don’t know who first made this joke.) Why does the “translation” aspects of learning (“urge to translate”, as defined by Tim Swift) matter so much in mathematics? Because mathematics is itself a Babel of separate but closely intertwined mathematical languages. I had a chance to write about mathematical languages in my previous book, Mathematics under the Microscope [107]. Here I reproduce only a very illuminating quote from the late Israel Gelfand, one of the greatest mathematicians of the 20th century. I had the privilege to work with him and closely observe his idiosyncratic ways of doing mathematics. Once he told me something remarkably consonant with Tim Swift’s words: Many people think that I am slow, almost stupid. Yes, it takes time for me to understand what people are saying to me. To understand a mathematical fact, you have to translate it into a mathematical language which you know. Most mathematicians use three, four languages. But I am an old man and know too many languages. When you tell me something from combinatorics, I have to translate what you say in the languages of representation theory, integral geometry, hypergeometric functions, cohomology, and so on, in too many languages. This takes time. It was amusing to watch how fellow mathematicians, not accustomed to the peculiarities of Gelfand’s style, spoke to him the first time. Very soon they became bewildered at why he insisted on their giving him really basic, everyone-always-knew-it kinds of definitions; then they were taken aback when he became furious at the merest suggestion that the definition was easier to write down than to say orally (“I know, you want to cheat me; do not try to cheat me!”). The next morning, their second conversation usually was even more entertaining, because Gelfand started it with the demand to repeat all the definitions; then he proceeded by questioning everything which was agreed upon yesterday, and eventually settled for a definition given in a completely different language. I observed such scenes many times and came to the conclusion that, for Gelfand, a definition of some simple basic concept, or a clear formulation of a very simple example, was a kind of synchronization marker which aligned together many different languages and made possible the translation of much more complex mathematics. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 3 Units of measurement In this chapter, I wish to explain my personal obsession with dimensional analysis. It serves as a justification for the next chapter, where I turn to some examples from the history of physics and the natural sciences. I would like to start with a warning wisely formulated by Jonathan Crabtree.1 The definition of the unit is in the eye of the beholder. My mum would say “One more spoonful” yet to me, there were four more peas! Using this as an excuse, I turn to my own memories. 3.1 Fantasy units of measurement L’eau, qui valait au début du siège deux késitah le bât, se vendait maintenant un shekel d’argent ; les provisions de viande et de blé s’épuisaient aussi ; on avait peur de la faim ; quelques-uns même parlaient des bouches inutiles, ce qui effrayait tout le monde. Gustave Flaubert, Salammbô As a boy I was a voracious reader with, unsurprisingly, a strong bent towards all kinds of fantasy, adventure and exoticism. I remember being enchanted by Flaubert’s Salammbô (in Russian translation) and being amused by the translator’s comment about the line in the novel that I used as the epigraph to this chapter: The water which was worth two kesitahs per bath at the opening of the siege was now sold for a shekel of silver . . . 1 JC is male, Australian ... 19 20 3 Units of measurement The translator explained that Flaubert was much criticized for using ancient units of value and measure with a blatant disregard to their actual meaning and value. “So what?”—thought I—“it does not matter what le bât, kesitah, or shekel were, it just sounds great.” Reading Jules Verne’s Vingt mille lieues sous les mers and Alexandre Dumas’ Les Trois Mousquetaires introduced into my language an exotic unit of distance, lieue. I had a rather vague understanding of how long a lieue was; apparently, it was sufficiently long since D’Artagnan’s ability to ride eight lieues on horseback deserved a special and very respectful mention in Dumas’ book. This exposure to French literature led me to use lieue as a unit of distance in solving an arithmetic problem. It so happened that, in Year 4 or 5 (which meant that I was 10 or 11 years old) I was suddenly called to take part in a district mathematics competition. I had to solve some kind of a problem about a river boat going upstream and downstream—I do not remember the problem but I am pretty confident that it was close in spirit to the following one, which I use here for illustrative purposes. It takes five days for a steamboat to get from St Louis to New Orleans, and seven days to return from New Orleans to St Louis. How long will it take for a raft to drift from St Louis to New Orleans? This was my solution. We need to somehow handle the speeds of the steamboat and the river current. Let us introduce a new measure of distance, called—why not?—lieue, so that the speeds of the steamboat downstream and upstream can be easily calculated. This can be achieved by choosing the distance from St Louis to New Orleans equal to 5 × 7 = 35 lieue. Then the speed of the steamboat downstream is lieue 35 =7 , 5 day while the speed upstream is 35 lieue =5 . 7 day Since the speed of the current gets added to, or subtracted from, the speed of the steamship in still water, the speed of the current is lieue 7−5 =1 2 day and a raft will drift from St Louis to New Orleans for 35/1 = 35 days. Alas, my solution was instantly dismissed as meaningless by a teacher in charge of the competition and I was sternly reprimanded for my use of a silly word: lieue. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 3.2 Discussion 21 3.2 Discussion Obviously, an introduction and use of an artificial unit of measurement is logically equivalent to an introduction of an (intermediate) unknown and subsequent use of de-facto algebraic manipulations in disguise. When you attempt to reconstruct your way of thinking when you were a child, you inevitably make some guesses and then try to find supporting evidence. In this case, I cannot dismiss my conjecture that, perhaps, my play with lieues was subconsciously motivated by ideas from algebra, even if these ideas had barely started to take hold in my mind. At the time, we had not yet started algebra at school, but over the summer vacations I read Vladimir Levshin’s books Three Days in Karlikania and Black Mask of Al-Jabr, an introduction to elementary algebra written in the form of a fantasy tale. The fairy tale narrative around mathematics was a bit too childish for me, and I was not sufficiently interested to follow the mathematical entertainments of the book with a pencil on paper; very soon, I forgot about the books. However, at the same mathematics competition where I introduced the fantasy lieue I discovered that another problem looked like being open to a Karlikania treatment. It was something about money: two things together cost that much, etc. I do not remember the problem; but I remember that I looked at it and decided to try Karlikania’s approach. So, I denoted some quantity appearing in the problem by a—I chose the quantity more or less at random— and started to rewrite the problem as a balanced equality. Crucially, I remembered Karlikania’s key idea: when you move something to another side of the equality, you change its sign. I remembered that principle because, at the time of reading, it struck me as slightly paradoxical. To my surprise, everything in my first attempt at doing algebra worked out in the smoothest possible way, and I fairly quickly got the answer. Alas, this my solution was also dismissed as going ahead of the curriculum, and I was again reprimanded for wrongly labeling the unknown by a, when everyone knew that it had to be x. I did not argue with the teacher and did not tell her that I actually thought about the choice of a letter, and picked a exactly because I was not certain that what I was doing was a canonical school method, where, I had heard, x’s were used. Perhaps I have to reassure the reader that this unfortunate incident did not have any negative psychological effect on me at all. I was safely inoculated against any potential pedagogical trauma. My mother was a teacher herself (and a very good one), and by that S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 22 3 Units of measurement time I had already received from her and deeply interiorized her lesson: “Teachers are human beings; do you really think that they have no right to be stupid? If you know that your teacher is thick, live with it, keep your mouth shut and treat her with respect, as you would treat any other person”. My first school teacher—and I dearly loved her—was a life long best friend of my mother; half a dozen of my classmates were children of teachers from our school. I grew up surrounded by teachers who were also friends of my family, or our neighbors, or parents of my friends, and I retained for the rest of my life a very close affinity with the teaching community. So, I was inoculated—but more frequently an unexpected intellectual conflict with a teacher could be quite difficult for a child. Here is a testimony from Pierre Arnoux2 ; because of it mathematical content, I quote it here, not in Chapter 18 where more sad stories are assembled. The story dates back from my first year in college (“sixième” in French), I was ten years old. The teacher gave us as homework a problem of the type “John is three times older than Peter, and if you subtract the age of Peter from twice the age of John, you get 40” (I do not remember the exact statement). We were supposed to solve this by words, but I could not make it, and I asked my father. He explained to me that we could give name to the ages, so it came to J = 3P , 2J − P = 40, then replace everywhere J by 3P and solve the problem, obtaining 5P = 40, P = 8, J = 24. This seemed rather complicated, and very fuzzy, but it clearly worked since it gave the solution. Next day, when I came back, the teacher asked me to give my solution on the blackboard; I was very unsure of myself, because the solution seemed somewhat “illegal”, since it was clearly outside of what we had been taught. I said so, then explained my solution (which seemed slightly strange to the other students, as far as I remember). When I had finished, the teacher said “well, this is the solution, why do you see a problem? Why all this fuss?” and dismissed me to my place. I remember very clearly my shock at feeling that I had found something completely new (to me, at least!) and rather difficult, and that the teacher was completely unable to perceive the difference between this (in fact, the beginning of algebra) and what he had previously taught us. And I also remember that my respect for that teacher was very much decreased after that. 2 PA is male, French, a professor of mathematics in a French university. Another story from him is on page 70, and this one continues on page 181. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 3.3 History 23 3.3 History When writing this book, I sent to a few historians of mathematics the following questions: Had the manipulations with units of measurement that I did aged 11 actually been done in history of arithmetic? Are there traces of them in historic sources? In response, Roy Wagner wrote to me that there were solution “algorithms” from the abacist textbooks concerning travel problems that could be reconstructed as similar to the one that I considered, and sent me his rough literal translation of Problem 108 from Paolo dell’Abbaco’s (1282 – 1374), Trattato d’aritmetica: From here to Florence is 60 miles, and there’s one who walks it in 8 days, and another in 5 days. It is asked: Departing at the same time, one from here and the other from there, in how many days will they meet? Do the following: multiply 5 by 8, makes 40, and say thus: in 40 days one will make the trip 8 times, and the other 5 times, so both together will make it 13 times. Now say: if 40 days equal 13 trips, for one trip how many days will it have? And so multiply 1 times 40, makes 40, and divide this 40 by 13, which makes 3 days and 1/3 of a day; and so I say that in 3 days and 1/3 of a day they will find themselves together. And this is done, so all similar problems are done. It is a very interesting solution, since it is based on introduction of a convenient dimensionless unit, a trip. For abacists who lived in a strictly regimented traditional society, it was psychologically difficult to move away from established units of measurement. Notice that the problem starts with declaring the distance “from here to Florence”, 60 miles, but this datum is not used in the solution, and the word “mile” does not appear in the solution. Albrecht Heeffer wrote to me with further examples: To answer your question, is this mechanism of an artificial unit been done in the history of algebra? Yes indeed, if have found it in several abbaco manuscripts, and it functions as an intermediate unknown is some sense. I have seen it used in problems for finding three numbers, a, b and c in geometrical progression, given some extra conditions. The “cosa” or unknown is used for, let us say, the largest number, c. Then one supposes that the smaller, a is 1. This allows to derive that c = b2 and hence to derive a value for b and c. In the last stage, the value of a is derived to meet the extra conditions. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 24 3 Units of measurement Fig. 3.1. This woodcut from the Margarita Philosophica of Gregorius Reisch, published in Freiburg in 1503, shows “Arithmetica” watching a competition between an “abacist” and an “algorist”. Image source: Wikipedia Commons. Public domain. Text is quoted from Matthias Tomczak, [871]. Again, it appears that the artificial unit is dimensionless. To freely use arbitrary, made on-the-fly units of measurement, one has to be conditioned in a cultural relativism. The latter was a relatively late phenomenon of the human civilization, and Flaubert was one of its proponents in the literature. This brief excursion into history justifies devoting the whole of the next chapter to history of the dimensional analysis. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 3.3 History 25 Exercises The following three exercises are from historic sources, they were kindly sent to me by Albrecht Heeffer as a comment on the problem discussed in this chapter: It takes five days for a steamboat to get from St Louis to New Orleans, and seven days to return back from New Orleans to St Louis. How long will it take for a raft to drift from St Louis to New Orleans? Exercise 3.1 Albrecht Heeffer wrote to me: While meeting and overtaking problems are abundant in Renaissance and early modern books on arithmetic and algebra, problems involving a current or a wind appear rather late. The earliest occurrence of such problems I could find is from the end of the eighteenth century: “If during the Tide of Ebb, a Wherry should set out from London westward, and at the same instant another should put off at Chertsey for London, taking the distance by water at 34 miles : The stream forwards one and retards the other, say 2 21 Miles an Hour : The Boats are equally laden, the Rowers equally good, and in the ordinary Way of working, in still Water, would proceed at the rate of 5 Miles an Hour : The Question is, where in the River the two Boats would meet ?” [From The Tutors Guide: Being a Complete System of Arithmetic, by Charles Vyse, 1772, p. 211.] Solve this problem. Exercise 3.2 Albrecht Heeffer further comments that Such problems are usually dealt with in a chapter on two equations with two unknowns: Horatio N. Robinson. New Elementary Algebra: Containing the Rudiments of the Science for Schools and Academies. Ivison, Blakeman, Taylor & Co., New York, 1875. Prob. 90, p. 305. “A person residing on the bank of the Ohio, 15 miles above Cincinnati, can row his boat to the city in 2 12 hours, but it requires 7 12 hours to return. With what force can he row his boat in still water, and what is the velocity of the river ?” Exercise 3.3 And here is Albrecht Heeffer’s final remark: Although in your problem the distance is not being given, the answer can easily be given, if you know how to calculate the speed of the current or tide. This is nicely explained in the following book, and the earliest occurrence of this kind of problems I could find is in Algebra Made Easy : Chiefly Intended for the Use of Schools, by Thomas Tate (London : Longman, Brown, Green, and Longmans, 1847), p. 86, no. 11. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 26 3 Units of measurement “A rower goes a mph with the tide and b mph against the tide. What is the rate of the tide?” Again, solve it. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 4 History of Dimensional Analysis Der würde nicht gelebt haben, der nur in der Gegenwart lebte. Heinrich Mann It is time to formulate the key principle of dimensional analysis: two physical quantities have the same dimensions if their measurements change in exactly the same way under all possible choices of units of measurement. For example, the value of acceleration is more sensitive than velocity to a change of units of time, but behaves similarly to velocity under a change of units of length. But, instead of change of units, we can vary the size and other parameters of our objects and processes—then the same scaling considerations will give us important insights into their behavior. In our first example, it makes sense to stay strictly within mathematics. A bit later we shall turn wonderful applications of the scaling principles to physical laws. We start with the basic observation: area of geometric figure changes as squares of its characteristic linear measurement. For example, if we increase a side of a square by factor of two, its area increases by factor of four. This observation allows us to prove Pythagoras’ Theorem: If a and b are sides and c the hypothenuse of a right-angled triangle, then a2 + b2 = c2 . Indeed, consider all triangles similar to the given triangle with sides a and b and the hypotenuse c; the corresponding sides a′ , b′ and the hypothenuse c′ are obtained from a, b, c by stretching with the same coefficient. We can take the hypotenuse c for characteristic linear measurements, then area S of the triangle changes as square of c, that is, S = kc2 for some coefficient k which is the same for all similar triangles. The key step now is that the right-angled triangle can be cut in 27 28 4 History of Dimensional Analysis two similar triangles in which a and c become hypotenuses, and the area S becomes the sum of the areas of the two smaller triangles, kc2 = ka2 + kb2 , which instantly gives us Pythagoras’ Theorem: c2 = a2 + b 2 . This argument apparently has a long history. Alexander Givental [194] comments: In fact the foregoing argument is not new. Moreover, it is so close to the original proof of [Euclid’s] Proposition VI.31 that G. Polya [79] attributes it to Euclid himself. At this point, we turn to some wonderful examples from the history of physics and the natural sciences. 4.1 Galileo Galilei and the first of his two “New Sciences” Dimensional analysis has a fascinating history indeed. Its origins can be traced back to the seminal book by Galileo Galilei Two New Sciences [192]. In a modern day assessment (Peterson [226]), Galileo’s last book was the Two New Sciences, a dialogue in four days. The third and fourth days describe his solution to the longstanding problem of projectile motion, a result of obvious importance and the birth of physics as we know it. But this was only the second of his two new sciences. What was the first one? Two New Sciences begins in the Venetian Arsenal, the shipyard of the Republic of Venice, with a discussion of the effect of scaling up or scaling down in practical construction projects, like shipbuilding. [. . . ] According to the publisher’s foreword, it is this topic that should be understood as the first of the two new sciences. Galileo’s observations on scaling in general are ingenious and elegant, and entirely deserving of the prominent place he gives them. These ideas are basic in physics, and are introduced in most introductory physics texts under the heading of dimensional analysis. We could even say that modern renormalization group methods are just our most recent way to deal with problems of scale, still recognizably in a tradition pioneered by Galileo. Just read this unbelievable quote from Galileo, a formulation of the scaling problem by one of the discussants in the book, Sagredo: S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 4.1 Galileo Galilei 29 “Now, because mechanics has its foundation in geometry, where mere size cuts no figure, I do not see that the properties of circles, triangles, cylinders, cones and other solid figures will change with their size. If therefore a large machine be constructed in such a way that its parts bear to one another the same ratio as in a smaller one, and if the smaller is sufficiently strong for the purpose for which it was designed, I do not see why the larger also should not be [sufficiently strong] . . . ” Sagredo clearly formulates the principle that geometric properties are invariant under scaling transformations, and he wants to understand why this invariance is broken in the real world. The answer is proposed by Salviati: “. . . these forces, resistances, moments, figures, etc. may be considered in the abstract, dissociated from matter, or in the concrete, associated with matter. Hence the properties which belong to figures that are merely geometrical and nonmaterial must be modified when we fill these figures with matter and therefore give them weight.” In modern parlance, change of properties results in a change of the group of permitted transformations and breaks scale invariance, but the new theory still allows a development in terms of invariants. Salviati continues: “Since I assume matter to be unchangeable and always the same, it is clear that we are no less able to treat this constant and invariable property in a rigid manner than if it belonged to simple and pure mathematics.” Galileo then uses these general principles, for example, to explain why smaller objects fall more slowly than big ones: their area varies as the square of linear dimension, while their weight varies as the cube of linear dimension, and therefore for smaller objects surface forces—such as air drag—become more significant compared to the force of weight. He considers how fibrous materials, like wood beams, break under their weight and makes a now famous remark that if we scale an animal up, its bones should become thicker and thicker in comparison with their length. We shall return to the discussion of this in the exercises. The reader who wants to learn more about how Galileo’s ideas continue to live in physics can find a concise and clear outline of the modern understanding of dimensional analysis in a beautiful little book by Yuri Manin [63]. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 30 4 History of Dimensional Analysis And I conclude this brief introduction to principles of dimensionality by a childhood testimony from my friend Hovik Khudaverdyan1 : I was about 10 when my parents bought me as gift a scaled plastic model of Ilyushin IL-18 plane; it was made to 1:100 scale and its weight was, according to the label, 154 gram. I was asking my parents: how could this happen, the empty weight of IL-18 was 35 tons, 1/100 of it had to be 350 kilograms? Even if plastic was lighter than metal, this could not explain the discrepancy! 4.2 Froude’s Law of Steamship Comparisons The next significant step in the development of dimensional analysis appears to be Froude’s Law of Steamship Comparisons: the maximal speed of similarly designed steamships is proportional to the square root of their length. William Froude (1810–1879) was the first to formulate reliable laws for the resistance that water offers to ships and for predicting their stability. In this section, we give a deduction of this law adapted from D’Arcy Thompson [870, p. 24]. But first we have to discuss some difficulties of the mathematical modeling of physical phenomena and the limitations of dimensional analysis. 4.2.1 Difficulty of making physical models We need to understand first how the drag F (force of resistance) offered by water to a ship depends on the speed of the ship. Surprisingly, it is is easier to do for high speeds than for lower ones. At high speeds we can assume that the drag is offered by water being violently thrown away from the course of the ship, ignoring a finer picture of what is happening with the water. Obviously, the drag F should depend on the crossection area S of the ship, its speed V , and density of water ρ: the heavier the water, the harder it is to throw it aside. Therefore we are looking for an equation in the form F = cρx S y V z with a dimensionless coefficient c. Substituting dimensions, we have x   y  length z mass mass · length 2 = · length · , time time2 length3 1 HK is male, Armenian, initial mathematics education was in Armenian. He is a physicist by education who moved to research in physics-inspired geometry, holds a PhD, teaches in a British university. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 4.2 Froude’s Law of Steamship Comparisons 31 and immediately see that z = 2, x = y = 1, and the so-called Formula of Quadratic Drag takes the form F = cρSV 2 . According to Wikipedia, this equation is attributed to Lord Rayleigh. At lower speeds, water flows smoothly around the hull of the ship, and a much more delicate analysis is needed. This has been implemented by Sir George Stokes2 (1819–1903) for the special case of small spherical objects moving slowly through a viscous fluid. It is F = −6πηrV, where r is the effective radius of the object, η is the viscosity of the fluid. In short, for small objects like bacteria the drag is proportional to their length and velocity; we shall denote this symbolically as F ∼ LV. Notice that the proportionality coefficient is no longer dimensionless. Sadly, Stokes’s beautiful mathematical deduction does not apply to real ships. 4.2.2 Deduction of Froude’s Law Instead of Raleigh’s and Stokes’ drag formulae, we have to use the property (and we shall treat it simply as an experimental fact, as Froude did after a number of experiments) that, at small speeds, the drag F is proportional to the area of cross section of the ship (and hence to the square L2 of its typical linear size L) and to the velocity V of the ship. We shall write this symbolically as F ∼ L2 V and call it Froude’s drag. To sustain constant speed, the ship has to produce power F V . Now we have to recall that we are talking about steamships, powered by an engine inside, this engine working on steam produced by burning coal, this coal being carried by the ship, etc. Therefore 2 Stokes is likely to be known to the reader as the author of the Stokes Theorem for surface integrals: Z Z dω = ω. M S HADOWS OF THE T RUTH V ER . 0.822 ∂M 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 32 4 History of Dimensional Analysis it is reasonable to assume that the power produced by the ship is proportional to its volume, and therefore F V ∼ L3 . Combining the two proportions together, we have L2 V 2 ∼ L3 and therefore V ∼ L1/2 . This is the promised Froude’s Law of Steamship Comparisons. 4.3 The triumph of “named numbers”: Kolmogorov’s “5/3” Law To demonstrate the power of “named numbers”, I borrow this section from my book Mathematics under the Microscope [107]. The deduction of Kolmogorov’ seminal “5/3” law for the energy distribution in the turbulent fluid [55] is so simple that it can be done in a few lines. It remains the most striking and beautiful example of dimensional analysis in mathematics. I was lucky to study at a good secondary school where my physics teacher (Anatoly Mikhailovich Trubachov, to whom I express my eternal gratitude) derived the “5/3” law in one of his improvised lectures. In my exposition, I borrow some details from Arnold [100] and Ball [3] (where I also picked up the idea of using a woodcut by Katsushika Hokusai, Figure 4.1, as an illustration). 4.3.1 Turbulent flows: basic setup The turbulent flow of a liquid consists of vortices; the flow in every vortex is made of smaller vortices, all the way down the scale to the point when the viscosity of the fluid turns the kinetic energy of motion into heat (Figure 4.1). If there is no influx of energy (like the wind whipping up a storm in Hokusai’s woodcut), the energy of the motion will eventually dissipate and the water will stand still. So, assume that we have a balanced energy flow, the storm is already at full strength and stays that way. The motion of a liquid is made of waves of different lengths; Kolmogorov asked the question, what is the share of energy carried by waves of a particular length? Here is a somewhat simplified description of his analysis. We start by making a list of the quantities involved and their dimensions. First, we have the energy flow (let me recall, in our setup it is the same as the dissipation of energy). The dimension of energy is S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 4.3 Kolmogorov’s “5/3” Law 33 Fig. 4.1. Multiple scales in the motion of a fluid, from a woodcut by Katsushika Hokusai The Great Wave off Kanagawa (from the series Thirty-six Views of Mount Fuji, 1823–29). This image is much beloved by chaos scientists. Source: Wikimedia Commons. Public domain. mass · length2 time2 (remember the formula K = mv 2 /2 for the kinetic energy of a moving material point). It will be convenient to make all calculations per unit of mass. Then the energy flow ǫ has dimension energy length2 = mass · time time3 For counting waves, it is convenient to use the wave number, that is, the number of waves fitting into the unit of length. Therefore the wave number k has dimension 1 . length Finally, the energy spectrum E(k) is the quantity such that, given the interval ∆k = k1 − k2 between the two wave numbers, the energy (per unit of mass) carried by waves in this interval should be approximately equal to E(k1 )∆k. Hence the dimension of E is S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 34 4 History of Dimensional Analysis energy length3 . = mass · wave number time2 4.3.2 Subtler analysis To make the next crucial calculations, Kolmogorov made the major assumption that amounted to saying that3 The way bigger vortices are made from smaller ones is the same throughout the range of wave numbers, from the biggest vortices (say, like a cyclone covering the whole continent) to a smaller one (like a whirl of dust on a street corner). Then we can assume that the energy spectrum E, the energy flow ǫ and the wave number k are linked by an equation which does not involve anything else. Since the three quantities involved have completely different dimensions, we can combine them only by means of an equation of the form E(k) ≈ Cǫx · k y . And now the all-important scaling considerations come into the play. In the equation above, C is a constant. Since the equation should remain the same for small scale and for global scale events, the shape of the equation should not depend on the choice of units of measurements, hence the constant C should be dimensionless. Let us now check how the equation looks in terms of dimensions: !x  y length2 1 length3 = · . length time2 time3 After equating lengths with lengths and times with times, we have length3 = length2x · length−y time2 = time3x , which leads to a system of two simultaneous linear equations in x and y, 3 = 2x − y 2 = 3x This can be solved with ease and gives us x= 3 5 2 and y = − . 3 3 This formulation is a bit cruder than most experts would accept; I borrow it from Arnold [100]. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 4.3 Kolmogorov’s “5/3” Law 35 Therefore we come to Kolmogorov’s “5/3” law: E(k) ≈ Cǫ2/3 k −5/3 . We have done it! It is claimed in the specialist literature that the dimensionless constant C can be determined from experiments and happens to be pretty close to 1. Fig. 4.2. The vortex nature of the Universe. Vincent van Gogh, Starry Night, 1889. Source: Wikimedia Commons. Public domain. 4.3.3 Discussion The status of Kolmogorov’s celebrated result is quite remarkable. In the words of an expert on turbulence, Alexander Chorin [13], Nothing illustrates better the way in which turbulence is suspended between ignorance and light than the Kolmogorov theory of turbulence, which is both the cornerstone of what we know and a mystery that has not been fathomed. The same spectrum [. . . ] appears in the sun, in the oceans, and in man-made machinery. The 5/3 law is well verified experimentally and, by suggesting that not all scales must be computed anew in each problem, opens the door to practical modelling. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 36 4 History of Dimensional Analysis Vladimir Arnold [100] reminds us that the main premises of Kolmogorov’s argument remain unproven—after more than 60 years! Even worse, Chorin points to the rather disturbing fact that Kolmogorov’s spectrum often appears in problems where his assumptions clearly fail. [. . . ] The 5/3 law can now be derived in many ways, often under assumptions that are antithetical to Kolmogorov’s. Turbulence theory finds itself in the odd situation of having to build on its main result while still struggling to understand it. 4.4 Dimension of Lagrange multipliers By contrast with stories from my childhood, this one comes from my very mature age. Only working over this book, I finally managed to get over my old incomprehension. It was about the nature of Lagrange multipliers. Recall that Lagrange multipliers are used in search of maxima and minima of functions subject to a constraint. For example, we may wish to maximize a function of two variables f (x, y), where x and y cannot take arbitrary values but are constrained by condition g(x, y) = c. Fig. 4.3. Constrained optimization. Source: Wikipedia. Public domain. Lagrange proposed to introduce an extra variable λ and consider a function Λ(x, y, λ) = f (x, y) + λ · (g(x, y) − c). At the point of constrained extremum, all three partial derivative of Λ vanish: ∂Λ ∂Λ ∂Λ = = = 0. ∂x ∂y ∂λ It is obvious for S HADOWS OF THE T RUTH ∂Λ = g(x, y) − c = 0 ∂λ V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 4.4 Dimension of Lagrange multipliers 37 because of the imposed constraint. For the other two partial derivatives, we have a simple geometric explanation: when we look at the contour lines (level curves) of the function f (x, y), we notice that at the point of constrained extremum it has to be tangent to the curve g(x, y) = c; compare Figures 4.3 and 4.4. We have a kind of duality here: the curve g(x, y) = c is a level curve of the function g(x, y). Since level curves of functions are perpendicular to gradients, we have that the two gradients " # " # ∂f (x,y) ∂x ∂f (x,y) ∂y and ∂g(x,y) ∂x ∂g(x,y) ∂y are collinear at the point where level curves are tangent. Hence " # " #   ∂f (x,y) ∂g(x,y) 0 ∂x ∂x + λ ∂g(x,y) = ∂f (x,y) 0 ∂y ∂y for some λ, which immediately translates into ∂Λ ∂Λ = = 0. ∂x ∂y Fig. 4.4. Contour lines are tangent at the point of constrained extremum. Source: Wikipedia. Public domain. But what is the meaning of λ? What is dimension of the Lagrange multiplier, when we maximise, say, energy, mass × length2 time2 S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 38 4 History of Dimensional Analysis by varying constrained space coordinates x and y which have dimension of length, while the constraint function g(x, y) has dimension of time and measures how much time it takes to get our system into the state (position) (x, y)? What is the meaning of this dimension? The correct answer: price. It is the shadow price of mathematical economics: if everything is expressed in money (which are dimensionless because of their universality), and, with energy priced at the current market rate, λ gets the dimension 1 time of the price of unit of time. And this is indeed price. Assume that you have to hire an hourly paid workers, and that it takes him g(x, y) man-hours to tune up your energy-producing installation up to the state (x, y) at which point the installation will produce your a certain amount of energy which you sell for f (x, y) dollars. You have limited funds c, hence you are constrained by equation g(x, y) = c. But assume you got an extra dollar to invest; then the shadow price λ tells you how much you have to be prepared to pay for extra workers. If you hire new workers at hourly rate below λ, you will increase your profit; if you have to pay more than λ, expansion of the work force leads to decline in your profit. Disturbingly, when I started to type in into G OOGLE: money is the measure . . . G OOGLE automatically suggested to continue the phrase as money is the measure of success. Well, there is some mathematical justification for this bland truism. Alas, the greatest flaw of mathematics as a cultural system is that it is morally neutral. 4.5 Length and area And back to Dmitri Burago: You know, I recently published a paper Area Spaces: The First Steps [11]. The main statement there is that a Riemannian manifold of dimension at least 3 is uniquely described by an abstract object; a formal group of closed curves (oh, sure, 2-currents) with a norm: the minimal filling area. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 4.5 Length and area 39 Fig. 4.5. Euclid VI.31: Generalization of Pythagoras’ Theorem. After A. Givental [194]. Exercises Pythagorean Theorem Exercise 4.1 Alexander Givental [194] points to a relatively unknown fact: It turns out that Book VI of the Elements contains a generalization of the Pythagorean theorem that seems much less famous. Namely, Euclid VI.31 asserts (see Figure 4.5) that A + B = C for the areas A, B and C of similar figures of any shape built on the sides of a right triangle. The Pythagorean theorem is clearly the special case where the shape is the square. Prove this generalization of the Pythagorean Theorem. Givental’s wonderful paper [194] shows that Euclid was in fact using some form of scaling arguments. Froude’s Law of Steamship Comparisons Froude’s Law of Steamship Comparisons was used to great effect in D’Arcy Thompson’s book On Growth and Form [870, p. 24] to analyze the speed of animals. D’Arcy Thompson applied the same derivation as in Section 4.2.2 to fish because of the validity of the principal assumption: the energy is produced internally. This leads us to a natural question: Exercise 4.2 How would Froude’s Law look for solar powered ships? (Use Froude’s drag.) Exercise 4.3 Try to relate the previous exercise to the fact that all known living organisms able to move in water and capable of photosynthesis (and thus able to use solar power directly) consist of a single cell, see Figure 4.6. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 40 4 History of Dimensional Analysis Exercise 4.4 Prove the following corollary of Froude’s Law: The relative speed of a fish (that is, speed measured in numbers of its lengths covered by the fish per unit of time) is inversely proportional to the square root of its length. This explains a well-known phenomenon: little fish in a stream appear to be very quick. Exercise 4.5 Estimate, which is relatively faster: an ant or a racehorse? Exercise 4.6 Compare the expected maximal speeds of photosynthesising (that is, solar powered) and phagocytosising (that is, fuel powered) organisms of comparable size. Use Stokes’ drag. Exercise 4.7 Why does a mouse have (relatively) a slimmer body build than an elephant? Fig. 4.6. Euglena, one of many single cell solar powered living organisms able to move in water, it propels itself using its tail (flagellum). c BIODIhttp://biodidac.bio.uottawa.ca/info/regles.htm, DAC. I refer the reader to [869] for more on this and other biological problems from this section. Exercise 4.8 Deduce a version of Froude’s Law for high speeds and quadratic drag. Legs and paces Exercise 4.9 Analyze the following quote from [28, p. 233]: S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 4.5 Length and area 41 Fig. 4.7. Walking on stilts increases speed (for Exercise 4.9). Habitants des Landes. Jean Louis Gintrac (1808–1886). Source: Wikipedia, http://upload.wikimedia.org/wikipedia/commons/2/ 22/GintracLandes.jpg. Public domain. Thus treating a man’s leg as swinging like a pendulum through a fixed definite angle, the length of a pace is proportional to his height, but the number of paces a minute is inversely as the square root of the height; so that his pace of walking and getting over the ground will vary as the square root of his height, as in Froude’s Law again. Using dimensional analysis, prove: • the number of swings of a pendulum per minute varies inversely as the square root of the length of the pendulum; and use this fact to derive Greenhill’s conclusion: • his pace of walking and getting over the ground will vary as the square root of his height (or of length of legs, see Figure 4.7). Exercise 4.10 Investigate further: • How do the frequency of a pendulum’s swings and the pace of a man walking depend on the acceleration due to gravity? (Read more on that in [236]; see Figure 4.8.) And even further: • Read an article [873] which proves that desert ants count their paces: S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 42 4 History of Dimensional Analysis Fig. 4.8. Buzz Aldrin walking on the Moon (for Exercise 4.10). Source: NASA, http://www.hq.nasa.gov/alsj/a11/ AS11-40-5903HR.jpg. Public domain. Ants walked on normal legs from their nest to a feeder, where they were placed on stilts (made of pig bristles glued on to their legs) and released. (See Figure 4.9.) On their back home ants on stilts misjudged the 10-meter distance back and overshot the nest entrance. When the experiment was repeated with the short-legged ants (their legs cruelly amputated), they stopped and were looking for nest entrance too early. In this experiment, it is crucial to distinguish between counting steps and measuring time—how would you handle this issue if you were an experimenter? Exercise 4.11 Dominoes falling make a wave; see Figure ??. We may conjecture that the wave speed v depends on the the spacing d between the dominoes, the height h of each domino, and the acceleration due to gravity g [?]. Show that if h/d is fixed then the propagation speed is proportional to the square root of d. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 4.5 Length and area 43 Fig. 4.9. An ant walks with stilts glued on to its legs. Source: National Geographic News, http://news. nationalgeographic.com/news/bigphotos/images/ c Science. 060629-ants-stilts_big.jpg. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 5 Adding One by One 5.1 Adding one by one It appears that addition comes from counting. In words of Iain Currie1 , My Mother told me the following story. When I was about two and a half a small flock of birds flew overhead. I said: “Look, there are two and three birds”. I didn’t yet know the number five but I understood simple counting. So far so good; but challenges arise fairly soon. My colleague Elizabeth Kimber2 told me about a difficulty she experienced in her first encounter with arithmetic, aged 6. She could easily solve “put a number in the box” problems of the type 7 +  = 12, by counting how many 1’s she had to add to 7 in order to get 12 but struggled with  + 6 = 11, because she did not know where to start. Worse, she felt that she could not communicate her difficulty to adults. Her teacher forgot to explain to her that addition was commutative. Another one of my colleagues, AB3 , told me how afraid she was of subtraction. She could easily visualize subtraction of 4 from 100, say, as a stack of 100 books; after removing 4 books from the top (by 1 2 3 IC is male, British, has a PhD in mathematics, teaches statistics and actuarial science at an university. For the record: EHK is English, female, has a PhD in Mathematics, teaches mathematics at a highly selective secondary school. AB is Turkish, female, has a PhD in Mathematics, teaches mathematics in a research-led university. 45 46 5 Adding One by One reverse counting: 100, 99, 98, 97), 96 are left. But what will happen if you remove 4 books from the bottom of the stack? This is a phenomenon well known and well studied in the mainstream pedagogical research. James Hiebert reports from a field study [688]: Fifty-five percent of the responses to the verbal problem a+b= and to the verbal problem a−b= included modeling the sets with cubes. This percentage drops to about 40% for the a+ = c problems and to about 18% for the +b=c problems. Statistics is always instructive, but I would rather understand the intrinsic logic of individual personal stories. I find an ally in the neurologist Vilayanur Ramachandran who said about statistical analysis [381, pp. xi–xii]: There is also a tension in the field of neurology between the ‘single case study’ approach, the intensive study of just one or two patients with a syndrome, and sifting through a large number of patients and doing a statistical analysis. The criticism is sometimes made that it’s easy to be misled by single strange cases, but this is nonsense. Most of the syndromes in neurology that have stood the test of time [. . . ] were initially discovered by a careful study of single case and I don’t know of even one that was discovered by averaging results from a large sample. My approach is different from the traditional pedagogical research: instead of looking at pedagogical and psychological explanation for EHK’s and AB’s problems, I am asking a question: what were hidden mathematical reasons for their difficulties? For an answer, I look at Peano arithmetic and some elementary category theory. I will be happy if my educationalist colleagues recognize that my approach complements, not contradicts that one of more traditional educational research. I am open to any criticism and comments; I am prepared to accept that my project is far-fetched. However, the remarkable level of support that I have already received from my correspondents—mostly professional research mathematicians— leads me to argue confidently that my project is worth trying. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 5.2 Dedekind-Peano axioms 47 So, let us look at axioms for arithmetic introduced by Richard Dedekind (but commonly called Peano axioms) in the hope that they provide some insight in the nature of EHK’s and AB’s difficulties. 5.2 Dedekind-Peano axioms Recall that the Dedekind-Peano axioms describe the properties of natural numbers N in terms of a “successor” function S(n). There is no canonical notation for the successor function, in various books it is denoted s(n), σ(n), n′ , or even n++ , as in popular computer languages C and C++ . Axiom 1 1 is a natural number. Axiom 2 For every natural number n, S(n) is a natural number. Axioms 1 and 2 define a unary representation of the natural numbers: the number 2 is is another name for S(1), and, in general, any natural number n is S n−1 (1) = S(S(· · · S(1) · · · )) (n − 1 times). As we shall soon see, the next two axioms deserve to be treated separately; they define the properties of this representation. Axiom 3 For every natural number n other than 1, S(n) 6= 1. That is, there is no natural number whose successor is 1. Axiom 4 For all natural numbers m and n, if S(m) = S(n), then m = n. That is, S is an injection. The final axiom (Axiom of Induction) has a very different nature and is best understood as a method of reasoning about all natural numbers. Axiom 5 If K is a set such that: • 1 is in K, and • for every natural number n, if n is in K, then S(n) is in K, then K contains every natural number. Thus, Dedekind-Peano arithmetic is a formalisation of that very counting by one that little EHK did, and addition is defined in precisely the same way as EHK learned to do it: by a recursion m + 1 = S(m) m + S(n) = S(m + n). Commutativity of addition is a non-trivial theorem (although still accessible to a beginner). To force you to sympathize with poor little EHK and to poor little AB, I will prove it to you in Section 5.5.2. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 48 5 Adding One by One 5.3 A brief digression: is 1 a number? Having postponed more serious work, we can spend a few minutes discussing Axiom 1: 1 is a natural number. Even this axiom is not as self-evident as it appears to be. In many languages, including English, the word “number” can denote some collection or ensemble of objects with the tacit understanding that it contains at least a few, and in any case more than one, objects. For example, “A number of people feel that 1 is not a number” makes sense and suggests that more than one person thinks that 1 is not a number. Such usage reflects an earlier stage of development of the system of numerals in which 1 was not a number; numbers were made of ones, of basic units; but 1 is not made of ones. What is very important for the history of mathematics, it appears that, for similar reasons, 1 was not a number for ancient Greek mathematicians, as evidenced in Euclid’s Elements: Euclid carefully separated the use of the words “number” and “unit”. And, as a digression within digression, I want to mention the issue of collective nouns—I shall discuss them again in later chapters, so the present digression is not waste of time. The English language has a peculiar tendency to form or find a special word to denote groups of particular animals. For example, Englishmen say a herd of cows, a flock of sheep, a pack of dogs, a school of fish. To illustrate how far things go, it will suffice to mention that ducks on water form a paddling, but when in flight they are a flush. Some nouns are obscure; for example, I found in Wikipedia a sedge of bitterns, but I did not even know what bitterns are. A dictionary told me that bittern was a familiar vypь—which puzzled me even more because I thought that vypь was a rather solitary bird; see Figure 5.1. The invention of collective nouns for groups of people from various professional groups is an (admittedly, fringe) genre of English humor; to my taste, a number of mathematicians appears to be one of the more obvious solutions. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 5.4 How much mathematics can a child see at the level of basic counting? Fig. 5.1. Bittern (Botaurus stellaris). 5.4 How much mathematics can a child see at the level of basic counting? We postpone the promised systematic development of the concept of addition once more and will first look at how much child can see at the level of basic counting and the most primitive form of addition, by adding one by one. Quite a lot, apparently, as confirmed by a striking testimony from Roy Stewart Roberts4 : On a large sheet of paper I made a triangle of numbers and addition signs as below. Down the right side I made a list of the results of the additions. It was clear that this process could continue as long as I wanted and my attention went to the vertical sequence on the right. . 1 2+2 3+3+3 4+4+4+4 5+5+5+5+5 6+6+6+6+6+6 7+7+7+7+7+7+7 ... 0 1 4 9 16 25 36 49 It was clear that the numbers in the sequence increased more rapidly as you went down so I formed the sequence of first differences. Of course I obtained the odd numbers. So I thought, “Is this true in general? Does the sequence continue always to generate the odd numbers no matter how far we go?” I also thought, 4 RSR tells about himself: “ As an adult I obtained a PhD in mathematics, differential topology–group actions on manifolds and fibre bundles, and now am retired if mathematicians ever retire. My mother tongue is English and the above mathematics was all in English.” The episode took place before he went to school. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 49 50 5 Adding One by One “Can I prove it?” and asked my father, who had a PhD in chemistry. He confirmed that the odd numbers were indeed correct and mentioned algebra. I wondered how can he know and can I prove it? I think I thought in terms of a proof based on counters; I did not know my addition tables and certainly not my multiplication tables, and performed the additions by counting, mainly in my head but possibly also using my fingers. I did not properly formulate a proof based on counters until grown up, as I later had algebra that made the result obvious anyway. A proof based on counters is quite easy and possibly I got near to it at the time. Perhaps I did not continue thinking about the matter to the point of constructing a proof because I became aware of the question, “Even if I get a proof, how will I know the proof is correct?” This question bothered me. I think I was aged four at the time, coming up to five, just after the Second World War was ended. Roy Roberts’s story mentions further interesting insights: The point at the top of the triangle denoted zero zeroes added together. The symbol “0” would not have been correct and I had a little difficulty deciding what I should put at the top. Evidently I understood zero. At some point, probably earlier than the research, I had discovered that you can continue counting forever, using the usual representation of numbers if one ran out of names. Roy Roberts mentioned “a proof based on counters” for the formula 1 + 3 + 5 + · · · + (2n + 1) = (n + 1)2 (5.1) (where, we have to remember, (n + 1)2 means “n + 1 added with itself n + 1 times”). He added an explanation: In order to subtract a square number from the next in sequence, think of a square array of counters for the larger and remove a square array for the smaller, necessarily based in a corner. The result is an “L” shape of counters with the two legs of the “L” of equal length. This clearly generates the odd numbers but to see this just repeat the above argument, remove from the “L” the smaller “L” in sequence and the result is two separate counters, hence generating the odd numbers. We can also easily see that this proof is an oral description of the famous visual proof; see Figure 5.2. It is a pity that the picture was not shown to 4 years old Roy. It is wort noticing that a proof of formula (5.1) by mathematical induction is interesting for historic reasons: it appears to be a very first proof in print based on explicitly formulated principle of mathematical induction. It was done by Augustus De Morgan in 1838 in a article in The Penny Cyclopedia of the Society for the Diffusion of S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 5.4 How much mathematics can a child see at the level of basic counting? Fig. 5.2. A visual proof of 1 + 3 + 5 + · · · + (2n + 1) = (n + 1)2 . Image source: [860]. Useful Knowledge [225]. I quote it at length; notice that De Morgan uses the term successive induction: INDUCTION (Mathematics). The method of induction, in the sense in which the word is used in natural philosophy, is not known in pure mathematics. There certainly are instances in which a general proposition is proved by a collection of the demonstrations of different cases, which may remind the investigator of the inductive process, or the collection of the general from the particular. Such instances however must not be taken as permanent, for it usually happens that a general demonstration is discovered as soon as attention is turned to the subject. There is however one particular method of proceeding which is extremely common in mathematical reasoning, and to which we propose to give the name of successive induction. It has the main character of induction in physics, because it is really the collection of a general truth from a demonstration which implies the examination of every particular case; but it differs from the process of physics inasmuch as each case depends upon one which precedes. Substituting however demonstration for observation, the mathematical process bears an analogy to the experimental one, which, in our opinion, is a sufficient justification of the term ‘successive induction.’ A couple of instances of the method will enable the mathematical reader to recognise a mode of investigation with which he is already familiar. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 51 52 5 Adding One by One Example 1.—The sum of any number of successive odd numbers, beginning from unity, is a square number, namely, the square of half the even number which follows I he last odd number. Let this proposition be true in any one single instance; that is, n being some whole number, let 1, 3, 5 up to 2n + 1 put together give (n + 1)2 . Then the next odd number being 2n + 3, the sum of all the odd numbers up to 2n + 3 will be (n + 1)2 + 2n + 3, or n2 + 4n + 4, or (n + 2)2 . But n + 2 is the half of the even number next following 2n + 3: consequently, if the proposition be true of any one set of odd numbers, it is true of one more. But it is true of the first odd number 1, for this is the square of half the even number next following. Consequently, being true of 1, it is true of 1 + 3; being true of 1 + 3, it is true of 1 + 3 + 5; being true of 1 + 3 + 5, it is true of 1 + 3 + 5 + 7, and so on, ad infinitum. And a touching final comment comes from Michael Breen5 : My clearest memory is creating a sequence for my father—one of those “guess the next term” problems, which are probably not logically sound. He was sick, so I thought I would give him something to think about to occupy his time, because he liked numbers. So I took the perfect squares, 1, 4, 9, 16, 25, 36, 49, . . . , subtracted adjacent terms, and asked him to guess the next term. Of course, the sequence was 3, 5, 7, 9, 11, 13 the odd numbers, but I completely missed that and was shocked when he solved it so quickly! It is a lesson in how I often think: Concentrating too much on the details and missing the big picture– a bad trait for a mathematician to have. 5.5 Properties of addition After all the delays and detours we finally come to rigorous mathematical exploration of addition. I have already mentioned that there are several alternative forms of notation for the successor function: S(n), s(n), σ(n), n′ and even n++ . I shall use notation n′ ; as the reader will soon see, it is very convenient—and natural—to write a symbol for the successor function after the number that has to be incremented. In this new notation, the recursive rule for addition looks like 5 MB received his PhD at the University of Arkansas. After teaching at the university level for 12 years, he became a Public Awareness Officer at the American Mathematical Society. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 5.5 Properties of addition 53 (5.2) n + 1 = n′ ′ ′ n + m = (n + m) . (5.3) I will also use Axiom 5 in a more conventional form, which is obviously equivalent to the original one. Axiom 5 Assume that a certain statement about numbers is such that: • the statement is true for number 1 (Basis of Induction); • if the statement is true for a natural number n (Inductive Assumption) then it is true for the next number n′ (Inductive Step). Then the statement is true for all natural numbers. I now prove two canonical properties of addition. 5.5.1 Associativity of addition Theorem 1. Assume that + is a binary operation which satisfies conditions (5.2) and (5.3). Then + is associative, that is, (a + b) + c = a + (b + c) for all a, b, c. Proof. The proof uses induction on c. Basis of Induction. (a + b) + 1 by (5.2) = (a + b)′ by (5.3) = a + b′ by (5.2) = a + (b + 1). Inductive Assumption: (a + b) + c = a + (b + c). Inductive Step. (a + b) + c′ by (5.3) = ((a + b) + c)′ by inductive assumption = (a + (b + c))′ by (5.3) = a + (b + c)′ by (5.3) = a + (b + c′ ).  S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 54 5 Adding One by One 5.5.2 Commutativity of addition We start with a very special, but crucially important case. Theorem 2. Assume that + is a binary operation which satisfies conditions (5.2) and (5.3). Then 1+a=a+1 for all a. Proof. We shall prove the theorem by induction on a. Basis of Induction. 1 + 1 = 1 + 1. There is nothing to prove here. Inductive Assumption: 1 + a = a + 1. Inductive Step. 1 + a′ by (5.3) = (1 + a)′ by inductive assumption = (a + 1)′ by (5.2) = (a′ )′ by (5.2) = a′ + 1.  Theorem 3. Assume that + is a binary operation which satisfies conditions (5.2) and (5.3). Then + is commutative, that is, a+b=b+a for all a and b. Proof. We prove the theorem by induction on b. Basis of Induction: Theorem 2. Inductive Assumption: a + b = b + a. Inductive Step. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 5.6 Dark clouds 55 a + b′ by (5.3) = (a + b)′ by inductive assumption = (b + a)′ by (5.2) = (b + a) + 1 by Theorem 1 = b + (a + 1) by Theorem 2 = b + (1 + a) by Theorem 1 = (b + 1) + a by (5.2) = b′ + a.  5.6 Dark clouds Notice that I was careful to formulate Theorems 1–3 in the most cautious way, by emphasizing their conditional nature: if + is a binary operation which satisfies conditions (5.2) and (5.3) then . . . The reason for my restraint is that writing down conditions (5.2) and (5.3) does not mean they define a function. Another problem is that if you look at the proofs of Theorems 1– 3, you notice that they do not refer to Axioms 3 and 4 and are based entirely on the Induction Axiom, Axiom 5. Therefore if we can exhibit a “toy version” of a system of natural numbers where we have a distinguished element 1, and the successor function S, and the Induction Axiom, but have no Axioms 3 and 4, we shall still be able to define addition by conditions (5.2) and (5.3), and perhaps some other functions. David Pierce [77] suggests that we take for such a “toy model” a system of residues Z/nZ modulo n, with residue 1 in the role of the distinguished element, and with a successor function x 7→ x + 1 mod n. David Pierce makes an incisive comment: Indeed, if one thinks that the recursive definitions of addition and multiplication— n + 0 = n, n + (k + 1) = (n + k) + 1; n · 0 = 0, n · (k + 1) = n · k + n S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 56 5 Adding One by One —are obviously justified by induction alone, then one may think the same for exponentiation, with n0 = 1 nk+1 = nk · n. However, while addition and multiplication are well-defined on Z/nZ (which admits induction), exponentiation is not. Indeed, the recursive definition of exponentiation fails in Z/3Z: n n2 n3 n3 × n n4 2 1 2 1 2 (so n3 × n is not equal n4 , what is disappointing). But the recursive definition of exponentiation holds in Z/6Z: n 1 2 3 4 5 6 n2 1 4 3 4 1 6 n3 1 2 3 4 5 6 n4 1 4 3 4 1 6 n5 1 2 3 4 5 6 n6 1 4 3 4 1 6 n7 1 2 3 4 5 6 The former is an exception rather than rule, as clarified by the following theorem. Theorem 4 (Don Zagier [97]). On Z/nZ, the recursive definition of exponentiation n0 = 1 nk+1 = nk · n. is valid if and only if n ∈ {0, 1, 2, 6, 42, 1806}. The desired n are such that xn+1 ≡ x (mod n); such n were found by John Dyer-Bennet [12] and Don Zagier [97]. Indeed the function known in modular arithmetic as exponentiation is a map (Z/nZ)∗ × Z/φ(n)Z → Z/nZ (x, y) 7→ xy , where (Z/nZ)∗ , as usual, denotes the group of invertible elements of the residue ring Z/nZ. Its order is φ(n) for Euler’s function φ, S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 5.7 Induction and recursion 57 and by Lagrange’s Theorem (known in this special case as Euler’s Theorem, aφ(n) = 1 for every a ∈ (Z/nZ)∗ , hence raising a to powers from Z/φ(n)Z make sense. This modular exponentiation is widely used in cryptography; we shall return to it later in the book. Notice that modular exponentiation is not a function Z/nZ × Z/nZ → Z/nZ. When we state Fermat’s Theorem ap ≡ a mod p the exponent p is not to be understood modulo p. I share David Pierce’s indignation at the state of affairs [77]: Yet the confusion continues to be made, even in textbooks intended for students of mathematics and computer science who ought to be able to understand the distinction. Textbooks also perpetuate related confusions, such as suggestions that induction and ‘strong’ induction (or else the ‘well-ordering principle’) are logically equivalent, and that either one is sufficient to axiomatize the natural numbers. [. . . ] This is one example to suggest that getting things straight may make a pedagogical difference. But I have to admit that I shared the widespread ignorance until David Pierce brought my attention to the issue—despite the fact that, in a calculus course that I took in the first year of my university studies, the lecturer (Gleb Pavlovich Akilov) explicitly proved the existence of a function of natural argument defined by a recursive scheme [1]. To save our theory from collapse, we shall prove the existence of addition after a brief pedagogical digression. 5.7 Induction and recursion From a pedagogical point of view, recursion is simpler than induction— because it goes back, to smaller numbers and simpler cases. This is supported by another childhood story, from BB6 : I would like to point out that the stories are provided by the people who happen to stumble on this blog. I am sure that the readership of this blog is atypical in terms of mathematical thinking 6 BB was 11 or 12 years old at the time of the story. He is male, Russian, currently a PhD student in pure mathematics. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 58 5 Adding One by One Fig. 5.3. A cyclic Gray code discussed by BB in his story is a cyclic path on edges of the n-cube passing exactly once through each vertex of the cube. In this picture, n = 3. and learning. Most people who have responded have not only gone to deal with unusually abstract concepts in their career, but actually do mathematics. So, the examples here might represent not so much the major difficulties that need to be overcome before an understanding can be reached (as in finding the correct way of thinking of division of apples by apples), but the signs that understanding has already been reached, and that the difficulty is purely semantic, i.e. how to express it. My own story: In our math circle we covered induction (domino analogy, proofs of summation formulae such as ! n+1 , 1 + ··· + n = 2 and varied other examples). I did passably well on the problems, but still I did not understand what induction was really for, until the end-of-year competition. I failed to solve a single problem: arrange all binary strings of length 10 around the circle so that two adjacent strings differ in precisely one position (it is known as a cyclic Gray code of size 10). It was when I was told the solution that I felt that I finally understood the induction. The missing element was probably the fact that I did not realize that the statement proved by induction is an honest mathematical statement that pertains to concrete numbers like 10, and not only to x, y, n, m and 1996, among which only the latter is a number, but so big and arbitrary that it could as well be denoted by n. And another story, from RTC:7 I do remember that when I was about 12 or 13 at school we were taught mathematical induction for the first time. Despite the fact 7 RTC is male, British, a professor of mathematics in a British university. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 5.8 Digression into infinite descent 59 Fig. 5.4. Infinite descent. This poster gave birth to the term Droste effect. Source: W IKIMEDIA C OMMONS . Public domain. Günter Törner writes about a similar image: At the age of 7 I get aware of some advertising of a brandy (gin) showing the bottle placed into a nice landscape. And on the label on the bottle there was the same drawing, however smaller (and on this picture on the bottle was a smaller bottle etc.). Ad infinitum—I thought and I was fascinated by the idea. that I was supposed to be the maths whizz who would catch on to any new concept very quickly, I distinctly remember feeling cheated by induction. At that age we had only ever proved things directly, and with induction we seemed to be side-stepping the issue and not proving anything. 5.8 Digression into infinite descent Let us have a look at the historic predecessor of induction, principle of infinite decent. Its use can be traced to Proposition XIII.6 of Euclid’s Elements8 . 8 http://www.claymath.org/library/historical/euclid/ files/elem.13.6.html. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 60 5 Adding One by One Fig. 5.5. Golden Rectangle. The quickest proof by infinite descent is perhaps proof of irrationality of the Golden Ratio (I found it in a beautiful little book by Tim Gowers [126, p. 44]). Theorem 5. The Golden Ratio is irrational. Proof. By definition, the rectangle ABCD is a golden rectangle, if after cutting off the square BBCC, the remaining rectangle ADCB is similar to the original one, ABCD. The ratio of lengths of sides of a golden rectangle is called the golden ratio. If the golden ratio were rational, a “golden rectangle” could be drawn on square grid paper; see Figure 5.5. After cutting a square from it we get a smaller “golden rectangle” drawn on square grid paper. By principle of infinite descent, this is impossible—hence the golden ratio is irrational. Hmm, we did not even care about the numeric value of the golden ratio . . .  And, to conclude this section, I give one more example of an √ argument by infinite descent: proof of irrationality of 2. √ Theorem 6. 2 is not rational. Proof. We √ shall prove an equivalent, but a stranger looking statement: if 2 is rational then it is an integer. Set √ p 2= q where p and q are natural numbers; select such representation for √ 2 with the smallest possible q. Our aim is to prove that q = 1; if this is not the case then observe that q < p < 2q and S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 5.9 Landau’s proof of the existence of addition 61 2q − p p = q p−q (check this identity!) is a representation with a smaller denominator. A contradiction.  5.9 Landau’s proof of the existence of addition I decided to borrow verbatim a proof of the existence of addition from Edmund Landau’s famous book Grundlagen der Analysis [57]. It is from his book that I picked up the notation S(n) = n′ which I had already used in my proofs. Although this is not emphasized by Landau, the proof of consistency of addition does not use Axioms 3 and 4. Are these axioms of any use at all? We shall return to this question later. Theorem 7. [57, Theorem 4] To every pair of numbers x, y, we may assign in exactly one way a natural number, called x + y, such that (1) (2) x + 1 = x′ for every x, x + y ′ = (x + y)′ for every x and every y. Proof. (A) First we will show that for each fixed x there is at most one possibility of defining x + y for all y in such a way that x + 1 = x′ and x + y ′ = (x + y)′ for every y. Let ay and by be defined for all y and be such that a1 = x′ , b1 = x′ , ay′ = (ay )′ , by′ = (by )′ for every y. Let M be the set of all y for which ay = b y . (I) (II) a1 = x′ = b1 ; hence 1 belongs to M. If y belongs to M, then ay = by , hence by Axiom 2, (ay )′ = (by )′ , therefore ay′ = (ay )′ = (by )′ = by′ , so that y ′ belongs to M. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 62 5 Adding One by One Hence M is the set of all natural numbers; i.e. for every y we have ay = b y . (B) Now we will show that for each x it is actually possible to define x + y for all y in such a way that x + 1 = x′ and x + y ′ = (x + y)′ for every y. Let M be the set of all x for which this is possible (in exactly one way, by (A)). (I) For x = 1, the number x + y = y ′ is as required, since x + 1 = 1′ = x′ , x + y ′ = (y ′ )′ = (x + y)′ . Hence 1 belongs to M. (II) Let x belong to M, so that there exists an x + y for all y. Then the number x′ + y = (x + y)′ is the required number for x′ , since x′ + 1 = (x + 1)′ = (x′ )′ and x′ + y ′ = (x + y ′ )′ = ((x + y)′ )′ = (x′ + y)′ . Hence x′ belongs to M. Therefore M contains all x.  Landau’s book is characterized by a specific austere beauty of entirely formal axiomatic development, dry, cut to the bone, streamlined. Not surprisingly, it is claimed that logical austerity and precision were Landau’s characteristic personal traits.9 Grundlagen der Analysis opens with two prefaces, one intended for the student and the other for the teacher; we shall return to Preface for the Teacher in Section 7.3.1, it is a remarkable pedagogical document. The preface for the student is very short and begins thus: 1. Please don’t read the preface for the teacher. 2. I will ask of you only the ability to read English and to think logically-no high school mathematics, and certainly no higher mathematics. [. . . ] 3. Please forget everything you have learned in school; for you haven’t learned it. Please keep in mind at all times the corresponding portions of your school curriculum; for you haven’t actually forgotten them. 9 Asked for a testimony to the effect that Emmy Noether was a great woman mathematician, Landau famously said: “I can testify that she is a great mathematician, but that she is a woman, I cannot swear.” S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 5.9 Landau’s proof of the existence of addition 63 4. The multiplication table will not occur in this book, not even the theorem, 2 × 2 = 4, but I would recommend, as an exercise for Chap. I, section 4, that you define 2 = 1 + 1, 4 = (((1 + 1) + 1) + 1), and then prove the theorem. Our discussion of natural numbers continues in Chapter 7. Exercises Exercise 5.1 Prove Theorem 4 for prime values of n. You may wish to use Fermat’s Theorem: If p is a prime integer and 0 < a < p then ap ≡ a mod p. Exercise 5.2 Then try to prove Theorem 4 in full generality. Exercise 5.3 Follow Edmund Landau’s advice and prove from the DedekindPeano axioms that 2 × 2 = 4. Exercise 5.4 Assuming the existence of multiplication on N, prove the existence and uniqueness of the exponentiation N × N → N, (n, k) 7→ nk satisfying the recursive relations n0 = 1, nk+1 = nk · n. Exercise 5.5 Make an infinite descent proof of irrationality of on this diagram: √ 2 based Exercise 5.6 Modify the proof of Theorem 6 to prove irrationality of S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 √ 3. c A LEXANDRE V. B OROVIK 6 What is a Minus Sign Anyway? 6.1 Fuzziness of the rules After our study of associativity of addition (Chapter 5, Theorem 1), we can turn our attention to subtraction. This is a story from Jo French.1 At school, I was confused by the order of operations: BODMAS (brackets, “powers of”, division, multiplication, addition and subtraction). It obviously doesn’t work: You don’t add numbers before you subtract them, e.g. a − b + c is not a − (b + c). This very issue has been addressed John Baldwin in a paper “What is a minus sign anyway?” placed on the Internet. John Baldwin asks a seemingly naive question: in school mathematics teaching, how do we justify manipulations like (40 + 20) − (12 + 5) = (40 − 12) + (20 − 5)? John Baldwin writes: The associative law can only work on two applications of the plus sign. We generalize it to say we can regroup any sequences of additions. It is quite plausible that this is a distinction one should not make for teachers or at least the question should be at what level you want them to be aware of it. The second problem is that when minus signs are interspersed this gets more complicated. Since associativity fails for subtraction some further rules are required. Indeed, notice that (3 − 2) − 1 1 JF is female, English, a mathematics PhD student in a leading university. At the time of this episode she was about 8 or 9 years old. This her story continues on Page 66. 65 66 6 What is a Minus Sign Anyway? is not the same as 3 − (2 − 1). Subtraction is not associative! This is a classical example of a “fuzziness” in mathematics teaching, a phenomenon especially noticeable at earlier stages of school education. Very frequently, children are placed in a position where they have to figure out rules which have not been made explicit. Of course, children learn the grammar of their mother tongue exactly that way, by absorption. Unfortunately, by the time they are taught mathematics, their natural ability to extract grammar rules from adult’s speech is already significantly suppressed. Striking the right balance of rigor and computational fluency is a hard task. Still, what struck me in John Baldwin’s paper was his reluctance to bring this subtle distinction between the unary operation of negation and the binary operation of subtraction to the attention of (American) teachers: It is quite plausible that this is a distinction one should not make for teachers or at least the question should be at what level you want them to be aware of it. John Baldwin commented further in my blog: You are well to wonder when I question whether or maybe better how teachers should be taught this. One of the faults of a preliminary version is that certain facts understood by the writer and immediate audience are not spelled out. Thus we were talking in the seminar for whom that was written about future elementary school teachers. And the issue that I was alluding to was at what stage you can make such students self-aware of subtle matters. One can’t just tell them that one operation is binary and the other unary. Because they have never heard either word. One of the members of the audience had been extremely successful in teaching this group of students without making such formal distinction but by giving them lots of opportunity by operating on lattices of numbers to develop the understanding —analogously to, as you mentioned, children learning their native tongue. In fact, I haven’t really resolved this issue in my own mind and recently have been working with high school teachers where the more explicit understanding is essential (and with some effort) communicable. But return to Jo French. She also wrote: I didn’t really feel happy with what subtraction meant generally– I would just add negative numbers. I only really felt happy with this after learning what a group is. (Obviously the same is true for division and multiplication, but I remember being unhappy with subtraction more clearly.) S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 6.2 A formal treatment of subtraction 67 It appears that little Jo were happier with much more formal treatment of subtraction as addition of negative numbers. As we shall see in the next section, she was not alone. 6.2 A formal treatment of subtraction My friend and colleague Tuna Altınel2 learnt “new math” style mathematics in a Turkish school where sciences had been taught in French and in accordance with French curricula3 . He surprised me by telling me that his teachers and textbooks made a clear distinction between “minus” as a binary symbol for subtraction and minus as an unary symbol for taking the opposite. Actually, the opposite of x, which we would denote −x, was written as opp(x), and, after skipping a few intermediate applications of the commutative and associative laws for addition, the example above would look like (40 + 20) − (12 + 5) = (40 + 20) + opp(12 + 5) = (40 + 20) + (opp(12) + opp(5)) = (40 + opp(12)) + (20 + opp(5)) = (40 − 12) + (20 − 5). A very clean but excessively detailed calculation. Tuna Altınel wrote to me recently: Just in passing, you may mention that it took some time before comfortably replacing the use of opp with that of the − sign. Certainly, it is another question whether this is related to the way teaching was done, or a lack of personal mathematical vocation. I have just found among my books my sixth grade math book [670] and also the famous definition: a − b = a + opp(b) This is written in a pink colored rectangle as a rule for integers and followed in the same rectangle by an explanation: “soustraire un entier, c’est ajouter son opposé”. The word soustraire is bold face, and above the first part of the sentence there is an upward arrow pointing in the direction of the “−” sign and an oblique double arrow from “entier” towards “b”. The verb “c’est” is above the equality sign. Then there is another upward vertical arrow from “ajouter” to “+” and an oblique double arrow from “opposé” to “opp”. 2 3 TA is male, Turkish, has a PhD in mathematics from an American university and teaches in a French university. Galatasaray Lisesi, http://www.gsl.gsu.edu.tr/html_tr; a page in English: http://www.gsl.gsu.edu.tr/html_eng/000/000. html. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 68 6 What is a Minus Sign Anyway? I must say I found this definition very easy to grasp. I have a vague recollection that this was not the case for many of my classmates. Now that I am looking at the pages I also see that there is another notation used for negative numbers: 5− meaning minus five. By the way, this appears to be a common theme in childhood memories of mathematicians: those who were affected by the “new maths” easily survived it—and frequently enjoyed the experience. Tuna Altınel continues: I don’t know if this is relevant to what I wrote above but the book I quoted from has had an enormous place in my path to mathematicianship. We had very bad teachers in the 6th and 7th grades. The first teacher I had in the sixth grade had to quit for administrative reasons, then for a long time we didn’t have anyone, and finally someone who knew some math but not teaching, and whose French was worse than the lowest level student in my class. My brother pushed me to studying the book myself and finishing in summer the chapters left undone during the school year. Luckily, my brother’s4 and sister’s5 notes were available also. Themselves as well, but as a rule my brother would force me to “fry myself in my own oil”. 6.3 A formal treatment of negative numbers The next two stories are also from “new maths” era, French style; The first story is told by Olivier Gerard6 . You have already a few examples of new math courses in your book. In my case it was in 1979–1980 with a very motherly female teacher that I remember very vividly. In hindsight, it is evident she mastered the principles as well as the way to impose it on young pupils. But many of her colleagues did not. Here is an account of a point when I was ten about negative numbers. It doesn’t add much if anything to what you already recount in your book, but let’s say it is some corroborative evidence. As I have recently explained this to an English speaking friend in response to some inquiries and also as a counterpoint to some 4 5 6 TA’s big brother Kuban Altınel is a professor of computer science. Evren Altınel is a medical doctor, but, in her own words, she enjoyed mathematics at school. OG tells about himself: “To match the kind of characterization of people you use, I am a French-speaking male of 40 years, educated only in French during the first seventeen years of his life, now a mathematical researcher and computer scientist in the private industry. I have been interested in mathematics for an early age, at least 7 by my own recollections and papers and if I take my parents word, at least 4 years old, asking questions and reasoning about quantities and counts of things.” S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 6.3 A formal treatment of negative numbers 69 theoretical discussions between us about concept formation in the history of mathematics, Concerning this problem of introducing negative numbers, a little personal recollection. When I was in junior high school (French College 6e and 5e, for 11 and 12 years old students) 30 years ago, our mathematics teacher took a lot of care to distinguish for us during two weeks negative signs as affixed to numbers to make them negative, and on the other hand negative and plus signs as operations between numbers, giving us precise rules of operations and developments in which minus ⊖ and plus ⊕ where circled for operations and simple +, − when attached to numbers. This was without any geometric imagery (the first introduction I remember of such was in 4e and 3e for introducing coordinate systems and vectors, distinguishing direction and sign, first only in one dimension for weeks then in two dimensions). For two weeks we made only progressive exercises in this way, reducing in small steps an expression with a lot of parentheses and visually clashing signs to a final result where we were at least authorized to make the final arithmetical operation that we had practiced in primary school. At first we were told exactly what kind of simplification to do from one line to the next, then we could do it in the order we found the most convenient. It was a lot of effort and demanded good focus and systematic application of rules before they began to be internalized. And then after this time, she told us to consider, “as a convenient abbreviation”, a number without a sign + as a positive number, and to treat any minus sign in front of a parenthesis as the “subtraction” or “minus” operation. She then rewrote the rules in a more direct and compact form as the projected shadow of a more rigorous and explicit “reality” we had been experiencing for days. We were (at least the few boys and girls I used to chat with) so happy to write and treat now the same exercises quickly and efficiently (and a little mindlessly or automatically ) that we felt a little superior to our former selves and began to look at complicated expressions with a sense of familiarity. When discussing this episode with a former schoolmate, say ten years later while at the University, I was shocked that the only thing he remembered about this is that in math, they were changing rules all the time. I told him that it was on the contrary a very close simile to what was done in natural languages all the time: you learned quick but not very correct or not very precise ways to say things when speaking orally with your parents, friends and radio, but you learned also in school a full and scholarly way of saying the same thing in much longer form which could be analyzed with grammar and could be used for more varied purposes than the quick and dirty way. We had learned years before to subtract numbers as a converse of addition and we had then learn in high school a fuller story where the origin and quality of things S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 70 6 What is a Minus Sign Anyway? were more explicit, we learned of new distinctions that we did not imagine before. The second “new maths” story comes from Pierre Arnoux7 . My story goes back to the last year of high school (“terminale”). We were studying modern maths, and the lecture of the day was the construction of the relative integers (Z) from natural (positive) integers (N) by taking the cartesian product and factoring it by an equivalence relation; that is, the classical construction of the symmetrization of the commutative monoid. We could follow, and understand, all the steps in the construction, but the goal of the whole exercise eluded us completely; what were we doing? I think nobody in the class understood what was going on. After the class, during the 10 minutes break, I went to see the teacher, and asked her what was the point of the construction: after all, we had known negative integers for years, and there was absolutely nothing new in this. Fortunately, the teacher understood very well the subject (she had participated in writing the book), and was able to answer. She explained that we should consider this as a game, and this proved that, to obtain negative integers, we did not need to invent something completely new coming from outside: just playing with what we knew, the positive integers, and using a few tool, we could build negative integers, fractions, real numbers, complex numbers . . . The explanation was very convincing, and put a new light on the course; I was very happy to understand that, and very surprised that it had never been mentioned anywhere: we were, in effect, giving complicated answers to tricky, and quite philosophical, questions that had never been asked! I think that this short conversation had a decisive impact in turning me towards mathematics. What surprised me also is that, a few years later, after my Ph. D., I met again this high school teacher, and told her this story; she had absolutely no memory of this discussion! When I remember this moment, It evokes two ideas for me: • the first one is that fundamental understanding often comes in flash; a ten-minutes discussion allows you to make a big progress, while you were stuck for hours of hard work without going anywhere (but, you should not forget that, probably, without the hours of hard work, the flash would not come . . . ). • the second one is that it can happen that the same discussion seems completely mundane and without particular interest to one participant, and illuminating to the other one; in particular, as a teacher, we tend to forget that, because most of the time, we know quite well what we are speaking about. 7 PA is male, French, a professor of mathematics in a French university. Another story from him is on Page 22. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 6.4 Testimonies 71 And finally, Ted Eisenberg8 expresses (perhaps with the benefit of hindsight) a wish for a more formal approach: I recall that in the classroom we were talking about removing nested parentheses that were preceded by a negative sign. Things like: −(3 − 5), −(5 − (3 − 5)), −4(−2(5 − 2(3 − 5))). The class was having trouble and Mr. Pribnow asked if we could somehow reverse the order of (a − b). I realized that we could look at it as: a − b = a + (− 1)b = (− 1)(− 1)a + (− 1)b = (−1)[(− 1)a + b] = (−1)[b + (−1)a] = (−1)[b − a] = −(b − a). By doing this the whole notion of differentiating between the natural numbers N, the integers Z as being “signed” numbers, and that the operation of subtraction can be written as a sum of signed numbers a − b = a + (−b), became crystal clear to me. I remember being terribly excited about this observation and I went on to explain it to several friends. A large door in algebra had opened for us, and to this day I think of the subtraction of natural numbers in its equivalent form of adding two signed numbers. It amazes me that this equivalence is not drilled into children. I am convinced that so many of their problems in beginning algebra would disappear if it were. 6.4 Testimonies At this point, a few more childhood testimonies could be instructive. BS, aged 6 Each morning in class we spent about 90 minutes on arithmetic, and were usually issued with postcards containing three additions (of 3 two or three digit numbers) and three subtractions (of three digit numbers). I never got to the subtractions, being too slow: but finally worked out that they must be easier, as they involved only two numbers. So I started doing the three subtractions first; of 8 TE is male, American, a professor of mathematics education. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 72 6 What is a Minus Sign Anyway? course I got them all wrong because I just added the two numbers. I hadn’t realised it was a different sort of “sum”. About three months after the above, I’d mastered the algorithms for addition and subtraction in simple cases. We progressed to subtractions of the sort 72−55. The teacher asked “Now if we try to take 5 from 2 we can’t, so what can we do? Does anyone know?” I offered “Take the 2 from the 5”, and was not pleased to be told that wasn’t right. I was very indignant, even after being told the “correct” process; mine was shorter, easier and gave an answer of the required sort! I had no idea that “Addition” and “Subtraction” were other than formal algorithmic processes to write down an answer that was “right”, really no idea at all that there was “applied arithmetic” so as to speak. I don’t think that I got over this difficulty until very late in my maths education. I am pretty sure that as a third year student at university I still found it difficult to disentangle conceptually numbers and numerals, addition/multiplication from the usual algorithms to compute them. Clearly that’s got a lot to do with the way I learned arithmetic.9 BS’s story is echoed in a testimony from AH10 : AH, aged 9 My clearest memory of learning mathematics was when I was nine years of age. We had a blackboard-full of subtractions using decomposition to do. Because I had always previously subtracted the smaller ‘number’ from the larger ‘number’ (actually, digits but noone ever told me that) in each column, when I had to borrow ‘1’ from the ‘number’ in the next column and then continue with the subtraction, I remember feeling that this was time consuming and although I knew it was going to produce a page of wrong answers, I didn’t feel inclined to do it. Therefore I remember still subtracting the smallest digit from the largest, which was incorrect. The next day I think we moved on to the next aspects of mathematics. I don’t recall ever doing them again in that class, but I suppose we did. RE, aged 6 When I was—I suppose—about 6, my mother told me about negative numbers. We had not covered them at school, and she is not very mathematical. (I think she even expressed some scepticism 9 10 BS is male, English, has a doctorate in mathematics and a lifetime teaching mathematics in university setting. AH is female, a New Zealander living in England. She has a PhD in Mathematics Education. She is a mathematics teacher educator. This episode occurred during her schooling in New Zealand. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 6.5 Multivalued groups 73 that these were “proper numbers” rather than something that scientists had “made up”.) All the same, I got the basic idea, and was very excited about these strange new numbers. Next time we were studying subtraction in school, I got all the questions wrong! Probably I wanted to show off my new knowledge, so for “7 − 4 =?” I answered “−3”, and so on. Looking back, I had not understood that while addition is commutative, subtraction is not!11 Karen Petrie, aged 7 My confusion came at about the age of 7. We were sent home with a worksheet with many problems of the form a−b = c, where a and b were given integers and we had to provide c. One of the problems was something like 6 − 7 =? The rest had all had positive answers so this stumped me. My father, took some time to explain to me why the answer was −1. I went back into school to tell the teacher the next day, she was horrified that my father had explained this ‘mistake’ to me. She did so by saying you do not get a negative number of apples do you there are only 0 or more than 0. I took this to heat in such a manner that later on in my schooling when negative numbers were introduced, I was convinced they were always the result of mistakes. It took me a long time to realize that they were valid mathematical concepts, that could be useful, they just do not relate very well to apples.12 6.5 Multivalued groups The stories told by BS (Page 71) and AH (Page 72) are yet another confirmation of what increasingly appears to be a general principle: almost every error made by children because they thought that “it was natural to do it that way” is developed in the “grown up” mathematics into a serious theory. In BS’ and AH’s cases, a child did not care much about what is addition and what is subtraction, and, moreover, which is which. A systematic application of this approach means that non-negative integers N ∪ { 0 } can be endowed with a 2-valued binary operation (x, y) 7→ [x + y, |x − y|]. This turns out to be one of the basic examples of the so-called nvalued groups as introduced by Sergei Novikov and Victor Buchstaber; see [10] and Exercise 6.4 below for a more detailed discussion. To express the confusion which operation is which, [ ] denotes 11 12 RE is male, English, holds a PhD in mathematics. KP is female, British, holds a PhD, is researcher is computer science. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 74 6 What is a Minus Sign Anyway? a multiset, that is, an unordered set with possibly repeated elements. Multiset is a very natural concept; for example, the roots of the equations (x − 1)2 = 0 and (x − 1)3 = 0 form multisets [1, 1] and [1, 1, 1]. The origins of the theory of n-valued groups lie in algebraic topology. I do not know why this is happening; maybe mathematics is rich enough to contain an analogue of everything—or maybe there are some intrinsic reasons when every “simple and natural” alternative to “canonical” mathematics generates a rich mathematical theory. Exercises Exercise 6.1 Gwen Fisher13 contributed this old chestnut: There was a problem that stumped me as a child. I think my 3rd grade teacher told it to me, and I didn’t fully understand it until I was in grad school. It goes like this. . . 3 men go to stay at a hotel. The cost of 1 room is $30, so they decide to split the room 3 ways, each paying $10. Later, the hotel clerk realizes that he overcharged the men by $5 total, so he gives the porter 5 one dollar bills to give back to the men. The porter steals $2 and gives the men back the other $3. Thus, each man paid $9 for the room. So, 3 men, times $9 per man is $27, plus the $2 stolen by the porter is $29. But the room was originally $30, so where did the other $1 go? Exercise 6.2 Expand the system of natural numbers N by taking a disjoint union of two isomorphic copies of N (denoted N and −N) and a set { 0 } containing of a new symbol 0: Z = N ⊔ { 0 } ⊔ −N. Define on Z an operation of addition “+”, such that: • • • its restriction to N and −N coincides with the earlier defined operation + on N and −N. 0 is the identity element for +. hZ; 0, +i is a commutative group, thus making Z into the familiar additive group of integers. 13 GF is female, American; she has a MA in mathematics and a PhD in mathematics education, teaches mathematics and creates mathematically inspired beadwoven art—see http://www.beadinfinitum. com/ and http://www.gwenbeads.etsy.com/. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 6.5 Multivalued groups 75 Exercise 6.3 After that define on Z an operation of subtraction “−” and prove the identity discussed by John Baldwin: (a + b) − (c + d) = (a − c) + (b − d). You may be interested to know that this exercise is Theorem 237 of Landau’s Grundlagen der Analysis [57, p. 101]. But Landau was not in hurry to introduce subtraction; he first developed the theory of fractions, real and complex numbers, and only then proved the properties of subtraction, simultaneously for all these number systems. Exercise 6.4 [10] For a set X, we denote by (X)n be the set of unordered n-tuples of elements of X, possibly with repetitions. We shall use square brackets [ ] to write unordered n-tuples. For example, [1, 2, 1] and [1, 1, 2] represent the same element of (N)3 , while [1, 1, 2] and [1, 2, 2] are different elements of (N)3 . We shall call (X)n the n-multiset of X. An n-valued multiplication on X is a map ∗ : X × X → (X)n . x ∗ y = [z1 , z2 , . . . , zn ], zk = (x ∗ y)k . The following three axioms define n-valued group structure. • Associativity. The n2 -sets: [x ∗ (y ∗ z)1 , x ∗ (y ∗ z)2 , . . . , x ∗ (y ∗ z)n ], [(x ∗ y)1 ∗ z, (x ∗ y)2 ∗ z, . . . , (x ∗ y)n ∗ z] are equal for all x, y, z ∈ X. • Unit. An element e ∈ X such that e ∗ x = x ∗ e = [x, x, . . . , x] for all x ∈ X. • Inverse. A map inv : X → X such that for all x ∈ X. e ∈ inv(x) ∗ x and e ∈ x ∗ inv(x) Check that a 2-valued operation x ∗ y = [x + y, |x − y|] defines a 2-valued group structure on R+ = {x ∈ R | x > 0} because it satisfies the axioms of the unit, inverse and associativity: • • • Unit: e = 0. Inverse: inv(x) = x. Associativity: 4-subsets of R+ [x + y + z, |x − y − z|, x + |y − z|, |x − |y − z||] [x + y + z, |x + y − z|, |x − y| + z, are equal for all nonnegative x, y, z. ||x − y| − z|] and S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 7 Counting Sheep 7.1 Numbers in computer science An anonymous contributor to my blog1 , a professional computer scientist, once left the following comment: I would caution everyone . . . not to confuse “mathematical thinking" with “The thinking done by computer scientists and programmers". Unfortunately, most people who are not computer scientists believe these two modes of thinking to be the same. The purposes, nature, frequency and levels of abstraction commonly used in programming are very different from those in mathematics. This statement may appear to be extreme, but let us not to jump to conclusions and look first at a very simple example. I suggest to have a look at M ATLAB, an industry standard software package for mathematical (mostly numerical) computations.2 The following fragment of text is a screen dump of me playing with natural numbers in M ATLAB . >> t= 2 t = 2 >> 1/t ans = 1 2 0.5000 Mathematics under the Microscope, http://micromath.wordpress.com/. I apologize to my computer scientist colleagues who on a number of occasions explained to me that M ATLAB is nothing more but a glorified calculator. I choose M ATLAB because, for a lay person, it provides an easy, even if limited, insight into what is going on in computer realizations of natural numbers. 77 78 7 Counting Sheep What you see here is a basic calculation which uses floating point arithmetic for computations with rounding; lines starting with the prompt » are my input; unmarked lines are M ATLAB ’s response. Next, let us make the same calculation with a different kind of integers: >> s=sym(’2’) s = 2 >> 1/s ans = 1/2 Here we use “symbolic integers”, designed for use as coefficients in symbolic expressions. You can see that in the first example 1/2 was rounded as 0.5000, in the second case 1/2 is written as it is, as a fraction. Since M ATLAB keeps in its memory the values of the variables s and t, we may force it to combine the two kinds of integers in a single calculation: >> 1/(s+t) ans = 1/4 We observe that the sum s + t of a floating point number t and a symbolic integer s is treated by M ATLAB a symbolic integer. Examples involving analytic functions are even more striking: >> sqrt(t) ans = 1.4142 >> sqrt(s) ans = 2^(1/2) >> sqrt(t)*sqrt(s) ans = 2 We see that M ATLAB can handle two absolutely different representations of integers, remembering, however, the intimate relation between them. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 7.2 Counting sheep 79 M ATLAB is written in C++ . When represented in C++ , even the simplest mathematical objects and structures appear in the form of (a potentially infinite variety of) classes linked by mechanisms of inheritance and polymorphism. This is a manifestation of one of the paradigms of the computer science: if mathematicians instinctively seek to build their discipline around a small number of “canonical” structures, computer scientists frequently prefer to work with a host of similarly looking structures, each one adapted for a specific purpose. We shall look in the next chapters at how they keep control of this bestiary. For the time being, we have only to take note that we have to be prepared to look at many different number systems satisfying the Dedekind-Peano axioms. 7.2 Counting sheep Fig. 7.1. The shortening winter’s day is near a close. Joseph Farquharson, 1846–1935. Source: Wikimedia Commons. Public domain. The observation that concluded the previous section is nothing new if we turn our attention from computer languages to the natural human lore: we already dealt with “named” numbers. But “named” numbers can come in a much more extreme form, as numerals used for counting specific types of objects (most likely, they historically precede the emergence of the universal number system as we know it). In England, a popular slander about Yorkshiremen is that they use special numerals for counting sheep. Judging by S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 80 7 Counting Sheep the Lakeland Dialect Society website [868], local people proudly admit to sticking to the old ways. In Wensleydale, for example, the first ten sheep numerals are said to be yan, tean, tither, mither, pip, teaser, leaser, catra, horna, dick. If we turn to more modern times, it is entertaining to compare sheep numerals with Richard Feynman’s joke [22]: You see, the chemists have a complicated way of counting: instead of saying “one, two, three, four, five protons”, they say, “hydrogen, helium, lithium, beryllium, boron.” This a joke but we have to learn some lessons from it. One lesson is that we have to distinguish between ordinal numerals, which express relative order of objects, first, second, third, . . . and cardinal numerals which express the cardinality of a set, the number of elements: one, two, three, . . . To my eye, in Feynman’s joke the words hydrogen, helium, lithium,. . . look more like ordinal numerals. In languages around the world, there is a remarkable diversity of systems of numerals, both ordinal and cardinal. We have already discussed in Chapter 1 a special class of distributive numerals in the Turkish language. The Japanese language provides one of more striking examples of diversity of numerals. Here, different numerals are used for counting, for example, flat objects (like sheets of paper) and long slender objects (like pencils). I give a table of some of them: regular numbers 1 2 3 4 5 6 7 8 9 10 simple objects flat things long slender things ichi hitotsu ichimai ippo ni futatsu nimai nihon san mittsu sanmai sanbon shi or yon yottsu yonmai yohon go itsutsu gomai gohon roku muttsu rokumai roppon shichi or nana nanatsu nanamai nanahon hachi yatsu hachimai happon ku or kyu kokonotsu kyumai kyuhon ju or jyu tou jumai jyuppon S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 7.3 Abstract nonsense 81 7.3 Abstract nonsense: universal algebra and category theory 7.3.1 Existence and uniqueness It is time to pause and ask a natural question: why is the proof of consistency of the inductive definition of addition so difficult? We shall soon see that this is because, as frequently happens in mathematics, existence of nice additional structures on a mathematical object is closely related to the uniqueness of this object. Indeed, we had seen in Sections 7.1 and 7.2 that there exists a bewildering variety of systems of numerals and computer implementations of natural numbers. In the same time, we for some reason believe that they all are the same object: a unique canonical object which we call the system of natural numbers and denote by letter N. In his paper [77] David Pierce summarised difficulties demonstrated in Section 5.6 and Theorem 4: A set with an initial element and a successor-operation may admit proof by induction without admitting inductive or rather recursive definition of functions. Historically, this observation was made explicit by Dedekind [17, II.130] but overlooked by Peano [76]. Edmund Landau himself [57, Preface for the Teacher] confesses to committing an error— detected by his teaching assistant—in the first version of his lectures. In a more algebraic language, the issue is clarified by Henkin in [37]. In this section I shall present some slightly modified theorems from Henkin’s paper. 7.3.2 Unary algebras We shall need some algebraic terminology of a very general nature; similar terms are applied, for example, to groups and rings. A unary algebra is an algebraic structure A = hA; 1, Si consisting of a ground set A together with a distinguished element 1 and a unary function S : A −→ A. Notice that every unary algebra automatically satisfies Axioms 1 and 2. Theorem 8. A unary algebra A satisfies the Axiom of Induction (Axiom 5) if and only if A is generated by 1; S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 82 7 Counting Sheep Landau implicitly (and Henkin explicitly) shows that addition can be defined by induction alone. But, as we have just seen, the argument takes some work. Notice that it follows from Landau’s proof that addition is defined and commutative on any 1-generated unary algebra, in particular, on the algebra shown on Figure 7.2: 1 2 3 b b b n ··· b b n+1 Fig. 7.2. A unary algebra with commutative and associative addition (with thanks to David Pierce). And would not the reader agree that little Elizabeth Kimber and little AB had good reasons to be confused? 7.3.3 Proofs If a unary algebra hA; 1, Si satisfies the Induction Axiom (Axiom 5), we shall call it an induction algebra. If, in addition, an induction algebra satisfies Axioms 3 and 4, we shall call it a Peano algebra. Theorem 9. Let A = hA; 1A , Si be an arbitrary Peano algebra. If B = hB; 1B , Si is a unary algebra then there exists a unique homomorphism α : A −→ B. Proof. to be included  7.4 Induction on systems other than N David Pierce brought my attention to a beautiful example of the use of induction on a unary algebra other than N. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 7.4 Induction on systems other than N 83 Fig. 7.3. The Sun Peeped o’er yon Southland Hills. Joseph Farquharson 1846–1935. Source: Wikimedia Commons, http://commons. wikimedia.org/wiki/File:Farquharson_Joseph_Southland_ Hills.jpg. Public domain. Theorem 10 (Fermat’s Little Theorem). Let p be a prime. Then ap − a = 0 for all a ∈ Z/pZ. The following proof belongs to Leonard Euler [188]. It had been presented by Euler to the St. Petersburg Academy of Sciences on 2 August 1936 [204, p. 164]. Gauss reproduced it in his Disquisitiones Arithmeticae [193, art. 50]. Both Euler and Gauss were running induction in Z, but we can do the same in Z/pZ. We start by observation that the binomial coefficients   p! p = k!(p − k)! k for k = 1, 2, . . . , p−1 have denominators not divisible by p and therefore are divisible by p. Now it follows from the Binomial Theorem that p   X p p−k k p (a + b) = a b ≡ ap + bp mod p. k k=0 Now we are proving theorem by induction on a. The basis of induction is obvious: 1p = 1. Assume now that ap − a = 0 in Z/pZ and compute S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 84 7 Counting Sheep (a + 1)p − (a + 1) = ap + 1p − a − 1 = ap − 1 = 0, which proves the inductive step. 7.5 Categories I am in general leery of a remark like “Here’s what the axioms really mean” when it prefaces a category-theoretic discussion. David Pierce A category C consists of • • • a class ob(C) of objects: a class hom(C) of morphisms, or arrows between the objects. Each morphism f has a unique source object a and target object b where a and b are in ob(C). We write f : a −→ b, and we say “f is a morphism (or arrow) from a to b”. We write hom(a, b) to denote the class of all morphisms from a to b. for every three objects a, b and c, a binary operation hom(a, b) × hom(b, c) −→ hom(a, c) called composition of morphisms; the composition of f : a −→ b and g : b −→ c is written as g ◦ f or gf such that the following axioms hold: • (associativity) if f : a −→ b, g : b −→ c and h : c −→ d then h ◦ (g ◦ f ) = (h ◦ g) ◦ f, • and (identity) for every object x, there exists a morphism Idx : x −→ x called the identity morphism for x, such that for every morphism f : a −→ b, we have Idb ◦ f = f = f ◦ Ida . S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 7.6 Digression:Natural numbers in Ancient Greece 85 An initial object of a category C is an object i in C such that for every object a in C, there exists precisely one morphism i −→ a. Now we can reformulate Theorem 9 in the language of categories. Theorem 11. Any Peano algebra is an initial object in the category of unary algebras. In particular, any two Peano algebras are isomorphic. 7.6 Digression: Natural numbers in Ancient Greece A commentary to Plato’s dialogue Charmides by an unknown scholar describes a clear distinction between ideal natural numbers and numbers used for mundane everyday counting: Logistic is the science that treats of numbered objects, not of numbers; it does not consider number in the true sense, but it works with 1 as unit and the numbered object as number, e.g. it regards 3 as a triad and 10 as a decad, and applies the theorems of arithmetic to such cases. It is, then, logistic which treats on the one hand the problem called by Archimedes the cattle-problem, and on the other hand melite and phialite numbers, the latter appertaining to bowls, the former to flocks; in other types of problem too it has regard to the number of sensible bodies, treating them as absolute. Its subject-matter is everything that is numbered; its branches include the so-called Greek and Egyptian methods in multiplications and divisions, as well as the addition and splitting up of fractions, whereby it explores the secrets lurking in the subject-matter of the problems by means of the theory of triangular and polygonal numbers. Its aim is to provide a common ground in the relations of life and to be useful in making contracts, but it appears to regard sensible objects as though they were absolute. (Quoted from [235, pp. 17–19], with thanks to David Pierce who brought this quotation to my attention.) Exercises Exercise 7.1 For which values of n does the unary algebra of Figure 7.2 allow the same recursive definition of addition and multiplication as in Dedekind-Peano arithmetic? If the definitions are consistent, are the corresponding binary operations commutative? associative? Will the distributive law of multiplication with respect to addition hold? S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 86 7 Counting Sheep Exercise 7.2 Prove that if a unary algebra hA; 1, Si satisfies the Induction Axiom (Axiom 5) then it satisfies one of the Axioms 3 and 4. Exercise 7.3 For an inexperienced learner of mathematics, it is sometime difficult to accept that any two empty sets are equal. Indeed, how can we claim that the set of all flying pigs equals the set of all mermaids? There is nothing in common between pigs and mermaids. Prove that any two empty sets are equal by considering a category with sets as objects and appropriately defined morphisms, and such that empty sets are initial objects in that category. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 8 Fractions Customer: How big is a portion of half roast chicken? Does it come with legs? Waiter: It comes with one leg. A conversation in a diner. Adding fractions is a notorious issue in mathematics education, and appears to grow in its notoriety. We start our discussion by quoting a testimony from Simon J. Shepherd1: My main problem while quite young was adding fractions. I was quite happy multiplying them, since you simply multiplied the two top numbers to get the answer top number, then multiplied the two bottom numbers to get the answer bottom number. But with adding you had to find a common denominator by cross-multiplying and I remember it took me years to crack this technique! Funnily enough, many of our foundation year students have exactly the same problem. Perhaps learning the multiplication before the addition actually inhibits learning the addition rule? My personal opinion is that, indeed, addition should come before the multiplication. I will try to explain that in the next section. 8.1 Fractions as “named” numbers In response to my call for personal stories about difficulties in studying (early) mathematics Alex Grad2 sent me the following e-mail: 1 2 SJS is male, British, a professor of computational mathematics. AG is male, Romanian, a student of computer science. His other stories are on Pages 10 and 14. 87 88 8 Fractions When I was about 9 years old, I first learned at school about fractions, and understood them quite well, but I had difficulties in understanding the concept of fractions that were bigger than 1, because you see we were thought that fractions are part of something, so I could understand the concept of, for example 1/3 (you a take a piece of something you divided in 3 equal pieces and you take one), but I couldn’t understand what meant 4/3 (how can you take 4 pieces when there are only 3?). Of course I got it in several days, but I remember that I was baffled at first. I am surprised how frequently such memories are related to the subtle interplay of hidden mathematical structures, like the dance of shadows in a moonlit garden; these shadows can both fascinate and scare an imaginative child. As a child, I myself was puzzled by expressions like 5/4; but it appears that my worries were resolved by pedagogical guidance: I was taught to think about fractions as named numbers of a special kind: quarter apples. Fractions like 5/4 are not the result of dividing 5 apples between 4 people, since this operation of division is not yet defined; they come from making a sufficient number of material objects of a new kind, “quarter apples” and then counting five “quarter apples”. AG was less fortunate: The word for fraction in Romanian is fractie, and that was the terminology used by my teacher and in textbooks. The word is used frequently in common language to express a part of something bigger, like fraction is used in English (I think . . . ), so I think that it’s very likely that my difficulty could be of a linguistic nature, I can’t eliminate also the fact that actually that was how fractions were introduced to us—pupils (like parts of an object) and only later the notion was extended, and so maybe I had problems accommodating to the new notion. Or maybe both reasons . . . I was luckier: in Russian the word for “fraction” is reserved for arithmetic and is not used in everyday language3 . Also, I was clearly told to think about simple fractions 1/n as “named numbers” of a special kind. In effect, when dealing with quarter apples, we are working in the additive semigroup 14 N generated by 41 . What happens next is much more interesting and sophisticated: we have to learn how to add half apples with quarter apples. This is done, of course, by dividing each half in two quarters, which amounts to constructing a homomorphism 1 1 N −→ N. 2 4 3 Unless you are a hunter and use the same word “drobь” for lead shot pellets. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 8.2 Inductive limit 89 Since both 21 N and 14 N are canonically isomorphic to N, we, being adults now, can make a shortcut in notation and write this homomorhism simply as N −→ N, In effect, we have a direct system k× N −→ N, z 7→ 2 × z. k = 2, 3, 4, . . . —or, if you prefer less abstract notation— 1 1 Id N −→ N. n kn Then we do something outrageous: we take the inductive limit of this direct system. In primary school, of course, taking the inductive limit is called bringing fractions to a common denominator. Fig. 8.1. Fraction-of-an-inch Adding Machine, 1952, Patent no. 169941. Photo by Windell H. Oskay. Source: http://www.evilmadscientist. c 2009 Evil Mad Scientist Laboratories, com/article.php/inchadder. used with permission. 8.2 Inductive limit Here are the formal definitions of a direct set, direct system, direct (inductive) limit taken from Wikipedia. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 90 8 Fractions A directed set is a nonempty set A together with a reflexive and transitive binary relation 6 (that is, a preorder), with the additional property that every pair of elements has an upper bound. Notice that the directed set in our definition is the set N ordered by the divisibility relation. It is not linearly ordered and has a pretty sophisticated structure by itself! Let (I, 6) be a directed set. Let {Ai | i ∈ I} be a family of objects indexed by I and suppose we have a family of homomorphisms fij : Ai → Aj for all i 6 j with the following properties: fii is the identity in Ai , fik = fjk ◦ fij for all i 6 j 6 k. Then the pair ({Ai }, {fij }) is called a direct system over I. In our case, the directed set is the set N (or rather the set {1/n | n ∈ N}) ordered by divisibility; the direct system is formed by semigroups k1 N for k ∈ N with natural embeddings 1 1 N → N if n divides m. n m 1 N) is mn (or The upper bound of n and m (or of n1 N and m Of course, taking the upper bound in the directed set   1 |n∈N n 1 mn N). is nothing more but bringing the fractions to a common denominator. It is not a simple and straightforward operation. In words of SJS4 : My main problem while quite young was adding fractions. I was quite happy multiplying them, since you simply multiplied the two top numbers to get the answer top number, then multiplied the two bottom numbers to get the answer bottom number. But with adding you had to find a common denominator by cross-multiplying and I remember it took me years to crack this technique!!! But let us return to construction of an inductive limit. The underlying set of the direct (inductive) limit, A, of the direct system ({Ai }, {fij }) is defined as the disjoint union of the Ai ’s modulo a certain equivalence relation ∼: G A= Ai / ∼ Here, if xi is in Ai and xj is in Aj we set xi ∼ xj if there is some k in I such that fik (xi ) = fjk (xj ). The sum of xi ∈ Ai and xj ∈ Aj is defined as the equivalence class of fik (xi ) + fjk (xj ) for some k such that i 6 k and j 6 k. 4 SJS is male, British, a professor of computational mathematics. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 8.2 Inductive limit 91 The inductive limit of the direct system 1 1 N → N if n divides m. n m is, of course, the additive semigroup of positive rational numbers Q+ . For a child, this may look like a magic, as a story from Tony D. Gilbert5 confirms: I greatly liked arithmetic at primary school. In my 3rd year (age 9/10, now more than 50 years ago), I had an inspirational teacher (across all subjects) by the name of Miss Jesse (I think that’s the correct spelling, but her name was pronounces ‘Jessie’). Among other things she taught us fractions (it seems amazing now that she taught us this, very successfully as far as I was concerned, whereas I regularly see first year students of Science who are quite unable to cope with these). My recollection is that she used dividing up a cake as her model for fractions (just as I have done with many a student since!). Having established fractions in lowest terms, she then went on to deal with canceling down, multiplication and division, and also addition and subtraction of fractions with the same denominator. . . but addition and subtraction of fractions with distinct denominators were a mystery that awaited us. Then we had a detour into highest common factors (HCF) and lowest common multiples (LCM). Finally, she at last explained how if you wanted, say, to do the addition 1/2 + 1/3, the trick was to put each fraction over the LCM of the two denominators and then like magic, the problem had been reduced to an earlier one, the problem of addition or subtraction of fractions with the same denominator. The point of this story was my reaction to that final explanation. I can still remember my puzzlement before the explanation as to how to one would deal with the problem of distinct denominators, and my really wanting to have the problem resolved. Then once we had patiently (and in my case enjoyably) gone through the detour of HCFs and LCMs, my pleasure and appreciation of the cleverness (or so it seemed to me) of the resolution of the problem, once the explanation was given. Similarly, one may try to define multiplication on Q+ using the same inductive limit; I leave this construction of multiplication as an exercise for the reader. The definition of division on Q+ is much helped by the fact that endomorphisms of the additive semigroup Q+ are invertible and form a group (the multiplicative group of positive rational numbers Q× ) which acts on Q+ simply transitively: for every non-zero r, s ∈ Q+ there exists a unique q ∈ Q× such that s = q · r. 5 ADG is male, British, has PhD in mathematics, teaches in a Scottish university. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 92 8 Fractions And here is a story from BC6 which illustrates this point: I could not understand the “invert and multiply” rule for dividing fractions. I could obey the rule, but why was multiplying by 4/3 the same as dividing by 3/4? My teachers could not explain, but I was used to that. I couldn’t work it out for myself either, which was less usual. Finally I asked my father, who was an accountant. He said: if you divide everything into halves, you have twice as many things. Suddenly not just fractions but the whole of algebra made sense for the first time. 8.3 Field of fractions of an integral domain There is, of course, a canonical construction of the field of fractions of a commutative integral domain; it works equally well for the integers Z and for polynomials R[x]. Some my correspondents found this, at a first glance more formal and purely symbolic approach to fractions and rational functions easier to grasp. Listen to a testimony from RW7 : I recall very few difficulties with mathematics in my formative years: on the whole it was a case of learning a simple rule and doing exactly what you were told. I do recall being introduced to fractions at age about 7, and wanting to know how to add them together. The teacher refused to tell me, saying it was difficult and we’d come to that later. I think I managed to get the formula a c ad + bc + = b d bd from an older pupil at the school (it was a very small school with several classes of different age-groups being taught in the same room, so this was not difficult), and I remember being totally mystified by it. At least I think it was an algebraic formula, but this might be a false memory implanted by my later understanding: it might really have been an algorithm, or even an example. Presumably my difficulties went away once I was taught it properly. Victor Maltcev8 preferred a purely axiomatic approach: When I was 8 we learned at school (2nd grade) rational numbers. I had very big problems in grasping them the whole 2nd and 3rd. Eventually I started feeling I will never get along with them and every time before classes in Maths I felt depressed that again we 6 7 8 BC was 10 years old. He is male, a non-English Western European. RW is male, British, a professor of mathematics. VM is male, Ukranian, a PhD student in a British university. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 8.4 Back to commutativity of multiplication 93 will do something with rational numbers and I still do not understand what are they. When I was 10, 5th grade (omitting the 4th), in one of the classes I realized that in order to understand rational numbers we should not understand them!—just impose the axioms on symbolic [expressions] “a over b” and deal with them. This gave really a big push to studying Maths after school classes. Alexey Muranov: I also remember that probably in the 5th grade I added up fractions with different denominators without being taught or asked :). Ah, here is something more interesting: to understand that the quantity expressed as the fraction m/n (m pieces each of which is obtained by dividing 1 into n equal parts) is the same as m divided by n (the result of taking m whole pieces and dividing this into n equal parts) was not at all obvious, I have difficulty to explain this even now. 8.4 Back to commutativity of multiplication For the construction of the field of fractions of an integral domain to be straightforward, we need commutativity of multiplication in the domain. And this is also tricky. Bernhard Baumgartner9 : Learning multiplication of entire numbers, like 3 × 4, we were confronted with pictures showing three lines with four objects and four lines with three objects. This should demonstrate the commutativity. I had a vague feeling, that this demonstration is not really a proof, but did not dare to say aloud such a nasty thinking to the teacher. Today I would subsume this epistemological problem under The Unreasonable Effectiveness of Mathematics . . . (Eugene P. Wigner)10 . MW: I think the last time we spoke, I was employed at XX College in YY but I now find myself back in ZZ teaching mathematics and statistics to Higher Education students at ZZ College. I’d like to share an experience I had only last week with one of my private tuition students rather than my day-to-day students: My student J is female and 27. J is an intriguing case as she has a severe aversion to all things mathematical and, in particular, division. In fact, from what I have seen, it is beyond me as to how 9 10 BB is male, Austrian, a professor of mathematical physics. See [165]—AVB. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 94 8 Fractions she managed to be accepted on a Higher Education course in the first place! Here is how part of our conversation went: MW: “What is 4 times 5?” J: (slight pause)“Twenty”. MW: “Correct. Ok, how many 5s are there in 20?” J: (longer pause) “Four” MW: "Correct. Ok, how many 4s are there in 20?” J: (very long pause, counting on fingers) “Five”. I found this fascinating. The idea that J was unable to make the connection between 4, 5 and 20 and how the fact that 4 × 5 equals 20 determines the answer to the quotients 20/5 and 20/4. I decided to try again. . . MW: “What is 2 times 9?” J: (slight pause)“Eighteen”. MW: “Ok, how many 9s are there in 18?" J: (longer pause)“Two”. MW: “Ok, how many 2s are there in 18?” J: (very long pause, counting on fingers) “Not sure. . . Eight?” MW: “Not quite”. J: “Nine?" MW: “Correct". Yet again, the multiplication was no real problem, the first division only a minor problem, but the second division caused major problems. MW: “What is 4 × 11?” J: (slight pause)“Forty-four”. MW: “Ok, how many 11s are there in 44?" J: (longer pause)“Four”. MW: “Ok, how many 4s are there in 44?” J: (very long pause, counting on fingers) “Not sure. . . ” We did make progress in the end but I must admit to being shocked at this lack of understanding from someone of my own age! By contrast, a story from Jonathan Kirby: Here is my favourite [true!] story of how I came to love mathematics. When I was 6, every day I had to do a few homework problems like: 2 + 7 = . . . ; 4 × 5 = . . . One day I was sitting at home doing these, and couldn’t remember the answer to 5 × 3, so I asked my mother: J. What is five times three? M. 15. J. No, that’s three times five. M. It’s the same. It doesn’t matter which way round they are. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 8.4 Back to commutativity of multiplication 95 J. Really? Are you sure? To me, five times three meant adding five copies of three, which was clearly different from adding three copies of five. I was puzzling over why they should be the same for the next few days, then suddenly I realised that I could see why they were the same. On squared paper, you could colour in five rows of three on top of each other, and the total number of coloured squares was the product, 15. But then you could rotate the paper and have three rows of five, and the same number of coloured squares. I have loved mathematics [almost] ever since. And Alexey Muranov reminds us about the role of visualization: I also vaguely remember that some of the algebraic laws studied in the second grade, such as associativity and distributivity, were posing problems. I do not remember now which exactly, but it could have been the associative laws for multiplication. I am not sure, but this could be because the other laws could be explained on 1or 2-dimensional pictures, and in class we only used 2-dimensional pictures, but the best way to explain the associativity for multiplication would be to arrange objects in a 3-dimensional parallelepiped. I did not think in terms of pictures, I somehow was able to imagine those laws, but probably I used geometry subconsciously. Exercises Exercise 8.1 Define multiplication on the inductive limit of the system of additive semigroups k× N −→ N, k = 2, 3, 4, . . . Exercise 8.2 Prove that the endomorphism ring of the additive group of rational numbers is a field. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 9 Pedagogical Intermission: Didactic Transformation In Chapter 8, we had a roller coaster ride from exceptionally elementary to highly abstract mathematics. Let us take a short break, to pause and reflect. I wish to devote a few pages to a discussion of the structure of this book. Have you noticed that I started with multiplication of natural numbers and only then moved to addition? This is not the natural logical order for the two topics. I deviated from the natural logic of development of mathematical theory for a simple reason: for me, it made pedagogical sense to postpone the application of severe rigor and the axiomatic method to later chapters. The choice of the structure of the lecture course1 which preceded this book—and that determined the structure of the book—was a quite explicit and deliberate decision. I spent at least an hour thinking about it (while slowly walking up a winding dusty road through olive groves on the hills above Sirince ¸ in Asia Minor). I would be delighted to have comments from my readers: do they agree that it was right choice? Notice that in this chapter I will discuss mostly the issue of university mathematics education. Even if the topic of the book is elementary mathematics, it is discussed at a university mathematics level. 9.1 Didactic transformation It is time to introduce some definitions. A remarkably compact formulation of what makes mathematics education so special can be found in a paper by a prominent mathematician Hyman Bass [618]: Upon his retirement in 1990 as president of the International Commission on Mathematical Instruction, Jean-Pierre Kahane 1 In the Nesin Mathematics Village, Sirince, ¸ Turkey. 97 98 9 Pedagogical Intermission:Didactic Transformation Fig. 9.1. Guido Reni. A fragment of The Rape of Helena, 1631. Musée du Louvre. Source: Wikipedia Commons. Public domain. described the connection between mathematics and mathematics education in the following terms: • In no other living science is the part of presentation, of the transformation of disciplinary knowledge to knowledge as it is to be taught (transformation didactique) so important at a research level. • In no other discipline, however, is the distance between the taught and the new so large. • In no other science has teaching and learning such social importance. • In no other science is there such an old tradition of scientists’ commitment to educational questions. The concept of didactic transformation is fairly old and can be traced back to Auguste Comte [113, preface]: A discourse, then, which is in the full sense didactic, ought to differ essentially from one [that is] simply logical, in which the thinker freely follows his own course, paying no attention to the natural conditions of all communication. [. . . ] On the other hand, this transformation for the purposes of teaching is only practicable where the doctrines are sufficiently worked out for us to be able to distinctly compare the different methods of expanding them as a whole and to easily foresee the objections which they will naturally elicit. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 9.1 Didactic transformation 99 However, a G OOGLE S CHOLAR search shows that the concept is used in the (English language) literature on mathematics education less widely than one would anticipate. In French literature, there is a concept of transposition didactique; the term has been devised by Yves Chevallard [645] and has become a mainstream element of the French mathematics education theory. I prefer the original term “didactic transformation” as coined by Comte; the word “transposition” invites for a narrow understanding of the concept as mere selection, adaptation and re-ordering of contents to be taught. The word “transformation” has a wider meaning, and, as this chapter will show, is more suitable. We have to remember that even a simple change in order of exposition could have dramatic effects and usually requires a serious rethinking of the underlying logical development of the material. Here is an example suggested by Roy Stewart Roberts: I suggest that a course in calculus and analysis should define the logarithm function by the usual integral—a simple substitution will give a proof that ln xy − ln x = ln y. This depends on having a theory of integration that enables the integration of continuous functions, but we would want this anyway. Then define xy = exp(y ln x), where of course exp is defined to be the inverse function to ln. To develop the theory based on the extension of ax from x rational to x real would need some subtle analysis, possibly including uniform continuity. I therefore favour the definition of ln as an integral. The reader would perhaps agree that a practical realization of this interesting proposal requires some careful mathematical work—and this chapter contains much more dramatic examples of possible alternative approaches to calculus. We have to accept that, in mathematics, didactic transformation is indeed a form of mathematical practice. Moreover, it is in a sense applied research since it is aimed at a specific application of mathematics: teaching. It remains mostly unpublished, underrated and ignored because it is frequently confined to early stages of course development or to ephemera of classroom practice. Didactic transformation could, and should be informed by advice from researchers in education and cognitive psychologists— but methodologically it remains a part of hardcore mathematics. Indeed, returning to Comte’s words: . . . the doctrines are sufficiently worked out for us to be able to distinctly compare the different methods of expanding them . . . S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 100 9 Pedagogical Intermission:Didactic Transformation we see that, in the context of university level mathematics teaching, the expressions “work out” and “expand” refer to purely mathematical activities: essentially, they mean “to prove, or check mathematically”. The situation with the words “distinctly compare” is even more interesting: here we see in action the reflexive power of mathematics as a precise and flexible tool for the study of the structure and function of mathematics itself. Unfortunately, the principles of didactic transformation are frequently neglected in the mainstream mass mathematics instruction. Hans Freudenthal even coined the expression “anti-didactic inversion” [666], to describe the regrettable situation when purely procedural aspects of mathematics dominate the teaching at the expense of explaining where these procedures came from and how they are justified. No mathematical idea has ever been published in the way it was discovered. Techniques have been developed and are used, if a problem has been solved, to turn the solution procedure upside down, or if it is a larger complex of statements and theories, to turn definitions into propositions, and propositions into definitions, the hot invention into icy beauty. This then if if has affected teaching matter, is the didactical inversion . . . [However] one should recognise that the young learner is entitled to recapitulate in a fashion the learning process of mankind. Not in the trivial abridged version, but equally we cannot require the new generation to start just at the point where their predecessors left off. [666] But I would rather avoid criticism. Instead, I wish to turn to a case study. Its choice is motivated mostly by references to the concepts of convergence and limit in the later chapters of this book. 9.2 Continuity, limit, derivatives Few topics in undergraduate education generate more controversy than the classical epsilon-delta approach to limits and continuity. I quote Raphael Núnez [371, p. 179]: Formal definitions and axioms in mathematics are themselves created by human ideas [. . . ] and they only capture very limited aspects of the richness of mathematical ideas. Moreover, definitions and axioms often neither formalize nor generalize human everyday concepts. A clear example is provided by the modern definitions of limits and continuity, which were coined after the work by Cauchy, Weierstrass, Dedekind, and others in the 19th century. These definitions are at odds with the inferential organization of natural continuity provided by cognitive mechanisms such S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 9.3 Continuity, limit, derivatives:the Zoo of alternative approaches as fictive and metaphorical motion. Anyone who has taught calculus to new students can tell how counter-intuitive and hard to understand the epsilon-delta definitions of limits and continuity are (and this is an extremely well-documented fact in the mathematics education literature). The reason is (cognitively) simple. Static epsilon-delta formalisms neither formalize nor generalize the rich human dynamic concepts underlying continuity and the “approaching” the location. This thesis is fully developed in a book by Lakoff and Núnez (2000) and is very representative of a school of thought in mathematics education largely informed by neurophysiological research. The “motion” metaphor for a limit has its drawbacks and can be confusing for a child. Here is a story from Azadeh Neman: As for the limits, we were told to imagine an animal getting closer to point on the plane but not really arriving at it. This is a very poor description and gave me many years of unnecessary pain while dealing with limits. The thing is this animal might actually decompose itself into many different animals who have nothing to do with the first one and whoever acts more dramatically close to a certain point will guide the limit close to that point.2 Moreover, a striking feature of Nunéz’s thesis (and the debate on the role of the concept of limit in mathematics education in general) is that it ignores an impressive variety of alternative treatments of calculus available in the (research) mathematical literature, some of them being remarkably intuitive and elementary. 9.3 Continuity, limit, derivatives: the Zoo of alternative approaches o-M INIMAL STRUCTURES. We shall start our excursion by visiting a well established area of research on the boundary of real analysis and mathematical logic—the so-called theory of o-minimal structures—where all (definable) functions of a single variable happen to be piecewise monotone and take all intermediate values. In naive terms, these are functions whose graphs can be drawn with a pencil, with the concept “can be drawn” being made explicit and rigorous [93]. Historically, the theory of o-minimal structures is a direct descendant of Euclid’s Elements—it originates in Tarski’s work on a decision procedure for Euclidean geometry [92]. N ON - STANDARD ANALYSIS, APRÈS R OBINSON. There is another approach, due to the famous logician Abraham Robinson [84], 2 AN is female, Iranian, learned mathematics in Persian and later in English; at the time of this episode she was 15 and taught in Persian. AN holds a PhD and is a postdoctoral researcher in mathematics. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 101 102 9 Pedagogical Intermission:Didactic Transformation which places infinitesimals (quantities and variables which are bigger than zero but smaller than any positive real number), purged from calculus in 19th century, back at the core of the subject. So far, this approach has made only relatively modest inroads into mainstream teaching; see Keisler [43, 44, 46, 47]. Even at that subtle point, I can quote a childhood testimony; it comes from Herbert Gangl3 : When the teacher had introduced the notion of “nested intervals” to prove that there always must lie one number and only one, I had to suspend my disbelief quite a bit—why couldn’t there be more inside that limit? At the time I could imagine a kind of “cloud” around such a limit point (maybe not necessarily embedded in the real line) which still contains many more points in the limit, although I couldn’t formulate it in any way. Later on this “disbelief” was sort of redeemed when I was confronted with infinitesimals, but my problem here (or at least one of them) was that I had not understood the notion of continuity as we are taught nowadays. (Later I discovered Keisler’s book http:// www.math.wisc.edu/~keisler/calc.html4 and found it very intuitive.) Eventually I adopted/accepted the “theorem” (about a unique number), but felt for quite some time that I had taken a “leap of faith” here. I suppose that this must have been around the age of 10–12, it was taught to me in my mother tongue (German), the notion is “Intervallschachtelung”. N ON - STANDARD ANALYSIS, APRÈS N ELSON. Internal set theory proposed by Nelson [73, 74] blurs the difference between finite and infinite in a very simple, effective and controlled way. A brilliant little book Nonstandard Analysis by Alain Robert [83] demonstrates the didactic power of this approach. The Introduction to the book starts with a quote from Euler: Since L. Euler was among the most inspired users of infinitesimals, let him have the first word . . . Here is how he deduces the expansion of the cosine function in his book [E] (Caput VII, Section V 133.). He starts from the Moivre formula 1 [(cos z + i sin z)n + (cos z − i sin z)n ] 2 n(n − 1) cosn−2 z sin2 z = cosn z − 1·2 n(n − 1)(n − 2)(n − 3) cosn−4 z sin4 z + · · · + 1·2·3·4 cos nz = 3 4 HG is male, German, has a PhD in mathematics, teaches in a British university. H. Jerome Keisler [46]. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 9.3 Continuity, limit, derivatives:the Zoo of alternative approaches and writes (loc. cit., p. 141) sit arcus z infinite parvus; erit cos ·z = 1, sin ·z = z; sit autem n numerus infinite magnus, ut sit arcus nz finitae magnitudinis, puta nz = v cos ·v = 1 − v 2 /2! + v 4 /4!—etc. In the context of Robert’s book, this argument makes perfect sense. S YNTHETIC DIFFERENTIAL GEOMETRY. My correspondent Frederick Ross has brought my attention to yet another [. . . ] approach eschewing ǫs and δs for calculus called synthetic differential geometry (due, like so much, to Lawvere). It is distinct from Robinson’s approach in kind of an interesting way. Robinson’s approach is via set theory. Synthetic differential geometry wanders in via intuitionist logic via the observation that ¬(¬(dx = 0)), says that one could construct a structure wherein there are numbers which could be equal to zero, but cannot be proven to be so, thus leaving enough room to play in the cracks. This is [. . . ] an interesting way to do topology which divests it of much of its pointset metaphor—which makes it particularly nice for computer science purposes where that metaphor is so much baggage attached to an otherwise essential tool. The interested reader can find a comprehensive exposition of synthetic differential geometry in Anders Kock [54]. C ALCULUS IN O- NOTATION, APRÈS K NUTH . Next, there is a lecture course by Donald Knuth on calculus in O-notation, taught by Knuth for may years and exceptionally well polished pedagogically. It is available from Knuth’s website and is partially published in [27].5 Donald Knuth’s approach to calculus via o-notation leads, however, to a different range of difficulties. A testimony from Mikhail Katz: I was studying at the FeMeSha no. 18 in Moscow around 1973. I recall being comfortable with the definition of derivative as a limit. On the other hand, the alternative definition that the instructor 5 One has to know the cult status of Donald Knuth in the mathematical and computer science communities to fully appreciate his influence: when I placed the text of his letter about teaching undergraduate calculus [50] on my blog http://micromath.wordpress.com/2008/ 04/14/donald-knuth-calculus-via-o-notation/, the post got 25,000 hits in 24 hours. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 103 104 9 Pedagogical Intermission:Didactic Transformation provided caused me no end of anxiety. Namely, he said the derivative is a number D such that f (x) = f (a) + D(x − a) + o(x − a). As you correctly point out, it takes a considerable amount of mathematical training to formulate precisely what the problem was. The problem was that the definition says absolutely nothing about how one could find such a “o()”, or how to go about simultaneously (in what sequence?) finding D and “o()”. In retrospect, what I must have been bothered by is the non-constructive nature of this definition. Actually I am currently writing a text on constructivism, and it could be that even after all these years I would still be unable to identify the source of the anxiety were it not for the fact of having understood constructivism better recently. It is a timely reminder: didactic transformation is necessarily a very subtle balancing act. U NIFORM L IPSCHITZ BOUNDS. There is also a very promising approach to calculus based on eliminating the concept of a limit and replacing it by uniform Lipschitz bounds [60, 64]. It is close in its spirit to Knuth’s calculus in O-notation but differs in some important aspects and notation. For example, the definition of derivative f ′ (a) of the function y = f (x) at point x = a becomes |f (x) − f (a) − f ′ (a)(x − a)| 6 K(x − a)2 ; for a non-mathematician, it suffices to notice that the notorious epsilons and deltas are not present in the formula. The “rich human dynamic concepts underlying continuity” so loved by Núnez have also gone, having been replaced by a closely related—but different—concept, that of approximation. This definition can be applied only to a narrower class of functions than the one usually called differentiable. It is natural to call a function f defined on a real segment [A, B] uniformly Lipschitz differentiable on [A, B] if for some constant K |f (x) − f (a) − f ′ (a)(x − a)| 6 K(x − a)2 . However, the class of uniformly Lipschitz differentiable functions suffices for most engineering applications and is open to a rigorous and didactically efficient treatment in teaching. The uniform bounds approach allows to introduce yet another simplification: at an entry level, differentiation can be introduced as a mere factoring. I borrowed the following example √ from Michael Livshits [60]—the derivative of the function y = x can be found by a simple manipulation from the toolbox of high school algebra. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 9.4 Some practical issues 105 We start with computing the slope of the√secant line√ to the curve √ y = x that passes through the points (a, a) and (x, x): √ √ √ √ x− a x− a = √ √ 2 x−a ( x)2 − ( a √ √ x− a √ √ √ = √ ( x − a) · ( x + a) 1 √ , = √ ( x + a) where the last expression can be evaluated at x = a yielding dy 1 = √ . dx x=a 2 a Our brief walk around the calculus Zoo results in a discovery of the most peculiar phenomenon: in order to be able to discuss and compare alternative treatments of undergraduate calculus one has to be able to see it in a much wider mathematical perspective; in particular, some basic understanding of set-theoretic topology, functional analysis and model theory is really useful—even if we are talking about Donald Knuth’s method formulated in a rather traditional and elementary language. 9.4 Some practical issues The volume and crucial role of didactic transformation, or hidden preparatory work of a teacher, is a specific feature of mathematics teaching at the university level. It is only natural to suggest that didactic transformation should form a part of a professional toolbox of a mathematics lecturer. This modest thesis, however, has serious implications for lecturers’ training and professional development. Mathematics provides a bewildering variety of apparently incomparable approaches to the same topic. We also have to remember that we cannot freely bend them into the desired shape or pick and mix elements of different approaches: each of them has its own internal logic which cannot be interfered with. In that case, how do we predict and assess the relative advantages and disadvantages of a particular approach in teaching to a given group of students in a given course? As a practicing teacher of mathematics, I can rely on my colleagues’ collective wisdom, but so far I could not find any usable advice in the literature on mathematics education. Every mathematician is aware of the existence of so-called mathematical folklore, the corpus of small problems, examples, brainteasers, jokes, etc., not properly documented and existing S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 106 9 Pedagogical Intermission:Didactic Transformation mostly in oral tradition. It is a small universe of its own, and mathematicians’ pedagogical observations form an important part thereof. Occasionally, these observations find their way to print. But in general the collective pedagogical experience of university mathematicians remains uncharted territory. One also has to take into consideration cultural differences between various countries and various university systems. In Britain, where I live and work, almost every course in university mathematics departments and most mathematics courses in service teaching are tailor made. This is different from the usual practice, say, in American universities where courses in so-called “precalculus” are frequently based on standard mass print textbooks and can be taught to mixed—and large—audiences of mathematics and engineering majors. My recent experience of close reading of three dozen final examination papers from a good Scottish university (in my role as an external examiner) yet again reminded me how much mathematical effort is invested in the development of courses and the design of examination problems—and how little of this effort is seen by outsiders. I argue that this hidden work of teachers is essentially a form of mathematical research; it uses the same methods and is based on the same value system. The difference is the form of output; instead of a peer reviewed academic publication (or a technical report for the customer, as it is frequently the case in applied and industrial mathematics) the output may take the form, say, of a detailed syllabus for a course which exposes classical theorems in an unusual order, or just a page of lecture notes with a new treatment of a particular mathematical topic. The criterion of success is the level of students’ understanding, not approval by peers. The mathematical problems solved by a lecturer in the process of course development and conversion of mathematical material into a form suitable for teaching are far from glamorous. They are not in the same league as the Poincaré Conjecture or the Riemann Hypothesis; they are more like (to give an example from my own practice) “find a way to explain to your students orthogonal diagonalization of quadratic forms without introducing the inner product and without ever mentioning orthogonal matrices—but make sure that the method works”. As mathematical research stands, this kind of work is perhaps unambitious, but it is nevertheless mathematical problem solving made very challenging by severe restrictions on the mathematical tools allowed. Why are mathematics lecturers readily engaging in this taxing and time consuming work? Their motivation comes mostly from various external factors, starting from time constraints to requests to cover particular material from colleagues who teach subsequent courses. One may wish to add to the list S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 9.4 Some practical issues 107 changes in school syllabi, the so-called “widening participation agenda”, etc. I argue that the intrinsic methodological link with research makes teaching of mathematics at the university level different from teaching many other university disciplines. Of course, it also makes university teaching very different from secondary school teaching. I believe that for that reason university level mathematics teaching deserves some special treatment in educational research. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 10 Carrying: Cinderella of Arithmetic 10.1 Palindromic decimals and palindromic polynomials My next case study is based on conversations with an 8 year old boy, DW, in May 2007. DW’s parents sent me a file of DW’s book. It included the following paragraph, reproduced here verbatim: I wrote to DW: Dear D, Indeed, there is something weird. I believe you have figured out that 109 110 10 Carrying: Cinderella of Arithmetic 1×1 = 1 11 × 11 = 121 111 × 111 = 12321 1111 × 1111 = 1234321 11111 × 11111 = 123454321 111111 × 111111 = 12345654321 1111111 × 1111111 = 1234567654321 11111111 × 11111111 = 123456787654321 111111111 × 111111111 = 12345678987654321 There is a wonderful palindromic pattern in the results. But mathematics is interested not so much in beautiful patterns but in the reasons why the patterns cannot be extended without loss of their beauty. In our case, the pattern breaks at the next step (judging by your book, you have already noticed that): 1111111111 × 1111111111 = 1234567900987654321 The result is no longer symmetric. Why? What is the difference from the previous 9 squares? Can you give any suggestions? I had some brief e-mail exchanges with DW which suggested that he might have an explanation, but could not clearly express himself. Our discussion continued when he visited me (with his mother) in Manchester on 8 May 2007. I wrote on the whiteboard in my office (this is a photograph of actual writing on the board): and asked DW whether the symmetric pattern of results continued indefinitely. DW instantly answered “No” and also instantly wrote on the board, apparently from memory: S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 10.1 Palindromic decimals and palindromic polynomials 111 “Good”—said I—but let us try to figure out why this is happening”, and wrote on the board: “Yes”—said DW—“this is column multiplication”. “And what are the sums of columns’?” “1, 2, 3, 4, 3, 2, 1”—dictated DW to me, and I have written down the result: “Will the symmetric pattern continue indefinitely?”—asked I. “No”—was DW’s answer—“when there are 10 1’s in a column, 1 is added on the left and there is no symmetry.” “Yes!”—said I—carries break the symmetry. But let us look at another example”—and I wrote: S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 112 10 Carrying: Cinderella of Arithmetic DW was intrigued and made a couple of experiments (and it appeared from his behavior that he was using mostly mental arithmetic, writing down the result, term by term, with pauses): and said with obvious enthusiasm: “Yes, it is the same pattern!” “Wonderful”—answered I—“let us see why this is happening. I’ll give you a hint: multiplication of polynomials can be written as column multiplication”, and started to write: DW did not let me finish, grabbed the marker from my hand and insisted on doing it himself: He stopped after he barely started the second line and said very firmly: “Yes, it is like with numbers”. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 10.2 DW: a discussion 113 “Well”—said I—“but will the pattern break down or will it continue forever?” That was the first time when DW fell in deep thought (and I was a bit uncomfortable about the degree of his concentration and retraction from the real world). This was also the first time when his response was not instant—perhaps, a whole 20 seconds passed in silence. Then he suddenly smiled happily and answered: “No, it will not break down!” “Why?”—inquired I. “Because when you add polynomials, the coefficients just add up, there are no carries.” At that point I decided to stop the session on the pretext that it was late and the boy was perhaps tired, but, to round up the discussion, made a general comment: “You know, in mathematics polynomials are sometimes used to explain what is happening with numbers”. The last word, however, belonged to DW: “Yes, 10 is x.” 10.2 DW: a discussion DW is a classical example of what is usually called a “mathematically able child” (although I prefer to avoid this much compromised expression and say instead “mathematically inclined child”). He mastered, more or less on his own, some mathematical routines— multiplication of decimals and polynomials—which are normally taught to children at a much later age. He also showed instinctive interest in detecting beautiful patterns in the behavior of numbers, and, which is even more important, in the limits of applicability of patterns, in their breaking points. DW understands what is generalization and, moreover, loves making quick, I would say recklessly quick generalizations. Vadim Krutetsky [826] lists this trait among characteristic traits of “mathematically able” children: very frequently, they are children, who, after solving just one problem, already know how to solve any problem of the same type. But let us return to our principal theme: the hidden structures of elementary mathematics. In our conversation, DW was shown— I emphasize, for the first time in his life—a beautiful but hidden connection between decimals and polynomials—and was able to see it! In our little exercise, DW advanced (a tiny step) in conceptual understanding of mathematics: he had seen an example of how one mathematical structure (polynomials) may hide inside another mathematical structure (decimals). S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 114 10 Carrying: Cinderella of Arithmetic My final comment is that although DW made a small, but important step towards deeper understanding of mathematics, this step is not necessarily visible in the usual mathematics education framework. It is unlikely that a school assignment will detect him making this small step. Procedurally, in this small exercise DW learned next to nothing—he had multiplied numbers and polynomials before, and he will multiply them with the same speed afterwards. One should not conclude, however, that the “procedural” aspect of mathematics is of no importance. DW’s ability to do this tiny bit of “conceptual” mathematics would be impossible without him mastering the standard routines (in this case, column multiplication of decimals and addition and multiplication of polynomials). 10.3 Decimals and polynomials: an epiphany DW’s words “10 is x” is a formulation of analogy between decimals and polynomials which is not frequently emphasized in schools but, when discovered by children on their own, is experienced as an epiphany. This expression is taken from another childhood story, by LW.1 When I was in the fourth grade (about 9 years old), we learned long division. I had enormous difficulty learning the method, though I could divide 3 and 4 digit numbers by 1–2 digit numbers in my head. I don’t recall exactly how I did the divisions in my head, though I suspect that the method was similar to long division. I recall that it was broadly based on “seeing how much I needed to add to the result to move on.” However, I couldn’t seem to remember long division, despite being able to follow a list of instructions on homework. My homework on long division took hours to finish. I think the issue was that my teacher and parents never explained why long division actually worked, so it seemed like a disconnected list of steps that had little relation to one another. On a quiz, my teacher thought I had cheated, since I had written no work on any of the problems. I only became able to do long division in high school when I learned how to do long division with polynomials. At that time, the teacher went to great pains to explain why it worked. While doing a homework, I had an epiphany: long division for polynomials was a close cousin of long division for numbers. Suddenly I could do long division of numbers. I now take pains when teaching calculus to try and explain why formulas are true whenever possible. This has lead to mixed 1 LW was 9 years at the time of his story, he is male, learned mathematics in American English (his native tongue), currently is a second year student in a PhD program in pure mathematics. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 10.4 Carrying: Cinderella of arithmetic 115 reactions on my evaluations, with some students saying that they enjoy knowing why the formulas are true and others saying that they have enough trouble learning formulas in the first place without having to also learn why they’re true. For me, the two are deeply connected. If I don’t know why something is true (at least in broad outline), I find it significantly harder to remember. As we shall see in the next section, LW had every reason indeed to ask “Why?”—decimals is an immensely rich structure. Also, LW’s story is an evidence in support of another observation made by Krutetskii [826]: “mathematically inclined children” may appear to be slow because they are frequently trying to solve a more general problem than the one given, and they understand the underlying reasons for a rule they are told to apply unquestioningly. 10.4 Carrying: Cinderella of arithmetic The deceptive simplicity of elementary school arithmetic is especially transparent when we take a closer look at carries in the addition of decimals. 10.4.1 Cohomology In Molière’s play Le Bourgeois Gentilhomme, Monsieur Jourdain was surprised to learn that he had been speaking prose all his life: P HILOSOPHY M ASTER: Is it verse that you wish to write her? M ONSIEUR J OURDAIN : No, no. No verse. P HILOSOPHY M ASTER: Do you want only prose? M ONSIEUR J OURDAIN : No, I don’t want either prose or verse. P HILOSOPHY M ASTER: It must be one or the other. M ONSIEUR J OURDAIN : Why? P HILOSOPHY M ASTER: Because, sir, there is no other way to express oneself than with prose or verse. M ONSIEUR J OURDAIN : There is nothing but prose or verse? P HILOSOPHY M ASTER: No, sir, everything that is not prose is verse, and everything that is not verse is prose. M ONSIEUR J OURDAIN : And when one speaks, what is that then? P HILOSOPHY M ASTER: Prose. M ONSIEUR J OURDAIN : What! When I say, “Nicole, bring me my slippers, and give me my nightcap," that’s prose? P HILOSOPHY M ASTER: Yes, Sir. M ONSIEUR J OURDAIN : By my faith! For more than forty years I have been speaking prose without knowing anything about it, and I am much obliged to you for having taught me that. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 116 10 Carrying: Cinderella of Arithmetic I was recently reminded that, starting from my elementary school and then all my life, I was calculating 2-cocycles. Indeed, a carry in elementary arithmetic, a digit that is transferred from one column of digits to another column of more significant digits during addition of two decimals, is defined by the rule  1 if a + b > 9 c(a, b) = . 0 otherwise One can easily check that this is a 2-cocycle from Z/10Z to Z and is responsible for the extension of additive groups 0 −→ 10Z −→ Z −→ Z/10Z −→ 0. DW discovered (without knowing the words “2-cocycle” and “cohomology”) that carry is doing what cocycles frequently do: they are responsible for breaking symmetry. Carry is difficult in itself, but psychological difficulties in its computation are aggravated by lower level cognitive hindrances, as a brief comment from Alex Cook 2 illustrates: When doing basic mental addition (eg of exam scores) I find it harder to do A: 23 + 8 = than B: 28 + 3 = Ditto for other sums when the second number is between 6 and 9. Not sure if this has plagued me since childhood or not. Children may develop their own idiosyncratic procedures for handling carries and stick to them for all their lives. This is a comment from Toby Howard3 : Whenever I add numbers, I partially subtract from the next multiple of ten. Example: for 7 + 8, I would think “7 needs 3 to make 10, so we have 5 left over, to add to the 10, so 10 + 5 = 15”. I seem to have started this only in my teens. And, as Logan Zoellner4 reminded us, the problems start with the very basics: concepts of decimal place and decimal point: I remember having difficulty grasping multiplication and long division with decimals. The principle of “move the decimal over and 2 3 4 ARC is male, British, was raised in Scotland, has a PhD in mathematical statistics, teaches statistics in an university. TB is male, British, teaches computer science at an university. LZ is male, American, university graduate, works as a mathematician. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 10.4 Carrying: Cinderella of arithmetic 117 move it back later” (eg .36 × .25 = 36 × 25 × .0001) confused me because I was never quite sure how many places to move it over. It was only later that I actually realized that what I was doing was multiplying everything by a power of ten and then dividing again later. In the problem above, for example, I would have been tempted to only move the decimal place back two places because that was how far I had moved it over. I was home-schooled, so I usually just looked in the back of the book for the right answer and eventually figured out the correct pattern, but didn’t understand it till much later. 10.4.2 A few formal definitions Let G be a group and A an abelian group with an action of G on A: G×A → A (g, a) 7→ g · a. A 2-cocycle is a map f :G×G →A such that g · f (h, k) + f (g, hk) = f (gh, k) + f (g, h) Let E be an extension of A by G, 1 −→ A −→ E −→ G −→ 1, S a system of coset representative of A in E, s:G→E a coset map. Then defined by f :G×G →A s(gh) = f (g, h)s(g)s(h) is a 2-cocycle. It measures the extent to which the collection of coset representatives fails to be closed under multiplication. A 2-coboundary for the action of G on A is a function f :G×G →A such that there exists a function φ:G→A S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 118 10 Carrying: Cinderella of Arithmetic such that: f (g, h) = g · φ(h) − φ(gh) + φ(g). Two 2-cocycles for the same extension E differ by a 2-coboundary. Therefore the extensions are described by the second cohomology group H 2 (G, A) = Z 2 (G, A)/B 2 (G, A), where Z 2 (G, A) is the group of 2-cocycles with respect to natural pointwise addition and B 2 (G, A) is the group of coboundaries. I leave it to the reader as an exercise to show that “carry” is a 2-cocycle. 10.4.3 Limits and series Carry has another interesting property: it contains a seed of a concept of limit leading to immensely rich p-adic analysis. In this context, it is interesting to analyze how p-adic analysis arises from the purely algebraic concept of carry and completely avoids all alleged psychological traps which imperil the study of real analysis. Very frequently, when we deal with a mathematical object and wish to modify it and make it “infinite” in some sense, we have several different ways for doing so. For example, the usual decimal numbers can be extended to infinite decimal expansions to the right: π = 3.1415926 . . . and to the left: . . . 987654321 In the second case, the operations of multiplication and addition on “infinite to the left” decimals (called 10-adic integers) are defined in the usual way, with the excess carried to the next position on the left. Carries march on and on, uninterruptedly, and this steadiness of their pace is the psychological basis of a very intuitive concept of limit. 10-adics are not frequently used in mathematics, but p-adic integers for prime values of the base p, defined in a similar way by expanding integers written to base p to the left, are quite useful and very popular. 10-adic integers are not so good as p-adic for prime p because they contain zero divisors, non-zero numbers x and y such that xy = 0. The following elementary example was provided by Hovik Khudaverdyan and Gábor Megyesi and nicely illustrates the concept of 10-adic limit. If you look at the sequence of iterated squares 5, 52 = 25, 252 = 625, 6252 = 390625, 3906252 = 152587890625 . . . S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 10.4 Carrying: Cinderella of arithmetic 119 you notice that consecutive numbers have in common an increasingly long sequences of the rightmost digits, that is, n+1 52 n ≡ 52 mod 10n , the fact which could be easily proven by induction. This freezing of rightmost digits means that the sequence converges to a 10-adic integer x = . . . 92256259918212890625. One can easily see that x has the property that x2 = x and hence x(x − 1) = 0. Therefore x and x − 1 are desired zero divisors. It can be shown that zero divisors appear in the ring of 10-adic integers because 10 is not a prime number. An exercise for the reader: prove that the ring of 2-adic integers has no zero divisors. Properties of 10-adic and p-adic integers are quite different from those of real numbers: to give one example, you cannot order 10adic integers in a way compatible with addition and multiplication (so that the usual rules of manipulating inequalities would hold). This can be seen already from one of the simplest instances of addition: . . . 99999 + 1 = . . . 00000 = 0. For lack of space, I will mention only one other property of p-adic integers made self-explanatory by application of simplest rules of operating with carries. 10.4.4 Euler’s sum Notice that a paradoxical summation of the infinite series 1 + 2 + 4 + · · · = −1, due to Euler, makes sense and is completely correct if one uses 2adic integers written by base 2 expansions. Indeed, it becomes an easy-to-check arithmetic calculation: 1 + 2 + 4 + 8 + · · · = 1 + 10 + 100 + 1000 + · · · = · · · 111111111 But S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 120 10 Carrying: Cinderella of Arithmetic + · · · 111111111 1 = · · · 000000000 Hence in 2-adic arithmetic 1 + 2 + 4 + 8 + · · · = −1. It is worth noting that Euler was most likely to know binary numbers. A first clear description of them in modern notation was published in 1703 by Gottfried Leibniz in his paper Explication de l’Arithmétique Binaire [216], see Figure 10.1: 0=0 1=1 2 = 10 3 = 11 4 = 100 5 = 101 6 = 110 7 = 111 .. . n 2 = 10 · · · 0 ( n zeroes) As Anton Glaser explains in [195], Leibniz was not the first to come up with binary numerals; Glaser finds the origins of binary numerals in unpublished manuscripts of the English mathematician and astronomer Thomas Harriot (c. 1560-1621) and in work of Bishop Juan Caramuel y Lobkowitz (1606-1682). 10.5 Unary number system I do not remember when I was first shown this unsophisticated party trick: you ask someone to write down, in secret, a one digit number. As a rule, lay persons write down 7. Also, as a rule, mathematicians write down 1. It is a part of a general tendency to look at simplest and/or most degenerate cases. And now listen to Alexey Muranov5 : 5 AM is male, Russian, has a PhD in mathematics, works in a French university. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 10.5 Unary number system 121 I remember that I could understand the algorithms for adding and multiplying numbers, but the algorithm of long division was much harder, I might have ended up just memorizing it. This must have been my 2nd or 3rd grade. Later it was a revelation to me, probably in 7th or 8th grade, that numbers are not strings of digits. This happened when I realized that the base-n number system has all the same rights to name numbers as the base-10 number system, and hence the number itself was not what I thought it was before, it existed independently of the number system, and if any number system had to be the preferred one, that should have been probably the unary number system (where 1 = “1”, 2 = “11”, 3 = “111” ... but I do not remember if I had this idea myself, or learned it somewhere later). Exercises Exercise 10.1 If you still believe that real numbers are the best of all worlds, try to find, without a calculator, the first three digits of the ratio 0.12345 · · · 484950 . 0.5152 · · · 99100 In principle, the first few digits of the numerator and the denominator should suffice for the computation. But how many of them do you need? Exercise 10.2 Check the 2-cocycle property of carries. Exercise 10.3 Compute H 2 (Z, Z/10Z) for the trivial action of Z/10Z on Z. Exercise 10.4 Prove by induction on n that, for all natural numbers n, n+1 52 n ≡ 52 mod 10n . Exercise 10.5 [671] The list of numbers 49, 4489, 444889, 44448889, 4444488889, . . . goes on for ever. What is the next number on the list? The first number 49 is a perfect square. Is the second number 4489 a perfect square? Is the tenth number on the list a perfect square? Which other numbers in the list are perfect squares? Exercise 10.6 Prove that a 10-adic integer has an inverse if and only if its last digit is 1, 3, 7 or 9. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 122 10 Carrying: Cinderella of Arithmetic Hint. If the last digit of x is 1, we can write it as x = 1 + 10y and (1 + 10y)−1 = 1 − 10y + 100y 2 − 100y 3 + · · · , with the expression on the right making sense because in the infinite sum in every position we sum up only finitely many digits. Exercise 10.7 Prove that a 2-adic integer has an inverse if and only if its last digit is 1. Exercise 10.8 Prove that every 2-adic integer can be written in the form 2k · x, where k is an (ordinary) non-negative integer and x is an invertible 2-adic integer. Exercise 10.9 Prove that the ring Z2 of 2-adic integers is a domain, that is, it has no zero divisors. Exercise 10.10 [90] Prove that the p-adic integer · · · 0100000000000000000100011 = ∞ X pk! k=1 is transcendental, that is, it is not a root of any polynomial with integer coefficients. Exercise 10.11 Study the arithmetic properties of the ring Z[[x]] of formal power series over integers; here, formal power series are expressions a0 + a1 x + a2 x 2 + · · · + an x n + · · · with all coefficients ai ∈ Z, with the usual operations of addition and multiplication. • • • Is Z[[x]] a domain? (Domain is a commutative ring which has no zero divisors.) Which elements of Z[[x]] are invertible? An invertible element in a domain R, that is, an element r ∈ R such that rs = 1 for some s ∈ R is also called an unit. Take a page from DW’s book and find in Z[[x]] the inverse of 1 + 2x + 3x2 + 4x3 + 5x4 + · · · . • Does Z[[x]] have unique factorization? See [7] for answers. Exercise 10.12 Try column (long) multiplication of unary numbers, say, 111 × 11. And what about long division? S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 10.5 Unary number system 123 Fig. 10.1. A page from Leibnitz’s paper about binary system [216]. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 11 Pedagogical Intermission: Nomination and Definition 11.1 Names We all know the old story about the rustic who said he wasn’t surprised that savants with all their instruments could figure out the size of stars and their course—what baffled him was how they found out their names. L. S. Vygotsky [415] The conversation with DW in Section 10.1 raises an interesting issue—children need names for the concepts, objects, and structures they meet with in their first encounters with arithmetic. Moreover, they need names for basic discriminating objects in the world, and naming is an integral part of infant’s cognitive strategy, starting with their first words [430]. Many my correspondents report the importance of availability of names for their understanding of mathematics: SUE ¸ had intuitive understanding of the concept of a prime number, but did not know the name. Elizabeth Kimber told me that her troubles would have been instantly resolved if her teacher had mentioned to her that addition is commutative. She suffered more from her inability to communicate her difficulties to adults than from the difficulties as such. And, of course, inventing names is a natural part of a child’s games. Here is a testimony from Jürgen Wolfart1 : Probably I was four years old when my mother still forced me to go to bed after lunch for a while and have a little sleep (children don’t need this rest after lunch, but parents need children’s sleep). Quite often, I couldn’t sleep and made some calculations with small integers to entertain myself, and afterwards I presented the results to 1 JW is male, German, a professor of mathematics. 125 126 11 Pedagogical Intermission:Nomination and Definition my mother. Soon, I did not restrict myself to addition (“und”) and invented by myself other arithmetic operations—unfortunately I don’t remember which, probably “minus”—but I invented also a name for it, of course not the usual one. I don’t remember which name, but I remember that my mother reconstructed from my results what operation I had in mind and told me what I did in official terminology. So I forgot my own words for it, but I had a new toy for the siesta time.2 Mathematics is a plethora of names, and even memorizing them all could already be a challenging intellectual task for a child. Not surprisingly, the following observation belongs to a poet; it is taken from Cahiers (Notebooks) Paul Valéry: Vu Estaumier, nommé Directeur de l’Ecole Supérieure des PTT. Me dit que, enfant, à 6 ans il avait appris à compter jusqu’à 6 – en 2 jours. Il comprit alors qu’il y avait 7, et ainsi de suite, et il prit peur qu’il fallût apprendre une infinité de noms. Cet infini l’épouvanta au point de refuser de continuer à apprendre les autres nombres.3 [872, Tome II, p. 798] Notice that a child was frightened not by infinity of numbers, but by infinity of names; he was afraid that the sequence of random words lacking any pattern or logic: un, deux, trois, quatre, cinq, six, . . . will drag on and on for ever. I agree—it is a scary thought. But mathematics can bring safety back by providing means for a systematic production of the infinity of names, as evidenced by Roy Stewart Roberts: At some point [. . . ] I had discovered that you can continue counting forever, using the usual representation of numbers if one ran out of names. 2 3 An enforced siesta—or other forms of confinement in a bed—feature prominently in childhood stories. What follows is a testimony from Mikhail Belolipetsky: I have fragmented memories of searching for various paths on a carpet that was hanging on the wall above my bed (that is, cycles in a graph). I was about four or five years old. MB is male, Russian, has a PhD in mathematics, he is a lecturer in mathematics in a British university. Seen Estaumier, appointed Manager of the School of PTT [this is the French Postal and Communication Service]. Tells me that, as a child of 6, he had learnt how to count up to 6—took him 2 days. He then understood that there was 7, and so on, and he was scared that he would have to learn an infinity of names. This infinity frightened him to the point that he refused to keep on learning the other numbers. Translation is by Adrien Deloro. The quote is kindly provided by Donald Preece. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 11.1 Names 127 The supply of names is all-important, but also important is a structural framework for their use, as it can be seen seen from another story, from Swiatoslaw G.4 About the age of eight I was told about the numbers like trillions, quadrillions etc. I was a pupil in a musical school so I was able to count to octillions, nonillions, duodecilions maybe. . . Then I started to wonder what are the limits. I.e. how many numbers can one name in Latin to give names to powers of 1.000.000. I knew the names of the intervals: tercja, kwarta, kwinta, seksta etc. (this was in Polish: third, forth, . . . it would not work in English). And I knew that they come form Latin, so when I heard quadrillion, quintillion I was able to figure out sextillion by myself. Until novendecillion. I did not know how to say twenty in Latin then, this made me puzzled how long would that go. And let us now turn to an opinion of a professional. Roger Howe said about his paper Taking place value seriously [692]: One of biggest surprises of that project was the realization that there was no standard short name for the basic building blocks of the decimal system, the numbers like 2 and 70 and 300 and 4, 000, that have only one non-zero digit. Such a number can be described as a “digit times a power of 10?”, but this is rather cumbersome; also, it assumes the concept of “power of a number”, while one would like to draw student attention to these numbers before that concept is available. So I gave them a name. In the first draft of the essay, I called them “very round number”s. However, some of my Park City colleagues found this insufficiently dignified, so the term that survived to the essay is “single place number”. After writing the essay, I took every opportunity I had to advocate more attention to place value in US math education. (I found that Singapore does a much better job with place value, which reassured me that I was not crazy to emphasize it.) But I gradually realized that I was not communicating what I wanted to, and that the reason was connected to the lack of a name for a “digit times a power of 10”?. Place value in the US is treated as a vocabulary issue: ones place, tens place, hundreds place. It is described procedurally rather than conceptually. Place value is treated as the rule that tells you how to read a number, not as the idea of expressing any number as a sum numbers of a very special form: “digit times a power of 10”?. [693] A discussion of this issue in my blog produced names raging from excessively academic “monomials” to “monoplace numbers” (or “monoplacers” to a colourful proposal from America: 4 SG is Polish, professor of mathematics. (In his words, his mathematics instruction at the time of the episode was perhaps in both Polish and Ukrainian; he just did not remember. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 128 11 Pedagogical Intermission:Nomination and Definition How about just “rounds”. Then if you have a number with several digits you can call it a “clip”, and beyond that a “magazine.” I’m sure this would be OK with everyone. As they say, jury is still out. 11.2 Nomination Fig. 11.1. Maria Montessori (1870 – 1952) giving a lesson in touching geometrical insets [722]. Nomination (that is, naming, giving a name to a thing) is an important but underestimated stage in development of a mathematical concept and in learning mathematics. I quote Semen Kutateladze [704]: Nomination is a principal ingredient of education and transfer of knowledge. Nomination differs from definition. The latter implies the description of something new with the already available notions. Nomination is the calling of something, which is the starting point of any definition. Of course, the frontiers between nomination and definition are misty and indefinite rather than rigid and clear-cut. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 11.2 Nomination 129 And here is another mathematician talking about this important, but underrated concept: Suppose that you want to teach the ‘cat’ concept to a very young child. Do you explain that a cat is a relatively small, primarily carnivorous mammal with retractible claws, a distinctive sonic output, etc.? I’ll bet not. You probably show the kid a lot of different cats, saying ‘kitty’ each time, until it gets the idea. [105] And back to Kutateladze: We are rarely aware of the fact that the secondary school arithmetic and geometry are the finest gems of the intellectual legacy of our forefathers. There is no literate who fails to recognize a triangle. However, just a few know an appropriate formal definition. This is not just an accident: since definitions of many fundamental objects of mathematics in the Elements are not definitions in our modern understanding of the word; they are descriptions. For example, Euclid (or a later editor of Elements) defines a straight line as a line that lies evenly with its points. It makes sense to interpret this definition as meaning that a line is straight if it collapses to a point when we hold one end up to our eye.5 We have to remember that most basic concepts of elementary mathematics are the result of nomination not supported by a formal definition: number, set, curve, figure, etc. Basically, mathematics starts with nomination. I have already written [107] about Vladimir Radzivilovsky and his method of teaching mathematics to very young children. It involved a systematic use of an idea— borrowed from the 19th century Italian educator Maria Montessori— of teaching children to recognize and name basic geometric shapes: triangle, square, circle, etc. and comparing them by placing shapes into similarly shaped (but perhaps differently oriented) pockets in a board, Figure 11.2. Nomenclature is a key component of the Montessori Method. It is important to emphasize that not only vision, but also locomotor and tactile sensory systems were engaged in this exercise— Radzivilovsky trained his children to recognize and name shapes with eyes closed. 5 The interpretation of Euclid’s definition of a straight line as a line of sight was suggested to me by David Pierce and supported by Alexander Jones. See a detailed discussion of “straightness” in a book by David Henderson and Daina Taimina [35, Chapter 1]. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 130 11 Pedagogical Intermission:Nomination and Definition Fig. 11.2. Insets of geometrical shapes, as used by Maria Montessori (1870 – 1952) [722]. Perhaps, I would suggest introducing a name for an even more elementary didactic act: pointing, like pointing a finger at a thing before naming it. A teacher dealing with a mathematically perceptive child should point to interesting mathematical objects; if a child is prepared to grasp the object and play with it, a name has to be introduced— and, in most cases, there is no need to rush ahead and introduce formal definitions. But, of course, one has to remember that names should relate to things, since, using semiotics terminology, there is no signifier without signified. Use of uprooted terms, torn away from any con- S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 11.2 Nomination 131 text, can disorient not only a child, but a mature learner. To that effect, read a testimony from CW6 : For me, maths was always easy, until I came to two subjects in my undergraduate courses: ring theory and category theory. With these I simply could not remember all the names of the different types of things. I’m not a stamp collector, I’m a model maker. I could cope with the theory, but when someone said—“Consider a Noetherian ring such that . . . ” I was lost. I couldn’t remember—still can’t—what a Noetherian ring might be. Using the name lost me every time. There was no problem with the maths itself, it was the use of names as labels that lost me. I could do the work, I couldn’t work out what the work was to do. Once I realized that I had trouble in remembering names for things, I then turned it into a tool. Naming things is an incredibly important action, so there were several occasions when I specifically put it to use. How? As follows. When discussing a problem with colleagues we would offer a definition, then make up a random name for it. “Consider a disk with three holes and a point removed from its boundary. We call this a Wunkle.” “Consider a Wunkle with two intervals in the interior identified. Call this a Rissmuck.” and so on. Every time we defined something new we would see if it was a refinement of a previous item, and then name it. There were so many names that some concepts got different names, and we were constantly referring to the definitions and their names. However, after some months of working on the same problem we found that the important concepts corresponded to the remembered names. Concepts whose names were not remembered turned out to be less important. It was as if, and perhaps it really was, that the linguistic process of remembering the name was itself finding the important concepts for us. 6 CW is male, British, has a Ph.D. in Pure Maths (graph theory and combinatorics). S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 12 The Towers of Hanoi and Binary Trees Ritchie Flick I think I was 9 or so, and I found the problem: “The tower of Hanoi”. I think it was at a puzzle shop or something like that. On the product description, there was the formula for the puzzle and I didn’t understand a single word at that time and the fact that it was in French didn’t make things better (my mother language is Luxembourgish). When I was trying to ask some of my teachers at that time, I couldn’t find anybody who could explain it to me. For some reason, I remembered this story, and I finally investigated again the “The tower of Hanoi” and now, after 10 years, I finally understand the formula of back then. 133 13 Mathematics of Finger-Pointing 13.1 John Baez: a taste of lambda calculus The mathematician and theoretical physicist John Baez is famous for his website This Week’s Finds in Mathematical Physics [2]. For those not in the know: the first entry there appeared in 1993, this website was one of the earliest and influential precursors of the blog movement. His Week 240 contains a cute introduction to the area of mathematics called lambda calculus. It is important in the context of this book because it demonstrates one of the most remarkable powers of mathematics: capacity for explicating, in precise mathematical terms, of informal concepts and procedures of mathematical practice.1 Indeed, some basic concepts of the lambda calculus explicate a basic act of mathematical thinking: pointing a finger at a mathematical object from a variety of similar objects and saying: “Let’s take this one”. But, instead of giving my own explanations, I prefer to reproduce verbatim a long quote from John Baez—with his kind permission: Let’s play a game. I have a set X in my pocket, and I’m not telling you what it is. Can you pick an element of X in a systematic way?2 No, of course not: you don’t have enough information. X could even be empty, in which case you’re clearly doomed! But even if it’s nonempty, if you don’t know anything about it, you can’t pick an element in a systematic way. So, you lose. Okay, let’s play another game. Can you pick an element of 1 2 Week 240 post (http://math.ucr.edu/home/baez/week240.html) is based on recent work by James Dolan and Todd Trimble; it covers much more than one of its more elementary fragment used in my book. In my naive terminology, this means pointing finger at an element in the set.—AB 135 136 13 Mathematics of Finger-Pointing XX in a systematic way? Here AB means the set of functions from B to A. So, I’m asking if you can pick a function from X to itself in a systematic way. Yes! You can pick the identity function! This sends each element of X to itself: x 7→ x. You don’t need to know anything about X to describe this function. X can even be empty. So, you win. Are there any other ways to win? No. Now let’s play another game. Can you pick an element of X X (X ) in a systematic way? An element in here takes functions from X to itself and turns them into elements of X. When X is the set of real numbers, people call this sort of thing a “functional”, so let’s use that term. A functional eats functions and spits out elements. You can scratch your head for a while trying to dream up a systematic way to pick a functional for any set X. But, there’s no way. So, you lose. Let’s play another game. Can you pick an element of  XX (X X ) in a systematic way? An element in here eats functions and spits out functions. When X is the set of real numbers, people often call this sort of thing an “operator”, so let’s use that term. Given an unknown set X, can you pick an operator in a systematic way? Sure! You can pick the identity operator. This operator eats any function from X to itself and spits out the same function: f 7→ f. Anyway: you win. Are there any other ways to win? Yes! There’s an operator that takes any function and spits out the identity function: f 7→ (x 7→ x). This is a bit funny-looking, but I hope you get what it means: you put in any function f , and out pops the identity function x 7→ x. This arrow notation is very powerful. It’s usually called the “lambda calculus”, since when Church invented it in the 1930s, he wrote it using the Greek letter lambda instead of an arrow: instead of x 7→ y S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 13.2 Here it is 137 he wrote λx.y. But this just makes things more confusing, so let’s not do it. Are there more ways to win this game? Yes! There’s also an operator called “squaring”, which takes any function f from X to itself and "squares" it—in other words, does it twice. If we write the result as f 2 , this operator is f 7→ f 2 But, we can express this operator without using any special symbol for squaring. The function f is the same as the function x 7→ f (x) so the function f is the same as 2 x 7→ f (f (x)) and the operator “squaring” is the same as f 7→ (x 7→ f (f (x))). This looks pretty complicated. But, it shows that our systematic way of choosing an element of  XX (X X ) can still be expressed using just the lambda calculus. Now that you know “squaring” is a way to win this particular game, you’ll immediately guess a bunch of other ways: “cubing”, and so on. It turns out all the winning strategies are of this form! We can list them all using the lambda calculus: f 7→ (x 7→ x) f 7→ (x 7→ f (x)) f 7→ (x 7→ f (f (x))) f 7→ (x 7→ f (f (f (x)))) etc. Note that the second one is just a longer name for the identity operator. The longer name makes the pattern clear. Notice that this recursive construction of operators, gives us a way to construct, within lambda calculus, natural numbers. Fingerpointing leads to counting! 13.2 Here it is Now let us listen to some former children. Here is a testimony from Alan Hutchinson3 : 3 AH is male, British, a lecturer in computer science at an university. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 138 13 Mathematics of Finger-Pointing (Age 9?) First exposure to algebra: I would write “x, where x is the answer” because I did not understand the questions. It distressed me. The following is a piece of mathematical folklore floating all over the Internet and blogosphere, claimed to be taken from an actual student’s work: The reader would probably agree that this solution “by finger pointing” is suspiciously close to the principles of lambda calculus. The student, by inexperience, made a shortcut frequently (and also knowingly and deliberately) taken by mathematicians. When I was an undergraduate student, I was lucky that the teacher of my tutorials class in analysis was Semen Kutateladze— I quoted him in Chapter 11. At that time, he was a mildly eccentric young man who loved to conduct his classes while reclining on a desk and smoking a pipe; he was a Bourbakist and an expert in functional analysis. Kutateladze told us in the first class, with an obvious disgust: “Apparently you expect me to teach you to evaluate integrals?” “No, we don’t—responded we reassuringly—we did that last year”. “Oh great! In that case we better do the real stuff. You know, I hate that brainless calculus”. “But perhaps, in your work, you occasionally have to evaluate an integral?” “Of course, I do them every day—that way: if I have to integrate a function f with respect to a Lebesgue measure µ, I write . . . ”— and he wrote on the blackboard: Z f dµ. Taking both answers, the student’s and my teacher’s, at their face value, do you see a difference? I have to admit that I no longer do. I wish to record that as a question for researchers in methodology of mathematics education: indeed, what is the difference between the two responses? S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 13.3 A dialogue with Peter McBride 139 13.3 A dialogue with Peter McBride The previous section first appeared as a post in my blog, and Peter McBride responded to it with the following comment: The examples of student “errors” shown on the post you linked to are, in fact, examples of the medieval-craft nature of pure mathematics, as noted 30 years ago by the computer scientist, Edsger Dikstra. The discipline still has no systematic, agreed, industrialstrength, notion of semantics: everything is still ad hoc. For instance, in some cases it is correct to cancel the symbol “n” above and below a fraction line; in other cases (when, for example, the symbol “n” is embedded inside a “sine” function), it is not. Why there is difference here is never explained, but bright students somehow master it implicitly. It is easy, very easy, for people who long ago mastered the unwritten and non-formal semantics of pure mathematics to forget how hard in fact this mastery is to acquire. For the rest of humanity, the lack of systematic treatment of the meaning and treatment of apparently-random scratches on paper is part of what makes the subject so difficult to learn. And that was my response: Dear Peter, what is actually taught to children when we think that we teach them mathematics? An analogy with (much younger) children mastering their mother tongue could be helpful: we do not teach kids grammar, they somehow pick the grammar themselves. Moreover, it is likely to be a disaster to try to teach formal grammar to three or four years old kids. The same is happening in mathematics: even at the elementary (primary) school level, mathematics is saturated with sophisticated structures and rules which are not made explicit to children. Children have to somehow pick these rules from their teachers’ talk and their textbooks. (Their teachers, as a rule, also have no explicit knowledge.) However, “systematic, agreed, industrial-strength, notion of semantics” (and lambda calculus could be a useful fragment of such formalization) will not help children. I am not even certain that it could help teachers. It is of no use to most working mathematicians, either—because mathematicians have been selected by their ability to pick and follow unstated rules of formal systems and conditioned do not care much about people who cannot follow “the rules of the game”. But we have to understand the “atomic” (and frequently unspoken) structures and procedures of mathematical cognition for the sake of health of our profession and our professional community. In effect, the problem as described by you is about relations between the mathematical community and non-mathematicians. Peter McBride: S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 140 13 Mathematics of Finger-Pointing Well, Alexandre, I guess we disagree profoundly about this. I believe students of mathematics, at whatever age, would benefit from all aspects of what they are being taught being made explicit, from the outset. I am sure, however (and this despite my University Medal in the subject), that on this issue I am in a minority among people with a university mathematics education. As I said, it seems very easy for people who have somehow absorbed the hidden rules without being taught them explicitly, to think that everyone can do likewise. Everyone can’t, and no one should have to. The rules, and all of them, should be made explicit. But that would mean mathematicians leaving their craft-based processes in the past where they belong, and actually systematizing the discipline. Hand-waving is easier. And, as a social anthropologist would note, having unwritten rules allows an elite to keep its club small and select. What should be my response to that? In Chapter 6, What is a minus sign anyway? I quoted there John Baldwin who said about the ambiguity of the “minus” symbol as used in elementary mathematics: It is quite plausible that this is a distinction one should not make for teachers or at least the question should be at what level you want them to be aware of it. And I agree with John Baldwin: it is unlikely that, as Peter McBride states, students of mathematics, at whatever age, would benefit from all aspects of what they are being taught being made explicit, from the outset. We have to be realists: we have first to teach the teachers—and this is already a highly non-trivial task. I will put it that way: to make “all aspects of what children are taught explicit” is possible; but even if it was desirable it would require time and a huge investment of political will and money. It is absolutely unrealistic in the present sociopolitical climate. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 14 Numbers and Functions 14.1 Chinese Remainder Theorem 14.1.1 History Some stories about mathematics are repeated again and again, and, to save time, I simply quote from Cut the Knot1 : “According to D. Wells [95], the following problem was posed by Sun Tsu Suan-Ching (4th century AD): There are certain things whose number is unknown. Repeatedly divided by 3, the remainder is 2; by 5 the remainder is 3; and by 7 the remainder is 2. What will be the number? Oystein Ore [75] mentions another puzzle with a dramatic element from Brahma-Sphuta-Siddhanta (Brahma’s Correct System) by Brahmagupta (born 598 AD): An old woman goes to market and a horse steps on her basket and crushes the eggs. The rider offers to pay for the damages and asks her how many eggs she had brought. She does not remember the exact number, but when she had taken them out two at a time, there was one egg left. The same happened when she picked them out three, four, five, and six at a time, but when she took them seven at a time they came out even. What is the smallest number of eggs she could have had? Problems of this kind are all examples of what universally became known as the Chinese Remainder Theorem. In mathematical parlance the problems can be stated as finding x, given its remainders of division by several numbers m1 , m2 , . . . , mk :” 1 http://www.cut-the-knot.org/blue/chinese.shtml 141 142 14 Numbers and Functions  x ≡ b1       x ≡ bk (mod m1 ) .. . (14.1) (mod mk ) In this book, we shall treat the case when m1 , m2 , . . . , mk are pairwise relatively prime: (mi , mj ) = 1 for i 6= j. Notice that Brahmagupta’s problem is not covered by this case; I leave that problem as an exercise to my readers. 14.1.2 Simultaneous Congruences We aim to prove one of the oldest theorems of Number Theory. But first we need a simple property of simultaneous congruences. Theorem 12. x ≡ y (mod mi ) for i = 1, 2, . . . , r if and only if x≡y (mod [m1 , m2 , . . . , mr ]). Here [m1 , m2 , . . . , mr ] denotes the least common multiple of m1 , . . . , mr . Proof. If x ≡ y (mod mi ) for i = 1, 2, . . . , r, then mi | (y − x) for i = 1, 2, . . . , r. That is, y − x is a common multiple of m1 , m2 , . . . , mr , and therefore [m1 , m2 , . . . , mr ] | (y − x). This implies x≡y (mod [m1 , m2 , . . . , mr ]). On the other hand, if x ≡ y mod [m1 , m2 , . . . , mr ] then since x≡y (mod mi ), mi | [m1 , m2 , . . . , mr ] | x − y. 2 S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 14.1 Chinese Remainder Theorem 143 14.1.3 Algorithm Theorem 13. Chinese Remainder Theorem. Let m1 , m2 , . . . , mk denote k positive integers that are relatively prime in pairs and let b1 , b2 , . . . , bk denote any k integers. Then the congruences (14.1) have common solutions. Any two solutions are congruent modulo m1 m2 · · · mk . Proof. Writing M = m1 m2 · · · mk M we see that is an integer and that mj   M , mj = 1. mj Therefore, by the Euclid’s Algorithm [EXPAND!], there are integers aj such that   M aj ≡ 1 mod mj . mj Clearly, if i 6= j, then  M mj  aj ≡ 0 (mod mi ). Now we define x0 as x0 = Then we have x0 ≡ k X M aj b j . m j j=1 k X M M aj b j ≡ ai b i ≡ ai m m j i j=1 (14.2) (mod mi ), so that x0 is a common solution of the original congruences. If x0 and x1 are both common solutions of x ≡ bi (mod mi ), i = 1, 2, . . . , k, then x0 ≡ x1 (mod mi ) for i = 1, 2, . . . , k and hence x0 ≡ x1 (mod M ) by Theorem 12. This completes the proof. 2 S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 144 14 Numbers and Functions 14.1.4 Example Returning to Sun Tsu Suan-Ching, we can now write a general formula for solving systems of simultaneous congruences  x ≡ b1 (mod 3)   x ≡ b2 (mod 5) (14.3)   x ≡ b3 (mod 7) Our general solution in this case is just x ≡ 5 · 7 · a1 · b 1 + 3 · 7 · a2 · b 2 + 3 · 5 · a3 · b 3 (mod 3 · 5 · 7), where a1 , a2 , a3 are chosen to satisfy  5 · 7 · a1 ≡ 1 (mod 3)   3 · 7 · a2 ≡ 1 (mod 5) ,   3 · 5 · a3 ≡ 1 (mod 7) so we can take (in this simple example, by direct inspection, in a general case—using Euclid’s Algorithm) a1 = 2, a2 = 1, a3 = 1 and have the answer x ≡ 70b1 + 21b2 + 15b3 (mod 105). With the original values b1 = 2, b2 = 3, b3 = 2 we get x ≡ 70 · 2 + 21 · 3 + 15 · 2 = 233 (mod 105). The smallest of infinitely many positive solutions is x = 23. 14.2 The Lagrange Interpolation Formula Let us now perform some notational tricks. The formula   M aj ≡ 1 (mod mj ) mj means that aj is the multiplicative inverse of M/mj modulo mj ; let us denote it −1  M . aj = mj mod mj Then the solution for the Chinese Remainder Problem (14.3) can be written as S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 14.2 The Lagrange Interpolation Formula x0 = 145 −1  k X M M · bj . · mj mj mod mj j=1 (14.4) Now let us turn our attention to the Lagrange Interpolation Problem: given k distinct points a1 , . . . , ak , k > 2, find a polynomial f (t) of degree k − 1 which takes at these points given values b1 , . . . , bk , correspondingly:  f (a1 ) = b1    .. (14.5) .    f (ak ) = bk And here, remainders appear at the scene. If we divide a polynomial f (t) by degree one polynomial t − a with remainder, f (t) = (t − a)g(t) + b, we instantly see that the value f (a) of f (t) at t = a is b: f (a) = (a − a)g(a) + b = b. Therefore, using the “modulus” notation for equality of remainders borrowed from number theory, we can rewrite our problem (14.5) as  f (t) ≡ b1 (mod t − a1 )    .. (14.6) .    f (t) ≡ bk (mod t − ak ) We rewrote the Lagrange Interpolation Problem exactly in the same form as the Chinese Reminder Problem. You would agree that it is natural to try to solve the former by emulating the solution (14.4) of the latter. To that end, denote m1 (t) = t − a1 , . . . , mk (t) = t − ak and We have M (t) = m1 (t) · · · mk (t) = (t − a1 ) · · · (t − ak ). M (t) = m1 (t) · · · mj−1 (t) · mj+1 (t) · mk (t) mj (t) = (t − a1 ) · · · (t − aj−1 ) · (t − aj+1 ) · · · (t − ak ) (notice that, in the expressions on the right hand side, the terms mj (t) and (t − aj ) are absent). S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 146 14 Numbers and Functions The remainder of the polynomial at t = aj :   M (t) M (t) ≡ mj (t) mj (t) t=aj M (t) modulo t−aj is its value mj (t) (mod t − aj ) = [(t − a1 ) · · · (t − aj−1 ) · (t − aj+1 ) · · · (t − ak )]t=aj = (aj − a1 ) · · · (aj − aj−1 ) · (aj − aj+1 ) · · · (aj − ak ) Moreover, the remainders upon division by t − aj are just real numbers, and their multiplicative inverses are nothing but ordinary inverses; hence  −1 M (t) 1 = . mj (t) mod t−aj (aj − a1 ) · · · (aj − aj−1 ) · (aj − aj+1 ) · · · (aj − ak ) Now we can assemble all bits of formula (14.4) together: f (t) = k X j=1 (t − a1 ) · · · (t − aj−1 ) · (t − aj+1 ) · · · (t − ak ) · bj (aj − a1 ) · · · (aj − aj−1 ) · (aj − aj+1 ) · · · (aj − ak ) (14.7) This is the famous Lagrange Interpolation Formula. 14.3 Numbers as functions We have seen that the Chinese Remainder Theorem and the Lagrange Interpolation Formula are the same procedure applied to two different rings: the ring Z of integers in the former and the ring R[t] of polynomials in one variable in the latter. It is essential in the set-up of the Lagrange Interpolation Formula that polynomials are treated as functions from R to R: for every point a ∈ R, there is a value f (a) of polynomial f (t) at t = a. The map va : R[t] → R f (t) 7→ f (a) is a homomorphism: it preserves addition and multiplication: va (f (t) + g(t)) = va (f (t)) + va (g(t)) va (f (t) · g(t)) = va (f (t)) · va (g(t)), or, in more elementary notation, [f + g](a) = f (a) + g(a), S HADOWS OF THE T RUTH V ER . 0.822 [f · g](a) = f (a) · g(a). 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 14.3 Numbers as functions 147 It is important for modern understanding of algebra and number theory that integers can be treated as a kind of function. For the domain, it is convenient to take the set of all prime numbers, it even has special notation Spec Z. The range is trickier: there is no single range, but at every “point” p ∈ Spec Z , the “function” m ∈ Z takes values in the field of remainders Z/pZ modulo p: m : p 7→ “m (mod p)”. It does not matter much that fields vary from one “point” to another: we multiply and add functions pointwise! [In a later chapter, I will try to use this analogy between numbers and functions and try to explain what NonStandard Real Numbers and Non-Standard Analysis are.] Exercises We can use the Chinese Remainder Theorem to prove an important property of Euler’s function φ(n). Recall that φ(n) is defined for natural numbers n and is equal to the number of natural numbers which are smaller or equal than n and coprime to n. In particular, φ(1) = 1, φ(2) = 1, φ(3) = 2, φ(4) = 2, φ(5) = 4, φ(6) = 2. Exercise 14.1 Let m and n be any two positive, relatively prime numbers. Then φ(nm) = φ(n)φ(m). Exercise 14.2 Give the following interpretation of the Chinese Remainder Theorem: If m1 and m2 are relatively prime then Z/m1 m2 Z ≃ Z/m1 Z × Z/m2 Z (direct product of rings). Exercise 14.3 Give examples showing that if the moduli m1 , . . . , mk are not relatively prime in pairs, then there may be not any solution of the congruences. Exercise 14.4 Generalise the Chinese Remainder Theorem to the case of non-relatively prime moduli m1 , . . . , mk and solve Brahmagupta’s Problem. Exercise 14.5 (David Pierce) Make Figure 14.1 into a proof of the Chinese Remainder Theorem for two moduli: If (m1 , m2 ) = 1 then every system of two congruences ( x ≡ b1 (mod m1 ) has a solution. S HADOWS OF THE T RUTH x ≡ b2 V ER . 0.822 (mod m2 ) 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 148 14 Numbers and Functions Fig. 14.1. Make this diagram into a proof of the Chinese Remainder Theorem (Exercise 14.5). S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 15 Graph Paper and the Arithmetic of Complex Numbers Fig. 15.1. Graph paper. 15.1 Graph paper Some time ago I placed a question in my blog: who, when and where printed the first sheet of graphed paper—and what was the motivation? 149 150 15 Graph Paper and the Arithmetic of Complex Numbers This was what my anonymous readers told me. In this section, I quote at length from the blog. He refers to the USA; in Britain, introduction of graph paper in mainstream mathematics education is associated with the name of John Perry [174]. Graph paper, a math class staple, was developed between 1890 and 1910. During this period the number of high school students in the U.S. quadrupled, and by 1920, according to E. L. Thorndike, one of every three teenagers in America “enters High School”, compared to one in ten in 1890. The population of “high school age” people had also grown so that the total number of people entering High School was six times as large as only three decades before. Research mathematicians and educators took an active interest in improving high school education. E. H. Moore, a distinguished mathematician at the University of Chicago, served on mathematics education panels and wrote at length on the advantages of teaching students to graph curves using paper with “squared lines.” When the University of Chicago opened in 1892 E. H. Moore was the acting head of the mathematics department. [. . . ] The Fourth Yearbook of the NCTM, Significant Changes and Trends in the Teaching of Mathematics Throughout the World Since 1910, published in 1929, has on page 159, “The graph, of great and growing importance, began to receive the attention of mathematics teachers during the first decade of the present century (20th)”.normalsize Later on page 160 they continue, “The graph appeared somewhat prior to 1908, and although used to excess for a time, has held its position about as long and as successfully as any proposed reform. Owing to the prominence of the statistical graph, and the increased interest in educational statistics, graphic work is assured a permanent place in our courses in mathematics.” Hall and Stevens A School Arithmetic, printed in 1919, has a chapter on graphing on “squared paper”. Some more notes on graph paper can be found here.1 The actual date of the first commercially published “coordinate paper” is usually attributed to Dr. Buxton of England in 1795 (if you know more about this man, let me know). The earliest record I know of the use of coordinate paper in published research was in 1800. Luke Howard (who is remembered for creating the names of clouds: cumulus, nimbus, and such) included a graph of barometric variations [201]. 1 Notes on the History of Math Teaching and Math Books, http://www. pballew.net/mathbooks.html. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 15.2 Pizza, logarithms and graph paper 151 The increased use of graphs and graph paper around the turn of the century is supported by a Preface to the “New Edition” of Algebra for Beginners by Hall and Knight. The book, which was reprinted yearly between the original edition and 1904 had no graphs appearing anywhere. When the “New Edition” appeared in 1906 it had an appendix on “Easy Graphs”, and the cover had been changed to include the subhead, “Including Easy Graphs”. The preface includes a strong statement that “the squared paper should be of good quality and accurately ruled to inches and tenths of an inch. Experience shows that anything on a smaller scale (such as “millimeter” paper) is practically worthless in the hands of beginners.” [. . . ] The term “graph paper” seems not to have caught on quickly. I have a Hall (the same H S Hall as before) and Stevens A School Arithmetic, printed in 1919 that has a chapter on graphing on “squared paper”. Even later is a 1937 D. C. Heath text, Analytic Geometry by W. A. Wilson and J. A. Tracey, that uses the phrase “coordinate paper” (page 223, topic 153). Even in 1919 Practical Mathematics for Home Study by Claude Irwin Palmer introduced a section on “Area Found by the Use of Squared Paper” and then defined “paper accurately ruled into small squares” (pg 183). It may be that the term squared paper hung on much longer in England than in the US. I have a 1961 copy of Public School Arithmetic (“Thirty-sixth impression, First published in 1910”) by Baker and Bourne published in London that still uses the term “squared paper” but uses graphs extensively. 15.2 Pizza, logarithms and graph paper To illustrate our excursion into the history of graph paper, it would be timely to quote a story told by SB 2 . It does not need much comment. One of my favorite stories to tell is how my parents taught me logarithms on a pizza box. One night when I was in maybe sixth grade, we were eating pizza for dinner and I decided to share an interesting fact I had picked up somewhere: “Did you know that as the pizza cools, the temperature drops in half every ten minutes?” Leaving aside the fact that this is an oversimplification (it’s the difference between the temperature of the pizza and the air that decays in this way, and I’ve no idea if the 10 minute figure is correct), this little tidbit sparked a dinner-long discussion about logarithms and exponential decay. My parents got out a marker 2 SB is female, American, a mathematics PhD student. A writer of a wonderful blog. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 152 15 Graph Paper and the Arithmetic of Complex Numbers and drew axes for time and temperature on the pizza box, then plotted the temperature at ten-minute intervals. When the points were connected I could see that they didn’t form a straight line, but rather a curve that got closer and closer to 0 and never quite touched. My parents then explained that when graphed on a special kind of paper, the curve would become a straight line. My dad got out a piece of log paper, with equally-spaced vertical lines and strangely spaced horizontal lines, where every group of ten was spread out at the bottom, but bunched up at the top of the interval. My parents explained that on a normal axis, each line represented one unit more than the one below. But on a logarithmic scale, each line (actually, each group of ten lines) represented ten times as much as the previous one. When we plotted our pizza temperature vs. time graph on the log paper, sure enough, we got a straight line! On the other hand, a graph that would have been a line on the original axes now became a curve that looked very similar to the original temperature graph. Fig. 15.2. Log-log paper. My dad also pulled out some log-log paper, where both axes have a logarithmic scale. (I must be the only person of my generation who’s ever seen real log-log paper—who needs it when we can plot things so easily on computers these days?) I didn’t understand why you would need logs on both axes—it wouldn’t straighten out the temperature graph, and lines on normal axes would still be straight on these (my parents weren’t about to try to explain exponents and properties of logarithms in a mathematically rigorous way to me at age 11!) Nevertheless, I thought it was interesting, and the odd-looking distribution of the lines had a kind of nice flow to it. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 15.3 Multiplication of squares 153 Now I don’t expect every parent to teach their kids about advanced math concepts at the dinner table. Most adults have never learned even this much math, or learned it but forgot it as soon as the final exam was over. I was extremely lucky to have parents with both the capability and the interest in math to teach me these kinds of things just for fun. (It is fun—how wonderfully geeky is it to draw mathematical diagrams on a pizza box? I wish I’d saved it.) And I don’t expect every kid to be capable of understanding this while in elementary school; I was pretty exceptional. But mathematics doesn’t have to be hard, and it doesn’t have to be boring—I firmly believe every high school student is capable of understanding the concepts my parents taught me that night. More than that, I believe every student is capable of having fun playing with ideas, discovering patterns and trying to figure out why they work. 15.3 Multiplication of squares Preparing once for a chat with my young friend DW I was thinking about sums of two integer squares (an archetypal mathematical problem: representation of numbers by forms) and a remarkable fact that if n and m are sums of two integer squares (let us call them bisquare numbers) then their product mn also has the same property. What is the most natural way to explain this property to a child? One may of course use the Fibonacci identity (a2 + b2 )(c2 + d2 ) = (ac − bd)2 + (ad + bc)2 , but it is not very convincing. Actually bisquare numbers are exactly areas of squares with vertices at nodes of graphed paper (with unit grid step)—this immediately follows from the Pythagoras Theorem. And the multiplication property of bisquare numbers follows from a simple rule for multiplication of squares drawn on graph paper (with one vertex marked), see Figure 15.3. EXPLAIN Of course, this is multiplication of complex numbers—restricted to the ring of Gaussian integers Z[i], the red square and the blue square being Gaussian integer 3+2i and 2+i, correspondingly, their product being 4 + 7i (check—it is the new big blue square!). What becomes immediately obvious is the addition rule for arguments: arg(uv) = arg(u) + arg(v), as well as the multiplicativity of absolute values: S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 154 15 Graph Paper and the Arithmetic of Complex Numbers Fig. 15.3. Multiplication of squares drawn on the squared paper. |uv| = |u||v| (which was our aim in the first instance, since |u|2 is the area of the square representing u). Fig. 15.4. As Mitchell Harris has pointed to me, the Fibonacci identity (a2 + b2 )(c2 + d2 ) = (ac − bd)2 + (ad + bc)2 has a simple graphical proof based on Pythagoras Theorem. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 15.4 Pythagorean triples 155 15.4 Pythagorean triples The sums of two squares are archetypal objects of number theory, and, not surprisingly, they easily attract the attention of children. Here is a testimony to that effect from JW3 : [. . . in the sixth form] I proved most of the easy results on expressions of integers as sums of two squares for myself; I was fascinated by algebraic relations between squares and, for example, the half-angle formulae (expressing sin θ and cos θ in terms of tan(θ/2). My friend and colleague Hovik Khudaverdyan explained to me the right way of solving the classical Pythagorean Triples equation x2 + y 2 = z 2 in integers. Of course, after change of variables u= x , z v= y z we have to solve a slightly simpler equation u2 + v 2 = 1 in rational numbers u and v. Now draw the unit circle in the plane R2 : Then we observe that the line through point (0, 1) and point (u, v) has rational coefficients and intersects the x-axis in a point (t, 0) with rational coordinate t. Moreover, this projection establishes a one-to-one correspondence between rational points on the circle (without the north pole (0, 1)) and the rational points on the real line. This observation is fairly obvious in one direction, (u, v) 7→ (t, 0), and very subtle in another: the straight line through points (0, 1) and (t, 0) intersects the unit circle 3 JW is male, a professor of mathematics and an editor of a research mathematics journal. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 156 15 Graph Paper and the Arithmetic of Complex Numbers x2 + y 2 = 1 in a rational point (0, 1); the coordinates of the second point of intersection are the second roots of quadratic equations with rational coefficients and are therefore also rational. So we take t for a parameter. The equation of the straight line through the point (0, 1) and (t, 0) is, obviously, x + y = 1. t Hence we have to solve the simultaneous system x +y = 1 t x2 + y 2 = 1 which is fairly easy and quickly yields 2t 1 + t2 t2 − 1 y= . 1 + t2 x= After expressing a rational number t as a ratio of two integers: t= p , q we have, as a general formula for Pythagorean triples (x, y, z), (2pq)2 + (p2 − q 2 )2 = (p2 + q 2 )2 ; that is, x = 2pq, y = p2 − q 2 , z = p2 + q 2 . As simple as that. Of course, all that is just a birational parametrization of the circle by stereographic projection, a well-known folklore fact frequently mentioned in introductory courses on algebraic geometry. But when I was a boy no-one explained that to me, and the change of variable always appeared to be a mystery or, even worse, an artificial manipulation with symbols. I was not the only one puzzled by change of variables in Pythagorean triples. Listen to a testimony from RA.4 4 RA is male, American, a professor of computer science. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 15.4 Pythagorean triples 157 I don’t consider myself a mathematician, but your letter reminds me of what I think of as my first mathematical failure. I must have been in my early teens. My math instruction was in my native language, English. I always did well my math classes and was one of the school “stars.” I wanted to find a formula to generate all the Pythagorean triples. I was very confident that would be able to do it. I struggled with it for a long time before giving up. What never occurred to me at the time was that the formula could have two free variables. I was looking for something with one! When I saw the answer (and was surprised that others had even thought to ask the question, which shows what little contact I had had with math at the time) I felt embarrassed about my failure to think more openly about the problem. Exercises Exercise 15.1 Use stereographic projection to explain the universal trigonometric substitution (also known as Weierstrass substitution formulae) for evaluation of integrals of the rational expressions in cos x, sin x: Z P (cos x, sin x) dx. Q(cos x, sin x) If one sets t = tan then substitutions x 2 , 2t , 1 + t2 2 1−t , cos x = 1 + t2 2 dx = 1 + t2 sin x = convert the integral to that of a rational expression in t. At this point, it is worth quoting Vadim Tropashko: I was caught with the nonfundamental nature of trigonometric identities at PhysTech entrance exam. At the time I knew the equation for the circle and formulas for cos(x + y) and sin(x + y)— these have some structure making them easy to remember. When asked to exhibit some other formula I murmured “this can be easily derived, can you please give me couple of minutes?” The examiner didn’t appreciate it; his reply: “Well, when you are attending the lecture, you are supposed to know all the stuff lecturer can throw at you—you will have no time to derive anything”. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 16 Uniqueness of Factorization I start this chapter with a question which intrigued me when I was a student: is there a meaningful mathematical statement about finite fields which is proven by induction on the characteristic of a field? Finally and many years later I got a partial answer: a meaningful (and exceptionally important) statement about prime numbers proven by induction on the size of prime number in question. The following proof of a classical result belongs to Peter Walker [94]. His proof uses division with remainder, but not the Euclidean algorithm for finding the greatest common divisor of two integers. It is an elegant illustration to our discussion of induction and recursion in Chapter 5 since division with remainder, in the form of long division, remains the most important example of a recursive algorithm in the whole elementary mathematics. 16.1 Uniqueness of factorization So, we are concerned with the following well known property of prime numbers. Theorem 14. For all primes p, p | ab implies p | a or p | b. Proof. We say that a prime p is genuine if it satisfies the condition p | ab implies p | a or p | b for all integers a and b. We shall prove by induction on p that all primes p are genuine. B ASIS OF INDUCTION. Number 2 is a genuine prime number. Indeed if a product of two integers is even then at least one of the multiplicands is even. I NDUCTIVE STEP. Let p be the least non-genuine prime number so that p | ab but p does not divide a and does not divide b. 159 160 16 Uniqueness of Factorization Using division with remainder, we can write a = mp + c and b = np + d where 0 6 c, d < p. Of course, p | ab, so p | cd. If either c = 0 or d = 0 then p divides a or b as required. If not, then both c and d are at least 1 and can be factorized into primes: c = p1 · · · pk and d = q1 · · · ql (existence of factorization can be proved earlier). Now p | cd and for some integer u we have up = p1 · · · pk q1 · · · ql where all pi , qj are less than p and are therefore genuine prime numbers. Since none of pi , qj can divide p, they divide u and can be canceled out one by one from the equation, leaving an obviously contradictory equality vp = 1.  Following the standard development, Theorem 14 has the wellknown corollary. Theorem 15. The Fundamental Theorem of Arithmetic. If a natural number n is written as product of primes in two ways, n = p1 · · · pk and n = q1 · · · ql then k = l and each pi equals one of qj . Children tend to treat this fact as an experimental one—because that is how this is happening in nature. Uniqueness of factorization of polynomials, however, is far less obvious. Barry Mazur emphasizes the role of Theorem 14 and points out [67] that Not only is this property a characterization of prime number, but it reflects a fundamental feature of prime numbers; in fact, such an important characterizing feature of primality that there is much to be gained in our understanding if we simply turn the tables on the the way we introduce primes into our discussion, and make the following new Turn-Around Definition: A prime number is a number that has the property that whenever it divides a product of two numbers, it divides one (or both) of the numbers. It is a special case of a well known maxim of working mathematicians’ philosophy: Really good theorems are those that end their life as definitions. But it is time to turn to discussion of a child’s perception of primality and uniqueness of factorization. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 16.2 Dialog with AL 161 16.2 Dialog with AL AL1 : I read about the Fundamental Theorem of Arithmetic when I was about 10 or 11. I felt very uncomfortable about the proof of the Theorem. The statement seemed so obvious that the proof appeared to be absolutely redundant, not adding anything to understanding and making me worry that I was missing something. I would then have been much happier to accept the Theorem as a self-evident axiom. AB (the author of this book): Perhaps, there is indeed an intrinsic danger in proving obvious things. But, at the same age of 10 or 11, would you accept as obvious another, and closely related fact: If a prime number divides a product of two integers, it divides one of them? For example, 13 divides 46189 and 46189 = 209 × 221. Hence 13 should divide at least one of 209 or 221. (Indeed, 13 × 17 = 221 and 11 × 19 = 209.) And a more general question: what was approximately the boundary where you started to feel that a proof could be useful? I feel that this elusive boundary is one of the most intriguing things in mathematical education. AL: I would consider the statement “If a prime number divides a product of two integers, it divides one of them” as equally obvious to FTA and not requiring any proof. For me then, these statements were approximately at the same level of transparency as commutativity of multiplication: you don’t need to prove it, you just put coins into a rectangular shape on the desk and you see that, indeed, multiplication is commutative, it is a “law of nature”. Of course to get a similar feeling for FTA (or your statement on product of two numbers) you need more playing with numbers and that was what I was doing at the time. The proof of FTA appeared then as an attempt to reduce one obvious thing to other equally obvious things. For the contrast, approximately at the same time I found the proof of the irrationality of square root of 2 very exciting, it was a real proof! The question about boundaries is a quite delicate one. In a sense I still have this strange feeling about the proof of FTA, but only when I think about it as a statement about “proper”, or “real”, 1 AL is male, Russian, was taught mathematics in Russian. Graduated from the Mechanical and Mathematical Department of Moscow State University. PhD in (theoretical) computer science. He is now a lecturer in computer science. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 162 16 Uniqueness of Factorization or “just” natural numbers in a semi-formal setting. In university, of course I learned in a broader algebraic and logical context and that is then when I started to see that the proof of FTA does make a sense. One may have Euclidean and non-Euclidean rings, and FTA may have its explanation via more fundamental concepts. Or, you may fix some axiomatic system and ask whether FTA is derivable, or independent—not that I was thinking about the particular status of FTA at the time, but at least I saw the framework where the proof of FTA does make sense. AB: And what would you say about an intentionally elaborate and beautiful proof of a self-evident statement, like Theorem 14? AL: Concerning the proof of the divisibility lemma by induction over prime numbers, if I saw it when I was a schoolboy I would definitely have the same reaction. The basis of induction is very similar to the statement, so, again, it is a reduction of “obvious” to “obvious”. And I still feel a bit uncomfortable about the proof now, because of some vagueness in the distinction between what is taken for granted and what is to be proven. Ideally, I would like to see an explicit list of principles (axioms) needed to carry out the proof. 16.3 Generalizations It is instructive to transfer the inductive proof Theorem 14 to polynomials. Let F be a field and F [x] the ring of polynomials over F in single variable x. A polynomial is called irreducible if it cannot be written as a product of two polynomials, both of which are of strictly smaller degree. Notice that we can divide polynomials with remainder and use induction on degree of polynomials to repeat the proof of Theorem 14 in this new context. Theorem 16. For all irreducible polynomials p(x), p(x) | a(x)b(x) implies p(x) | a(x) or p(x) | b(x). Of course, this observation can be generalized further. Indeed, degree of polynomial is an Euclidean function on the ring of polynomials F [x] over a field F . Therefore the previous observation is a special case of a more general standard result of algebra: Euclidean domain is a unique factorization domain. I leave details (with a brief reminder of definitions) to the reader as Exercise 16.3 at the end of this chapter. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 16.4 The Fermat Theorem for polynomials 163 16.4 The Fermat Theorem for polynomials To demonstrate the role of the uniqueness of factorization, we shall prove a version of the Fermat Theorem—it will immediately continue the theme started in Section 15.4 where we successfully solved the Pythagorean Equation a2 + b 2 = c2 in terms of polynomials a, b, c ∈ Z[u, v]. Theorem 17. Let n > 2. There is no a, b, c in C[x] such that an + b n = cn with coprime a and b, at least one of which is nonconstant. Proof. The following proof was communicated to me by Oleg Belegradek. Suppose not. Choose a counterexample with max{deg(a), deg(b)} > 0 and as small as possible. Let ǫ1 , . . . , ǫn be all nth complex roots of −1. Then cn = (a + ǫ1 b)(a + ǫ2 b) · · · (a + ǫn b). As a and b are coprime, the polynomials a + ǫi b are nonzero and pairwise coprime. Since C[x] is a unique factorization domain, a + ǫi b = fin for some fi ∈ C[x]. For i 6= j, at least one of fi , fj is nonconstant; otherwise a, b would be constant; also 0 < max{deg(fi ), deg(fj )} < max{deg(a); deg(b)}. Since all fin are in Ca+Cb, any three of them are linearly dependent over C. Let, say, αf1n + βf2n = f3n . Then α 6= 0 or β 6= 0, and (γf1 )n + (δf2 )n = f3n , where γ n = α, δ n = β, which contradicts to the choice of a, b.  I leave it as an exercise to the reader to expand the statement to the 2-variable case. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 164 16 Uniqueness of Factorization Exercises Exercise 16.1 Prove Theorem 14 by induction on the value of the expression pab + min(a, b). (Such a proof was discovered by Marc Bezem [6].) Exercise 16.2 Prove Theorem 16 using induction on degree of the polynomial p(x). Exercise 16.3 Recall that a Euclidean function on a ring R is a function g : R → {0} ∪ N such that for any two elements a and b 6= 0 in R, we can write a = qb + r with q, r ∈ R in such a way that we have either r = 0 or g(r) < g(b). An Euclidean domain is a commutative ring without zero divisors (integral domain) and with an Euclidean function. A definition of a unique factorization domain is a bit longer since it should take into account behavior of the ring of polynomial where non-zero constant polynomials divide any other polynomial. To that end, we say an element of a commutative ring R a unit if it is invertible. An element p ∈ R is called irreducible if p cannot be written as a product of two non-units. for a and b arbitrary elements of R. Two irreducible elements a and b are associated if a divides b and b divides a (like, for example, polynomials x +1 and 2x + 2 in R[x]). An integral domain R is a unique factorization domain if every nonzero non-unit r ∈ R can be written as a product of irreducible elements of R: r = p1 p2 · · · pk and this representation is unique in the following sense: if q1 , . . . , qm are irreducible and x = q1 q2 · · · ql , then k = l and each pi is associated to some qj . Prove that an Euclidean domain is a unique factorization domain. Exercise 16.4 Generalize Theorem 17 to polynomials over arbitrary fields [82, Lecture XIII]: Let n > 2 and let K be a field of characteristic not dividing n. There is no a, b, c in K[x] such that an + bn = cn with coprime a and b, at least one of which is nonconstant. Exercise 16.5 Prove Fermat’s Theorem in the 2-variable case: there are no non-constant polynomials x, y, z ∈ C[u, v] with coprime x and y that satisfy xn + y n = z n for some n > 2. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 17 Pedagogical Intermission: Factorization One of my correspondents, Frances Bell kindly contributed the following observation: One of the things that I learned from my fairly brief experience in teaching maths in school was that many children need to switch between rote and constructive learning, and to rehearse what they know in different settings. For example, it would be very difficult for a child to factorize in algebra without fluency in arithmetic factors. My aunt studied algebra in her late seventies (having left school at 14) and found factorization fairly easy because of her excellent arithmetic skills unlike her young classmates who lack proficiency with their times tables. My husband told me how the old UK money system LSD1 with 240 pennies in a pound (and even half pennies and farthings) contained so many useful factors that children could relate to real coins. Frances Bell refers to an intimate relation between factorization in algebra and in arithmetic which manifests itself at a subconscious level in schoolchildren’s mathematical activities. The issue becomes even more interesting when children have to explicate this relation. Of course, it could be a very challenging task. Here are two examples from “olympiad” or “extension” type problems: (a) The number 100000000003000000000000700000000021 is the product of two smaller natural numbers. Find them. (b) How many of the integers 11, 1001, 100001, 10000001, . . . are primes? 1 Due to one of the many idiosyncrasies of the traditional British English, LSD is an abbreviation for “pounds, shillings, pence”. 165 166 17 Pedagogical Intermission: Factorization Potentially able students need to be challenged to think flexibly and to make connections—such as the simple (but logically subtle and elusive) connection here between factorization in algebra and in integer arithmetic. The first of the two problems was chosen from a first year university course on number theory and cryptography that I taught for some years in Manchester. When building their own toy implementations of the RSA cryptographic system, students have to produce, with the help of the M ATLAB software package, products n = pq of two large prime numbers p and q, sufficiently big so that n cannot be factorized by the standard routines of M ATLAB. A surprising number of students end up with numbers like the one in the problem. When explicitly prompted, only two or three students in a class of 30 could see why integers such as that in (a) are instantly factorizable into two smaller factors. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 18 Being in Control Science is most fun when it’s still in its infancy. Vilayanur Ramachandran [381, p. xiv] To avoid any possible misunderstanding, I feel that I have to make the following qualifying remarks: • • • • I am neither a philosopher nor a psychologist. This book is not about philosophy of mathematics, it is about mathematics. This book is not about psychology of mathematics, it is about mathematics. This book is not about mathematical education, it is about mathematics. In this chapter, I encroach onto the sacred grounds of developmental psychology, which was not my original intention. The chapter was not part of the draft plan of the book. But when I started to receive childhood testimonies from my correspondents, I discovered a recurrent theme in the stories which I tentatively labeled as being in control (I use the words suggested by Leo Harrington, see the next section). I decided to simply report these stories with minimal comment on my part. 18.1 Leo Harrington: Who is in control? As I have already said, my thanks go to Leo Harrington1 who helped me to crystallize a theme which goes through many childhood stories: a child’s instinctive urge to be in control of the math1 LH is male, American, professor of mathematical logic in an American university. 167 168 18 Being in Control ematics he or she is being taught. But LH’s story is best told in his own words: I have three stories that I think of as related. My stories may not be of the kind you want, since they involve no mathematics; but for me they involve some very primitive meta-mathematics, namely: who is in control of the meta-mathematics. This is my mother’s memory, not mine. When I was three my mother pushed my baby cart and me to a store. At the time prices were given by little plastic numbers below the item. When we got home my clothes revealed lots of plastic numbers. My mother made me return them. When I was nine in third grade I came home from school with a card full of numbers which I had been told to memorize by tomorrow. I sat on my bed crying; the only time I ever cried over an assignment. The card was the multiplication table. When I was around twelve, in grade school, a magazine article said how many (as a decimal number) babies were born in the world each second. The teacher asked for someone to calculate how many babies were born each year. I volunteered and went to the blackboard. When I had found how many babies were born each day, the number had a .5 at the end. The teacher said to forget the .5. I erased the .5. One of the components of the emotional and aesthetic attractiveness of mathematics is the specific feeling of being in control of something intangible, elusive, complex, and the self-confidence that grows from that feeling. In this chapter, I assemble stories about being, or not being, in control of mathematics and of relations with other people arising from mathematics. Children can be surprisingly sensitive to issues of control. A partial explanation of this phenomenon touches the autistic spectrum and the peculiarly skewed positioning of many mathematicians on it. Simon Baron-Cohen (famous for his suggestion that mathematicians should avoid marriages between them because of a higher risk of producing autistic offspring) emphasizes that [253, p. 139]: People with autism not only notice such small details and sometimes can retrieve this information in an exact manner, but they also love to predict and control the world. Remarkably, the memories that are described by mathematicians as their first memories of mathematics are frequently memories of attempting to control the world. Listen to Victor Maltcev: I remember three episodes of a mathematical nature in my childhood: S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 18.1 Leo Harrington: Who is in control? 169 (1) When I was 5, I was once playing with toys early in the morning while all the rest were sleeping. When my mother saw me alone, she thought I would like some company and asked my elder brother to go to me. He came and immediately started messing around soldiers. I shouted at him and asked to put everything were it used to be. He put them back but I said he did not. He asked why but I could not find any explanation for the feeling that the probability of that is 1. [. . . ] Or to Jakub Gismatullin2 I lived in Kazan with my parents during the first 10 years of my life (except some holidays, which we have spend in Poland). The wallpaper near my crib has been put up in a very carelessly way. I remember, when I was at around 4 or 5 years old, I got up every morning in a bad mood because of this. Every morning I was trying to move some parts of the wallpaper in my mind. However, after some time (2–3 year) I got used to see inexact pattern. Some time ago I even imagined the hole plane covered by the wallpaper, however not in a perfect way, but in a way that looks around my crib. That is, I think I have found a certain shape on a wallpaper near my crib, that can be used to tessellate the whole plane (of course this shape contains some inaccuracies). In our flat in Poland we decided no to have a wallpaper, but just a plain wall. My personal reason to make this choice was not to irritate my 2-years old daughter. Or to SC: As a very young lad walking home from school in the Euclidean grid of streets that are the suburbs of Chicago, I thought about avoiding sidewalk cracks: “Step on a crack, break you back.” Somehow I knew, or had been told by an older brother that lines were infinite. I reasoned that I didn’t know if sidewalk cracks were perfectly regular, and the cracks running north to south from a block to the east might extend to my path. It was Chicago and from the point of view of a child, virtually infinite, so there clearly was no way to avoid sidewalk cracks. I missed an opportunity to become obsessive compulsive. Yes, SC missed a chance. In their extreme cases, mathematical obsessions could become clinical: “One of our first patients, just 17 years old, was brought to us in a wheelchair,” says Professor Christer Lindquist, a pioneer in the use of a brain surgery technique for people with Obsessive Compulsive Disorder (OCD), known as Gamma Knife. “This boy would set himself maths problems, which he had to solve before he 2 JK is male, has a PhD in mathematics, is a professional research mathematician. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 170 18 Being in Control could eat. His OCD had become so severe, and the maths problems he set himself so complex, that he couldn’t solve them any more, so he couldn’t eat.” [247] Computer programming could be a natural outlet for a child’s drive to control the world by means of mathematics. Here is a testimony from IH3 : As a child, I wanted to write a computer program which draws a chessboard (this was on an A PPLE II). I wanted to do it using two nested loops both going from 1 to 8 and then having some condition which decides, depending on x and y, whether the corresponding square should be black or white. I knew already how to check for the parity of an integer, but it did not know how to use this to get the correct condition on x and y. (I could have treated the cases when x is even and when x is odd separately and then check the parity of y. But I had the feeling that there should be a simpler solution.) So I tried to test for the parity of x · y. Somehow, this seemed like the most natural thing to do. (I guess that I thought that adding is a too simple operation to yield interesting results like chessboards.) When I noticed that this did not work, I probably started thinking a bit more, trying by hand to check which operation would do the right thing. At some point, I must have found out that after all, x+y works, but I guess that before, I tried several other non-linear polynomials in x and y. And another story, from Antonio Jose Di Scala 4 : I do not remember any concrete stories about numbers or mathematics in the period of my life before I was 15. From that period what I do remember is that I had problems to understand the mathematics from school. When I was 15 (in 1987) I met my first computer (Commodore 64) and my life changed forever. When I told these stories to my wife Anna, she instantly responded by telling me how she, aged 9, was using the Russian word priuiqitь, “to tame”, to describe accommodation of new concepts that she learnt at school: the concept had to become tame, obedient like a well trained dog. The word was her secret, she never mentioned it to parents or teachers. Anna was not alone in her invention; here is a story from Ya˘gmur Denizhan: 3 4 IH is male, German, has a PhD in mathematics, teaches mathematics at an university; the episode told took place when he was about 9 years old. AJDS is male, Argentinean, has a PhD in mathematics, holds a research position in an Italian university. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 18.2 The quest for truth 171 Although I obviously knew the word before, my real encounter with and comprehension of the concept of “taming” is connected with my reading The Little Prince. As far as I can figure out I must have been nearly 12 years old. Saint-Exupéry offered me a good framework for my potential critiques in face of the world of grown-ups that I was going to enter. I also must have embraced the concept “taming” so readily that it became part of my inner language. Some years ago a friend of mine told me of a scene from our university years: One day when he entered the canteen he saw me sitting at a table with notebooks spread in front of me but seemingly doing nothing. He asked me what I was doing and I said (though I do not remember having said it, it sounds very much like me) “I am taming the formulae”. (Having heard this story I can recall the feeling. Most probably I must have been studying quantum physics.) For a young children, one of the first and most basics way to control consciously their actions is talking to themselves. This what Alison Price5 tells about he experiences: I do not have memories of a particular content of mathematics but do have vivid memories of myself as an 8 year old being taught in a very formal teaching context. The teacher would show us how to do a calculation on the board and we were then expected to work through examples in our maths books in silence. My problem was that I could not learn new mathematics without being able to talk about what I was doing. As a result I frequently was hit with a plastic ruler (this was allowed in the 1950s) for talking in mathematics class. As you can imagine this was not conducive to learning. Later in this chapter we shall have a chance to see what does it mean mathematically to be in control of one’s personal image of mathematics. 18.2 The quest for truth Podlinna istori vaxego soznani naqinaets s pervoi lжi. Svo  pomn. The genuine history of your consciousness starts with the first lie. I remember mine. Iosif Brodsky 5 The Reverend Dr Alison J Price is English, female, Principal Lecturer in Mathematics and Religious Education in a British university. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 172 18 Being in Control Mathematics is an art of truth. Not surprisingly, it appears that some mathematicians first discover a possibility of conscious attitude to life and the world when they first recognise other people’s lies. 18.3 The quest for logic Solomon Garfunkel6 : This one may be too far afield, as it relates to my first memory of constructing a logical argument. But as my research field was mathematical logic, you may allow it. I was probably no more than 5 or 6. I came to the conclusion that adults were insane. My reasoning went like this—ice cream was the most wonderful food ever invented. Any sane person who could would eat ice cream all day every day. Adults could do whatever they wanted (as opposed to us kids). They could in fact eat ice cream all of the time. They didn’t, ergo they were insane. By the way, I still feel that this argument holds up. 18.4 The quest for understanding VG7 : At 11 I had just entered a Comprehensive ‘Middle’ School where, due to our Education system in Kent, I had to remain before progressing to the Grammar School at 13. The standard of teaching was very poor and a lady, who didn’t seem to know too much mathematics, attempted to teach us how manipulate equations. I was bemused by her instructions (only verbal, I seem to remember) to, for example, ‘take the 6 over to the other side and turn the plus to a minus’. My maternal grandmother, an intelligent but not particularly well educated lady, happened to be visiting. When I explained my problem she thought about it for a moment and explained that an equation was like scales for baking, if it balances and you want it to stay that way, if you take 6 ounces off one side then you must take six ounces off the other. After that I got the idea and school algebra was never difficult. But the experience made me realise how much a clear explanation as to WHY one is following a mathematical procedure gives one confidence in doing the subject at an early age. 6 7 SG is male, American, a well-known expert in mathematical education. VG is female, British, has a PhD in mathematics, teaches in a British univeristy. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 18.4 The quest for understanding 173 The story is echoed in Vadim Tropashko’s account of how the “balancing the scales” explanation was missed in his first encounter with equations: The introduction of variable x in the first grade! (Public school in Minsk). The story went like this: Teacher: “And now I’ll show you something that you’ll carry over to the 10th grade (at least)—the mysterious thing called x”. Then, she carried over how to solve equations, e.g. x + 3 = 5. Her method was algorithmic: if there is something added to x then the solution is subtracting it from RHS, whereas if it was subtracted. . . Memorizing this simple rules seems to be easy, but not for me. I vividly remember struggling with this until much later when I was shown that one can add/subtract the same value on both sides of an equation! Ulrich Kortenkamp8 : I was four years old and sitting on our sofa in the living room, moving cushions from one place to the other. When my mother asked me what I was doing (apparently creating a mess in the living room), I proudly told her that I finally understood the reason why 6+7 equals 13. I do remember that I found out that I can split the 7 cushions into 3 and 4, and that I could use the 4 to make 10 out of 6. I still fall back to this visualization all the time, 36 years later, and I cannot think of 6+7 without splitting the 7 in 3 and 4. Natasha Alechina9 : When I was, I guess 8 or 9, I remember reading a book about school children where the hero struggled with mathematics, and a friend was explaining how to solve a problem to him, which I remember I appreciated as a good explanation, but it was not a problem I struggled with myself. I think it was something like “a boy and a girl were collecting hazelnuts, and together they collected 30, the boy collected twice more than the girl, how many did each of them collect”? The friend explained it with pictures, the boy had two pockets and the girl one, each pocket held the same number of hazelnuts, etc. The book was called Vitya Maleev at school and at home, don’t remember the author,10 and this is also the only thing I remember from the book. . . Nicola Arcozzi11 : I do not whether the short story I am going to tell you fits the requirements of a story about math education, since it takes place 8 9 10 11 UK is male, German, professor of mathematics and education. NA is female, Russian, holds a PhD, teaches computer science in a British university. Nikolay Nosov, a wonderful writer for children.—AVB NA is male, Italian, has a PhD in mathematics, teaches at an university. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 174 18 Being in Control in the family. I was about eight-nine years old (Italian third-fourth grade) and I was learning about continents. “Is Australia a continent or an island?” I asked my father. He answered it was both a continent and an island; an answer I found deeply unsatisfactory. I thought for a while about islands and what makes them different from continents, until—weeks later—I reached the conclusion that, by stretching and contracting, Eurasia could be an island of the Oceans as well as an island of the Como lake (“all its shores are on the Como lake”). A sunny day right after rain I was walking with my mother, I pointed to a puddle and I said: “we are on the island of that puddle”. She shrugged and replied “why do you always say such stupid things”. (Only many years afterwards I learned that was part of something called topology). Teresa Patten12 : When I was a girl of about 7, being instructed in English (my native language), I remember asking my teacher to explain, “What is two plus one?” She told me the answer is three, and explained that if I have two oranges and my friend gives me one orange, I will then have three oranges. “Yes,” I said, “but what is it?” What I was really trying to ask was “What is the nature of number?” I wanted to know how this abstract concept can apply universally to any unit we determine to be a unit, and how this correlates to our sensory experience of things as individual items. In particular, I wanted to know which is more ‘real’, the abstract concept or the thing itself? But of course as a 7-year-old I did not have the mastery of language to express this, and even if I could have done so I sincerely doubt my teacher would have understood the question. So instead of pursuing the idea she concluded that I could not do simple addition and put me into the lowest math group (this was California in the 1970s, and at that place and time children were taught in groups determined by aptitude), which is where I remained until I was about 11. I think the saddest part of this is that until my late teens I believed I had no ability to do math whatsoever. I simply assumed that my classmates all understood the nature of number and other theoretical questions that seemed so difficult to me. It never occurred to me that the other children probably never even thought to ask them. Jens Høyrup13 : I remember being puzzled about certain physical questions at ages 3 to 5, including one pertaining to geometrical optics: at age 5, I believe, before schooling, I had the impression that things when becoming more distant did not diminish as much as they “should” 12 13 TP is female, has an undergraduate degree in mathematics. JH is male, Danish, has a PhD in physics, a historian of mathematics. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 18.4 The quest for understanding 175 (that is, proportionally to distance, as predicted by a model with linear visual rays), and tried to solve the apparent puzzle by drawing a model where visual rays were conic in shape; I had a feeling that my model did not work satisfactorily, but neither made experiments checking my initial erroneous intuition nor the model, nor did I really see my mistake; so I gave up. Johan Swanljung14 : I do remember one story, and it wasn’t in school at all—it was simply something that I found so perplexing that it stayed with me. I was 11 or 12 and I was passionate about Dungeons and Dragons, which meant I had a lot of dice. If you roll a large number of dice, you don’t expect them all to come up the same. If you roll, say 10 six-sided dice and they all come up sixes, you’re very likely to suspect trickery. This seemed obviously true, but there was a logical problem. If the dice were fair, shouldn’t every possible combination be equally likely? In that case, why is getting 10 sixes strange? Logically I could convince myself that there was nothing special about that outcome, but something still felt deeply wrong about that conclusion. After all, rolling a pair of sixes isn’t more likely than rolling a 3 and a 4, is it? So why would that be different when you roll ten sixes? It wasn’t until several years later that it became clear to me that it had to do with how we count outcomes (and that yes, two sixes is half as likely as a 3 and a 4 if you roll two dice). There are many ways to get 5 sixes out of 10, but only one way to get 10 sixes. The outcomes are only equally likely if we keep track of which die is which. I don’t remember the moment of understanding, unfortunately. Perhaps it was in school. I have no memory of studying probability in school, although I’m certain I must have. Emily Cliff15 : I must have been about 8, because we learned long division in school when I was 9. I don’t remember why I decided I wanted to learn to do long division—perhaps I had heard my brother (who is 18 months older) talking about it—but I remember being in the kitchen and asking my father to teach me. He didn’t want to, because he thought I would be too bored in school when they taught us, so I took a calculator (yellow with red buttons, I think, except for the equals sign button, which was maybe green or blue) and 14 15 JS is male, Swedish, a teacher of mathematics. He says about this episode: Language of mathematical instruction: English. I am bilingual (Swedish/English) and at that age English was my stronger language. EC is female, Canadian, a mathematics PhD student in a leading British university. Her first language was English, but education was in French until high school, when she switched to English. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 176 18 Being in Control some scrap paper (we always had a pile of paper with math papers printed on only one side—my father brings it home from the recycling bins at the university) and hid behind the sofa until I figured it out. I don’t think I did anything very complicated, probably only one, maybe two digits into three or four, and I don’t remember whether I was calculating decimal places or was happy with remainders. Or maybe I was multiplying two numbers together with the calculator and then working backwards. But I remember being pleased a year later when we learned it in school and my method matched up. I think my notation was a bit different, and I think I had some trouble remembering which order to put the two numbers with respect to the division symbol on the calculator—that never made sense to me, even though we had obviously covered in it school for basic division. I think I probably never told my father about all this, because I talked to him on the phone this morning and he didn’t remember anything about it. 18.5 The quest for power Joseph Lauri16 : My childhood memories of mathematics are almost nightmares. We had a very competitive 11+ exam to pass from primary to secondary (our school system is basically the same as the English one). The maths exam used to have problems like: Tom, Dick and Harry have so much marbles. Tom has three more marbles than Dick and Harry together but if they give him so much each he will have twice as much. Find how many marbles. . . I still have a vague recollection of a similar type of problem involving three jugs of liquid and if one pours so much into the next one then. . . At that age these problems were simply beyond me. They had some way of teaching us how to solve them using "shares" or something. I did not even sit for the 11+ exam but managed to enter a secondary school which had an easy maths qualifying test. In the first few months we we taught algebra. I discovered that I could do the 11+ exam using algebra in about half the time allotted. It was then that I felt the sense of power that mathematics gives you, and I was hooked. Now I have a son and a daughter in the last years of primary school. The pendulum has swung the other way. They try to avoid anything which is intrinsically difficult. The difficulties are created by the modern insistence of avoiding algorithms or any efficient way to work out a problem. Constructivism and all that 16 JM is male, Maltese, Professor of Mathematics at the University of Malta. His initial mathematics education was in English, with Maltese being used in the classroom as a meta-language. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 18.6 The quest for rigour 177 leading to the situation that kids finish primary without knowing how to add 1/2 plus 1/3 (it is officially so in our National Curriculum) and they never actually get round to doing it because in the first year of secondary they start using calculators and spreadsheet (even this is in the curriculum if you read between the lines). Now I am not for one moment suggesting going back to my nightmare days. But today, children are not shown how powerful maths is, so they cannot feel the kick and the buzz I felt when I was 11. 18.6 The quest for rigour We are usually convinced more easily by reasons we have found ourselves than by those which have occurred to others. Blaise Pascal, Pensées (1670) NP17 : Here is one thing which I remember well when playing with a compass as a child: after having drawn a circle with the compass, I started at a particular point P on the circle and - not changing the radius of the compass—tried to create successively points on the original circle by getting to the next point clockwise with the help of the compass by using the previous point as centre. After doing this six times, I always seemed to get back almost to the original point P , but never precisely. At that time, I wasn’t sure whether this was based on my drawing not being precise (all my points and lines had some width) or whether it really was something which was only almost true. I cannot precisely say at what age I encountered this puzzle. I am certain that no foreign language was involved at that time. I wrote to NP asking him: What would be your reaction if, instead of a compass, you were using a software package like G EO G EBRA or C INDERELLA which automatically adjusts the picture and helps the user to achieve formal mathematical perfection of the picture? His response: Hypothetically assuming I had this tool available at the time of my compass experiments, I might have found out that this fact holds precisely and it might have encouraged me to find some theoretical reason for this to be true. But I can only guess that this would have happened. 17 NP is male, German, has a PhD in mathematics, teaches in a British university. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 178 18 Being in Control MR18 : I was about 12 years old, at secondary school. We had been taught about ruler and compasses constructions, including angle bisections, and had spent time practicing various constructions using the geometrical instruments which I enjoyed. The teacher made a remark in class about it being impossible to trisect an angle. Hmm I though to myself—it was very easy to bisect an angle, why not trisect? I spent a long time working on ways to trisect an angle with ruler and compasses later at home. But, I remember feeling that I could never be sure whether I had done it and my execution of the instruments was not accurate, hence explaining the error I detected by measuring with the protractor, or whether I had not done it and the small error measured did indicate the fact of nontrisection. PS I was delighted to find that angles in origami-geometry can be trisected! RL19 : The one memory that is still strong, for whatever reason, was of an interchange with my father when I was 6. (He was a lawyer, but he liked the math he knew and had been head of the math team, he told me, at Stuyvesant High School in NYC—he graduated in 1920, I believe. I don’t think he ever learned calculus.) One afternoon I had been doing a page of drill on adding and subtracting one-digit integers and I noticed that if you added 5 or subtracted 5 from the same number, the results were the same in the units digit. I was quite delighted with this observation, and told him about it when he came home from work. His response was “Oh yes, and that’s because . . . ”, giving a reason. I remember thinking—who cares why? it’s just cool, here was something that had made me happy to discover. So you might say that at that time I was challenged in the analytical part of mathematical curiosity. But the deepest meaning of this memory is no doubt of the emotion and family dynamic. Solomon Garfunkel: One experience immediately comes to mind. I was in third grade, so likely about 8 years old. My teacher asked the class how far the moon was from the sun. I raced to the board and drew a picture with the earth at the bottom, the moon directly overhead in a circular orbit and the sun directly above both. I reasoned that since 18 19 MR is female, English, has a PhD, an university lecturer and a researcher in mathematical education. RL is male, American, and professor of mathematics in an American university. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 18.6 The quest for rigour 179 the earth was 93 million miles from the sun and the moon was 250 thousand miles from the earth, all we had to do was to subtract and voila the moon was 92.75 million miles from the sun. The teacher was effusive in her praise, lauding my reasoning ability. When I got home and began to tell my father about the problem (and the praise) I quickly realized how foolish I had been to oversimplify the model as I did and not take into account the movement of the bodies involved. I also realized that , even assuming circular orbits, I didn’t have the technical knowledge to actually solve the problem and that there were many (I couldn’t have thought infinite)different solutions. I also realized that my teacher thought I was correct, which led to a healthy skepticism of teachers ever after. Lawrence Braden 20 : When I was twelve years old, Mr. P, my math (maths) teacher, told the class that π equalled 22/7. Also that π equalled 3.1416. He was speaking to me in Ojai, California, in a language we mistakenly called English but which was roughly equivalent to it under most circumstances. Excited, I went home with the grandiose notion of finding π to a hundred decimal places by the process of long division, and wondered if anyone had ever done that sort of thing before. Well, of course, I kept getting the wrong answer! Not 3.1416 at all! I did it over four or five times, and was really disturbed. The book said that π equaled 3.1416 and the teacher said that π equaled 3.1416 so I logically came to the conclusion that I did not know how to do long division! A truly disturbing notion; I thought I was pretty good at it and here I couldn’t even do this simple problem! “Oh”, Mr. P said the next day. “I didn’t mean that π was exactly equal to 22/7." It was at that point that I learned not to take everything a maths teacher (or any sort of teacher) said as hewn in stone. Years later I came upon Niven’s truly beautiful and elementary proof of the irrationality of π, and Lindemann’s proof of its transcendence. Teachers should be careful what they say. They will be believed by their children. I myself taught at least two generations of students that parabolas came in different shapes before learning differently. But that was out of ignorance. Mr. P knew better. RW21 : I became unpopular with my class mates and some teachers because sometimes, maths questions appeared ambiguous to me. When a question was hidden in the text and you had to think about 20 21 LB is male, American, a recently-retired math teacher of 40 years experience. RW is female, German, a professor of mathematics in a German university. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 180 18 Being in Control how to translate it into a mathematical problem, I frequently came up with at least one solution different from what the teacher expected, referring to different ways of understanding what was written. I used to be very stubborn when a teacher would try to “sell me” that only one unique answer was correct. Of course I was wrong sometimes and just misunderstood what was said, but sometimes I was right and some teachers made me respect them a lot by discussing my views in an open way (independent of whether I was right or wrong). Robin Harte22 : I went to physics classes (and was told that I had to do extra maths if I wanted to do physics) and at one point our physics teacher was explaining something later called flux, which he described as “the number of lines of force”. What I want to recall is a subconscious awareness that something was out of kilter: nothing was said, but this quantity seemed not necessarily to be a whole number and that seemed to bother me. So in the sixth form at school I believe I had some kind of mathematical sense. I can also report small hairs standing up on the back of my neck the first time I met differentiation. As Tanya Khovanova’s 23 story confirms, it could be easier to accept that it is physics that is non-rigorous, thus protecting the sacred status of mathematical rigour: I had problems with physics rather than mathematics. I remember when I was in fifth grade (I lived in Moscow at that time) my father told me about the relativity theory and the fact that the speed of light is constant. When I started studying mechanics in my 8th grade I was completely confused. The vector addition of speeds contradicted to all I new about mechanics. I was bugging the teacher to give me the proof why speed were added this way. He couldn’t give a proof and I couldn’t move forward. For half a year I was stuck and was getting bad grades until someone realized what my problems and explained to me that physics is not mathematics and that physics is based on experiments rather than proofs. Also, that vector speed addition is an approximation. After that my grades immediately got back to my usual A’s and physics became a breeze. 22 23 RH is male, British. He says about himself: I did go on to do mathematics in Trinity College Dublin, and a PhD in Cambridge, and never really to escape the system. TK is female, Russian, has a PhD in Mathematics, holds a research position in an American university. A popularizer of mathematics, see her website http://tanyakhovanova.com. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 18.6 The quest for rigour 181 GCS24 : Age 6, a state primary school in a working class London area. I always enjoyed playing with numbers. A teacher tried to tell us that when you broke a 12 inch ruler into two pieces that were the same, each would be 6 inches long. I went to see her, because I couldn’t see how you could break the ruler into equal pieces, because of the point at 6; it wouldn’t know which piece to join. In adult notation [0, 6] and [6, 12] are not disjoint, but [0, 6), [6, 12] are not isometric. No matter how I tried to explain the problem, she didn’t understand. It was a valuable lesson, because from that point on my expectations of schoolteachers were much reduced. Another aspect of mathematical thinking manifested itself in a very literal frame of mind. I can remember when the head teacher told us one day to ‘run round the playground’, all the other children (correctly) interpreted his instruction as an invitation to go outside and play. I, of course, stuck to the perimeter and performed laps. Mihai Putinar25 : I think my mind was shaped by the generous and beautiful nature (mountain region) where I grew up, and mainly by playing all my childhood along a river which passed in the proximity of our house. The flowing water imposes a grammar of thought which I feel remains deeply rooted inside me. Why mathematics? At 12–14 years of age I had serious troubles understanding the natural sciences the way were presented in school (physics, biology, chemistry). I could not accept the dogmatic approach, with ad-hoc working definitions (a la “the energy is the quantity numerically equal to . . . ”), all melted into a descriptive story, sparkled with many examples and without a structure. Mathematics came as the unique salvation: crystal clear, simple, founded on a few principles, totally mental yet so effective. The deductive method, well exposed in the fifth-sixth grade by planar geometry, the implacable power of algebraic identities, the endless mysteries of numbers won my heart. Since then I could not stop doing math with great pleasure and with all my energy. And Pierre Arnoux26 returns to an earlier told story (page 22): Concerning the first story I told you, when I was 10 years old, I remember very clearly the feeling I had when I first learnt the idea of a variable. The best comparison is that I felt I walked on 24 25 26 GCS is male, English, a lecturer of mathematics in a British university. MP is male, Romanian, a professor of mathematics in an American university. PA is male, French, a professor of mathematics in a French university. Another story from him is on pages 22 and 70. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 182 18 Being in Control very thin ice, which could break at any moment, and I only felt safe when I arrived at the solution. Olivier Gerard: I concur with Pierre Arnoux with a similar image. When I started solving equations in junior high school, I had the feeling that the variable was a very precious thing, that could become ugly if badly handled and following the rules such as “passing numbers from one side to the other and inverting signs” carefully was like walking softly or stepping slowly when moving a large stacks of things or books in one’s hands. If something was done too roughly the things ended on the floor, some of them bruised or broken. As many children experience, the broken glass or the damaged object ended to be found missing by their parents. With potential bad consequences. 18.7 Suspicion of easy options Olivier Gerard raises another issues which I have never seen discussed in literature on mathematics education. In most exercises we had to do, there were miraculous simplifications, especially with fractions, giving round numbers and “palatable” expressions at the end as the solution. This was a good indication that we succeeded in following the rules correctly but then it was also misleading if used too systematically by teachers. I understood more and more with time that this was in great part a convenience for teachers when designing exercises: they could easily grade that way. I remember saying to myself: they must start from the result and then they go backwards to the question to make it look complicated. They are so good, they probably can do that in their head. With hindsight these round number solutions can have perverse consequences. Inside math teaching, there is the fact that pupils can infer incorrect informal rules for themselves about what the steps and results should be, based on the very special subsets of applications and numbers they see doing exercises. Some develop a kind of mysticism of what the value of x or f (x) should look like across exercises. They are not ready nor willing to start with a blank slate about them with every problem. When interviewing some young pupils that had a lot of difficulty with mathematics in high school, I discovered that some of them actually thought that they could learn more about x or f each time they were doing another exercise, and the harder they tried, the better memory they had, the more confused they got. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 18.7 Suspicion of easy options 183 It is worth noticing that, with benefit of hindsight, Olivier Gerard sees danger in artificial easy options open to an inquisitive child—but he himself did not succumb to temptation of adopting a grossly simplified approach to mathematics. As a child of 10 or 11, I could easily guess answers to exercises in the arithmetic textbook that required evaluation of complex expressions involving brackets and all four arithmetic operations. I quickly realized the artificial nature of expressions—all numbers were integers and all divisions were exact, without remainders. This revealed too much, and crude estimates of dividends and divisors frequently produced almost instant result. I remember developing a bit more sophisticated tricks for more complex exercises, but I cannot recall them now. In any case, even as a child, I was uneasy about the pedagogical wisdom immortalized in the famous Russian painting, see Figure 18.1, and I still cannot decide whether it is good or bad. But I agree with Olivier Gerard’s further comments: In relation with physics, where “numerical applications” are messier or are intended as approximations, that should be truncated as appropriate, it will give or prepare a false intuition of simplicity or correctness, more based on numerology than relative dimensions and units. In relation with a little more advanced mathematics, when you discover some of the increasing complexity say of solutions to polynomials equation like the 4th degree (and of course the impossibility of solving it generally for degree greater than 4), the fact that many important constants such as π, e, are both very simple but not that much if you do not know integrals and infinite series, the difficulty of constructing exact polygons of odd order, the fact that triplexes do not exist, etc. you can have a deception if you anticipated mathematics to be always cutting through complexity with very clean and minimal closed formulae. I know of some of my former classmates who felt deceived in that way and decided to turn more fully to other disciplines such as biology, chemistry, history,. . . Perhaps to be further disappointed in their expectations. I cannot say. And, last but not least, child’s suspicion that teachers cheat and build exercises from ready answers is already an important mathematical discovery. Olivier Gerard’s concludes his reminiscences with a charming episode: I remember that when 11 I wondered how you could make a labyrinth puzzle like the one I liked to solve on activity and puzzle books. So I took squared paper and tried to design one myself in order to ask my sister to solve it. I had a very hard time trying to do it without success, I was always walling myself into it. It was S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 184 18 Being in Control Fig. 18.1. Mental arithmetic. Nikolai Petrovich Bogdanov-Belskii, 1896. Tretyakov Gallery, Moscow. Public domain. The teacher is Sergei Aleksandrovich Rachinskii (1833–1902), biology professor and a philanthropist; Bogdanov-Belskii (1868–1945) was his pupil and protege. An exercise for the reader: solve the example by mental arithmetic. also because I wanted to design the most difficult labyrinth puzzle possible, that only me could solve. So I did not try to do first something simple. Then one year later while going home from school, I thought “I have first to draw the path I want, then I build the S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 18.8 “Everything had to be proven” 185 labyrinth around it”. I started doing it that way, tracing a solution with lead pencil in order to erase it and beginning walls with black ink. But I soon stopped. I was confident I had found the way to do it, it was not interested anymore, I was not motivated anymore either by challenging others to solve the resulting puzzle. I switched to play with something else. 18.8 “Everything had to be proven” A child’s quest for rigor in mathematics can be supported by his or her intellectual environment. The following story, from Azadeh Neman27 , comes from her childhood in the political hotspot of Iran: I was talking to a colleague here and he mentioned something which I thought might be useful for your book. It was about the idea of proof and how hard it is for students to grasp what is a rigorous proof. Since I assume most people in the field of mathematics to be a lot more clever than I am, I found it interesting that I did not experienced such a problem. After considering the matter a bit I realized the problem comes from the late introduction to the idea of rigorous proofs. Since I was brought up in an environment of constant political argument and everything one claimed had to be proven there, it was not so strange to see the same principle applied, more coherently and clearly of course, to two-dimensional geometry. Adding one sheer good luck in my first year in Highschool I ended up with a very methodical teacher who would give us marks for clear statement (and deduce for the absence) of the hypothesis and the goal. Her method kind of stayed in mind ever since. And now we move to another hotspot, Turkey. Ali Nesin28 tells about his childhood and about his father Aziz Nesin, a famous writer and political activist: I spent my early childhood in drawing, dreaming and reading. I loved to be left alone and be immersed in my thoughts. I remember looking at oil paintings for hours. I believe what made me a mathematician is this love of loneliness. Also, I loved to listen to the intense debates of intellectuals. Most often about politics and literature. Indeed every other night our home was full of writers, poets, painters, actors. . . I was trying to understand which one was the winner of the debate! 27 28 AN is female, Iranian, educated in Iran and Britain, holds a PhD in mathematics from a French University. AN is male, Turkish, professor of mathematics, an influential teacher, popularizer and promoter of mathematics. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 186 18 Being in Control Most people at these “raki tables” were highly educated and clever and excellent speakers. But none of them could equal my father. His flawless logic, the examples he picked, the analogies he found, strong images he offered, the eloquent and the soft way he formulated his ideas, and a witty but warm humor on top of all that was a real delight; not only to me but to everyone. I believe these conversations also played an immense role in my becoming a mathematician. 18.9 Raw emotions In our discussion so far, the issue of control was mostly mathematical. When classroom politics and relations with teachers and other students are involved, the stories told are sometimes bitter, sometimes good humored, but they are almost always quite emotional. Here is a testimony from Charles Leedham-Green 29 : The first time I went to school I was three years and seven months old. There were two other pupils at the school, girls rather older than me, so there was no streaming. The teacher was probably in her teens. We had a mathematics lesson. We were each given a piece of cardboard that had the outline of a one (1) with all the trimmings, thus ____ /__ | | | | | | | __| |_ |______| We were required to colour this in red, and after some effort we more or less achieved this task, and the best effort, far far better than mine, was held up for general admiration, and we were asked what this might be. The two girls answered ‘The figure one’ in unison. I had the impression that they had been coached to do this. However, I was not so sure and remained dumb. I had a suspicion that as even this high quality art work was inclined to go over the lines, and leave some parts of the interior uncoloured, it might not be quite a figure one. I am still not certain of whether I was right to be cautious, but I went right to the bottom of the class [of three]. Later of course I went to a big school, with some twelve pupils and two teachers. It was an excellent school; I think the teachers, who were sisters, were in their seventies; perhaps a little younger; unmarried because of the slaughter of the men in the first world 29 CLG is male, English, professor of mathematics in a British university. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 18.9 Raw emotions 187 war. I was at this school till the age of six or seven, and was taught a fair amount of French, Latin, History, Norse mythology, spelling, art and mathematics. We were taught arithmetic up to the extraction of square roots and the calculation of hcf’s and lcm’s by factorisation. I was reasonably competent, except perhaps for the art, but was inclined to think about things (or gaze out of the window) rather than getting on with my work, a tendency that the teacher regarded with some understanding. Sometimes, when doing a long multiplication, I would set it out the other way from the way we had been taught; so I started by multiplying by the unit entry of the second number instead of starting with the most significant digit. The teacher took this in her stride, and learnedly told me that I had set it out in the German style. She was a brilliant teacher, but they retired and closed the school as I was leaving, raising the fees for the last year from £2 per term to £2 10s per term. At my next school, which had 70 pupils in 6 classes, having rather lost track, through lack of practice, with how to set out multiplications, I set out a multiplication in some way unknown to God or man, but got the right answer, to the confusion of the teacher. I also surprised him by knowing the word ‘colloquy’ at the age of seven. Not infantile precocity, but spelling lessons at the previous school. In the next story, from FB30 , the most interesting point is the use of the word “eavesdropping” that gives it a distinctive emotional color. I was a competent at mental arithmetic but did it as a learned trick without real understanding. I understood multiplication / division eventually by eavesdropping on teacher explaining the concepts using (not real) squares of chocolate to a child who was struggling. This experience was useful to me when I taught secondary mathematics for a while. Similarly, the key word in Robin Harte’s story appears to be “escape”. I have a curious memory for you, I probably got into mathematics as an escape from physics which in turn was an escape from Latin and Greek—my father was a school teacher who taught Latin and Greek and it was important to me to stay as far as possible from that—I was at the same school where he taught Latin and Greek which was bad karma. [. . . ] I did go on to do mathematics in Trinity College Dublin, and a PhD in Cambridge, and never really to escape the system. 30 FB tells about herself: “I was 9 years old; I am female, from UK, studied mathematics at university but not very happily.” She teaches at a British university and actively promotes the use of information and communications technologies in teaching and learning. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 188 18 Being in Control Reuben Hersh: 31 The best I can do—not what you are looking for, I am afraid—was when my family moved from The Bronx to Mount Vernon (a nearby suburb) and my mother and I met the principal of my new school. (I was entering the 10th grade, but I was only 12 or 13 years old, and had a protective mother.) Anyway, my mother told the principal, “He loves mathematics,” and the principal said to me, “You’ll take geometry.” To which I replied (based on my previous two years of junior high school) “But I like algebra!” To which the principal simply replied, “You’ll take geometry.” Of course, I did OK in geometry. But I was always more inclined to algebra. Alexander Bogomolny32: [I was 12.] One day, my math teacher asked me to stay after the class as she wanted to show me something interesting. After every one left, she wrote on the blackboard an example of multiplication of two-digit numbers by 11. She asked whether I saw why it worked and whether I liked that. I was utterly embarrassed. Of course I knew that trick. I do not remember how it happened, but I was aware of many more, this particular one being probably the most trivial. I blushed, thanked her and ran out of the room with a realization that, if I wanted to learn anything, I had to do that on my own. Michael Breen: When we learned long division, I could do it in my head, so that is how I would do it in class. My fourth grade teacher scolded me very much for this, and insisted that I write things down. She was undoubtedly right, because as the numbers got larger, my brain would not have been able to hold everything, but her scolding was quite upsetting. Alan Hutchinson: I asked the head of maths at school what more maths there is to learn. He laughed at me. 31 32 RH is male, American, a professor of mathematics and a prominent author of books on methodology and philosophy of mathematics. He informed me, in particular: “The first chapter of our forthcoming book Loving and Hating Mathematics [131] has plenty of stories of childhood experiences of mathematicians, but they are mostly happy and positive experiences.” AB is male, his mother tongue is Russian. He has MSc in mathematics from the Moscow State University and a PhD in mathematics from the Hebrew University. AB runs a wonderful website http: //www.cut-the-knot.org/ctk/ErrDisc.shtml. The episode happened when he was 12 years old. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 18.9 Raw emotions 189 Gwen Fisher: I think my biggest general challenge with math as a kid was wondering what it was useful for. So many of my teachers taught stuff so abstractly, it was just moving symbols around on paper. But I was good at that, so I didn’t fuss about it much. One very memorable occasion for me was when I asked my junior high school math teacher why matrix multiplication was useful. He admonished me harshly for my question, and never answered it. The effect of this incident on me was that I never asked “Why do we do this?” in a math class again. . . until I was in graduate school! I was learning about finite fields and asked my teacher and then recoiled in anticipation of the tongue lashing that I thought would result from my question. Alas, he answered me, so I told him this story about matrix multiplication. And so he told me an application for matrix multiplication that an 8th grader could understand. That was a very important moment for me as a math teacher. Theresia Eisenkölbl: I know that this is not exactly what you are searching for with your demand for stories, but I have to comment on the “intrinsic competitiveness” of mathematics. In mathematics olympiads and in professional mathematics, competitiveness is ritualized and contained by ethical rules. In my olympiad training session, you were expected to explain things to newer participants even though they might well be better than you at the competition. In olympiads and real life, I have found that this kind of ethics almost always corresponded to mathematical capability. So here is my anecdote on competitiveness: In primary school (age 6–10), I had about half the running speed of any other child. On the other hand, I understood everything in maths immediately and was very quick with mental calculations. In sports, we had weekly running competitions of groups by four, where my group never had a chance at all. In maths, we had weekly speed calculating competitions where each pair of students were posed a question and the quicker one would remain standing until one student was left and got a little prize. I won every time and was only allowed to participate every other week. The lesson that good performance in math is bad for your social standing with your peers could not have been more evident. And I would like to round up this section with a bit of very happy memories. Here is a story from Anthony O’Farrell33 : I suppose you are referring to kindergarten and primary school? I never had any difficulty with any mathematical ideas presented to me at this level. Everything made immediate sense. However, 33 AO’F is male, Irish, a professor of mathematics. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 190 18 Being in Control most of it was rather unsurprising. I remember one or two things that took my fancy, such as the fast way to work out the squares of numbers such as 45. For the most part, school was prison, a place you had to go, where they told you stuff you knew, and made you memorise stuff you might otherwise ignore. But I went every day, without fail, and occasionally something nice happened. The one day that stands out in my memory was in 6th class (aged 11), when Brother Skehan explained how to complete the square and derive the formula for the solution of the quadratic ax2 + bx + c = 0. This was a revelation—the whole way of thinking was like nothing in my experience up to that point. I never forgot that elegant little argument. About 5 years ago, I met the brother again, and mentioned the incident to him. He remembered it, too. He told me that he was not required to do it, it was not on the curriculum, but it was a slow day and he just felt like doing something different. He observed that most of the class looked on in blank incomprehension, but that I sat up and drank it in, with shining eyes. 18.10 David Epstein: Give students problems that interest them David Epstein34 : I was about 9 years old when I was given as a present a book by H. E. Dudeney of puzzles. (See Wikipedia for an article about Dudeney.) I tried to do all the questions, which covered a wide variety of different kinds of problems. One particular type of problem “Alice was twice as old as Henry when she was as old as Henry is now, etc, etc” intrigued me but I couldn’t solve them. I somehow got the idea that “algebra” was the answer. I asked my teacher for a more advanced textbook, but she refused, so one lunch break, when everyone was outside playing, I took the book from the cupboard. The teacher denounced me to the class as a thief, and demanded the book’s return. But I didn’t return it. Instead, I studied it with great interest, and with a lot of difficulty in coming to terms with what it all meant. After a while I could solve these problems in Dudeney. Of course, there were many other types of Dudeney problems that I was unable to solve, but I was only excited by those for which I could find, by one means or another, a way of attacking them systematically. David Epstein continues from his perspective of an university teacher: 34 DE is male, British, learnt mathematics in English, a professor of mathematics. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 18.10 David Epstein: Give students problems that interest them Perhaps it’s appropriate for me to tell you my own views on the pedagogical principles related to this story. One of the points of mathematical pedagogy should be to give students problems that interest them. The problems must be of such a nature that the students can understand exactly what would be acceptable as an answer. In the case of problems such as those I was interested in, it would be possible, but very laborious, to solve them by brute force, trying lots of possibilities. In fact this is what I used to do before I met algebra. The teacher should encourage the student to solve a few such problems using the laborious methods. Then the teacher introduces the method, algebra and simultaneous linear equations, to help the student solve the interesting problem. If successful, the new material can be seen as a release and support, rather than as a daunting heap of new material to learn. I used this approach in a controversial course I gave on metric spaces for several years. I invented several problems that could be understood, but which could only be answered with metric space theory or topology. I restricted the theorems I proved in the course to those that could be used to solve my initial problems. The syllabus was rather different from the syllabus of the official course. Many university mathematics courses are based not on helping students to solve problems in which they may already be interested, but instead in terms of laying the foundations for some future course. The future course also consists of laying down the foundations for a future course, and so on . . . satisfaction infinitely delayed, except for the small minority who become research mathematicians. Of course I exaggerate. I wish I had time to write up my course for publication, as it was rather unusual, and the approach produced many interesting ideas, including some research problems. It is worth quoting from the autobiographic book Récoltes et Semailles by the great mathematician Alexandre Grothendieck [127]: Je passais pas mal de mon temps, même pendant les leçons (chut. . . ), à faire des problèmes de maths. Bientôt ceux qui se trouvaient dans le livre ne me suffisaient plus. Peut-être parce qu’ils avaient tendance, à force, à ressembler un peu trop les uns aux autres; mais surtout, je crois, parce qu’ils tombaient un peu trop du ciel, comme ça à la queue-leue-leue, sans dire d’où ils venaient ni où ils allaient. C’étaient les problèmes du livre, et pas mes problèmes. Pourtant, les questions vraiment naturelles ne manquaient pas. Ainsi, quand les longueurs a, b, c des trois cotés d’un triangle sont connues, ce triangle est connu (abstraction faite de sa position), donc il doit y avoir une “formule” explicite pour exprimer, par exemple, l’aire du triangle comme fonction de a, b, c. Pareil pour un tétraèdre dont on connaît la longueur des six arêtes — quel est le volume? Ce coup-là je crois que j’ai dû peiner, mais j’ai dû finir par y arriver, à force. De toutes façons, quand une chose me “tenait”, S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 191 192 18 Being in Control je ne comptais pas les heures ni les jours que j’y passais, quitte à oublier tout le reste! (Et il en est ainsi encore maintenant. . . ) An English translation (Adrien Deloro—with thanks): I would spend a lot of my time, even during the classes, solving math problems. Soon those from the book weren’t enough. Perhaps because on the long run they tended to look too much alike; but above all, I think, because they were a bit too much like godsent, just like that, one after the other, and they wouldn’t say whence they came from nor where they went. Those were the problems from the book, not my problems. And yet there were plenty of truly natural questions. For instance, when the lengths a, b, c of the three sides of a triangle are known, the triangle is known (apart from its position), so there must be an explicit “formula” yielding, for example, the area of the triangle as a function of a, b, c. Same thing for a tetrahedron when you know the lengths of the six edges—what is the volume? On this one I think I’ve suffered, but I must have eventually succeeded. Anyways when something really had “caught” me, I wouldn’t count the hours nor days that I would spend on it, should I forget all the rest! (And so it is still now. . . ) 18.11 Autodidact I continue this chapter with a story of a rebellion and dropping out of the mathematics education system altogether. In elementary-school (ca. age 6–9 years) I was fascinated by elementary geometry, in part because a popular TV series on astronomy and relativity caught my imagination and provided some fascinating statements, like the usual visualizations of strange noneuclidian things. I found the possibility of proving “obvious” statements by general principles absolutely fascinating, much more fascinating than proving more complicated statements. However, when I asked teachers about non-euclidian geometry, their negative reaction alienated me very much. Further I found the way some geometric objects were defined too ugly to accept. For example, an ugly definition for such a nice figure as a circle to prove interesting statements appeared to me very crude. On the other hand, my attempts to use nicer definitions did not work. At age ca. 14 I started learning analysis by myself, because I wanted to understand relativity, and the book I had proved every statement twice: first by elementary geometric constructions with the help of problem-specific thought experiments, then by analysis. That the later proofs were much shorter, more general and looked somehow better, made me interested in analysis. I had much trouble to understand what definitions are and to separate the Definition/Statement/Proof parts of the text. My feeling towards definitions was “Well, that’s O.K. and obvious, but what are S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 18.12 Blocking it out 193 these things really and why does one use this selection of features for the definition?” So, reading the first 20 pages needed several weeks, i.e. the same time as the rest of the book. I am Austrian, lack advanced school degrees, and after some years of tutoring students I was invited to do university entrance exam. My math education is completely autodidactic, and was from German books, but shifted—from age ca. 16 on, after I obtained access to the university library—to English ones (later to French and Russian too; unfortunately the library refuses to buy Japanese math books). This had a strange side effect, because I had then acquired from mathbooks apparently an English with strong traces of Latin grammar initially, which made people “accuse” me of being some weird upper class teenager playing the uneducated, but “obviously knows Latin”. A side effect of autodidactism may be that my mathematical interests are still strongly guided by aesthetical impressions, esp. if concepts allow visualizations. Some theories have very special aesthetical properties, e.g. class field theory and it’s connection with modular forms. A negative result of going to a university is that I study in a much more superficial way than earlier—formerly, I read something until I could not only solve the exercises, but derive the whole theory from a handful of basic ideas, which were often helpful in seemingly different math contexts too. Now I read more and faster, but restrict usually to understanding the techniques of the proofs. 18.12 Blocking it out And I conclude this chapter with posibly the most disturbing stories of all: about blocking out early mathematical memories entirely. The first testimony came from CC35 : I would very much like to contribute to your project but I am afraid I have no special recollection. In my school days mathematics was taught in Italian schools in such an abstract and purely technical way that I hated it, so I must have removed all my conscious recollections about it. I wish I could have read a book such as, say, Davis-Hersh, The Mathematical Experience, at that time. I would have loved it. JG36 : I’m afraid that I don’t have any recollections of my own to add to your collection—not because my learning of mathematics has been 35 36 CC is male, Italian, a professor of logic with research interests in logic, philosophy of mathematics and epistemology. JG is male, has a PhD in mathematics, teaches in a British university. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 194 18 Being in Control challenge-free but rather that my childhood memories are in very deep mental storage and I often have trouble recalling them. RSR: I don’t know what happened to my mathematics in early school. I don’t remember. Exercises Exercise 18.1 Calculate by mental arithmetic: 102 + 112 + 122 + 132 + 142 , 365 see Figure 18.1. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 19 Controlling Infinity The boy was small and the mountains were enormous. Heinrich Mann, Young Henry of Navarre. 19.1 Fear of infinity The first encounter with the infinity of the series of natural numbers, when counting goes on and on, is a classical and frequently mentioned case of of losing control in mathematics. I wrote about that in my book Mathematics under the Microscope [107], and mentioned my own story in Chapter 1 of this book. When I was 7 years old, the problem of distributing 10 apples between some people, 2 apples a person, was very disturbing to me. I tried to visualize the problem as an orderly distribution of apples to a queue of people, two apples to each person. Unfortunately, this necessitated dealing with a potentially unlimited number of recipients. In horror I saw an endless line of poor wretches, each stretching out his hand, begging for his two apples. I was not in control of the queue! Perhaps some backgrounds have to be explained. I lived in the Soviet Union. By the time I started my school, I had already had the unfortunate experience of the notorious bread queues of the summer of 1962. There was no famine as such: neither I nor anyone I knew was ever hungry, but every morning after breakfast (at about 9 o’clock) I had to go to the village bakery to replace my bigger brother who had been standing in the queue from 5 in the morning. A couple of hours later, when I was approaching the counter, another brother came to replace me. We were 6, 11, 15 years old, respectively. 195 196 19 Controlling Infinity These bread queues were one of the causes of the eventual fall from power of Nikita Khrushchev, the supreme Soviet leader of the time. His name was frequently mentioned in the queue, usually accompanied by expletives. This was the most useful lesson in politics that I ever got. But it also had an impact on my perception of mathematics. 19.2 Counting on and on I continue with stories less scary but perhaps no less emotionally charged. For a child, the world around him and her is big. Very big. And counting is a way of keeping it under control. I start with a short story from Archie McKerrell1 : About as soon as I could count I counted the steps to our flat at the top of a three storey tenement. (UK counting of storeys, ground, 1, 2, 3.) The number in each flight still seems almost part of the structure of the universe! 6 + 18 + 19 + 19 = 62 Michael Fried brought to my attention to the following passage from Piaget’s Genetic Epistemology [377, pp.16–17]. . . . I should like to give an example, just as primitive as that one, in which knowledge is abstracted from actions, from the coordination of actions, and not from objects. This example, one we have studied quite thoroughly with many children, was first suggested to me by a mathematician friend who quoted it as the point of departure of his interest in mathematics. When he was a small child, he was counting pebbles one day; he lined them up in a row, counted them from left to right, and got ten. Then, just for fun, he counted them from right to left to see what number he would get, and was astonished that he got ten again. He put the pebbles in a circle and counted them, and once again there were ten. He went around the circle in the other way and got ten again. And no matter how he put the pebbles down, when he counted them, the number came to ten. He discovered here what is known in mathematics as commutativity, that is, the sum is independent of the order. But how did he discover this? Is this commutativity a property of the pebbles? It is true that the pebbles, as it were, let him arrange them in various ways; he could not have done the same thing with drops of water. So in this sense there was a physical aspect to his knowledge. But the order was not in the pebbles; it 1 AMK is male, Scottish, has a PhD. He teaches at a a British university and is doing research in theoretical physics. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 19.2 Counting on and on 197 was he, the subject, who put the pebbles in a line and then in a circle. Moreover, the sum was not in the pebbles themselves; it was he who united them. The knowledge that this future mathematician discovered that day was drawn, then, not from the physical properties of the pebbles, but from the actions that he carried out on the pebbles. This knowledge is what I call logical mathematical knowledge and not physical knowledge. David Cariolaro2 : When I was 6 I went to school and learnt multiplication. When I went back home that day, instead of watching TV I calculated all the powers of 2 up to 216 —and memorized them, thereby getting acquainted with the behavior of the exponential function . . . PW3 : I remember one mathematical event which took place early in my life—I must have been 4 or 5 at the time. At that age I had the flexibility to keep a hula-hoop going round my waist, and I used to count the number of times the hoop went round: 1, 2, 3, 4, . . . I had trouble because I did not know what number came after 199, and so I used to do the hula-hoop 199 times and then stop. I think the problem was that someone had told me the name of the number 100 and how 101 came afterwards, but no one had thought to tell me the number 200. Its name did not seem obvious to me, because we have sequences like ten, twenty, thirty, . . . where you have to be taught the names, and I may have thought that 200 had a special name like these. I resolved this by asking someone what came after 199, and was told. The consequence was that I would then spend time doing the hula-hoop more than 199 times! I still wonder what the adult I asked thought of my question, but I suspect it was of greater significance to me. John Shackell4 : I don’t remember the incident myself, so I am relying on what my parents told me. I would have been three years old, getting towards four. My mother was confined for the birth of my sister and so I was being cared for by an aunt. I don’t think she had an easy task. I would stand on my head on the sofa and read the page numbers from an encyclopedia. I was very persistent. The conversation went approximately as follows: 2 3 4 DC is male, Italian, has a PhD in mathematics, holds a research position. PW is male, American, a professor of mathematics. JRS is male, English, a professor of mathematics. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 198 19 Controlling Infinity - “One thousand three hundred and twenty three, one thousand three hundred and twenty four.” “John, stop that counting.” - “One thousand three hundred and twenty five, one thousand three hundred and twenty six." - “Oh John DO stop that counting.” - “One thousand three hundred and twenty seven. I wish YOU were one thousand three hundred and twenty seven." - “Well you wouldn’t be so young yourself!” Viktor Verbovskiy5 : When I was 7 or 8 year old, ny uncle decided to teach me to count up to a million. I do not remember up to what number I could count at that time. Teaching was done in an abbreviated form: . . . after a thousand goes two thousand, then three thousand, . . . , ten thousand, eleven thousand, twenty thousand, thirty thousand, . . . , one hundred thousand, two hundred thousand, three hundred thousand, . . . , one million. I accepted that as a literal truth, and came to conclusion that one thousand multiplied by 37 gives one million. That is, it was assumed that one thousand is followed by one thousand and one, but I somehow missed that point, since it was not told to me explicitly. But when I told to my big brother about my discovery that one thousand multiplied by 37 makes a million he looked at me as if I was crazy, I was embarrassed, started to think and everything somehow fell into its place (although I do not remember how long it took). A summary: I accepted any new material absolutely uncritically and only after encountering an obvious error started to rethink it (on that point, I have not actually changed much). As a whole, at school I have easily handled all kind of routine manipulations, I could keep in mind a bunch of rules and check, on the top of my head, all variants of their consecutive application, but hated brainteaser that required ingenuity. Perhaps, it was when my love to logic was born, because in logic proofs are very consequential . . . Theresia Eisenkölbl6 : My brother and I had learned (presumably from our parents) how counting goes on and on without an end. We understood the construction but we were left with some doubt that you could really 5 6 VV is male, Russian, has a PhD in mathematics, holds a research position in mathematical logic. TE is female, learned mathematics in German. She has a PhD in Mathematics (and a Gold Medal of an International Mathematical Olympiad), teaches mathematics at an university. At the time of this episode she was 3 years old, her brother 5 years old. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 19.2 Counting on and on 199 count to high numbers, so we decided to count up to a million by dividing the work and doing it in the obligatory nap time in kindergarten in our heads. After a couple of days, we had to admit that it took too long, so we debated whether it was ok to count in steps of thousands or ten-thousands, now that we had counted to one thousand many times. We ended up being convinced that it is possible to count to a million but slightly unhappy that we could not really do so ourselves. SK: I recall from my childhood the following story. When I was 5–6 years old, definitely before I went to school, my elder sister told me several times and tried very hard to explain to me that there is no final / biggest number when you count. I remember I kept asking her “Where is the gate?”7 In my mind I imagined a closed gate that stops numbers from increasing! Surely everything should be finite in the finite world we live in. It was in Ukraine (at that time in the USSR), the language of “instruction’ ’/ communication obviously was my mother tongue. Admittedly, it did not help me to grasp the concept of infinity :) Ekaterina Komendantskaya: My four-year old daughter liked to exercise in counting. She would count 1, 2, 3, etc, until 20, and then would continue adding 21, 22; occasionally stopping every time she needed to know how a new decimal, such as 30, 40, etc. called, and then she would add 1, 2 and so on to it . . . Suddenly my daughter asked me: “Mum, and what is the last number?” I said: “There is no last number, you can always add one, to any number!” She made big and happy eyes and said: “And we were told at school that 100 is the last number!”. It crossed my mind that at school, children are not taught the concept of natural number, they only count “up to 10”, “up to 100”, but a child can naturally come to it at 4 years of age! I do not remember which language we were speaking (Russian or English), I always answer using the language she chooses for her question. She was brought as bilingual from birth; and switches from language to language easily. JM8 : When I was quite young—maybe 5—I asked what the biggest number was, and my father (a physician) told me there was none; 7 8 SK is Ukrainian, his mother tongue is Russian, he has a PhD in mathematics and teaches mathematics education at a university. In Russian (with an Odessian twist, in his own words), the phrase was “Gde жe vorota?” JM is male, American, professor of mathematics. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 200 19 Controlling Infinity they just went on and on. I remember trying to visualize this with a string of beads that vanished in a void beyond the view of thought. Some time around age 8, I think, my father posed to me the following question: “If a man walks half way from point A to point B on Monday, half the remaining distance on Tuesday, etc., will he ever reach B?” I still have a vivid memory of the precise diagram he had drawn for me, with several steps of the process marked. I puzzled over this trying to get the meaning, and finally offered the opinion that he would not ever get to B precisely, but he might wind up beyond it. I think the reasoning was, clearly he could not get to B, but he would have to wind up somewhere, and it could not be before B. Later, when I fully understood what the problem was about, it really bothered me that I had produced such a silly response. VS9 : I don’t remember where I encountered the statement that any number divided by zero gives infinity (I think that this was almost definitely not at school). Moreover at school I was taught that division by zero is just prohibited. I don’t remember precisely why I got so excited about this statement to the extent that I immediately went to see the grandmother of my friend (who was retired mathematics teacher) to discuss the issue. She told me in a didactic manner that indeed it is possible to divide by zero and any number divided by zero is infinity. My question “why” was not answered to my satisfaction. I was told that “this is just so, you will learn more when you get older”. The only explanation I’ve got was through a kind of limit process: if we decrease the denominator in a fraction, then the result becomes bigger and bigger. I had no problems with the idea of the limit or limit sequence. These concepts seem natural to me. The problem I couldn’t overcome was, that if we, say, take two different numerators we have two different limit sequences, their ratio doesn’t change, still we get seemingly the same answer: infinity. I remember that I found it very weird that for infinity you immediately get very strange relations which are considered to be correct. Jonathan McLaughlin10 I asked my mother: “What is the biggest number?” And she said: “There is no one.” I did not sleep that night. 9 10 VS is male, Russia, a professor of mathematics in a British university. JM is male, Irish, a PhD student in mathematics. Age at the time of this episode he was about 5 years old. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 19.3 Controlling infinity 201 19.3 Controlling infinity And I end this chapter with examples of a child’s victory over infinity. Again, control is the principal issue. We also see the the weapon used by children is reduction of the infinite to the finite or at least to a clearly described (even if infinite) procedure. And we have already familiar themes, like nomination and inventing names. Fig. 19.1. Vision to the Youth Bartholomew. Mikhail Vasilyevich Nesterov, 1890. The Tretyakov Gallery, Moscow. Source: Wikipedia Commons. Public domain. Bartholomew was to become St. Sergius of Radonezh, the most venerated spiritual leader of medieval Russia. David Jefferies11 I can vividly remember grasping the concept of infinity. I was walking along a road, with walls and hedges coming in periodically and at right angles to the road, that one could see into the distance. I had great wonderment when I realised that this road would never end, or indeed, get anywhere. I was younger than secondary school; I should say perhaps that this road was imaginary, in my daydreams. , 11 DJ is male, English, has a PhD in physics. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 202 19 Controlling Infinity I can remember also being told about decimals by my Dad, after showing off to him about my prowess in using integers and rational fractions. To this day, I am intrigued (my interests are in chaotic dynamics) by the combination of the infinite and the representation of irrational numbers.ixnumber!irrational It is clear that our knowledge about the world must be constrained, if the universe is indeed finite, for there is a maximum length to the expansion of an irrational number that can be stored in any imaginable memory device. Hillary Povey12 : I remember my mother lying in bed (I can visualise everything about the room, the weather, the time of day and so on). I decided to show her how far I could count—I imagine she went back to sleep. I suddenly realised that the counting numbers went on for ever - when you had exhausted all the columns available, you just opened up another column on the left and started, as it were, all over again. I loved this but also found it a bit frightening—like the idea of infinite space. I was 5 or 6 years old and this all happened in English. DD13 : When I was 4 2/3 I first went to nursery school. One day a girl came in with a pencil which had a sort of calculator on the back: a series of five or six wheels with 0–9 on each, allowing simple addition, counting, etc. This was in 1943 in NYC. Her calculator absolutely fascinated me, and I kept watching as, when the numbers got larger, there would be all 9s and then a new column on the left would pop up with a 1. I just got a feeling for how the whole system operated and it definitely made me feel really satisfied, though I did not know why or what I would do with this information. Also, no one of my friends seemed the least bit interested: I don’t think I explained it very well. That weekend, on the Sunday I got up at just before 6 AM and went into my parents bedroom, quietly, as I was allowed to do, went over to the window and looked out down the empty street, at the far end of which was East River Drive as it was then called, bordering the East River. After I bit I started thinking about those wheels. It seemed to me that more important than the 9s were numbers like 100, 1010, 110, 111 then 1000, 1001, 1010 and so on, and I played out these in longer and longer columns in my head until I was absolutely clear how it worked, and I just knew that what I would now call the placeinteger system fitted together in a completely satisfactory way. It was still early so I continued thinking about these numbers, and remembered that we used to argue over whether or not there was a 12 13 PV is female, English, professor of mathematics education. DD is male, American, mathematical physicist, works in a British university. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 19.3 Controlling infinity 203 largest number. We would make up peculiar names in these arguments (the boys in the nursery class, that is): so somebody would say that a zillion was largest, and someone else might say, no, a squillion was, and so on, nonsense on nonsense. But if these discussions meant anything, I thought, it should all clear itself up in the column pictures I now had in my mind. I then tried to picture the 0/1 arrangement of the largest number, and was tickled at the thought that if I then cranked everything up by rolling the smallest wheel round and then seeing (in my mind’s eye) the spreading effect it had, I would get an even larger number. Great. Then I got upset: I already had the largest number, according to nursery class arguments. So what was going on. I do not know where it came from, but I suddenly realized that there was no largest number, and I could say exactly why not: just roll on one more, or add 1 (I did know addition quite well by then.) Aha! So I woke up my Dad and excitedly told him that there was no largest number, I could show it, and recited what I had thought out. Poor fellow: it was the overtime season and he worked more than 8 hours a day six days a week—he was not impressed. I won’t tell you what he said. Later that day my mother was pleased that her first born son had done something, but I don’t believe that either of my parents, or even any of the others in the nursery class, ever really understood the point I was making, and certainly never got intense pleasure from thinking about numbers. Although this incident is filtered through my decades of doing mathematical physics, it remains clear in my mind and always has. John W. Neuberger14: At about the age of about seven, I lived with my family in an old two story wooden farm house in Iowa. On some very cold and windy nights (the house would seem to lean with the wind and then relax when the wind momentarily fell) I would be in my bed, not sleeping but thinking about processes which would generate very large numbers (start with a number, double it every second for a while, then double the result every half second for a while, then double that result every tenth second for a while, and so on. I was concerned with two matters, the first quite clear to me and the second one so vague that I had no hope to express it to myself or anyone else. The first was that for any large number generating scheme I had devised, there was always another that would grow faster. The second concern, the vague one, was that somehow there might be a scheme which had to terminate in finite time if only it were devised cleverly enough. Years later I saw this to be true. For example, double the starting number in one second, double again in the next 1/2 second, double again in the next 1/4 second, and so on. In two seconds one would have surpassed every number. My language of mathematical instruction was always English. 14 JWN is male, American, a professor of mathematics. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 204 19 Controlling Infinity Up until university, I had a rich intellectual life and I went to school, but I had no reason to connect the two activities. I was amazed and delighted at university to begin to make a connection. Nearly seventy years past seven years I still am fascinated by limiting problems and by the opportunity to deal with students in their thoughts on these matters. MM15 : I do not know if this is of interest to you or not but here is a thought-experiment of mine, probably around the age of 7-8, for sure after 6 and before 10. I started thinking about death and wanted to convince myself I would never die, instead of thinking about life after death . . . So I started thinking about an infinity in this way: first, I assumed that my entire life was only one dream in one night in another life where I am still the same person but could not fully realize that a full life goes on in each dream (an interesting point about personal identity, I guess). Now, that other life would be finite and have only a finite number of nights. So, I thought further that in each night there must be a finite number of dreams, encapsulating a finite number of lives. This was still short of infinity, so I started thinking that in each of these finitely many dreams of the finitely many nights, I would live a life that would in turn contain finitely many nights, which would contain finitely many dreams, and so on. I was not so sure that I was safe that way (i.e. that I would go on living forever), but I convinced myself that these were enough lives to live, so that even if the process would end, I would still have lived enough, and stopped thinking about it. LB16 commented on MM’s post in my blog: This is beautiful. I had the same “grand thought” at about the same age and I remember that I had the feeling of a true realization upon fully grasping its implication. The difference being that I did not look for a solution to a problem (death) rather I had been thinking about the nature of dreams and why dreams where as realistic as my “real” life. Then this thought developed on its own. Many years later I had one of my first lucid dreams. I was very excited and started to think about what I wished to do with the freedom I had gained knowing that I was now in dream state, only to wake up. What a disappointment. Some time later I woke up again, this time for real (?) only to find out that my prior waking up was just a change from lucid dreaming to normal dreaming. I had tricked myself out of the lucid state. Am I still dreaming? 15 16 MM is male, French, a professional philosopher with research interests in philosophy of mathematics. The episode took place at age 7 or 8. LB is male, Italian, an electronics engineer. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 19.4 Edge of an abyss 205 A reader of my blog, who signed his comment only as JT, remarked that little Mathiew was safe because of König’s Lemma: Every infinite finitely branching tree has an infinite path (with no repeated vertices). 19.4 Edge of an abyss I keep picturing all these little kids playing some game in this big field of rye and all. Thousands of little kids, and nobody’s around —nobody big, I mean—except me. And I’m standing on the edge of some crazy cliff. What I have to do, I have to catch everybody if they start to go over the cliff —I mean if they’re running and they don’t look where they’re going I have to come out from somewhere and catch them. That’s all I’d do all day. I’d just be the catcher in the rye and all. J. D. Salinger, Catcher in the Rye. One of the advantages of being a child is the blissful ignorance of dangers of the world; and the world of infinity is dead dangerous. Here is a story from Alexander Olshansky17 : In 1955 I was 9 years old. My father, Yuri Nikolaevich Olshansky, a lieutenant colonel-engineer in Russian Air Force, was transferred to a large air base in Engels. Every Sunday on the sport grounds of the base there were some sport competitions. A relay race of 800 meters + 400 meters + 200 meters + 100 meters was quite popular; it was called Swedish relay. After two or three races I have come to an obvious conclusion that the team wins which has the strongest runner on the first leg (or on the first two legs) because this runner stays in the race for longer. But the question that I asked to my father was in the spirit of Zeno’s paradoxes: if the race continues the same way, 50 meters + 25 meters + . . . will it be true that the runners will never reach the the end of the 4-th circle (one circle is 400 meters)? (My father was retelling my question to his fellow officers; before World War II, he graduated from the Mathematics Department of Saratov University). 17 AO is male, Russian, holds professorships in mathematics in Moscow and the USA. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 206 19 Controlling Infinity Little Sasha was walking on the edge of an abyss; being a trained mathematician, his father had a false sense of security because perhaps he believed that Zeno’s “arrow” paradox (of which Swedish relay is an obvious version) is resolved in elementary calculus by summation of the geometric progression 800 + 400 + 200 + 100 + 50 + 25 + · · · = 1600 = 4 × 400. This is true; the runners will indeed reach the end of the 4th circle, and fairly quickly. But if you think that Zeno’s paradox ends here, you are wrong; be prepared to face one of its most vicious forms. Indeed, the real trouble starts after the successful finish of the race: where is the baton? Indeed, the whole point of the relay is that each runner passes the baton to the runner on the next leg. After the race is over, each runner can honestly claim that he is no longer in possession of the baton because he passed it to the next runner. I repeat: can you explain where is the baton? The baton paradox is a version of a supertask invented by Jon Pérez Laraudogoitia [144] and is of serious importance for discussion of foundations of statistical mechanics. I borrow its compact description from Zurab Silagadze’s survey [154] of Zeno type paradoxes as they appear in modern science: In [144] Pérez Laraudogoitia constructed a beautifully simple supertask which demonstrates some weird things even in the context of classical mechanics. Imagine an infinite set of identical particles arranged in a straight line. The distance between the particles and their sizes decrease so that the whole system occupies an interval of unit length. Some other particle of the same mass approaches the system from the right with unit velocity. In elastic collision with identical particles the velocities are exchanged after the collision. Therefore a wave of elastic collisions goes through the system in unit time. And what then? Any particle of the system and the projectile particle comes to rest after colliding its left closest neighbor. Therefore all particles are at rest after the collision supertask is over and we are left with paradoxical conclusion [144] that the total initial energy (and momentum) of the system of particles can disappear by means of an infinitely denumerable number of elastic collisions, in each one of which the energy (and momentum) is conserved! S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 19.4 Edge of an abyss 207 Fig. 19.2. Energy and momentum are disappearing in Pérez Laraudogoitia’s infinite sequence of collisions. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 20 Pattern Hunting Michael Breen: About the same time [Fourth Grade], I discovered magic squares. They were so amazing to me that I would sit in the living room constructing them for a few months afterwards. Had I been a better mathematician, I would have spotted symmetries that escaped me until much later. Christopher Stephenson: I discovered for myself the following pattern: 2 × 2 = 4, 1 × 3 = 3 3 × 3 = 9, 2 × 4 = 8 4 × 4 = 16, 3 × 5 = 15 5 × 5 = 25, 4 × 6 = 24 .. . I thought (age 8) that this was a great discovery, and I explained it to everyone I could. No-one took the bait and used it to extend my knowledge. I was probably 10 before I got hold of an algebra book and found out how and why (x − 1)(x + 1) = (x2 − 1). That was enough to make me fall in love with algebra. But no-one, no-one, asked me questions like what about 3 × 3, 2 × 4, 1 × 5; 4 × 4, 3 × 5, 2 × 6; 5 × 5, 4 × 6, 3 × 7? 209 210 20 Pattern Hunting That could have led me on to (x − y)(x + y) = x2 − y 2 and also the differences between ascending squares being 1, 3, 5, 7 . . . I figured that out much later, maybe even at university. Richard Porter1 : It was probably the first time I became aware that I had a better than average ability in maths. I was about 8 years old and our teacher was giving us examples of how patterns emerge in the times tables. For example, 5, 10, 15, 20, 25 etc. alternate in the last digit between 5 and zero. Other examples given were the 2, 10 and 12 times tables. The teacher suggested that there was no such pattern for the 7 times table. I put up my hand and pointed out that there was: (7, 14, 21, 28, 35 etc.) the last digit was one more than the previous digit three numbers before. Of course, I was merely pointing out that any multiple of 7 plus 21 was another multiple of 7 whose last digit was increased by 1. Of course, I didn’t know that at the time. And one more story from Olivier Gerard: This one is both about square numbers, control, proto-algebra. I have been interested in squares from an early age. I did not call them “squares” but “numbers of the diagonal”. I started my investigations seriously when I was 7 years old because I was fascinated by multiplication tables. I had several examples of them, for instance at the back of draft booklets. Before that I was not confident enough in writing. I was not satisfied with the 10×10 model. I wanted to make the largest one possible. But this was well beyond my computations power as well as my logistical powers. At the time, I had only small paper sheets. Larger were reserved for older pupils. I asked my sister to lend me some of her sheets of 8 × 8 mm squared paper and took a lot of time to make preparations by drawing lines. As soon as I had a good start with one table, I wanted to make one bigger. I tried to do so by pasting sheets together in the two dimensions. But I was not very good at pasting, gluing, making alignments. I ended asking my mother to help me paste these large sheets together and make neat folds so that I could move it and pack it easily. I started with writing the number column and number rows, then the 1× and ×1 frame, then 2× and ×2. As I knew my 10 × 10 1 RP is male, British, has a PhD in mathematics, teaches in a British university. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 20 Pattern Hunting 211 table by heart I filled that part easily and then I usually started to work on the 10× and ×10 part and then the diagonal. I still have in my possession such a multiplication table in my own writing but unfinished, with the full diagonal done up to 30 × 30. Filling a multiplication table for integers requires only a knowledge of addition. In the 3× column you add 3 to the previous result. You can also transpose rows to columns if you do not last track of your copy. But actually, when I wanted to fill the diagonal one could say for aesthetical reasons, I usually multiplied to obtain the result, sometimes using other sheets of paper for drafting the multiplication. I did this because, I would have had to fill half a column or half a row with additions to fill the number in the diagonal by addition only. I had a simple rule for getting the next in line, not for going in diagonal. One day I noticed that 11 × 11 = 121 = 100 + 21 and 12 × 12 = 144 = 121 + 23. The fact was strange to me because 21 and 23 looked a lot alike and that I would have expected 22 in between. So I started making differences of squares. My only purpose at the time was to find a trick to fill in the diagonal quickly, ahead of the rest of the table, because I had begun to see the multiplication table as a compound of groups or series. I wanted to fill the table in a smart way. I remember that I formulated a “rule” in that way: to have the next term on the diagonal, make the difference of the previous two terms, add two and then add to the previous term. Something you could write in algebraic way (a + 1)(a + 1) = a × a + a × a − (a − 1) × (a − 1) + 2 and I used this rule by keeping track of the current odd number I last added so that I did not have to make the subtractions all over again each step. I remember I wanted to have something more practical i.e. something easy to start again when I was interrupted but I was not able when I was 7 to formulate the relation between the difference and the number being squared so I always started from 10 × 10 and 11 × 11 and then mentally added 2 each time I went one step further on the diagonal, until I had the correct difference to add for the next missing number on the diagonal. Often double checking or restarting when I had the feeling I might have lost track. I did not get that 21 = 2 × 10 + 1 or 2 × 11 − 1 for instance, and in a certain sense it would have been more difficult for me to apply a rule like this when making my multiplication tables than to keep track of the current difference to add. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 212 20 Pattern Hunting I did not discuss spontaneously mathematics with others when I was in primary school. I was mainly satisfied to have my large multiplication table, as a trophy. In cinquième (second year of junior high school, I was 11 year old) we were learning “produits remarquables” When I saw (a + b) × (a + b) = a2 + 2ab + b2 on the blackboard in class, I remember saying to myself “this cannot be right. I know really what is going on for years.” I had no time to check immediately but when I came back from school, I went to my room and I compared the formula from the lesson, by rewriting it for b = 1 and reworked my old rule for numbers on the diagonal and was really shocked and depressed to see they gave the same differences. It was clear to me that this formula was really superior to my rule, even though it was not more useful for filling a diagonal because usually you do not do that by large gaps. It was superior technology, I remember thinking this was “silent” technology because you did not need to make a sentence in your head to understand or apply it. In the end, I decided not to speak of all that in class or to my teacher. I was aware of large quantity of time I had spent constructing my understanding of multiplication tables which was just a tiny part of the formula, that now all my class would take for granted, without having worked so hard with multiplication tables. I slowly realized more vividly that what others were computing and discovering about numbers applied to the numbers I was using for my private purpose, that it was the same business. That it was not like the stories I could invent and that nobody could invent themselves identically or tamper with. That we were sharing the same numbers. It can seem odd that I could for years count in class and street, hear other people use the same words for numbers as I did and still consider certain aspects of numbers to be in a certain sense my personal property but I believe it was the way I was thinking. The fact of having been taught about numbers by other people and of being able to communicate (for instance for change and exercises) about numbers with others, many of them older or grown ups and at the same time having a kind of private knowledge, an intimacy about them, exclusive to me did not strike me as paradoxical. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 21 Visual Thinking vs Formal Logical Thinking LC: Thanks for this email and the chance to tell a story that I have never told anyone. In retrospect, this was an early sign of all my mathematical curiosity. I spent much of my childhood doing crochet with wool and making things with fabric. I loved the colours and the regular patterns, but also how the patterns could deviate and become a bit more random. This curiosity also extended to sketching things on paper, and I remember one year (age 8-9) where I persistently tried to do a certain type of drawing. I will explain it now using the language of mathematics, but at the time, it was a simple childhood drawing. I tried to discover and draw tessellations of the plane using different types of coloured regular polygons. That is, I tried to draw aperiodic coloured tessellations of the Euclidean plane by regular polygons. The existence of periodic regular polygons seemed obvious to me and I was exploring the boundaries of randomness in this type of construction. After many attempts, I gave up, resolving that I didn’t know what I was doing. This question did not continue to intrigue me, but I remember this playful exploration as a lot of fun. I was born and went to school in Australia, and my language of instruction was the same as my spoken language, English. It seems amusing now that this process is quite typical of a day of research as a professional mathematician. 21.1 Michael Fried1 : It was not a mathematics teacher who awakened in me an interest in mathematics, but an art teacher. In his 8th grade art class, Mr. 1 MF is male, Israeli, has a PhD in mathematics, teaches at an university. 213 214 21 Visual Thinking vs Formal Logical Thinking Spitz taught us perspective. I was completely enchanted with this and spent hours drawing perspective drawings on my own. I gave myself various problems, for example, how to draw a spiral staircase, and I asked myself questions such as whether it is possible to draw a tile floor in which the tiles were genuinely squares in perspective, and not just rectangles. Some of these problems and questions had genuine mathematical content, some of which, on hindsight, not entirely trivial. I had no idea, though, that any of this had anything to do with mathematics; I thought I was just doing art. But it is not perspective drawing that I really wanted to describe. It was something Mr. Spitz taught to me and only to me (at least it was not something he showed to the entire class). He showed me how to draw what he called a “logarithmic spiral.” I do not think he explained to me why it was called a logarithmic spiral—and it was a long time before I learned why indeed it was called that—but I was so taken with the shape and the “construction” he showed to me, that I almost became obsessed with it. The “construction” was as follows. Begin with a rectangle ABCD. Draw a diagonal AC, and then draw a line DF from D to the side BC, perpendicular to AC. From F , draw F G to AC parallel to AB, GH to F D parallel to AD, HI to AC parallel to AB, and so on. Then with these lines in place, one can sketch the spiral tangent to the lines BA, AD, DC, CF , F G, and so on. Of course there was this fudging at the end: he never explained, for instance, where the points of tangency should be, but this did not bother me. A little later, I saw another similar way of carry out the construction—maybe Mr. Spitz suggested it to me, but I do not think so. Instead of drawing DF perpendicular to AC, draw CF from C to the extended side AD and perpendicular to AC. Then draw F G to the extension of CD and perpendicular to CF , and continue this construction over and over. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 21.1 215 Again, I thought I was just doing art. But in the 9th grade, still drawing spirals, I started studying geometry. By the early spring, we were studying similar triangles. It was then that I realized, looking at my second method of drawing the logarithmic spiral, that each triangle I added was similar to the last. I think I also noticed that if AD = a and CD = b that DF = a( ab )2 , DG = a( ab )3 , and so on, that is, I began to grasp that this was a spiral growing (or rather shrinking) in a geometric sequence, though I did not have such terms yet. Certainly, the idea of a spiral similarity—turn and shrink, turn and shrink—was intuitively clear all at once, though, again, I did not know the term “spiral similarity” then. I also noticed that the original method of drawing the spiral was contained in the second method! This thrilled me even more than the discovery of the whirling similar triangles. I could also finally see more clearly the similar triangles spiraling in the first method: S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 216 21 Visual Thinking vs Formal Logical Thinking All this happened in the evening, lying in my bed, with the window open and the good early spring air filling the room. I add this because the experience making these little discoveries was a powerfully aesthetic one, and the smell in the air added to it. More importantly, I realized that its aesthetic power had as much to do with the mathematics involved as it had with the art with which it began. For the first time, I began to understand that these things Mr. Spitz had shown me, these things which fascinated me and which I loved, were, in fact, mathematics. And, conversely, no one afterward would ever have to persuade me that mathematics could be beautiful. This, I think, was the most significant discovery that evening, and without a doubt, it turned me towards mathematics. 21.2 EH: Visualisation Maths came easily to me until A-level, so I have no recollection of any issues. After GCSE a teacher did tell my parents that I should rethink my ambition to do a maths degree as I would soon hit my ceiling. He was partly right. I struggled for ages with the language of differentiation until I got the visual picture. Since then maths has been hard work to understand, though I usually get through with enough work. Making visualisations and images really helps me. Around the same time (GCSE) I came across the Penrose tiling for the first time and was fascinated. I remember working for ages trying to find a version with seven fold rotational symmetry rather than five. I drew lots of pictures, but did not get anything to work. I am now a mathematician, having done my PhD on the Penrose and related tilings. And the problem? I eventually solved it S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 21.3 Lego 217 (and the general case) and it became my second published paper. [33]. Jerry Swan: My recollection is that a lot of early mathematics was taught with reference to ‘scenes’, e.g. multiplication tables taught via the isomorphism with the corresponding rows and columns of numbers. I believe that early emphasis on this kind of visual/geometric nature de-emphasises more abstract pattern-matching that must then be re-introduced alongside these ‘scenic’ perspectives as one gains more mathematical maturity. The multiplication table as a matrix of objects example is the most concrete I can think of. My geometric intuition has always been rubbish (or at least I’ve intrinsically distrusted it) relative to my symbolic intuition. My guess is that this is the opposite of the case for the majority of the people who are likely to teach early maths, hence the geometric emphasis. I don’t really know what else to tell you on this—if you can prompt me a bit, I might be able to say more. On the “visual vs logical” issue, Roger Penrose uses an illustration of this general kind (some kind of hexagonal packing, if I recall correctly) as an example of something that human mathematicians can do that he claims is noncomputable (though as a lapsed Platonist, I personally believe that he’s confusing a lack of imagination with an insight into necessity). I think it appears in his book “Shadows of the Mind” (though it might possibly be “Emperors New Mind”). I strongly recommend that you give it a look. 21.3 Lego Fig. 21.1. S TICKLE B RICKS : A photograph showing how stickle bricks connect. Photographer: Malcolm Tyrrell. Wikimedia Commons, licensed under the Creative Commons Attribution-Share Alike 2.5 Generic license. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 218 21 Visual Thinking vs Formal Logical Thinking Dean Wyles2 : My earliest mathematical thinking, when I must have been just four or five years old, involved L EGO bricks. This was long before I became aware of formal mathematical thinking or language— my parents certainly didn’t speak any mathematics to me at all, not even simple arithmetic so I am not sure what prompted me to think in this way. I can remember wondering about possible combinations of colour sequences, when several different coloured bricks were aligned. I even remember including two reds and one blue, white and green; coming up with the answer sixty. This I later recognised as represented with factorials. Even now, when I teach to sixth form students, I cannot get combinations and permutations across to many students without reference to colour combinations of the kind I did myself, mentally, some forty years ago. I am always amazed when students appear to solve such problems directly without any obvious physical analogy and simply write down some factorial expression instinctively. My appreciation of space—volume and area—was enhanced with consideration of how many bricks could fit in as small a box as possible. I remember trying to estimate how more efficient the packing was if the bricks were tightly connected compared to being individually stored in a box. Not just noticing the efficiency but calculating it! Once I had exhausted my curiosity with L EGO I was delighted with the huge possibilities with S TICKLE B RICKS ; they can been linked in many more ways. Each brick has a rectangular array of stickles and so two m × n bricks can be stuck together in (m − 1) × (n − 1) ways, in addition to different alignments. I am afraid to say I was more interested in the possible constructions of a small number of bricks than trying to gather many many bricks to build something like a bridge or tower! This meant that I soon appreciated isomorphism and clearly many arrangements would be the same once you started to look at the pieces from different views. Once formal schooling kicked in and I became exposed to hours and hours of ‘sums’ I think I was quite put off by what most people call mathematics. I think I only started to excel when an Asian lad joined my class and a competitive streak began to come to me—I must have been about seven by this time; we would race each other to complete the hundred or so arithmetic or algebraic tasks on a page (large Beta books?). I suspect that my ‘approach’ to mathematical thinking has lead me to think that when I teach students mathematics, it is 2 DW is male, British, a teacher of mathematics in an academically selective college. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 21.3 Lego 219 their mathematics they are discovering, even with simple developments, and hope that I try to avoid the cramming approach so many of my school teachers adopted. Everyones’ mathematical world, mainly visual I guess, is a very personal thing. Surely that is the beautiful and wonderful thing about mathematics. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 22 Telling Left from Right The mirror is the master of painters. Leonardo Da Vinci 22.1 Why does the mirror change left and right but does not change up and down? The answer to this old chestnut appears to be well known. Here are two classical explanations.1 Blaise Pascal: Our notion of symmetry is derived from the human face. Hence, we demand symmetry horizontally and in breadth only, not vertically nor in depth.2 Immanuel Kant [133]: In physical space, on account of its three dimensions, we can conceive three planes which intersect one another at right angles. Since through the senses we know that which is outside us only 1 Thomas Ward brought my attention to a paper by his mother E. Honor Ward Lateral inversion ad all that [163] which contains interesting oservations. Honor Ward said, in particular: The image of an object in a plane mirror is virtual, erect, the same size as the object, and perverted. and concluded: My purpose in writing this article is [. . . ] to appeal for the adoption of the word PERVERTED into ordinary scientific, teaching, and examining terminology. 2 Quoted in W. H. Auden and L. Kronenberger, The Viking Book of Aphorisms, New York, 1966. 221 222 22 Telling Left from Right in so far as it stands in relation to ourselves, it is not surprising that in the relationship of these intersecting planes to our body we find the first ground from which to derive the concept of regions in space . . . One of these vertical planes divides the body into two outwardly similar parts and supplies the ground for the distinction between right and left; the other, which is perpendicular to it, makes it possible for us to have the concept before and behind. I believe that the actual explanation is both simpler and deeper: the left-right symmetry is built in the human neurophysiology at the level more primitive and elementary in relation to even most basic structures of human cognition. It is claimed that 1 in 6500 people possesses ability for mirror writing, when the non-dominant hand writes in direction opposite to the natural direction for a given language, producing the result that is a mirror image of the normal writing; intriguingly, this trait, although of no evolutionary significance, appears to be genetically predetermined [355]. Many more people can do a weaker version of mirror writing, when both hands write simultaneously, but in opposite directions. Figures 22.1 and 22.1 show my own experiment: I wrote the word “mirror” without any preparation or training. The sample produced by my left (non-dominant) hand bears all the specific features of my handwriting—but in mirror reflection. I have no time to discuss the huge literature on mirror writing; see, for example, bibliography in [329]. What matters for the purposes of this paper is the existence of built-in symmetry of our mind that, as children’s stories show, could be a mixed blessing in mastering mathematics. First of all, I was surprised to discover that a number of my correspondents told me about their difficulties of learning the left from the right. I could be biased on this point because I myself remember, and in surprisingly clear detail, how, in a kindergarten, my friend taught me to distinguish between the left shoe and the right shoe, by placing them on the floor in such a way that they “face” each other. I could not see the real reason why it should be that way. The two boots, for me, were logically equivalent. From my kindergarten times, I hardly remember anything else. Perhaps, the smell of burned milk from the kitchen . . . Anna, my wife, who is a mathematician, tells similar stories. Even now she gives me directions, when I drive, in the form “turn in S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 22.2 Pons Asinorum 223 your direction” instead of “turn right”. She explains that it takes too much time for her to figure out that “driver’s side” is the “right hand side”. Anna told me recently that when, aged four, she was eating sweets, she developed a custom of moving the candy from one side of the mouth to another, with a rationalization for this rule that both cheeks were equal and none of them had to be disadvantaged. 22.2 Pons Asinorum I eventually learned to tell a left boot from the right one, and was able to drive, without accidents, both in the USA (on the right hand side of the road) and in Britain (on the left hand side). But I suspect that I eventually benefited from my slightly awkward relationship with the left and the right when, at school, I was confronted with the famous Pons AsinorumixPons Asinorum@Pons Asinorum theorem of Euclidean geometry: the base angles ∠B and ∠C of an isosceles triangle △ABC are equal. In my school textbooks the Pons AsinorumixPons Asinorum@Pons Asinorum was proven outside the axiomatic system, by a direct application of the bilateral symmetry of the triangle viewed as a cardboard cutout. This approach was introduced in the canonical Russian textbook by Kiselev [703] and promoted by Hadamard in his highly influential Leçons de géométrie élementaire [31]. I remember that I have instantly seen a proof based on the formal symmetry of premises: • • • • • AB = AC AC = AB ∠BAC = ∠CAB (since the angle is equal to itself). △BAC = △CAB (by the Side-Angle-Side criterion of congruence). Therefore, ∠B = ∠C. A more detailed discussion of Pons Asinorum (including its role in history of automated theorem proving) can be found in my book [106, Section 7.7]. I remember being puzzled why my teacher was referring to a cardboard cutout when the theorem was obvious on its own. And I also remember not being able to explain to my teacher what I meant and why the theorem was obvious. Much later I happened to teach Euclidean geometry (in a Foundation Year course in a Russian university), and I used this chance to try the “formal symmetry” proof. The result was interesting: there were students who accepted it instantly, as something self-evident, while other students S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 224 22 Telling Left from Right Fig. 22.1. Kiselev’s proof of Pons Asinorum by “cardboard cutout” symmetry: he first proves Theorem 32: the vertex bisector of an equilateral triangle is also it median and altitude and then formulates as an easy corollary [703, pp. 21–22]. struggled. The reason for their difficulty is obvious: in the “formal symmetry” proof, the same triangle plays two different roles emphasized by two different notations: △BAC and △CAB. In effect, the symmetry of the premises—but not the physical, real life symmetry of the cardboard cutout!—forces the triangle to split, like an amoeba, into two twins. It is a subtle play with the concept of identity of a mathematical object. Euclid in his Elements gave a complicated proof of Pons Asinorum and used construction lines to make the two triangles to which the Side-Angle-Side criterion of congruence (the key point of the proof) is applied to triangles that are actually different; see Figure 22.2. In mathematics, the identity of objects is not set in black and white; it is more like a scale of grey. Some children are more sensitive than their peers and are better at recognizing shades of identity. I would not be surprised at all if this sensitivity sometimes affects a child’s ability to distinguish between the left boot and the right boot, too. 22.3 TB Age: 10 or so. I think. Gender: Male. Language: Finnish. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 22.4 Maria Zaturska 225 Fig. 22.2. Euclid proved Pons Asinorum by a sophisticated auxiliary construction, creating and using not just one, but two pairs of different but congruent triangles: △DAC = △EAB and △CBD = △BCE. I am an undergraduate mathematics student right now and will continue studying further. I had trouble remembering if a : b means a divided by b or b divided by a. There’s no way to tell. It is arbitrary. Likewise, some years earlier, remembering which side is left and which is right. I should also mention the notation a | b, as in a divides b. Or maybe it was b divides a. Probably the first one. For example: 2 | 4 but not 2 | 3. I learned a reliable mnemonic for left and right when learning to drive at the age of 18 (about). Before that, it was guesswork. I’ve never had much problem with it, aside from difficulties when communicating directions to somewhere. I still have to confirm that right is indeed where it is. I need to do the same thing with East and West (Sweden is to the West of Finland, Russia to the East). I’ll have to mention that I don’t remember dates or years, either. I generally remember what year it currently is and what year I was born and calculate my age from them, as an example. I’ve assumed this comes from ignoring what I consider irrelevant or from being lazy and not memorising a bunch of nigh-meaningless data. 22.4 Maria Zaturska 3 I think my experience may be more a matter of language than mathematics. I was brought up in the UK speaking Polish until I went to school aged 5. I picked up English as I went along - there were no Polish speaking children in my schools. I always (probably from age 7 up) had problems with subtraction when it was worded ‘what is x from y?’ as I thought symbolically, i.e. y − x, and so the order was confusing. (In later life I also 3 MZ is femail, Polish, studied in England, has a PhD in mathematics, teaches in a British Univesity. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 226 22 Telling Left from Right had problems with ‘x divides y’.) At the same sort of age I remember people saying it was ’‘five and twenty past six o’clock’ and again I couldn’t get my head around ‘five and twenty’ - why weren’t they using twenty five? 22.5 MP Your question does bring to mind two specific, somewhat mathematical, intellectual difficulties from (some time at) primary school: one was the issue of including/excluding endpoints (eg in computing the number of days between, say, 7th and 23rd of June: so whether to use 23−7 or 23−7+1; I think my usual strategy was to compute a similar example which could be counted on my (mental) fingers in order to determine whether or not it was correct (which would depend on the nature of the exact question being asked) to add 1). The other was the problem of reconciling the personal right / left frame of reference with the view-from-“above” frame which is necessary for north / south . . . (the latter won and I still have problems in quickly telling right from left). 22.6 Digression into ethnography [338, p. 68] Speakers of English and other Indo-European languages favor the use of an egocentric (relative) system to represent the location of objects—that is, relative to the self (e.g., “the man is on the right side of the flagpole”). In contrast, many if not most languages favor an allocentric frame, which comes in two flavors. Some allocentric languages such as Guugu Yimithirr (an Australian language) and Tzeltal (a Mayan language) favor a geocentric system in which absolute reference is based on cardinal directions (“the man is west of the house”). The other allocentric frame is an object-centered (intrinsic) approach that locates objects in space, relative to some coordinate system anchored to the object (“the man is behind the house”). One can add that Tenejapan Tzeltal people of Central America have no words or concepts for the left and the right—they live in square houses, with a door (made of two equal halves) positioned in the middle of the south wall [474, 473]; in their language, an egocentric reference system is virtually impossible. The role of cardinal directions [520] in the geocentric reference system of Guugu Yimithirr can be explained by the harsh environment in which they live—in the wilderness, it pays to remember S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 22.6 Digression into ethnography 227 where the north is; if you are lost, you are dead. Lera Boroditsky remarks [469] that, in tradition of another Aboriginal community in Queensland, Pormpuraaw,, cardinal directions dominate even the simplest formulae of polite small talk: to say hello, one asks, “Where are you going?”, and an appropriate response might be, “A long way to the south-southwest. How about you?” If you don’t know which way is which, you literally can’t get past hello. [469] Lera Boroditsky continues: People rely on their spatial knowledge to build many other more complex or abstract representations including time, number, musical pitch, kinship relations, morality and emotions. So if Pormpuraawans think differently about space, do they also think differently about other things, like time? To find out, my colleague Alice Gaby and I traveled to Australia and gave Pormpuraawans sets of pictures that showed temporal progressions (for example, pictures of a man at different ages, or a crocodile growing, or a banana being eaten). Their job was to arrange the shuffled photos on the ground to show the correct temporal order. We tested each person in two separate sittings, each time facing in a different cardinal direction. When asked to do this, English speakers arrange time from left to right. Hebrew speakers do it from right to left (because Hebrew is written from right to left). Pormpuraawans, we found, arranged time from east to west. That is, seated facing south, time went left to right. When facing north, right to left. When facing east, toward the body, and so on. Of course, we never told any of our participants which direction they faced. The Pormpuraawans not only knew that already, but they also spontaneously used this spatial orientation to construct their representations of time. A detailed account of this research is published in [470]. For a mathematician, the very existence of world outlook build around cardinal directions is a revelation: the approach to geometry in our (Western, that is) mathematics is strictly local! Of course, the value judgements—which of the two philosophies, local or global is best—but the “local” approach of traditional Euclidean geometry was perhaps conducive for the development of differential calculus and differential geometry. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 228 22 Telling Left from Right 22.7 BB And reluctance to distinguish between things that are undistinguishable from the formal logical point of view goes deeper and deeper. Here is what BB4 wrote to me: From the time I learned matrices (age 16 or so) I cannot remember which are the columns and which are the rows. Given that the arrangement of coefficients in a linear transformation can be written equally well in a matrix in two ways, it is something that always takes me 10–15 seconds to recall even now. Perhaps a few words of (mathematical) explanations are needed. Most mathematicians would agree that BB has reasons to be confused: for a mathematician, a matrix is an element in the tensor product V ⊗ V ∗ of a finite dimensional vector space V and its dual V ∗ . Since the dual of the dual of a finite dimensional space is the same as the original space, that is, (V ∗ )∗ ≃ V canonically, there is no intrinsic reason to distinguish between V and V ∗ , that is, between the rows and the columns of a square matrix. (Notice that PD also mentions psychological difficulties arising from duality.) 22.8 PD I had a short moment of bewilderment (around 14 or 15 or 16?) when I learned that the graphs of f (x − a) and f (nx) when compared to those of f (x) behave “in the opposite way” to one’s expectation. I got it very quickly and easily, but I remember being surprised. And this flows into similar adult experiences: Does the “transition matrix” transform the basis or the coordinates? (Actually, many books hide the appearance of the inverse of the transpose by suitably defining the transition matrix.) Given a matrix of a linear map, am I writing the map between the vector spaces or between their duals? Do the transition functions of a vector bundle transform the frame or the sections? Am I looking at the sheaf O(D) or O(−D) (D is a divisor on a variety), and do its sections have a pole or zero at D? This is all one and the same question and one learns to recognize it, but it is amazing how persistent it is. What I am going to say now probably won’t make sense. It seems that often there is a “fixed part” and a “moving part” in a 4 BB is male, Russian, has a PhD from an American university and holds a research position in a British university. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 22.8 PD 229 problem, and I have the feeling that my mind often gets confused by looking at the “moving part” and forgetting that it “moves” with respect to the “fixed part”. This refers both to mathematical and worldly experiences. In math this is often a question about inclusions vs natural inclusions (equalities). In daily life one example of a similar problem is learning the streets of a city. I have the feeling that often my brain stores some images separately, say, the direction in which I traverse a street for the first time gets stored with higher priority. Or there may be several images of the same place (from different viewpoints) which don’t exactly glue. As if the brain prefers to work with the moduli stack and has a problem when passing to the moduli space. In that slightly metaphorical phrase I meant exactly what you say. That it feels as if my brain often keeps track of the individual representatives of an equivalence class, or at least has a preferred one, instead of dealing with the class itself. For example, if there are several roads meeting at an intersection, it may happen that I perceive each of them slightly differently, they carry a different emotional charge, if you want. Of course, if I make a conscious effort, I can destroy this asymmetry. (I don’t think I am describing a pathology, I don’t get lost on campus, it’s a more subtle effect. Though I am pretty bad at orienting, and when I do it, it is more like a flow, rather than a logical sequential process: I prefer the right-brain orienting to the left-brain orienting.) I consider this to be a weakness of mine, of course. Mathematically, this has the effect that I can spend a lot of time trying to understand different equivalent formulations of the same result, then dash along a route, then back up and again look for equivalent formulations. . . This often leads to very slow progress. And at least when it comes to science, I have the feeling that this effect gets stronger with time. I definitely prefer writing a < b < c, as opposed to c > b > a, and my suspicion is that in the first case the numbers appear in the same order as on the real line—if we take the positive direction of the real line to be to the right of 0. I have a small empirical conjecture. I think a wide class of errors and confusions appear when there is some discrete (usually, Z/2Z) ambiguity/choice/convention in the problem, and one has to fix the choice and stick to it while solving. The issue is, however, if your brain decides to flip the choice halfway through the computation. For instance, if you think too much about conventions you may notice that certain choice is more convenient in some situations and less convenient in others, and you may decide to change your convention halfway through. Or, it may become purely involuntarily. One elementary example is subtraction: I’ve seen many students with weak arithmetic background getting the signs wrong in linear algebra, where they S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 230 22 Telling Left from Right start with the intent of subtracting one row from another . . . but when they reach a column where there is (2 − 5) they decide to compute it as −(5 − 2), and then forget where they were going . . . A similar example is multiplication (a + b)(c + d), where one has to make the choice about enumerating the terms: either as a(c + d) + b(c + d) or as (a + b)c + (a + b)d. After computing 2 of the 4 terms you may decide to change the choice . . . Of course, a similar problem appears in homological algebra: when you talk about a left/right exact functor/cofunctor, you know that the chirality is determined in the target category, but that works if you write all complexes from right to left. This somewhat ties with my feeling that there is often a “fixed” and a “moving” part of the problem . . . By the way, here is one other “chiral” kind of asymmetry—this time related to the multiplication table. Despite the commutativity of multiplication, I think certain products are computed faster than others: e.g., 6 × 7, 8 × 7, 7 × 9, 7 × 5 are faster to compute than 7 × 6, 7 × 8, 9 × 7, 5 × 7 etc. (Not all examples of such pairs include 7.) Of course, the difference in “computational time” is very small, but I’ve found several people who have noticed such a phenomenon (with mostly the same pairs of numbers, and with different native languages). Probably this has a fairly simple and prosaic explanation, and I can give a few myself, but I thought it is still worth mentioning. 22.9 Digression into Estonian language This quote from Urmas Sutrop [556] was brought to my attention by Yagmur Denizhan. Even the most ordinary everyday Estonian language contains numerous ancient expressions, possibly going back as far as the Ice Age. The Estonians say külma käes, päikese, tuule käes ‘in the hand of the cold, rain, sun, wind’, or ta sai koerte käest hammustada ‘he was bitten by the hand of dogs’, i.e. ‘he was bitten by dogs’, or ta sai nõgeste käest kõrvetada ‘he was stung from the hand of nettles’. In all the above Estonian expressions ‘hand’ occurs in singular. This is associated with the integral concept of the world of our ancestors. Everything formed a whole, a totality, also the paired parts of the body which were used only in singular. If one wanted to speak about one hand, one had to say pool kätt ‘half a hand’. Hence the division of the holistic world into the right and left halves, right and left sides. 22.10 Standing arches, hanging chains The symmetry between the mechanical behavior of a standing arch and a hanging chain is not obvious at all, and a genius on the scale S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 22.11 Orientation of surfaces 231 of Robert Hooke and Antoni Gaudi was needed for its discovery and a systematic exploitation. A mathematical justification of the surprising connection is very simple but subtle. I quote from Robert Osserman’s paper [71]: It was Robert Hooke who in 1675 made the connection between the ideal shape of an arch and that of a hanging chain in an aphorism that says, in abbreviated form, “As hangs the chain, so stands the arch.” In other words, the geometry of a standing arch should mirror that of a hanging chain. The horizontal and vertical forces in a hanging chain must add to a force directed along the chain, since any component perpendicular to the chain would cause it to move in that direction to gain equilibrium. Similarly, one wants the combined forces at each point of an arch to add up to a vector tangent to the arch. In both cases, the horizontal component of the force is constant and simply transmitted along the arch or chain, while the vertical forces are mirror images. Wikipedia provided a full quote from Hooke: Ut pendet continuum flexile, sic stabit contiguum rigidum inversum, meaning “As hangs a flexible cable so, inverted, stand the touching pieces of an arch.” Incidentally, the language of Tenejapan Tzeltal (which has no words for “left” and “right”) has special grammatical forms of verbs for expression of movement uphill or downhill [475] . . . 22.11 Orientation of surfaces Let S be a regular surface in R3 . Explain what it means to say that S is orientable. Show that the sphere S 2 is orientable. A Gauss map on a regular surface S is a smooth map N : S → S2 such that N (p) is normal to S at p for all p ∈ S. The surface S is orientable if there is a Gauss map on S. Let N : S 2 → S 2 be the identity map on S 2 . Then N is a Gauss map on S 2 . S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK References Mathematics and Exact Sciences 1. G. P. Akilov and V. N. Dyatlov, Elements of Functional Analysis, Novosibirsk State University, 1979. (In Russian.) 2. J. Baez, This Week’s Finds in Mathematical Physics. http:// math.ucr.edu/home/baez/TWF.html. 3. R. Ball, The Kolmogorov cascade, an entry in Encyclopedia of Nonlinear Science (A. Scott, ed.). Routledge, 2004. The text is availabe at http://wwwrsphysse.anu.edu.au/~rxb105/cascade.pdf 4. H. P. Barendregt, The Lambda Calculus: Its Syntax and Semantics. North-Holland, revised edition, 1984. 5. W. A. Beyer, J. D. Louck, and D. Zeilberger, A generalization of a curiosity that Feynman remembered all his life, Math. Magazine 69 no. 1 (1996), 43–44. Available from http://www.math.rutgers.edu/ ~zeilberg/mamarim/mamarimhtml/feynman.html. Accessed 28 February 2010. 6. M. Bezem, EuclidŠs lemma revisited, Normat 58 no. 4 (2010), 184. 7. D. Birmajer and J. B. Gill, Arithmetic in the ring of formal power series with integer coefficients, Amer. Math. Monthly 115, no. 6 (2008), 541–549. 8. A. Bogomolny, Cut The Knot! http://www.cut-the-knot.org/. 9. V. M. Buchstaber and S. P. Novikov, Formal groups, power systems and Adams operators, Russian Mat. Sb. (N.S.) 84 (1971), 81–118. 10. V. M. Buchstaber, n-valued groups: theory and applications, Mosc. Math. J. 6, no. 1 (2006), 57–84. 11. D. Burago and S. Ivanov, Area spaces: first steps. Geometric and Functional Analysis, 19 no. 3 (2009), 662–677, DOI: 10.1007/s00039009-0018-2 . 12. J. Dyer-Bennet, A theorem on partitions of the set of positive integers, Amer. Math. Monthly 47 (1940), 152–154. 13. A. J. Chorin, Book Review: Kolmogorov spectra of turbulence I: Wave turbulence, by V. E. Zakharov, V. S. Lvov, and G. Falkovich. Bull. Amer. Math. Soc. 29, no. 2 (1993), 304–306. 14. J. Copeland, Even Turing machines can compute uncomputable functions, in Unconventional Models of Computation (Discrete 233 234 References 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. Mathematics and Theoretical Computer Science) (C. Calude, J. Casti, and M. J. Dinneen, eds.). London, Springer-Verlag, 1998, pp. 150–164. J. Copeland, Accelerating Turing machines, Minds and Machines 12 (2002), 281–301. J. Copeland, Hypercomputation in the Chinese Room, in Unconventional Models of Computation 2002 (C.S. Calude et al. eds.). Lect. Notes Comp. Sci. 2509 (2002), 15–26. R. Dedekind, Essays on the Theory of Numbers. I: Continuity and Irrational Numbers. II: The Nature and Meaning of Numbers, Dover, 1963. P. Doyle and J. Conway, Division by three. http://arxiv.org/ abs/math.LO/0605779. M. Eastwood and R. Penrose, Drawing with complex numbers. arXiv:math/0001097v1. A. Enayat, δ as a Continuous Function of x and ǫ. Amer. Math. Monthly 107 no. 2 (2000), 151–155. Stable URL: http://www. jstor.org/stable/2589436. E. Engeler, Metamathematik der Elementarmathematik. Springer, 1983. ISBN-10: 354012151X. ISBN-13: 978-3540121510. R. Feynman, QED: The Strange Theory of Light and Matter. Princeton University Press, 1985. G. L. Fisher and B. Mellor, Three-dimensional finite point groups and the symmetry of beaded beads, J. Math. Arts 1, no. 2 (2007), 85–96. ISSN 1751–3472. M. M. France Stalking staggeringly large numbers, Annales de l’institut Fourier 56, no. 7 (2006) 1983–1985. I. M. Gelfand, A talk with Professor I. M. Gelfand (reminiscences of a mathematical boyhood), recorded by V. S. Retakh and A. B. Sosinsky. Kvant no. 1 (1989), 3–13. (In Russian.) J. Gleick, Genius: Richard Feynman and Modern Physics. Little, Brown, 1992. ISBN-10: 0316903167. ISBN-13: 978-0316903165. R.L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, Massachusetts. 1989. G. Greenhill, Mechanical similitude, Math. Gaz. 8, no. 122 (Mar. 1916), 229–233. N. Greenleaf, On Fermats equation in C(t), Amer. Math. Monthly 76 no. 7 (1969), 808–809. Stable URL: http://www.jstor.org/ stable/2317878. Accessed 05 August 2008. M. Gromov, Structures, Learning and Ergosystems: Chapters 1-4, 6. http://www.ihes.fr/~gromov/PDF/ergobrain.pdf. Accessed 22 December 2009. J. Hadamard, Leçons de géométrie élementaire. BiblioBazaar, 2009. ISBN-13 978-1110978281. B. Hansson, The existentce of group preference functions, Public Choice 28 (1976), 89–98. E. O. Harriss, Non-periodic rhomb substitution tilings that admit order n rotational symmetry, Discrete Comput. Geom. 34, no. 3 (2005), 523–536. DOI: 10.1007/s00454-005-1175-1. R. Hartshorne, Foundations of Projective Geometry. W. A. Benjamin, New York, 1967. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK References 235 35. D. W. Henderson and D. Taimina, Experiencing Geometry: Euclidean and Non-Euclidean with History. 3rd ed., Prentice-Hall, 2004. 36. D. W. Henderson and D. Taimina, Crochetting the hyperbolic plane, The Mathematical Intelligencer 23 no. 1 (2001), 17–28. 37. L. Henkin, On mathematical induction, Amer. Math. Monthly 67, no. 4 (1960), 323–338. 38. E. Hewitt, Rings of real-valued continuous functions. I. Trans. Amer. Math. Soc. 64 (1948), 45–99. Errata: Trans. Amer. Math. Soc. 64 (1948), 596. 39. K. E. Iverson, Notation as a tool of thought, Comm. ACM 23 (August 1980), 444–465. 40. V. Kanovei and M. Reeken, Nonstandard Analysis, Axiomatically. Springer-Verlag, 2004. 41. V. Kanovei and S. Shelah, A definable nonstandard model of the reals. J. Symbolic Logic 69 no. 1 (2004), 159–164. 42. K. U. Katz and M. G. Katz, When is .999 . . . less than 1? The Montana Mathematics Enthusiast 7 no. 1 (2010), 3–30. ISSN 1551–3440. 43. H. J. Keisler, Elementary Calculus: An Approach Using Infinitesimals (Experimental version). Bodgen & Quigley, 1971. 44. H. J. Keisler, Elementary Calculus: An Approach Using Infinitesimals. Prindle Weber & Schmidt, 1976. ISBN 9780871509116. 45. H. J. Keisler, Elementary Calculus: An Infinitesimal Approach. Second Edition. Prindle, Weber and Schimidt, Boston, 1986. 46. H. J. Keisler, Elementary Calculus: An Infinitesimal Approach. On-line edition licensed for non-commercial use under a Creative Commons License. http://www.math.wisc.edu/ ~keisler/calc.html. 47. H. J. Keisler, Foundations of Infinitesimal Calculus. On-line edition, http://www.math.wisc.edu/~keisler/foundations. html, retrieved 29 June 2009. 48. A. P. Kirman and D. Sondermann, Arrows theorem, many agents, and invisible dictators, J. Economic Theory 5 (1972), 267–277. 49. D. Knuth, The Art of Computer Programming. Vol. 2: Seminumerical Algorithms. 3rd edition, Addison-Wesley, 1998. 50. D. Knuth, Letter to the Editor, Notices Amer. Math. Soc. no. 3 (March 1998). 51. N. Koblitz, Algebraic Aspects of Cryptography. Springer-Verlag, 1998. 52. P. C. Kocher, Timing attacks on implementations of Diffie-Hellman, RSA, DSS, and other systems, in Advances in Cryptography: Proceedings of Crypto ’96, Springer-Verlag, 1996, pp. 104–113. 53. P. C. Kocher, J. Jaffe and B. Jun, Differential power analysis, in Advances in Cryptography: Proc. of Crypto ’99, Springer-Verlag, Lect. Notes Comp. Sci. 1666 (1999), pp. 388–397. 54. Synthetic Differential Geometry. Cambridge University Press, 2nd ed., 2006 (London Math. Society Lect. Notes Ser. vol. 333). ISBN10: 0521687381. ISBN-13: 978-0521687386. 55. A. N. Kolmogorov, Local structure of turbulence in an incompressible fluid for very large Reynolds numbers, Doklady Acad Sci. USSR 31 (1941), 301–305. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 236 References 56. J. Kramer, Is abstraction the key to computing? Communications of the ACM 50 no. 4 (2007), 37–42. 57. E. Landau, Foundations of Analysis. Chelsea, 1966. 58. R. P. Langlands, Mathematical retrospections, http:// publications.ias.edu/rpl_works/L12/shaw/photoshell. pdf. Accessed 5 September 2009. 59. A. H. Lightstone, Infinitesimals. Amer. Math. Monthly 79 (1972), 242–251. 60. M. Livshits, You could simplify calculus, http://arxiv.org/abs/ 0905.3611. Accessed 18 Janaury 2010. 61. J. Kemeny, Mathematics without numbers, Daedalus 88 (1959), 577– 591. 62. P. Maddy, How applied mathematics became pure, The Review Of Symbolic Logic 1, no. 1 (2008), 16–41. 63. Yu. I. Manin, Mathematics and Physics. Birkhäuser, 1981. 64. J. E. Marsden and A. Weinstein, Calculus Unlimited. BenjaminCummings Publishing Co., Longman, US, 1981. ISBN/ASIN: 0805369325. ISBN-13: 9780805369328. 65. J. Mason, L. Burton, and K. Stacey, Thinking Mathematically. London: Addison-Wesley, 1992. 66. B. Mazur, When is one thing equal to some other thing? http://www.math.harvard.edu/~mazur/preprints/when_ is_one.pdf. 67. B. Mazur, How did Theaetetus prove his theorem? http:// www.math.harvard.edu/~mazur/preprints/Eva.pdf Accessed 6 March 2010. 68. G. Michaelson, An Introduction to Functional Programming through Lambda Calculus. Addison-Wesley, 1989. 69. J. Michell and C. Ernst, The axioms of quantity and the theory of measurement. Translated from Part I of Otto Hölder’s German Text “Die Axiome der Quantitat¨ und die Lehre vom Mass”.5 J. Math. Psychology 40 no. 3 (1996), 235–252; no. 4, 345–356. 70. M. Minsky, Automated proof of Pons Asinorum, newsgroup post of 24 December 1999, http://www.math.niu.edu/~rusin/ known-math/99/minsky, accessed 12 July 2009. 71. R. Osserman, Mathematics of the Gateway Arch, Notices Amer. Math. Soc. 57 no. 2 (2010), 220–229. http://www.ams.org/ notices/201002/rtx100200220p.pdf. 72. M. Nakamura and S. Kakutani, Banach limits and the C(ech compactification of a countable discrete set Proc. Imp. Acad. 19 no. 5 (1943), 224–229. 73. E. Nelson, Internal set theory: A new approach to nonstandard analysis, Bull. Amer. Math. Soc. 83 (1977), 1165–1198. 74. E. Nelson, The syntax of non standard analysis, Ann. Pure and Appl. Logic 38 (1988), 123–134. 75. O. Ore, Number Theory and Its History. Dover Publications, 1976. 5 Originally published in 1901, Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig, Mathematisch- Physikaliche Classe 53 (1901), 1–64. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK References 237 76. G. Peano, The principles of arithmetic, presented by a new method, in From Frege to Gödel (Jean van Heijenoort, ed.). Harvard, 1967. 77. D. Pierce, Induction and recursion. A talk at Logic Colloquium 2008— ASL European Summer Meeting, Bern, Switzerland, 3–8 July 2008. 78. D. Pierce, Numbers. http://metu.edu.tr/~dpierce/ Mathematics/Numbers/. Accessed 16 April 2010. 79. G. Polya, Mathematics and Plausible Reasoning, vol. 1 (reprint of the 1954 original) Princeton University Press, Princeton, 1990. 80. M. O. Rabin, Probabilistic algorithm for testing primality, J. Number Theory 12 (1980), 128–138. 81. T. C. Ratliff, A comparison of Dodgson’s method and the Borda count, Economic Theory 20 (2002), 357–372. 82. P. Ribenboim, 13 Lectures on Fermat’s Last Theorem. Springer, 1979. ISBN-10: 0387904328; ISBN-13: 978-0387904320. 83. A. M. Robert, Nonstandard Analysis. Dover Publications, 2003. ISBN0-486-43279-3. 84. A. Robinson, Non-standard Analysis. Amsterdam, North-Holland, 1966. Later edition: A. Robinson, Non-standard Analysis (Princeton Landmarks in Mathematics and Physics). Princeton University Press, Princeton, 1996. 85. G. J. Simmons, How to insure that data acquired to verify treaty compliance are trustworthy, Proc. IEEE 76, no. 5 (May 1988), 621–627. 86. V. I. Smirnov, Course of Higher Mathematics. Vol. 1. Elsevier, 1964. 87. R. P. Stanley, Enumerative Combinatorics. Vol. 2, Cambridge University Press, 1999. 88. R. P. Stanley, Solutions to exercises on Catalan and related numbers, http://www-math.mit.edu/~rstan/ec/catsol.pdf, accessed 12 July 2009. 89. K. Sullivan, The teaching of elementary calculus using the nonstandard analysis approach, Amer. Math. Monthly 83 (1976), 370–375. 90. G. H. Suter, An elementary example of a transcendental p-adic number, Math. Magazine 49, no. 1 (1976), 42. Stable URL: http://www. jstor.org/stable/2689885. Accessed: 25/12/2008 07:22. 91. D. Taimina, Crocheting Adventures with Hyperbolic Planes. A K Peters, 2009. ISBN-13: 978-15688-145-20. 92. A. Tarski, A Decision Method for Elementary Algebra and Geometry. Rand, 1948. 93. L. van den Dries, Tame Topology and o-Minimal Structures, London Math. Soc. Lect. Notes Ser., vol. 248. Cambridge University Press, Cambridge, 1998. 94. P. Walker, A lemma on divisibility, Amer. Math. Monthly 115, no. 4 (2008), 338. 95. D. Wells, The Penguin Book of Curious and Interesting Puzzles, Penguin Books, 1992. 96. D. Wright, Mathematics and Music. Amer. Math. Soc., Providence R.I., 2009. ISBN 978-0-8218-4873-9. 97. D. Zagier, Problems posed at the St Andrews Colloquium, 1996. http://www-groups.dcs.st-and.ac.uk/~john/Zagier/ Problems.html. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 238 References 98. D. Zeilberger, “Real” analysis is a degenerate case of discrete analysis, 2001. http://www.math.rutgers.edu/~zeilberg/ . Philosophy and Methodology of Mathematics 99. A. Arana, On formally measuring and eliminating extraneous notions in proofs. Philosophia Mathematica 17 (2009), 208–219. 100. V. I. Arnold, What is Mathematics? Moscow, MTsNMO, 2004. 101. S. Awodey, An answer to Hellman’s question: Does Category theory provide a framework for mathematical structuralism? Philosophia Mathematica 12 (2004), 54–64. 102. J. D. Barrow, Pi in the Ski: Counting, Thinking and Being. Oxford, Oxford University Press, 1987. 103. J. P. van Bendegem, Alternative mathematics: the vague way, Synthese 125 (2000), 19–31. 104. G. Birkhoff, Mathematics and psychology, SIAM Review 11 no. 4 (1969), 429–469. 105. R. P. Boas, Can we make mathematics intelligible? Amer. Math. Monthly 88 no. 10 (1981), 727–731. 106. A. V. Borovik, Is mathematics special? In PhiMSAMP. Philosophy of Mathematics: Sociological Aspects and Mathematical Practice (Benedikt Löwe and Thomas Müller, eds.). College Publications, London, 2010. Texts in Philosophy 11, pp. 1–27. 107. A. V. Borovik, Mathematics under the Microscope, Amer. Math. Soc., Providence, R.I., 2010. 108. J. A. Bracken, A new look at time and eternity, Theology and Science 2, no. 1 (2004), 77–88. DOI: 10.1080/1474670042000196621. 109. P. Cameron, Autobiography, http://www.maths.qmul.ac.uk/ ~pjc/diaries/biog.pdf. Accessed 7 September 2009. 110. J. Carter, Individuation of objects—A problem for structuralism? Synthese 143 (2005) 291–307. 111. J. Carter, Categories for the working mathematician: making the impossible possible. Synthese 162 (2008), 1–13. DOI 10.1007/s11229– 007–9166–9. 112. J. van Cleve and R. E. Frederick, eds. The Philosophy of Right and Left: Incongruent Counterparts and the Nature of Space. Kluwer, 1991. ISBN-10: 0792308441. ISBN-13: 978-0792308447. 113. A. Comte, Catéchisme positivist. 1852. 114. D. Corfield, Towards a Philosophy of Real Mathematics. Cambridge University Press, 2003. ISBN 0521817226. 115. P. J. Davis, Fidelity in mathematical discourse: is one and one really two? Amer. Math. Monthly 79 no. 3 1972, 252–263. 116. J. W. Dawson, Jr. Why do mathematicians re-prove theorems? Philosophia Mathematica 14 no. 3 (2006), 269–286. ´ 117. S. Della Sala and R. Cubelli. S“Directional apraxia”: a unitary account of mirror writing following brain injury or as found in normal young children. J. Neuropsychol. 1 (2007), 3–26. DOI:10.1348/ 174866407X180783. 118. U. Dudley, What is mathematics for? Notices of the AMS 57 no. 5 (2010), 608–613. 119. D. Fallis, Intentional gaps in mathematical proofs, Synthese 134 (2003), 45–69. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK References 239 120. L. Feigenson, S. Dehaene, and L. Spelke, Core systems of number, Trends in Cognitive Sciences 8 (2004), 307–314. 121. D. Gale, Tracking The Automatic Ant: And Other Mathematical Explorations. Springer, 1998. ISBN-10: 0387982728. ISBN-13: 978-0387982724. 122. M. Gardner, Is Nature ambidextrous? Philosophy and Phenomenological Research 13 (December 1952), 200–211. 123. M. Gardner, The Ambidextrous Universe. W. H. Freeman, New York, 1989. 124. J. T. Gatto, Why schools donŠt educate, Nexus 4 no. 4 (1997). http://www.seniormanagementservices.com/Images/ JohnTaylorGattoTeacherAward.pdf. 125. J. T. Gatto, Dumbing Us Down: The Hidden Curriculum of Compulsory Schooling. New Society Publishers, 2002. ISBN-10: 0865714487. ISBN-13: 978-0865714489. 126. T. Gowers, Mathematics: A Very Short Introduction. Oxford Paperbacks, 2002. ISBN-10: 0192853619. ISBN-13: 978-0192853615. 127. A. Grothendieck, Récoltes et Semailles. http://acm.math. spbu.ru/RS/. Accessed 17 April 2010. 128. J. W. Hadamard, Essay on the Psychology of Invention in the Mathematical Field. Princeton, NJ: Princeton University Press, 1945. 129. M. Harris, “Why mathematics?” you might ask, Princeton Companion of Mathematics (T. Gowers, ed.), Princeton University Press, 2009, pp. 966–976. 130. R. Hersh, What is Mathematics, Really?. Vintage, London, 1998. 131. R. Hersh and V. John-Steiner, Loving and Hating Mathematics: Challenging the Myths of Mathematical Life. Princeton University Press, 2010. ISBN-13: 978-0691142470. 132. S. W. Hwang and W.-M. Roth, Scientific & Mathematical Bodies: The Interface of Culture and Mind. Sense Publishers, 2011. ISBN-10: 9460915655. ISBN-13: 978–9460915659. 133. I. Kant, On the first ground of the distinction of regions in space (1768), in The Philosophy of Right and Left: Incongruent Counterparts and the Nature of Space (J. van Cleve’s and R. E. Frederick, eds.). Kluwer, 1991, pp. 27–33. 134. E. C. W. Krabbe, Strategic maneuvering in mathematical proofs, Argumentation 22 (2008), 453–468. DOI: 10.1007/s10503-008-9098-7. 135. I. Lakatos, Proofs and Refutations. Cambridge University Press, 1976. 136. I. Lakatos, Cauchy and the continuum: the significance of nonstandard analysis for the history and philosophy of mathematics. Math. Intelligencer 1 no. 3 (1978), 151–161. 137. G. Lakoff and R. Núnez, Where Mathematics Comes From: How the Embodied Mind Brings Mathematics Into Being. Basic Books, New York, 2000. 138. B. Larvor, What is dialectical philosophy of mathematics? Philosophia Mathematica 9 no.2 (2001), 212–229. DOI: 10.1093/philmat/9.2.212. 139. M. Leng, Phenomenology and mathematical practice, Philosophia Mathematica 10 no. 1 (2002), 3–25. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 240 References 140. D. Lomas, Some initial steps toward a much-needed critical epistemological realism in mathematics and science education. In: Philosophy of Education 2001 (S. Rice, ed.). University of Illinois at Urbana-Champaign, 2001. http://www.ed.uiuc.edu/ EPS/PES-yearbook/2001/lomas%2001.pdf. 141. D. Lomas, What perception is doing, and what it is not doing, in mathematical reasoning. British J. Philosophy of Science 53 no. 2 (2002), 205–223. 142. Yu. I. Manin, Mathematics as Metaphor. Amer. Math. Soc., Providence, R.I., 2007. ISBN-13 978-0-8218-4331-4. 143. M. Otte, Does mathematics have objects? In what sense?, Synthese 134 (2003), 181–216. 144. J. Pérez Laraudogoitia, A beautiful supertask. Mind 105 (417) (1996), 81–83. 145. M. Potter, Reason’S Nearest Kin: Philosophy of Arithmetic from Kant to Carnap. Oxford University Press, 2000. 146. Y. Rav, Why do we prove theorems? Philosophia Mathematics (3) 7 (1999), 5–41. 147. G.-C. Rota, The pernicious influence of mathematics on philosophy, Synthese 88 (1991), 165–178. 148. G.-C. Rota, The phenomenology of mathematical beauty, Synthese 111 (1997), 171–182. 149. G.-C. Rota, The phenomenology of mathematical proof, Synthese 111 (1997), 183-196. 150. B. Rotman, Signifying Nothing: the Semiotics of Zero. Stanford University Press, 1993. 151. S. Shapiro, Philosophy of Mathematics: Structure and Ontology. Oxford University Press, 1997. ISBN 0-19-513930-5. 152. D. Sherry, The wake of Berkeley’s Analyst: rigor mathematicae? Stud. Hist. Philos. Sci. 18 no. 4 (1987), 455–480. 153. B. Shulman, What if we change our axioms? A feminist inquiry into the foundations of mathematics. Configurations 4 no. 3 (1996), 427– 451. DOI: 10.1353/con.1996.0022. 154. Z. K. Silagadze, Zeno meets modern science, Acta Physica Polonica B 36 no. 10 (2005), 2887–2929. 155. A. Sloman, The Computer Revolution in Philosophy:Philosophy, Science and Models of Mind. Humanities Press, 1978. ISBN 0-391-00830-7. 156. R. Sorensen, The aesthetics of mirror reversal, Philosophical Studies 100 no. 2 (2000), 175–191. 157. B. Sriraman, The influence of Platonism on mathematics research and theological beliefs, Theology and Science 2, no. 1 (2004), 131– 147. 158. D. Struik, The sociology of mathematics revisited: A personal note. Sci. and Society 50 (1986), 280–299. 159. N. Tennant, On the necessary existence of numbers. Nous 31 (1997), 307–336. 160. W. Thurston, On proof and progress in mathematics. Bulletin of the American Mathematical Society 30 no. 2 (1994), 161–177. 161. R. Tieszen,Phenomenology and mathematical knowledge, Synthese 75 (1988), 373–403. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK References 241 162. R. Tragesser, Gian-Carlo Rota and the phenomenological philosophy of mathematics, Philosophia Mathematica (3) 8 (2000), 3–8. 163. E. H. Ward, Lateral Inversion and all that, The School Science Review 143 (1959) 72–76. 164. C. Werndl, Justifying definitions in mathematics—going beyond Lakatos Philosophia Mathematica 17 no. 3 (2009) 313–340. 165. E. Wigner, The unreasonable effectiveness of mathematics in natural sciences. Communications in Pure and Applied Mathematics 13 no. 1 (1960), 1–14. 166. R. L. Wilder, Mathematics as a Cultural System. Pergamon, Oxford, 1981. ISBN-10: 0080257968. ISBN-13: 978-0080257969. History of Mathematics 167. W. F. Anderson, Arithmetical computations in Roman numerals. Classical Philology 51 no. 3 (1956), 145–150. 168. Archimedes, The Works of Archimedes (Sir Thomas Heath, transl.). Dover Publications, 2002. (ISBN 0486420841) 169. E. L. Bennett, Fractional quantities in Minoan bookkeeping. American J. Archaeology 54 (1950), 204–222. 170. J. L. Berggren, Episodes in the Mathematics of Medieval Islam. Springer, 2003. ISBN 0-387-40605-0. 171. H. J. M. Bos, Differentials, higher-order differentials and the derivative in the Leibnizian calculus. Arch. History Exact Sci. 14 (1974), 1–90. 172. C. B. Boyer, Fundamental steps in the development of numeration. Isis 35 (1944), 153–165. 173. C. B. Boyer, Note on Egyptian numeration. Mathematics Teacher 52 (1959), 127–129. 174. W. H. Brock and M. H. Price, Squared paper in the nineteenth century: Instrument of science and engineering, and symbol of reform in mathematical education. Ed. Stud. Math. 11 no. 4 (1980), 365–381. DOI: 10.1007/BF00231211. 175. G. Cantor, Contributions to the Founding of the Theory of Transfinite Numbers(P. E. B. Jourdain, transl.). Dover Publications, 1915. 176. G. Cantor, On the theory of the Transfinite. Correspondence of Georg Cantor and J. B. Cardinal Franzelin (1885–1886). Publication and afterword by L. LaRouche. Fidelio 3 no. 3 (1994). Available from http://www.schillerinstitute.org/fid_91-96/ 943_transfinite.html. Accessed 17 May 2011. 177. P. Cartier, A mad day’s work: from Grothendieck to Connes and Kontsevich. The evolution of concepts of space and symmetry. Bull. Amer. Math. Soc. 38, no. 4 (2001), 389–408. 178. J. Cleave, Cauchy, convergence and continuity, British. J. Philos. Sci. 22 (1971), 27–37. 179. P. C. Cohen, A Calculating People: the History of Numeracy in Early America. University of Chicago Press, 1982. 180. R. Cooke, The History of Mathematics: A Brief Course. Wiley-Interscience, 2005. ISBN-10: 0471444596. ISBN-13: 9780471444596. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 242 References 181. R. Cooke, Classical Algebra: Its Nature, Origins, and Uses. Wiley-Interscience, 2008. ISBN-10: 0470259523. ISBN-13: 9780470259528. 182. B. Datta, Ancient Hindu Geometry: The Science of the Sulba. Cosmo Publications, 2003. ISBN-10: 8170204607. ISBN-13: 9788170204602. 183. J. W. Dauben, Georg Cantor and Pope Leo XIII: Mathematics, theology, and the infinite, J. History of Ideas 38 no. 1 (1977), 85–108. Stable URL: http://www.jstor.org/stable/2708842 184. J. W. Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite. Princeton University Press, 1990. ISBN-10: 0691024472. ISBN-13: 978-0691024479. 185. J. W. Dauben, Abraham Robinson. The Creation of Nonstandard Analysis. The Personal and Mathematical Odyssey. Princeton University Press, 1995. 186. H. M. Edwards, Euler’s definition of the derivative, Bull. Amer. Math. Soc. 44 (2007), 575–580. 187. P. Ehrlich, The rise of non-Archimedean mathematics and the roots of a misconception. I. The emergence of non-Archimedean systems of magnitudes. Arch. Hist. Exact Sci. 60 no. 1 (2006), 1–121. 188. Theorematum quorandumad numeros primos spectantium demonstratio. Comment. acad. scient. Petrop. 8 (1736), 1741, p. 141–146. 189. W. Felscher, Bolzano, Cauchy, epsilon, delta. Amer. Math. Monthly 107 no. 9 (2000), 844–862. 190. J. V. Field, Rediscovering the Archimedean polyhedra: Piero della Francesca, Luca Pacioli, Leonardo da Vinci, Albrecht Dürer, Daniele Barbaro, and Johannes Kepler. Arch. Hist. Exact Sci. 50, nos. 3–4 (1997), 241–289. DOI: 10.1007/BF00374595. 191. J. Friberg, Numbers and measures in the earliest written records. Sci. Amer. 250 no. 2 (1984), 110–118. 192. Galileo Galilei, Two New Sciences, translated by Henry Crew and Alfonso de Salvio (Dover Publications, New York, NY, 1954). Originally published by Elzevir, 1638. 193. K. F. Gauss, Disquisitiones Arithmeticae, 1801. Available at http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN= PPN235993352, accessed 19 december 2010. 194. A. Givental, The Pythagorean Theorem: What is it about? Amer. Math. Monthly 113 no. 3 (2006), 261–265. 195. A. Glaser, History of Binary and Other Nondecimal Numeration. 1971, out of print, but available at http://www.eipiphiny. org/books/history-of-binary.pdf. 196. I. Grattan-Guinness, Numbers, magnitudes, ratios and proportions in Euclid’s Elements: how did he handle them? Hist. Math. 23 (1996), 355–375. [Printing correction: 24 (1997) 213.] 197. I. Grattan-Guinness, The mathematics of the past: distinguishing its history from its heritage, Historia Mathematica 31 (2004), 163–185. 198. I. Grattan-Guinness, On certain somewhat neglected features of the history of mathematics, Arch. Int. Hist. Sci. 55 (2005), 383—402. 199. I. Hacking, The Emergence of Probability: A Philosophical Study of Early Ideas About Probability, Induction and Statistical Inference. Cambridge University Press, 1975. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK References 243 200. A. Heeffer, Sources in the history of algebra before 1600, http:// logica.ugent.be/albrecht/math.php. Accessed 9 July 2010. 201. L. Howard, On a periodical variation of the barometer, apparently due to the influence of the sun and moon on the atmosphere, Philosophical Magazine 7 (1800), 355–363. 202. J. Høyrup, Lengths, Widths, Surfaces. A Portrait of Old Babylonian Algebra and Its Kin. Springer, 2002. ISBN-10: 0387953035. ISBN-13: 978-0387953038. 203. J. Høyrup, Jacopo Da Firenze’s Tractatus Algorismi and Early Italian Abbacus Culture. Birkhäuser, 2007. ISBN-10: 3764383909. ISBN-13: 978-3764383909. 204. V. J. M. Huguet and A. C. Hernández, Euler y la teoría de números, Cap. 6 in El Legado Matemático de Leonhard Euler a Trescieñtos Anos de su Nacimiento (A. A. Meneses, J. Delgado, and F. M. Pérez, Eds.). Universidad Autónoma Metropolitana, Mexico, 2007, pp. 147–179. Available at http://mx.oocities.com/ alfons_physikvier/EulerBuch.pdf, accessed on 19 December 2010. 205. D. Jesseph, Leibniz on the foundations of the calculus: the question of the reality of infinitesimal magnitudes. Leibniz and the sciences. Perspect. Sci. 6 no. 1–2 (1998), 6–40. 206. G. G. Joseph, The Crest of the Peacock: Non-European Roots of Mathematics. Tauris, London, 1991. ISBN-10: 1850432856. ISBN13: 978-1850432852. 207. G. G. Joseph, A Passage to Infinity: Medieval Indian Mathematics from Kerala and Its Impact. Sage Publications Pvt. Ltd, 2009. ISBN-10: 8132101685. ISBN-13: 978-8132101680. 208. I. Kleiner, History of the infinitely small and the infinitely large in calculus, Ed. Stud. Math. 48 (2001), 137–174. 209. J. Kozhamthadam, Jesuit contribution to the origin and development of modern science and mathematics. Indian J. Hist. Sci. 42 (2007), 13–30. 210. L. Kvasz, History of geometry and the development of the form of its language, Synthese 116 (1998), 141–186. 211. F. Lasserre, The Birth of Mathematics in the Age of Plato. Hutchinson, London, 1964. 212. D. Laugwitz, Definite values of infinite sums: aspects of the foundations of infinitesimal analysis around 1820. Arch. Hist. Exact Sci. 39 no. 3 (1989), 195–245. 213. D. Laugwitz, Cauchy-Zahlen als Grundlage der Infinitesimalmathematik. Mathematische Semesterberichte 38 (1991), 175–213. 214. D. Laugwitz, On the historical development of infinitesimal mathematics. Amer. Math. Monthly 104 no. 5 (1997), 447–455. 215. D. Laugwitz, On the historical development of infinitesimal mathematics. Parti II. the conceptual thinking of Cauchy. Amer. Math. Monthly 104 no. 7 (1997), 654–663. 216. G.-G. Leibnitz, Explication de l’arithmétique binaire, qui se sert des seuls caractéres O et I avec des remarques sur son utilité et sur ce qu’elle donne le sens des anciennes figures chinoises de Fohy, Mémoires de mathématique et de physique de l’Académie royale des sciences 1703, 85–89. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 244 References 217. A. H. Lighthouse, Infinitesemals, Amer. Math. Monthly 79 (1972), 242–251. 218. N. N. Luzin, Two letters of N. N. Luzin to M. Ya. Vygodski˘i (with an introduction by S. D. Demidov, translated by A. Shenitzer), Amer. Math. Monthly 107 (January 2000), 64–82. 219. K. O. May, Historiographic vices. I. Logical attribution. Hist. Math. 2 (1975), 185-187. 220. K. O. May, Historiographic vices. II. Priority chasing. Hist. Math. 2 (1975), 315–317. 221. K. O. May, What is good history and who should do it? Hist. Math. 3 (1976), 449–455. 222. F. A. Medvedev, Nonstandard Analysis and the History of Classical Analysis. Patterns in the Development of Modern Mathematics. “Nauka”, Moscow, 1987, pp. 75–84. (Russian.) 223. F. A. Medvedev, Nonstandard analysis and the history of classical analysis. Amer. Math. Monthly 105 no. 7 (1998), 659–664. 224. R. Netz, The Shaping of Deduction in Greek Mathematics: A Study in Cognitive History. Cambridge University Press, 1999. ISBN-10: 0521622794. ISBN-13: 978-0521622790. 225. The Penny Cyclopedia of the Society for the Diffusion of Useful Knowledge, vol. 12. London: Charles Knight and Co., 22, Ludgate Street, 1838. A scanned image of the original is available on G OOGLE B OOKS , http://books.google.com/books? id=DoyETlgz7QYC&printsec=frontcover&dq=editions: UCM5319406755&lr=#v=onepage&q&f=false. 226. M. A. Peterson, Galileo’s discovery of scaling laws, American J. Physics 70 no. 6 (June 2002), 575–580. arXiv:physics/ 0110031v3. 227. H. M. Pycior, Augustus De Morgan’s algebraic work: the three stages, Isis 74 no. 2 (1983), 211–226. 228. S. Restivo, Mathematics in Society and History. Kluwer, 1992. 229. M. Reynolds, The octagon in Leonardo’s drawings, Nexus Network J. 10, no. 1 (2008), 51–76. DOI: 10.1007/ S00004-007-0056-8. 230. J. L. Richards, Augustus De Morgan, the history of mathematics, and the foundations of algebra, Isis 78, no. 1 (1987), 7–30. Stable URL: http://www.jstor.org/stable/232727. 231. S. Rose, Mathematics and the art of navigation: the advance of scientific seamanship in Elizabethan England, Trans. Roy. Hist. Soc. 14 (2004) 175–184. 232. L. A. Sad, M. V. Teixeira, and R. R. Baldino, Cauchy and the problem of point-wise convergence. Archive Internationales d’Histoire des Sciences 51, no. 147 (2001), 277–308. 233. J. Sesiano, An Introduction to the History of Algebra: Solving Equations from Mesopotamian Times to the Renaissance. Amer. Math. Soc., Providence, R.I. 2009. ISBN 978-0-8218-4473-1. 234. S. Tanaka, Boltzmann on mathematics, Synthese 119 (1999), 203– 232. 235. I. Thomas, Selections Illustrating the History of Greek Mathematics with an English Translation by Ivor Thomas. Vol. 1: From Thales to Euclid. Harward University Press, Cambridge, Ma. 1938 (reprinted 1980). ISBN 0-674-99369-1. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK References 245 236. C. Vaughan and M. O’Malley, Froude and the contribution of naval architecture to our understanding of bipedal locomotion, Gait and Posture 21, no. 3 (2005), 350–362. 237. F. Viéte, The Analytic Art. Translated by T. Richard Witmer. Kent, Ohio: The Kent State University Press, 1983. 238. J. Vincent and C. Vincent, Japanese temple geometry, Austr. Senior Math. J. 18 no. 1 (2004), 8–20. Psychology, Psychophysiology, Cognitive Science, Linguistic, and Artificial Intelligence 239. E. Abravanel, The development of intersensory pattering with regard to selected spatial dimensions. Monographs of the Society for Research in Child Development 33, 2 whole no. 118 (1968). 240. A. Achim and M. Corballis, Mirror-image equivalence and the anterior commissure. Neuropsychologia 15 (1977), 475–788. 241. J. W. Adams, and G. J. Hitch, Working memory and children’s mental addition. J. Exp. Child Psychology 67 (1997), 21–38. 242. M. Alibali, R. Spencer, L.Knox, and S. Kita, Spontaneous gestures influence strategy choices in problem solving, Psychological Science, 2011. (To appear.) 243. G. L. Allen, Children’s control of reference systems in spatial tasks: Foundations of spatial cognitive skill? Spat. Cogn. Comput. 1 (1999), 413–429. 244. F. J. Allen, Mirror writing. Brain 19 (1896), 385–387. DOI: 10.1093/ brain/19.2-3.385. 245. M. H. Ashcraft and B. A. Fierman, Mental addition in third, fourth and sixth graders. J. Experimental Child Psychology, 33 (1982), 216– 234. 246. M. H. Ashcraft and J. A. Krause, Working memory, math performance, and math anxiety, Psychonomic Bulletin & Review 14, no. 2 (2007), 243–248. 247. L. Atkins, A radical treatment for obsessive-compulsive disorder patients, The Guardian, 12 December 2009. http: //www.guardian.co.uk/lifeandstyle/2009/dec/15/ obsessive-complusive-disorder-gamma-knife, accessed 15 December 2009. Author’s profile: http://www.guardian.co. uk/profile/lucy-atkins. 248. J. Atkinson, F. W. Campbell, and M. R. Francis, The magic number 4 ± 0: A new look at visual numerosity judgements, Perception 5, no. 3 (1976), 327—334. 249. P. L. Ayres, Systematic mathematical errors and cognitive load. Contemporary Educational Psychology 26 (2001), 227–248. 250. J. D. Balakrishnan and F. G. Ashby, Is subitizing a unique numerical ability, Perception and Psychophysics 50 (1991), 555–564. 251. S. Balfour, S. Borthwick, R. Cubelli, and S. Della Sala, Mirror writing and reversing single letters in stroke patients and normal elderly. J. Neurol. 254 (2007), 436–441. DOI: 10.1007/s00415-006-0384-8. 252. S. Baron-Cohen, P. Bolton, S. Wheelwright, V. Scahill, L. Short, G. Mead, and A. Smith, Autism occurs more often in families of physicists, engineers, and mathematicians, Autism 2 (1998), 296–301. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 246 References 253. S. Baron-Cohen, The Essential Difference. The Truth about the Male and Female Brain. Basic Book, New York, 2003. 254. J. Barth and J. Call, Tracking the displacement of objects: A series of studies with great apes and young children. J. Exp. Psychol. Anim. Behav. Process. 32 (2006), 239–252. 255. G. Basso, P. Nicheli, F. Frasinetti , and G. di Pellegrino, Time percep˝ tion in a neglected space. Neuroreport, 7 (1996), 2111U–2114. 256. D. B. Berch, E. J. Foley, R. J. Hill, and P. M. Ryan, Extracting parity and magnitude from Arabic numerals: Developmental changes in number processing and mental representation. J. Experimental Child Psychology, 74 (1999), 286–308. 257. L. Boroditsky, Metaphoric structuring: understanding time through spatial metaphors. Cognition 75 (2000), 1–28. 258. L. Boroditsky, Does language shape thought? English and Mandarin speakers’ conceptions of time. Cognitive Psychology 43 no. 1 (2001), 1–22. 259. L. Boroditsky and M. Ramscar, The roles of body and mind in abstract thought. Psychological Science 13 (2002) 185–188. 260. E. M. Brannon, The development of ordinal numerical knowledge in infancy, Cognition 83 (2002), 223–240. 261. G. Bremner, Space from infantŠs point of view. Bull. Br. Psychol. Soc. 31 (1978), 164. 262. G. Briggs and R. Nebes, Patterns of hand preference in a student population. Cortex 11 (1975) 230–238. 263. P. Bryant, C. Christie, and A. Rendu, Children’s understanding of the relation between addition and subtraction: Inversion, identity and decomposition. J. Experimental Child Psychology 74 (1999), 194–212. 264. P. Bryant and S. Squire, The influence of sharing on children’s initial concept of division, J. Exper. Child Psych. 81, no. 1 (Jan. 2002), 1–43. 265. V. V. Burdakov and E. B. Lyubchinskii, A rare case of “mirror writing”, Neuroscience and Behavioral Physiology 31 no. 2 (2001), 169– 170. 266. L. J. Buxbaum, H. B. Coslett, R. R. Schall, B. McNally, and G. Goldberg, Hemispatial factors in mirror writing. Neuropsychologia 31 (1993), 1417–1421 267. J. F. Cantlon, S. Cordes, M. E. Libertus, and E. M. Brannon, Comment on “Log or linear? Distinct intuitions of the number scale in Western and Amazonian indigene cultures.” Science, 323 (2009), 38. 268. L. Carmichael and H. Cashman, A study of mirror writing in relation to handedness and perceptual and motor habits. J. Gen. Psychol. 6 (1932), 296–329. 269. M. Carr, J. Alexander, and T. Folds-Bennett, Metacognition and mathematics strategy use. Applied Cognitive Psychology 8 (1994), 583–595. 270. D. Casasanto, Similarity and proximity: When does close in space mean close in mind? Memory and Cognition, 36 no. 6 (2008), 1047– 1056. 271. D. Casasanto and L. Boroditsky, . Time in the mind: Using space to think about time. Cognition 106 (2008), 579–593. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK References 247 272. D. Casasanto, O. Fotakopoulou, and L. Boroditsky, Space and time in the childŠs mind: Evidence for a cross-dimensional asymmetry. Cognitive Science 34 (2010), 387–405. DOI: 10.1111/j.15516709.2010.01094.x. 273. J.-L. Chan and E. Ross, Left-handed mirror writing following right anterior cerebral artery infarction: evidence for non-mirror transformation of motor programs by right supplementary motor area. Neurology 38 (1988), 59–63. 274. K. Cheng and N. S. Newcombe, Is there a geometric module for spatial orientation? Squaring theory and evidence. Psychonomic Bulletin and Review 12, no. 1 (2005), 1–23. 275. E. Claparède, E. 1919: La conscience de la ressemblance et de la difference chez l’enfant,Archives de Psychologie, 17 (1919), 67–80. 276. K. F. Condry and E. S. Spelke, The development of language and abstract concepts: The case of natural number. J. Exp. Psych.: General 137 (2008), 22–38. 277. M. Corballis and I. Beale, The Psychology of Left and Right. London: Lawrence Erlbaum, 1976. 278. J. M. Cornell, Spontaneous mirror-writing in children. Can. J. Psychol. 3 (1985), 174–179. DOI: 10.1037/h0080122. 279. N. Cowan, The magical number 4 in short-term memory: a reconsideration of mental storage capacity. Behavioral and Brain Sci. 24, no. 1 (2001), 87–154. 280. R. Cubelli and S. Della Sala, Mirror writing in pre-school children: a pilot study, Cogn. Process 10 (2009), 101–104. DOI: 10.1007/s10339008-0233-z. 281. M. Curd, Showing and telling: Can the difference between right and left be explained in words? Ratio 26 (1984), 63–66. 282. J. Davidoff and E. K. Warrington, A particular difficulty in discriminating between mirror images. Neuropsychologia 39 (2001), 1022– 1036. DOI:10.1016/S0028-3932(01)00039-2. 283. M. D. de Hevia, L. Girelli, E. Bricolo, and G. Vallar, The representational space of numerical magnitude: Illusions of length. Quarterly J. Experimental Psychology, 61 (2008), 1496–1514. 284. M. D. de Hevia and E. S. Spelke, Spontaneous mapping of number and space in adults and young children. Cognition, 110 (2009), 198– 207. 285. M. D. de Hevia and E. S. Spelke, Number-space mapping in human infants, Psychological Science, published online 19 March 2010. DOI: 10.1177/0956797610366091. http://pss.sagepub. com/content/early/2010/03/19/0956797610366091. 286. M. D. de Hevia, G. Vallar, and L. Girelli, Visualizing numbers in the mind’s eye: The role of visuo-spatial processing in numerical abilities. Neuroscience and Biobehavioral Reviews, 32 (2008), 1361–1372. 287. S. Dehaene, Varieties of numerical abilities. Cognition 44 (1992), 1– 42. 288. S. Dehaene, The Number Sense. Penguin Books, 2001. 289. S. Dehaene, The neural basis of the Weber-Fechner law: A logarithmic mental number line, Trends in Cognitive Sci. 7 (2003), 145–147. 290. S. Dehaene, S. Bossini, and P. Giraux, The mental representation of parity and number magnitude. J. Exp. Psychology: General 12 2(1993), 371–396. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 248 References 291. S. Dehaene and J .P. Changeux, Development of elementary numerical abilities: A neuronal model, J. Cognitive Neuroscience 5 (1993), 390–407. 292. S. Dehaene, V. Izard, P. Pica, and E. S. Spelke, Log or linear? Distinct intuitions of the number scale in Western and Amazonian indigene cultures. Science, 320 (2008), 1217–1220. 293. S. Dehaene, V. Izard, P. Pica, and E. S. Spelke, Response to comment on, “Log or linear? Distinct intuitions of the number scale in Western and Amazonian indigene cultures.” Science, 323 (2009), 38. 294. S. Dehaene, M. Piazza, P. Pinel and L. Cohen, Three parietal circuits for number processing, Cognitive Neuropsychology 20 (3–6) (2003)487–506. 295. S. Dehaene, E. Spelke, P. Pinel, R. Stanescu, and S. Tsivkin, Sources of mathematical thinking: Behavioral and brain-imaging evidence. Science 284 (1999), 970–974. 296. S. Della Sala and R. Cubelli, “Directional apraxia”: a unitary account of mirror writing following brain injury or as found in normal young children. J. Neuropsychol. 1 (2007),3–26. DOI: 10.1348/ 174866407X180783. 297. D. Delis, L. Robertson, and R. Efron, Hemispheric specialization of memory for visual hierarchical stimuli. Neuropsychologia 24 (1986), 205–214. 298. Y. Dong, H. Fukuyama, M. Honda, T. Okada, T. Hanakawa, K. Nakamura et al. Essential role of the right superior parietal cortex in Japanese kana mirror reading: an FMRI study. Brain 123 (2000), 790–799. 299. N. J. Enfield, The body as a cognitive artifact in kinship representations. Current Anthropology 46 no. 1 (2005), 51–81. 300. G. Ettlinger and A. Elithorn, Transfer between the hands of a mirrorimage tactile shape discrimination. Nature 194 (1962), 1101–1102. 301. R. Falk, Infinity: a cognitive challenge, Theory and Psychology 4 no. 1 (1994), 35–60. DOI: 10.1177/0959354394041002. 302. M. W. Faust, M. H. Ashcraft, and D. E. Fleck, Mathematics anxiety effects in simple and complex addition, Mathematical Cognition 2 (1996), 25–62. 303. L. Feigenson, S. Carey, and M. Hauser, The representations underlying infants’ choice of more: Object-files versus analog magnitudes. Psychological Science 13 (2002), 150–156. 304. L. Feigenson, S. Dehaene, and S. Spelke, Core systems of number. Trends in Cognitive Sciences 8 (2004), 307–314. 305. T. Feinberg and G. Jones, Object reversals after parietal lobe infarction—a case report. Cortex 21 (1985), 261–271. 306. F. Fischer, I. Liberman, and D. Shankweiler, Reading reversals and developmental dyslexia: a further study. Cortex 14 (1978), 496–510. 307. J. Flavell, Cognitive Development: Children’s Knowledge About the Mind. Annu. Rev. Psychol. 50 (1999), 21–45. 308. J. Freyd and B. Tversky, The force of symmetry in form perception. Amer. J. Psychology 97 (1984), 109–126. 309. A. J. Fürst and G. J. Hitch, Separate roles for executive and phonological components of working memory in mental arithmetic. Memory and Cognition 28 (2000), 774–782. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK References 249 310. M. Gaffron, Right and left in pictures, Art Quarterly 13 (1950), 312. 311. C. R. Gallistel, Language and spatial frames of reference in mind and brain. Trends in Cognitive Sciences 6, no. 8 (2002), 321–22. 312. C. R. Gallistel and R. Gelman, Preverbal and verbal counting and computation, Cognition 44 (1992), 43–74. 313. C. R. Gallistel and R. Gelman, Non-verbal numerical cognition: From reals to integers. Trends in Cognitive Science 4 (2000), 59–65. 314. W. R. Garner, Interaction of stimulus dimensions in concept and choice processes. Cognitive Psychology 8 (1976), 98–123. 315. D. C. Geary, C. C. Bow-Thomas, L. Fan, and R. S. Siegler, Even before formal instruction, Chinese children outperform American children in mental addition. Cognitive Development 8 (1993), 517–529. 316. D. C. Geary and S. C. Brown, Cognitive addition: Strategy choice and speed-of-processing differences in gifted, normal and mathematically disabled children. Developmental Psychology 27 (1991), 398–406. 317. R. Gelman and C. R. Gallistel, The Child’s Understanding of Number. Harvard University Press, 1978. 318. R. Gelman and C. R. Gallistel, Language and the origin of numerical concepts. Science 306 (2004), 441–443. 319. S. P. Gennari, S. A. Sloman, B. C. Malt, and W. T. Fitch, Motion events in language and cognition. Cognition 83, no. 1 (2002), 49–79. 320. D. Gentner, M. Imai, and L. Boroditsky. As time goes by: Evidence for two systems in processing space!time metaphors. Language and Cognitive Processes 17 no. 5 (2002), 537–565. DOI: 10.1080/01690960 14300031 7. 321. R. W. Gibbs, Why many concepts are metaphorical. Cognition 61 (1996), 309–319. 322. R. Goebel, D. Linden, H. Lanfermann, F. Zanella, and W. Singer, Functional imaging of mirror and inverse reading reveals separate coactivated networks for occulomotion and spatial transformations. Neuroreport 9 (1998), 713–719. 323. S. Goldin-Meadow, Hearing Gesture: How Our Hands Help Us Think. Harvard University Press, Cambridge, MA, 2003. 324. S. Goldin-Meadow, M. W. Alibali, and R. B. Church, Transitions in concept acquisition: Using the hand to read the mind. Psychological Review 100 (1993), 279–297. 325. S. Goldin-Meadow, S. Kim, and M. Singer, What the teacher’s hands tell the student’s mind about math. J. Ed. Psychology, 91 (1999), 720– 730. 326. S. Goldin-Meadow, H. Nusbaum, S. D. Kelly, and S. Wagner, Explaining math: Gesturing lightens the load. Psychological Science 12 (2001), 516–522. 327. S. Goldin-Meadow and M. A. Singer, From children’s hands to adults’ ears: Gesture’s role in teaching and learning. Developmental Psychology 39 (2003), 509–520. 328. A. Gopnik, The Philosophical Baby: What Children’s Minds Tell Us About Truth, Love, and the Meaning of Life. Farrar, Straus and Giroux, 2009, 288 pp. 329. J. A. Gottfried, F. Sancar, and A. Chatterjee, Acquired mirror writing and reading: evidence for reflected graphemic representations. Neuropsychologia 41 (2003), 96–107. DOI:10.1016/S0028-3932(02) 00130-6. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 250 References 330. E. L. Grigorenko, Developmental dyslexia: An update on genes, brains, and environments, J. Child Psych. and Psychiatry 42, no. 1 (2001), 91–125. DOI: 10.1017/S0021963001006564. 331. C. G. Gross and M. H. Bornstein, Left and right in science and Art, Leonardo 11 (1978), 29–38. 332. M. A. K. Halliday, Aspects of sociolinguistic research, UNESCO, Paris, 1974. A survey paper prepared for Symposium on the Interactions Between Linguistics and Mathematical Education, Nairobi, Sept. 1–11, 1974. UNESCO document ED74/CONF.808/COL.7. Paris: UNESCO, 1975. 333. M. S. Hamann and M. H. Ashcraft, Textbook presentations of the basic addition facts. Cognition and Instruction, 3 (1986), 173–192. 334. D. B. M. Haun, J. Call, G. Janzen, and S. C. Levinson, Evolutionary psychology of spatial representations in the hominidae. Current Biology 16 (2006), 1736–1740. DOI: 10.1016/j.cub.2006.07.049. 335. D. B. M. Haun, C. J. Rapold, J. Call, G. Janzen, and S. C. Levinson, Cognitive cladistics and cultural override in hominid spatial cognition. Proceedings of the National Academy of Sciences USA 103 no. 46 (2006), 17568–17573. 336. S. A. Hecht, Counting on working memory in simple arithmetic when counting is used for problem solving. Memory and Cognition 30 (2002), 447–455. 337. K. M. Heilman, G. Howell, E. Valenstein, and L. Rothi, Mirrorreading and writing in association with right-left spatial disorientation. J. Neurol. Neurosurg. Psychiatry 43 (1980), 774–780. 338. J. Henrich, S. J. Heine, and A. Norenzayan. The weirdest people in the world? Behavioral and Brain Sciences 33 (2010), 61–135. DOI: 10.1017/S0140525X0999152X. 339. A. Herskovits, Language and Spatial Cognition. Cambridge University Press, 1986. ISBN-10: 0521109183. ISBN-13: 9780521109185. 340. I. D. Holloway and D. Ansari, Mapping numerical magnitudes onto symbols: The numerical distance effect and individual differences in children’s math achievement, J. Experimental Child Psychology 103 (2009), 17–29. 341. H. C. Hughes, Sensory Exotica: a World Beyond Human Experience. Cambridge, MA: MIT Press, 1999. 342. B. Inhelder and J. Piaget, Mental imagery in the child. New York: Basic Books, 1971. 343. V. Izard, C. Sann, E. S. Spelke, and A. Streri, Newborn infants perceive abstract numbers. Proc. Nat. Acad. Sci. 106 (2009), 10382– 10385. 344. K. E. Jordan and E. M. Brannon, The multisensory representation of number in infancy. Proc. Nat. Acad. Sci. 103 (2006), 3486–3489. 345. K. Lander, A. Bevan, and B. Butterworth, Developmental dyscalculia and basic numerical capacities: A study of 8–9 year old students, Cognition 93 (2004), 99–125. 346. M. Lambon-Ralph, C. Jarvis, and A. Ellis, Life in a mirrored world: report of a case showing mirror reversal in reading and writing and for non-verbal materials. Neurocase 3 (1997), 249–258. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK References 251 347. J. Lave, Cognition in Practice: Mind, Mathematics and Culture in Everyday Life. Cambridge University Press, New York, 1988. 348. M. LeCorre and S. Carey, One, two, three, four, nothing more: An investigation of the conceptual sources of the verbal counting principles. Cognition 105 (2007), 395–438. 349. What counts in the development of young children’s number knowledge? S. C. Levine, L. W. Suriyakham, M. L. Rowe, J. Huttenlocher, and E. A. Gunderson, Developmental Psychology 46 no. 5 (Sep 2010), 1309–1319. DOI: 10.1037/a0019671. 350. P. Li and L. Gleitman, Turning the tables: language and spatial reasoning, Cognition 83, no. 3 (2002), 265–294. 351. M. E. Libertus, L. Feigenson, J. Halberda, Preschool acuity of the approximate number system correlates with school math ability, Developmental Science (2011). DOI: 10.1111/j.1467-7687.2011.01080.x. 352. J. S. Lipton and E. S. Spelke, Origins of number sense: Large number discrimination in human infants. Psychological Science, 14 (2003), 396–401. 353. A. Majid, M. Bowerman, S. Kita, Daniel B.M. Haun, and S. C. Levinson, Can language restructure cognition? The case for space. Trends in Cognitive Sci. 8, no. 3 (2004), 108–114. 354. B. Malt, S. Sloman, and S. Gennari, Universality and language specificity in object naming. J. Memory and Language 49, no. 1 (2003), 20–42. 355. I. Mathewson, Mirror writing ability is genetic and probably transmitted as a sex-linked dominant trait: it is hypothesised that mirror writers have bilateral language centres with a callosal interconnection. Med. Hypotheses 62 no. 5 (2004), 733–739. 356. T. Matlock, M. Ramscar, and L. Boroditsky, On the experiential link between spatial and temporal language, Cognitive Science 29 (2005) 655–664. 357. M. G. McGee, Human spatial abilities: Psychometric studies and environmental, genetic, hormonal, and neurological influences. Psychological Bulletin, 86 (1979), 889–918. 358. M. S. McGlone and J. L. Harding, Back (or forward?) to the future: The role of perspective in temporal language comprehension. J. Experimental Psychology: Learning, Memory, and Cognition 24 (1998), 1211–1223. 359. B. McNaughton, L. Chen, and E. Markus, ‘Dead reckoning’, landmark learning and the sense of direction: a neurophysiological and computational hypothesis. J. Cognitive Neuroscience 3 no. 2 (1990), 191–202. 360. B. P. Meier and M. D. Robinson, Why the sunny side is up. Associations between affect and vertical position. Psychological Science 15 (2004), 243–247. 361. N. K. Mello, Mirror image reversal in pigeons. Science 149 (1965), 1519–1520. 362. E. R. Michener, Understanding understanding mathematics. Cognitive Science, 2 (1978), 361–383. 363. K. S. Mix, Preschoolers’ recognition of numerical equivalence: Sequential sets. J. Experimental Child Psychology, 74 (1999), 309–332. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 252 References 364. K. S. Mix, The construction of number concepts. Cognitive Development 17 (2002), 1345–1363. 365. K. S. Mix, J. Huttenlocher, and S. C. Levine, Do preschool children recognize auditory-visual numerical correspondences? Child Development 67 (1996), 1592–1608. 366. A. Muir, The psychology of mathematical creativity, Math. Intelligencer 10, no. 1 (1988), 33–37. 367. E. Munnich, B. Landau, and B. Dosher, Spatial language and spatial representation: A cross-linguistic comparison. Cognition 81 no. 3 (2001), 171–207. 368. G. L. Murphy, On metaphoric representation. Cognition 60 (1996), 173–204. 369. J. Noble, Mirror-images and the forebrain commisure of the monkey. Nature 211 (1966), 1263–1266. 370. J. Noble, Paradoxical interocular transfer of mirror-image discriminations in the optic chiasm sectioned monkey. Brain Research 10 (1968), 120–126. 371. R. Núnez, Do real numbers really move? Language, thought and gesture: the embodied cognitive foundations of mathematics. In: 18 Unconventional Essays on the Nature of Mathematics (R. Hersh, ed.). Springer, 2005, pp. 160–181. 372. R. Núnez and E. Sweetser, Looking ahead to the past: Convergent evidence from Aymara language and gesture in the crosslinguistic comparison of spatial construals of time. Cognitive Science 30 no. 3(2006), 401–450. 373. D. R. Olson, Cognitive Development: The Child’s Acquisition of Diagonality. Lawrence Erlbaum Associates Inc., 1996. ISBN-10: 0805823026. ISBN-13: 978-0805823028. 374. A. Ortony, Why metaphors are necessary and not just nice. Educational Theory, 25 (1975), 45–53. 375. W. Paradowski and M. Ginzgerg, Mirror writing and hemiplegia. Perceptual and Motor Skills 32 (1971), 617–618. 376. J. Piaget and B. Inhelder, The Child’s Perception of Space. Routledge and Kegan Paul, 1956. 377. J. Piaget, Genetic Epistemology, Columbia University Press, 1970. ISBN-10: 0231033869. ISBN-13: 978-0231033862. 378. J. Piaget, Possibility and Necessity, University of Minnesota Press, 1987. ISBN-10: 0816613680. ISBN-13: 978-0816613687. 379. P. Pinel, D. Piazza, D. Le Bihan, and S. Dehaene, Distributed and overlapping cerebral representations of number, size, and luminance during comparative judgments. Neuron, 41 (2004), 1–20. 380. S. Pinker, How the Mind Works. Norton, New York, 1997. 381. V. Ramachandran, The Emerging Mind. Profile Books, 2005. ISBN 1-86197-303-9. 382. F. Restle, Speed of adding and comparing numbers. J. Experimental Psychology, 83 (1970), 274–278. 383. S. K. Revkin, M. I. Piazza, L. Cohen, and S. Dehaene, Does subitizing reflect numerical estimation? Psychological Science 19 (2008), 607– 614. 384. I. Rock, Orientation and Form. Academic Press, 1973. ISBN 0-12591250-1. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK References 253 385. R. Rodriguez, M. Aguilar, and G. Gonzalez,. Left non-dominant hand mirror writing. Brain and Language 37 (1989), 122–144. 386. E. Rusconi, B. Kwan, B. Giordano, C. Umilta, and B. Butterworth, Spatial representation of pitch height: The SMARC effect. Cognition 9 (2006), 113–129. 387. J. Sauvy and S. Suavy, The Child’s Discovery of Space: From Hopscotch to Mazes—An Introduction to Intuitive Topology. Penguin Books, 1974. ISBN-10: 014080384X. ISBN-13: 9780140803846. 388. W. Schaeken, P. N. Johnson-Laird, and G. dŠYdewalle, Mental models and temporal reasoning. Cognition 60 (1996), 205–234. 389. G. Schott, Some neurological observations on Leonardo da Vinci’s handwriting. J Neurol. Sci. 42 (1979), 321–329. DOI: 10.1016/ 0022510X(79)90165-5. 390. G. Schott, Mirror movements of the left arm following peripheral damage to the preferred right arm. J. Neurology, Neurosurgery, and Psychiatry 43 (1980), 768–773. 391. G. Schott. Mirror writing: Allen’s self observations, Lewis Carroll’s ‘looking glass’ letters, and Leonardo da Vinci’s maps. Lancet 354 (1999), 2158–2161. DOI:10.1016/S0140-6736(99)07016-6. 392. G. Schott, Mirror writing: neurological reflections on an unusual phenomenon. J. Neurol. Neurosurg. Psychiatry 78 (2008), 5–13. DOI:101136/jnnp2006094870. 393. T. Schubert, Your highness: Vertical positions as perceptual symbols of power. J. Personality and Social Psychology 89 no. 1 (2005), 1–21. 394. A. Scott, The vertical dimension and time in Mandarin. Australian J. Linguistics, 9 (1989), 295–314. 395. R. S. Siegler and J. E. Opfer, The development of numerical estimation: Evidence for multiple representations of numerical quantity. Psychological Science 14 (2003), 237–243. 396. M. A. Singer and S. Goldin-Meadow, Children learn when their teacher’s gestures and speech differ, Psychological Science 16 no. 2 (2005), 85–89. 397. A. Sloman, Kantian philosophy of mathematics and young robots. In Proceedings 7th International Conference on Mathematical Knowledge Management. Birmingham, UK, 28–30 July 2008. http://events.cs.bham.ac.uk/cicm08/mkm08/. 398. L. Smith, S. Jones, and B. Landau, Naming in young children: A dumb attentional mechanism? Cognition 60 (1996), 143–171. 399. R. Sorensen, Mirror imagery and biological selection, Biology and Philosophy 17, no. 3 (2002), 409–422. 400. E. S. Spelke, and S. Tsivkin, Language and number: A bilingual training study. Cognition 78, no. 1 (2001), 45–48. 401. J. C. Stevens and L. E. Marks, Cross-modality matching of brightness and loudness. Proc. Nat. Acad. Sci. 54 (1965) , 407–411. 402. S. S. Stevens, Psychophysics: An Introduction to its Perceptual, Neural, and Social Prospects. John Wiley and Sons, New York, 1975. 403. M. Streifler and S. Hofman, Sinistral mirror writing and reading after brain concussion in a bisystemic (oriento-occidental) polyglot. Cortex 12 (1976), 356–364. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 254 References 404. N. S. Sutherland, Visual discrimination of orientation and shape by octopus. Nature 179 (1957), 11–13. DOI:10.1038/179011a0. 405. L. Talmy, Force dynamics in language and cognition. Cognitive Science 12 (1988), 49–100. 406. R. Tankle and K. Heilman, Mirror writing in right-handers and lefthanders. Brain and Language 19 (1983), 115–123. 407. T. Tardif, Nouns are not always learned before verbs, but why? Evidence from Mandarin Chinese. Developmental Psychology 32 no. 3 (1996), 492–504. 408. E. L. Traugott, Spatial expressions of tense and temporal sequencing: A contribution to the study of semantic fields. Semiotica 15 (1975), 207–230. 409. L. Trick and Z. Pylyshyn, What enumeration studies can show us about spatial attention: Evidence for limited capacity preattentive processing, J. Exp. Psych.: Human Perception and Performance 19 (1993), 331–351. 410. R. Trueman, Mirror writing. Postgraduate Medicine 38 (1965), 469– 476. 411. S. W. Tuholski, R. W. Engle, and G. C. Baylis, Individual differences in working memory capacity and enumeration, Memory and Cognition 29 (2001), 484–492. 412. O. H. Turnbull and R. A. McCarthy, Failure to discriminate between mirror-image objects: a case of viewpoint-independent object recognition? Neurocase 2 (1996), 63–72. 413. B. Tversky, S. Kugelmass, and A. Winter, Cross-cultural and developmental trends in graphic productions. Cognitive Psychology 23 ˝ (1991), 515U-557. 414. E. Vurpillot, The Visual World of the Child. Allen and Unwin, London, 1976. 415. L. S. Vygotsky, Thought and Language. John Wiley Inc., New York, 1962. 416. J. Wade and R. Hart, Mirror phenomena in language and non-verbal activities: a case report. J Clin. Exp. Neuropsychol. 13 (1991), 299– 308. DOI:10.1080/01688639108401045. 417. S. Wagner, H. Nusbaum, and S. Goldin-Meadow, Probing the mental representation of gesture: Is handwaving spatial? J. Memory and Language 50 (2004), 395–407. 418. V. Walsh, A theory of magnitude: Common cortical metrics of time, space and quantity. Trends in Cognitive Science, 7 no. 11 (2003), 483– 488. 419. Wang X-de, Mirror writing of Chinese characters in children and neurological patients. Chin. Med. J. 105 (1992): 306–311. 420. S. Waxman and T. Kosowski, Nouns mark category relations: Toddlers’ and Preschoolers’ word-learning biases. Child Development 61 (1990), 1461–1473. 421. S. Wheelwright and S. Baron-Cohen, The link between autism and skills such as engineering, maths, physics, and computing: A reply to Jarrold and Routh. Autism 5 (2001), 223–227. 422. J. Winawer, N. Witthoft, M. C. Frank, L. Wu, A. R. Wade, and L. Boroditsky, Russian blues reveal effects of language on color discrimination. Proc. Nat. Acad. Sci. 104 no. 19 (May 8, 2007), 7780–7785. www.pnas.org/cgi/doi/10.1073/pnas.0701644104. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK References 255 423. J. Wood and E. Spelke, Infants’ enumeration of actions: Numerical discrimination and its signature limits. Developmental Science 8 (2005), 173–181. 424. K. Wynn, Children’s understanding of counting. Cognition 36 (1990), 155–193. 425. K. Wynn, Addition and subtraction by human infants. Nature 358 (1992), 749–750. 426. K. Wynn, Children’s acquisition of number words and the counting system. Cognitive Psychology 24 (1992), 220–251. 427. K. Wynn, Psychological foundations of number: Numerical competence in human infants. Trends in Cognitive Sciences 2 (1998), 296– 303. 428. K. Wynn and P. Bloom, Linguistic cues in the acquisition of number words. Journal of Child Language 24 (1997), 511–533. 429. F. Xu and R. I. Arriaga, Number discrimination in 10-month old infants. British J. Developmental Psychology 25 (2007), 103–108. 430. F. Xu, M. Cote, and A. Baker, Labelling guides object individuation in 12-month-old infants. Psychol. Sci. 16 (2005), 372–377. 431. F. Xu and S. Carey, Infants’ metaphysics: The case of numerical identity. Cognitive Psychology 30 (1996), 111–153. 432. F. Xu and E. S. Spelke, Large number discrimination in 6-month-old infants. Cognition 74 (2000), B1–B11. 433. O. Zangwill and C. Blakemore, Dyslexia: reversal of eye movements during reading. Neuropsychologia 10 (1972) 371–373. 434. S. Zebian, Linkages between number concepts, spatial thinking, and directionality of writing: The SNARC effect and the reverse SNARC effect in English and Arabic monoliterates, biliterates, and illiterate Arabic speakers. J. Cognition and Culture 5 (2005), 165–190. 435. J. Zhang, and D. A. Norman, A representational analysis of numeration systems. Cognition 57 (1995), 271–295. 436. Z. Zhou, S. T. Peverly, A. E. Boehm, and L. Chongde, American and Chinese childrenŠs understanding of distance, time, and speed interrelations. Cognitive Development 15 no. 2 (2000), 215–240. Linguistic and Anthropological Aspects of Mathematics 437. S. Adam, Ethnomathematical ideas in the curriculum. Math. Ed. Research J. 16, no. 2 (2004), 49–62. 438. S. Adam, W. Alangui, and B. Barton, A comment on: Rowlands and Carson ‘Where would formal, academic mathematics stand in a curriculum informed by ethnomathematics? A critical review’. Ed. Stud. Math. 52 (2003), 327–335. 439. L. R. Aiken, Language factors in learning mathematics. Review Ed. Research 42 (1972), 359–385. 440. R. Alexander, Culture and Pedagogy. International Comparisons in Primary Education.Blackwell Publishers, 2000. 441. G. Allouche, J.-P. Allouche, and J. Shallit, Kolam indiens, dessins sur le sable aux îles Vanuatu, courbe de Sierpinski et morphismes de monoïde, Annales de l’institut Fourier 56, no. 7 (2006), 2115–2130. MR 2290776, Zbl pre05176564. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 256 References 442. M. Ascher and R. Ascher, Ethnomathematics, Hist. Sci. 24 (1986), 125–144. 443. M. Ascher, Graphs in cultures: a study in ethnomathematics, Hist. Math. 55 (1988), 201–227. 444. M. Ascher, Ethnomathematics: A Multicultural View of Mathematical Ideas. New York: Chapman and Hall, 1991. 445. M. Ascher, Models and maps from the Marshall Islands: a case in ethnomathematics, Hist. Math. 22 (1995), 347–370. 446. M. Ascher, Malagasy Sikidy: A case in ethnomathematics, Hist. Math. 24, no. 4 (1997), 376–395. 447. M. Ascher, The Kolam tradition, Amer. Scientist 90, no. 1 (2002), 56– 63. 448. M. Ascher, Mathematics Elsewhere: An Exploration Of Ideas Across Cultures. Princeton University Press, 2004. ISBN-10: 0691120226. ISBN-13: 978-0691120225. 449. M. Ascher and R. Ascher, Mathematics of the Incas: Code of the Quipu. Dover Publications, 1997. ISBN 0-486-29554-0. 450. M. Ascher and U. D’Ambrosio, Ethnomathematics: A dialogue, For the Learning of Math. 14, no. 2 (1994), 36–43. DOI: 10.1006/hmat.1997.2159. 451. J. L. Austin and A. G. Howson, Language and mathematical education, Ed. Stud. Math. 10 (1979), 161–197. 452. J. Barta and T. Shockey, The mathematical ways of an aboriginal people: the Northern Ute, J. Math. and Culture 6, no. 1 (2006), 79–89. ISSN 1558-5336. 453. B. Barton, Making sense of ethnomathematics: Ethnomathematics is making sense. Ed. Stud. Math. 31, no. 1 (1996), 201–233. 454. B. Barton, Ethnomathematics: A political plaything, For the Learning of Math. 19, no. 1 (1999), 33–36. 455. B. Barton, Ethnomathematics and philosophy, Zentralblatt für Didaktik der Mathematik 31, no. 2 (1999), 54–58. 456. B. Barton, M. Domite, and C. Poisard, Cultural connections and mathematical manipulations. For the Learning of Math. 26, no. 2 (2006), 21–24. 457. B. Barton, F. Lichtenberk, and I. Reilly, The language of topology: A Turkish case study, Applied General Topology 6, no. 2 (2005), 107– 117. http://at.yorku.ca/i/a/a/k/24.htm. 458. A. J. C. Begg, Ethnomathematics: why, and what else? Zentralbl. für Didaktik der Math. 33, no. 3 (2001), 71–74. 459. G. Bennardo, Language and space in Tonga: ‘The front of the house is where the Chief sits!’ Anthropological Linguistics 42, no. 4 (2000), 499–544. 460. G. Bennardo, Map drawing in Tonga, Polynesia: Accessing mental representations of space. Field Methods 14, no. 4 (2002), 390–417. DOI: 10.1177/152582202237727. 461. J. Bennett, The difference between right and left, Amer. Philosophical Quart. 7 (1970), 175–191. 462. J. W. Berry, Learning mathematics in a second language: Some crosscultural issues, For the Learning of Math. 5, no. 2 (1985), 18–23. ISSN 1544-5879. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK References 257 463. M. Bierwisch E. Lang, Dimensional Adjectives: Grammatical Structure and Conceptual Interpretation. Springer-Verlag, 1989. ISBN-10: 0387506330. ISBN-3: 978-0387506333. 464. A. J. Bishop, Visualising and mathematics in a pre-technological culture. Ed. Stud. Math. 10 (1979), 135–146. 465. A. J. Bishop, The social construction of meaning: A significant development for mathematics education. For the Learning of Math. 5, no. 1 (1985), 24–28. 466. A. J. Bishop, Mathematics Enculturation: A Cultural Perspective on Mathematics Education. Kluwer Academic Publishers, Dordrecht, 1988. 467. A. J. Bishop, Cultural conflicts in mathematics education: Developing a research agenda. For the Learning of Math. 14, no. 2 (1994), 15–18. 468. M. C. Borba, Ethnomathematics and education, For the Learning of Math. 10, no. 1 (1990), 39–42. 469. L. Boroditsky, Lost in Translation. The Wall Street Journal, 24 July 2010. http://online.wsj.com/article/ SB10001424052748703467304575383131592767868.html? mod=fox_australian. Accessed 26 July 2010. 470. L. Boroditsky and A. Gaby, Absolute spatial representations of time in an Aboriginal Australian community, Psychological Science, 2010. http://www-psych.stanford.edu/~lera/papers/ absolute-time.pdf. 471. E. Broug, Islamic Geometric Patterns. Thames & Hudson, 2008. ISBN-10: 050028721X. ISBN-13: 978-0500287217. 472. T. Brown, Mathematics Education and Language: Interpreting Hermeneutics and Poststructuralism. Kluwer Academic Publishers, Dordrecht, 2001. 473. P. Brown and S. C. Levinson, Frames of spatial reference and their acquisition in Tenejapan Tzeltal, in Culture, Thought, and Development (L. Nucci, G. Saxe and E. Turiel, eds.). Erlbaum, Hillsdale, NJ, 2000, pp. 167–197. 474. P. Brown and S. C. Levinson, ‘Left’ and ‘right’ in Tenejapa: Investigating a linguistic and conceptual gap. Zeitschrift für Phonetik, Sprachwissenschaft und Kommunikationsforschung 45 (1992), 590–611. 475. P. Brown and S. C. Levinson, ‘Uphill’ and ‘downhill’ in Tzeltal. J. Linguistic Anthropology 3 (1993), 46–74. 476. J. Bourgoin, Arabic Geometrical Pattern and Design (Picture Archives). Dover Publications Inc., 1974. ISBN-10: 0486229246. ISBN-13: 978-0486229249. 477. T. N. Carraher, D. W. Carraher, and A. D. Schliemann, Mathematics in the street and in schools, British J. Developmental Psych. 3 (1985), 21-29. 478. M. Chemillier, Ethnomusicology, ethnomathematics. The logic underlying orally transmitted artistic practices. in Mathematics and Music (Gerard Assayag et al., eds). Springer-Verlag, 2002, pp. 161–183. 479. S. Chrisomalis, A cognitive typology for numerical notation. Cambridge Archaeological J. 14 no. 1 (2004), 37–52. 480. M. P. Closs, ed. Native American Mathematics. University of Texas Press, Austin, 1988. ISBN 0-292-75537-1. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 258 References 481. D. W. Crowe, Symmetry Comes of Age: The Role of Pattern in Culture. University of Washington Press, 2004. ISBN-10: 0295983663. ISBN-13: 978-0295983660. 482. T. Crump, The Anthropology of Numbers. Cambridge University Press (Cambridge Studies in Social and Cultural Anthropology, vol. 70), 1990. 483. M. Curd, “Showing and Telling”. Can the difference between right and left be explained in words? Ratio 26 (1984), 63–66. 484. U. D’Ambrosio, Ethnomathematics and its place in the history and pedagogy of mathematics, For the Learning of Math. 5, no. 1 (1985), 44–48. 485. U. D’Ambrosio, A research program and a course in the history of mathematics: Ethnomathematics, Hist. Math. 16 (1989), 285–288. 486. U. D’Ambrosio, Ethnomathematics: A research program on the history and philosophy of mathematics with pedagogical implications. Notices Amer. Math. Soc. 39, no. 10 (1992), 1183–1184. 487. U. D’Ambrosio and M. Ascher, Ethnomathematics: A dialogue, For the Learning of Math. 14, no. 2 (1994), 36–43. 488. B. Deacon, Geometrical drawings from Malekula and other islands of the New Hebrides, J. Royal Anthro. Institute, 64 (1934), 129–175. 489. S. Dehaene, V. Izard, P. Pica, and E. Spelke, Core knowledge of geometry in an Amazonian indigene group, Science 311 (2006), 381–384. 490. J. B. Deregowski, Pictorial perception and culture. Scientific Amer. 227 (1972), 82–88. 491. J. B. Deregowski, Teaching African children depth perception: In search of a method. Perception 3 (1974), 309–312. 492. J. B. Deregowski, Distortion in Art: The Eye and the Mind. Routledge and Kegan Paul, London, 1984. ISBN-10: 0710095163. ISBN13: 978-0710095169. 493. J. B. Deregowski, Wire toys in Africa south of the Sahara. Leonardo, 13 no. 3 (1980), 207–208. 494. H. Dwornik, A 2n -number system in the arithmetic of prehistoric societies. Organon 16–17 (1980–81), 199–222. 495. R. Eglash, Bamana sand divination: Recursion in ethnomathematics. Amer. Anthropologist 99, no. 1 (1997), 112–123. 496. R. Eglash, African Fractals: Modern Computing and Indigenous Design. Rutgers University Press, 1999. ISBN-10: 0813526140. ISBN-13: 978-0813526140. 497. D. B. Eyde, Recursive dualism in the Admiralty Islands, J. Soc. Océanistes 39 (1983), 3–12. 498. W. Esty, Language concepts of mathematics, Focus on Learning Problems in Math. 14, no. 4 (1992), 31–54. 499. A. Francois, Of men, hills, and winds: Space directionals in Mwotlap. Oceanic Linguistics 42, no. 2 (2003), 407–437. 500. R. Freeze, Existentials and other locatives, Language 68, no. 3 (1992), 553–95. Stable URL: http://www.jstor.org/stable/415794. 501. L. Galligan, Possible effects of English-Chinese language differences on the processing of mathematical text: A review. Math. Ed. Research J. 13, no. 2 (2001), 112–132. 502. J. Gay and M. Cole, The New Mathematics and an Old Culture: A Study of Learning Among the Kpelle of Liberia. Holt, Rinehart and Winston, 1967. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK References 259 503. P. Gerdes, How to recognise hidden geometrical thinking? A contribution to the development of anthropological mathematics. For the Learning of Math. 6, no. 2 (1986), 10–12, 17. 504. P. Gerdes, A widespread decorative motif and the Pythagorean theorem, For the Learning of Math. 8, no. 1 (1988), 35–39. 505. P. Gerdes, On possible uses of traditional angolan sand drawings in the mathematics classroom, Ed. Stud. Math. 19, no. 1 (1988), 3–22. 506. P. Gerdes, On culture, geometrical thinking and mathematics education, Ed. Stud. Math. 19 (1988), 137–162. 507. P. Gerdes, Reconstruction and extension of lost symmetries: Examples from the Tamil of South India, Computers and Math. with Applications 17, nos. 4–6 (1989), 791–813. 508. P. Gerdes, On mathematical elements in the Tchokwe ‘sona’ tradition, For the Learning of Math. 10, no. 1 (1990), 31–34. 509. P. Gerdes, Reflections on ethnomathematics, For the Learning of Math. 14, no. 2 (1994), 19–22. 510. P. Gerdes, Geometry from Africa. Mathematical and Educational Explorations. The Mathematical Association of America, 1999. ISBN-10: 0883857154. ISBN-13: 978-0883857151. 511. P. Gerdes, Drawings from Angola: Living Mathematics. Lulu.com, 2007. ISBN-10: 1430323132. ISBN-13: 978-1430323136. 512. P. Gerdes, Sona Geometry from Angola. Mathematics of an African Tradition. Polimetrica, 2007. ISBN-10: 8876990550. ISBN13: 978-8876990557. 513. P. Gerdes, Adventures in the World of Matrices. Nova Science Publishers Inc, 2007. ISBN-10: 1600217184. ISBN-13: 978-1600217180. 514. P. Gerdes, Lunda Geometry: Mirror Curves, Designs, Knots, Polyominoes, Patterns, Symmetries. Lulu.com, 2008. ISBN-10: 1435726294. ISBN-13: 978-1435726291. 515. P. Gerdes, African Basketry: A Gallery of Twill-Plaited Designs and Patterns. Lulu.com, 2008. ISBN-10: 1435726251. ISBN13: 978-1435726253. 516. T. Gladwin, East is a Big Bird: Navigation and Logic on Puluwat Atoll. Harvard University Press, 1974. ISBN-10: 0674224264. ISBN-13: 978-0674224261. 517. P. Gordon, Numerical cognition without words: Evidence from Amazonia, Science 306 no. 5695 (2004), 496–499. DOI: 10.1126/science.1094492. 518. B. Grünbaum, What symmetry groups are present in Alhambra? Notices Amer. Math. Soc. 53, no. 6 (2006), 670–673. 519. J. B. Haviland, Anchoring, iconicity, and orientation in Guugu Yimidhirr pointing gestures, J. Linguistic Anthropology 3, no. 10 (1993), 3–45. 520. J. B. Haviland, Guugu Yimithirr cardinal directions. Ethos 26, no. 1 (March 1998), 25–47. 521. A. Heron and M. Simonsson, Weight conservation in Zambian children: a non-verbal approach. Internat. J. Psychol. 4 no. 4 (1969), 281– 292. 522. K. Horsthemke, ‘Indigenous knowledge’—conceptions and misconceptions, J. Ed. 32 (2004), 31–48. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 260 References 523. K. Horsthemke, Knowledge, education and the limits of Africanisation, J. Philosophy Ed. 38, no. 4 (2004), 571–587. 524. K. Horsthemke and M. Schäfer, Does ‘African mathematics’ facilitate access to mathematics? Towards an ongoing critical analysis of ethnomathematics in a South African context. Pythagoras 65 (June 2007), 2–9. 525. G. G. Joseph, Foundations of Eurocentrism in mathematics, Race and Class 27, no. 3 (1987), 13–28. 526. B.-H. Juang and L. R. Rabiner, Speech Recognition, Automatic: History, in Encyclopedia of Language and Linguistics (K. Brown, ed.). 2nd edition, Elsevier, 2006, pp. 806–819. 527. M. R. Kelly, Some aspects of conservation of quantity and length in Papua/New Guinea in relation to language, sex and years at school. Papua and New Guinea J. Educ. 7, no. 1 (1971), 55–60. 528. G. H. Knight, The geometry of Maori Art—rafter patterns, New Zealand Math. Magazine 21, no. 2 (1984), 36–41. 529. G. Kubik, African space/time concepts with a discussion of possible cross-parallels in music, African Music 6, no. 4 (1987), 53–89. 530. B. Landau, and J. E. Hoffman, Parallels between spatial cognition and spatial language: Evidence from Williams syndrome. J. Memory and Language 53, no. 2 (2005), 163–185. 531. S. C. Levinson, Language and space. Annual Review of Anthropology 25 (1996), 353–382. 532. S. C. Levinson, S. Kita, D. B. M. Haun, and B. H. Rasch, Returning the tables: language affects spatial reasoning, Cognition 84 (2002), 155–188. 533. P. J. Lu and P. J. Steinhardt, Decagonal and quasi-crystalline tilings in medieval Islamic architecture. Science 315 (23 February 2003), 1106–1110. 534. T. Meaney, U. Fairhill, and T. Trinick, The role of language in ethnomathematics, J. Math. and Culture 3, no. 1 (2008). ISSN 1558–5336. 535. J. Mimica, Intimations of Infinity. The Cultural Meaning of the Iqwaye Counting and Number System. Berg Publishers, Oxford, 1992. 536. I. T. Miura, C. C. Kim, C.-M. Chang, and Y. Okamoto, Effects of language characteristics on childrenŠs cognitive representation of number: cross-national comparisons. Child Development 59 no. 6 (1988), 1445–1450. 537. I. T. Miura, Y. Okamoto, C. C. Kim, C-M. Chang, M. Steere, and M. Fayol, Comparisons of children’s cognitive representation of number: China, France, Japan, Korea, Sweden, and the United States. Internat. J. Behavioral Development 77 (1994), 401–411. 538. D. Nelson, G. G. Joseph, and J. Williams, Multicultural Mathematics. Oxford University Press, 1993. 539. K. Owens, The work of Glendon Lean on the counting systems of Papua New Guinea and Oceania, Math. Ed. Research J. 13, no. 1 (2001), 47–71. 540. E. Pederson, E. Danziger, D. Wilkins, S. Levinson, S. Kita, and G. Senft, Semantic typology and spatial conceptualization, Language 74, no. 3 (1998), 557–589. Stable URL: http://www.jstor.org/ stable/417793. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK References 261 541. P. Pica, C. Lemer, V. Izard, and S. Dehaene, Exact and approximate arithmetic in an Amazonian indigene group, Science 306 no. 5695 (2004), 499–503. DOI: 10.1126/science.1102085. 542. D. Pingree, The logic of non-western science: mathematical discoveries in medieval India, Daedalus 132, no. 4 (2003), 45–53. 543. R. Pinxten, I. van Dooren, and E. Soberon, Towards a Navajo Indian Geometry. Fonds voor Kollektief Fundamenteel Onderzoek, Gent, 1987. ISBN-10: 9070963256. ISBN-13: 978-9070963255. 544. A. B. Powell and M. Frankenstein, eds. Ethnomathematics: Challenging Eurocentrism in Mathematics Education State University of New York Press, Albany, NY, 1997. 545. N. C. Presmeg, Ethnomathematics in teacher education, J. Math. Teacher Ed. 1 (1998), 317–339. 546. J. R. Prince, Conceptions of time in New Guinea students. Papua and New Guinea J. Educ. 6, no. 5 (1970), 14–27. 547. T. Roberts, Mathematical registers in Aboriginal languages. For the Learning of Math. 18, no. 1 (1998), 10–16. 548. S. Rowlands and R. Carson, Where would formal, academic mathematics stand in a curriculum informed by ethnomathematics? A critical review. Ed. Stud. Math. 50 (2002), 79–102. 549. S. Rowlands and R. Carson, A comment on Adam, Alangui, and Barton’s “A comment on: Rowlands and Carson ‘Where would formal, academic mathematics stand in a curriculum informed by ethnomathematics? A critical review’ ”. Ed. Stud. Math. 56, nos. 2–3 (2004), 329–342. 550. G. Senft, ed. Referring to Space: Studies in Austronesian and Papuan languages. Clarendon Press, Oxford, 1997. ISBN-10: 0198236476. ISBN-13: 978-0198236474. 551. M. Setati and J. Adler, Between languages and discourses: language practices in primary multilingual mathematics classrooms in South Africa, Ed. Stud. Math. 43, no. 3 (2001), 243–269. 552. G. Siromnoey and R. Siromoney, Rosefeld’s cycle grammars and kolam, in Graph-Grammars and their Application to Computer Science. Third International Workshop, 1986 (H. Ehry et. al., eds.). Springer-Verlag, Berlin: Lect. Notes Comp. Sci. 291 (1987), 565–579. 553. G. Siromoney, R. Siromoney, and K. Krithivasan, Array grammars and kolam, Computer Graphics and Image Processing 3 (1974), 63– 82. MR 353725. 554. G. Siromoney, R. Siromoney and T. Robinsin, Kambi kolam and cycle grammars, in A Perspective in Theoretical Computer Science— Commemorative Volume for Gift Siromoney (R. Narasimhan ed.). Series in Computer Science, World Scientific, Singapore, 1989, pp. 267–300. 555. M. Sternstein, Mathematics and the Dan culture, J. Math. Culture 3, no. 1 (2008). ISSN 1558-5336. 556. U. Sutrop, Estonian Language. The Estonian Institute, Tsallinn, 2004. ISBN 9985-9341-9-9. www.einst.ee/failid/eestikeel. web_1.pdf. Accessed 21 June 2010. 557. R. Thomas, Proto-mathematics and/or real mathematics. For the Learning of Mathematics 16 no. 2 (1996), 11–18. Stable URL: http: //www.jstor.org/stable/40248200. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 262 References 558. U NESCO , Interactions between Linguistic and Mathematical Education. Final report of the Symposium sponsored by UNESCO, CEDO and ICMI. Nairobi, Kenya, Sep, 1–11, 1974. UNESCO document ED-74/CONF.808/COL.11. Paris: UNESCO, 1975. http:// unesdoc.unesco.org/images/0001/000149/014932eb.pdf. 559. R. Vithal and O. Skovsmose, The end of innocence: A critique of ethnomathematics, Ed. Stud. Math. 34 (1997), 131–157. 560. D. K. Washburn and D. W. Crowe, Symmetries of Culture: Theory and Practice of Plane Pattern Analysis. Seattle: University of Washington Press, 1988. ISBN-10: 0295970847. ISBN-13: 9780295970844. 561. K. Yanagisawa and S. Nagata, Fundamental study on design system of Kolam pattern, Forma 22 (2007), 31–46, 562. C. Zaslavsky, Africa Counts: Number and Pattern in African Culture. Prindle, Weber, and Schmidt, Boston, 1973. 563. C. Zaslavsky, World cultures in the mathematics class, For the Learning of Math. 11, no. 2 (1991), 32–35. 564. C. Zaslavsky, ‘Africa counts’ and ethnomathematics, For the Learning of Math. 14, no. 2 (1994), 3–8. 565. R. Zepp, Inclusive disjunction in West African languages, Perceptual and Motor Skills 56 (1983), 322. Mathematics of Kinship One year, at the Christmas party, all the five uncles were there; and among the uncles I include my father. A father is only a specialized kind of uncle anyway. Autobiography of the childhood of Gwen Raverat, daughter of Sir George Darwin, one of the five sons of Charles Darwin. (Quoted from [574].) 566. N. J. Allen, Tetradic theory: An approach to kinship. J. Anthropological Soc. Oxford 17 (1986), 87–109. 567. N. J. Allen, The evolution of kinship terminologies, Lingua 77 (1989), 173–185. 568. N. J. Allen, Assimilation of alternate generations. J. Anthropological Soc. Oxford 20 (1989), 45–55. 569. G. Bennardo and D. Read, Tongan kinship terminology: Insights from an algebraic analysis. Math. Anthropology and Cultural Theory 2, no. 1 (Dec. 2005), 1–52. 570. A. Biersack, Tongan exchange structures: Beyond descent and alliance. J. Polynesian Society 91, no. 2 (1982), 181–212. 571. G. De Meur and A. Gottcheiner, Prescriptive kinship systems, permutations, groups and graphs, Math. Anthropology and Cultural Theory 1, no. 1 (November 2000), 1–12. http://www. mathematicalanthropology.org/mact. 572. G. De Meur and P. Jorion, A possible genealogy of Australian marriage systems, Morphisms in “matrimonial class” systems’, Math. Social Sci. 2, no. 1 (1981), 9–21. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK References 263 573. W. W. Elmendorf, System change in Salish kinship terminologies. Southwestern J. Anthropology 17 (1961), 365–382. 574. S. H. Gould and P.-h. Liu, Fundamentals of kinship theory, Bull. Inst. Ethnology Acad. Sinica 57 (Spring 1984), 157–233. 575. A. Gottcheiner and G. De Meur, Prescriptive kinship systems, permutations, groups and graphs, Math. Anthropology and Cultural Theory 1, no. 1 (2000). http://www.mathematicalanthropology.org. 576. A. Gottcheiner, On some classes of kinship systems I: Abelian systems, Math. Anthropology and Cultural Theory 2, no. 4 (2008). http: //www.mathematicalanthropology.org. 577. L. Haddad and Y. Sureau, Les groups, les hypergroupes et l’énigme des Murngin, J. Pure Appl. Algebra 87(1993), 221–235. 578. P. Hage, Unthinkable categories and the fundamental laws of kinship. American Ethnologist 24 (1997), 652–667. 579. P. Hage and F. Harary, Arapesh sexual symbolism, primitive thought and Boolean groups. L’Homme 23 (1983), 57–77. 580. P. Hage and F. Harary, Structural Models in Anthropology. Cambridge University Press, 1984. ISBN-10: 0521273110. ISBN-13: 9780521273114. 581. P. Hage and F. Harary, Exchange in Oceania: A Graph Theoretic Analysis. Clarendon Press, Cambridge, 1991. ISBN-10: 0198277601. ISBN-13: 978-0198277606. 582. P. Hage and F. Harary, Island Networks: Communication, Kinship, and Classification Structures in Oceania. Clarendon Press, Cambridge, 1996. 583. J. H. T. Harvey and P.-h. Liu, Numerical kinship notation system: Mathematical model of genealogical space, Bull. Inst. Ethnology Acad. Sinica 23 (Spring 1967), 1–22. 584. D. Jenkins, Anthropology, mathematics, kinship: A tribute to the anthropologist Per Hage and his work with the mathematician Frank Harary. Math. Anthropology and Cultural Theory 2, no. 3 (JULY 2008), 1–28. ISSN 1544-5879. 585. P. Jorion, Relations généalogiques et catégories cosmologiques dans le mariage australien (de Howitt à Radcliffe-Brown), Revue Européenne des Sci. Sociales 20, 62 (1982), 63–83. 586. P. J. M. Jorion, Information flows in kinship networks, Math. Anthropology and Cultural Theory 1 no. 1 (2000). http://www. mathematicalanthropology.org/mact. 587. P. Jorion and G. De Meur, La question Murngin, un artefact de la littérature anthropologique, L’Homme 20, no. 2 (1980), 39–70. 588. P. Jorion, G. De Meur, and Trudeke Vuyk, Le mariage Pende, L’Homme 22, no. 1 (1982), 51–71. 589. F. Korn and R. Needham, Permutation models and prescriptive systems: the Tarau case, Man 5 (1970), 393–420. 590. D. Kronenfeld, Particularistic or universalistic analyses of Fanti kinterminology: the alternative goals of terminological analysis. Man (N.S.) 15 (1980), 151–169. 591. K. Kupka and A. Testart, À propos du problème Murngin: le système des sous-sections, L’Homme 22, no. 2 (1980), 71–90. 592. E. Leach, Jinghpaw kinship terminology: An experiment in ethnographic algebra. J. Royal Anthropological Inst. 75 (1945), 59–71. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 264 References 593. M. Leaf, The Punjabi kinship terminology as a semantic system. Amer. Anthropologist 73 (1971), 545–554. 594. F. K. Lehman, The relationship between genealogical and terminological structure in kinship terminologies, J. Quantitative Anthropology 4 (1993), 95–122. 595. C. Levi-Strauss, Les structures élémentaires de la parenté. Mouton, Paris, rééd. 1967. 596. P.-h. Liu, Theory of groups of permutations, matrices and kinship: A critique of mathematical approaches to prescriptive marriage systems, Bull. Inst. Ethnology Acad. Sinica 26 (Autumn 1968), 29–40. 597. G. P. Murdock, Patterns of sibling terminology. Ethnology 7 (1967), 1–24. 598. G. P. Murdock, Kin term patterns and their distribution, Ethnology 9 (1970), 165–208. 599. S. B. Nerlove and K. A. Romney, Sibling terminology and cross-sex behavior. Amer. Anthropologist 69 (1967), 179–187. 600. D. W. Read, An algebraic account of the American kinship terminology. Current Anthropology 25 (1984), 4l7–440. 601. D. W. Read, Formal analysis of kinship terminologies and its relationship to what constitutes kinship. Math. Anthropology and Cultural Theory 1, no. 1 (2000), 1–46. http://www. mathematicalanthropology.org/. 602. D. W. Read, Formal analysis of kinship terminologies and its relationship to what constitutes kinship. Anthropological Theory 1, no. 2 (2001), 239–267. 603. D. W. Read and C. A. Behrens, KAES, an Expert System for the Algebraic Analysis of Kinship Terminologies. J. Quantitative Anthropology 2 (1990), 353–393. 604. D. W. Read and M. D. Fischer, Kinship Algebra Expert System (KAES), 2004. http://kaes.anthrosciences.net/. 605. W. L. Warner, Morphology and functions of the Australian Murngin type of kinship, Part II, Amer. Anthropologist 3, no. 1 (1931), 172– 198. 606. D. R. White and P. Jorion, Representing and computing kinship: A new approach, Current Anthropology 33, no. 4 (1992), 454–463. 607. D. R. White and P. Jorion, Kinship networks and discrete structure theory: applications and implications, Social Networks 18 (1996), 267–314. 608. H. C. White, An Anatomy of Kinship. Prentice-Hall, 1965. Mathematical Education 609. R. Aharoni, What I learnt in primary school. Invited talk given at the BMC, Birmingham, April 2003. http://www.math.technion.ac. il/~ra/birminghamapril2003.doc. Accessed 20 March 2010. 610. R. Aharoni, Arithmetic for Parents. A Book for Grownups about Children’s Mathematics. Sumizdat, 2007. ISBN-10: 0977985253. ISBN-13: 978-0977985258. 611. P. Andrews, The curricular importance of mathematics: a comparison of English and Hungarian teachers’ espoused beliefs, J. Curriculum Stud. 39, no. 3 (2007), 317–338. ISSN 0022–0272. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK References 265 612. L. Aspinwall, K. Shaw, and N. Presmeg, Uncontrollable mental imagery: Graphical connections between a function and its derivative. Ed. Stud. Math. 33 (1997), 301—317. 613. D. L. Ball, C. Hoyles, H. N. Jahnke, and N. Movshovitz-Hadar, The teaching of proof, in Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002), Higher Ed. Press, Beijing, 2002, pp. 907–920. 614. A. J. Baroody, The case of Felicia: a young child’s strategies for reducing memory demands during mental addition, Cognition and Instruction 1 (1984) 109-116. 615. A. J. Baroody, The development of counting strategies for single-digit addition, J. Research in Mathematics Education, 18 (1987), 141–157. 616. A. J. Baroody and K. E. Gannon, The development of the commutativity principle and economical addition strategies. Cognition and Instruction 1 (1984), 321–339. 617. A. J. Baroody, H. P. Ginsburg, and B. Waxman, Children’s use of mathematical structure. J. Research Math. Ed. 14 (1983), 156–168. 618. H. Bass, Mathematics, mathematicians, and mathematics education, Bull. Amer. Math. Soc. 4, no. 4 (2005), 417–430. 619. M. T. Battista, Spatial visualization and gender differences in high school geometry. J. Res. Math. Ed. 21 (1990), 47–60. 620. M. T. Battista, Fifth graders’ enumeration of cubes in 3D arrays: Conceptual progress in an inquiry-based classroom. J. Research Math. Education 30 (1999), 417–449. 621. M. T. Battista, D. H. Clements, J. Arnoff, K. Battista, and C. Borrow, Students’ spatial structuring of 2D arrays of squares. J. Research Math. Education, 29 (1998), 503–532. 622. A. W. Bell, A study of pupils’ proof explanations in mathematical situations, Ed. Stud. Math. 7 (1976), 23–40. 623. A. Bell, E. Fischbein, and B. Greer, Choice of operation in verbal arithmetic problems: the effects of number size, problem structure and context, Ed. Stud. Math. 15, no. 2 (1984), 129–147. Stable URL: http://www.jstor.org/stable/3482245. 624. A. J. Bishop, Use of structural apparatus and spatial ability: A possible relationship. Research in Education 9 (1973), 43—49. 625. A. J. Bishop, Spatial abilities and mathematics education: A review. Ed. Stud. Math. 11 (1980), 257–269. 626. M. R. Boelkins, Removable ambiguities: making mathematics more intelligible, Primus 8 no. 4 (1998), 331–339. DOI: 10.1080/10511979808965908. 627. A. V. Borovik, Boyutsal analiz veya elmalari ınsanlara bülmek, Matematik Dünyası no. 2 (2008), 71–76. 628. G. M. Boulton-Lewis, Young children’s representations and strategies for subtraction. British J. Ed. Psychology 63 (1993), 441–456. 629. G. M. Boulton-Lewis and K. Tait, Young children’s representations and strategies for addition. British J. Ed. Psychology 64 (1994), 231– 242. 630. M. Boylan and H. Povey, Telling stories about mathematics. In Mathematical relationships: Identities and Participation (L. Black, H. Mendick, and Y. Solomon, eds.). Routledge, London, 2009, pp. 47– 57. ISBN-10: 0415996848. ISBN-13: 978-0415996846. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 266 References 631. T. Brown, Hermeneutics and mathematical activity, Ed. Stud. Math. 22, no. 5 (1991), 475–480. Stable URL: http://www.jstor.org/ stable/3482466. 632. L. Burton, Mathematical thinking: The struggle for meaning, J. Research Math. Ed. 15 (1984), 35–49. Stable URL: http://www. jstor.org/stable/748986. 633. L. Burton, The practices of mathematicians: What do they tell us about coming to know mathematics? Ed. Stud. Math. 37 (1999), 121– 143. 634. L. Burton and C. Morgan, Mathematicians writing, J. Research in Math. Ed. 31, no. 4 (2000), 429–453. Stable URL: http://www. jstor.org/stable/749652. 635. J. Cai, Assessing and understanding U.S. and Chinese students’ mathematical thinking. Zentralblatt für Didaktik der Mathematik 34 no. 6 (2002), 278–290. 636. K. H. Canobi, Individual differences in children’s addition and subtraction knowledge, Cognitive Development 19 (2004), 81–93. DOI: 10.1016/j.cogdev.2003.10.001. 637. T. P. Carpenter, J. Hiebert, and J. M. Moser, Problem structure and first-grade children’s initial solution processes for simple addition and subtraction problems, J. Research Math. Ed. 12 (1981), 27–39. 638. T. Carpenter and J. Moser, The acquisition of addition and subtraction concepts in grades one through three, J. Research Math. Ed. 15, no. 3 (1984), 179–202. 639. D. Carraher, B. M. Brizuela and D. Earnest, The reification of additive differences in early algebra: Viva La Diférence! In: Proceedings of the 12th ICMI study conference: The future of the teaching and learning of algebra (H. Chick, K. Stacey, J. Vincent, and J. Vicent, eds.). Melbourne, Australia: The University of Melbourne, 2001, pp. 163–170. 640. E. Carruthers and M. Worthington, Research uncovers children’s creative mathematical thinking, Primary Math. 7, no. 3 (2003), 21–25. 641. E. Carruthers and M. Worthington, Young children exploring early calculation, Math. Teaching 187 (2004), 21–25. ISSN 0025-5785. 642. E. Carruthers and M. Worthington, Children’s Mathematics: Making Marks, Making Meaning. Sage Publications, 2006. ISBN10: 1412922836. ISBN-13: 978-1412922838. 643. E. Carruthers and M. Worthington, Children’s mathematical graphics: understanding the key concept, Primary Math. 13, no. 3 (2009), 3–5. 644. E. Carruthers and M. Worthington, Understanding Children’ Mathematical Graphics: Beginnings in Play. Open University Press, 2011. ISBN-10: 0335237762. ISBN-13: 978-0335237760. 645. Y. Chevallard, La transposition didactique – Du savoir savant au savoir enseigné. La Pensée sauvage, Grenoble, 1985. 646. R. Cowan and S. Biddle, Children’s understanding of one-to-one correspondence in the context of sharing, Ed. Psych. 9, no. 2 (1989), 133– 140. 647. R. Cowan and M. Renton, Do they know what they are doing? Children’s use of economic addition strategies and knowledge of commutativity. Educational Psychology 16 (1996), 407–420. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK References 267 648. C. M. Diezmann and J. J. Watters, Identifying and supporting spatial intelligence in young children, Contemp. Issues in Early Childhood 1, no. 3 (2000), 299–313. 649. P. C. Dowling, Discipline and Mathematise: The myth of relevance in education, Perspectives in Education 16 no. 2 (1995), 209–226. 650. P. C. Dowling, A sociological analysis of school mathematics texts, Ed. Stud. Math. 31 (1996), 389–415. 651. P. C. Dowling, The Sociology of Mathematics Education: Mathematical Myths/Pedagogic Texts. Routledge, 1997. ISBN-10: 0750707917. ISBN-13: 978-0750707916. 652. E. Dubinsky, Constructive aspects of reflective abstraction in advanced mathematics, in Epistemological Foundations of Mathematical Experience (L. P. Steffe, ed.). Springer-Verlag, New York, 1991, pp. 160–187. 653. E. Dubinsky, K. Weller, M. McDonald, and A. Brown, Some historical issues and paradoxes regarding the infinity concept: an APOS analysis, Part 1, Ed. Stud. Math. 58 (2005), 335–359. 654. E. Dubinsky, K. Weller, M. McDonald, and A. Brown, Some historical issues and paradoxes regarding the infinity concept: an APOS analysis, Part 2, Ed. Stud. Math. 60 (2005), 253–266. 655. T. Eisenberg, On understanding the reluctance to visualize, Zentralblatt für Didaktik der Mathematik, 26, no. 4 (1994), 109–113. 656. T. Eisenberg, Prime knowledge about primes, Teaching Mathematics and its Applications 26 no. 2 (2007), 56–72. DOI: 10.1093/teamat/hrl018. 657. R. Ely, Nonstandard student conceptions about infinitesimals. J. Res. Math. Ed. no. 2 (2010) 117–146. 658. L. D. English, Young children’s combinatoric strategies, Ed. Stud. Math. 22, no. 5 (1991), 451–474. 659. L. D. English, Children’s strategies in solving two- and threedimensional combinatorial problems, J. Research Math. Ed. 24, no. 3 (1993), 255–273. Stable URL: http://www.jstor.org/stable/ 3482465. 660. P. Ernest, Mathematical induction: a pedagogical discussion, Ed. Stud. Math. 15, no. 2 (1984), 173–189. 661. P. Ernest, The Psychology of Learning Mathematics: The Cognitive, Affective and Contextual Domains of Mathematics Education, LAP LAMBERT Academic Publishing, 2011, 164 pp. ISBN10: 3844313060. ISBN-13: 978-3844313062. 662. P. Ernest, Mathematics and Special Educational Needs: Theories of Mathematical Ability and Effective Types of Intervention with Low and High Attainers in Mathematics, LAP LAMBERT Academic Publishing, 2011, 140 pp. ISBN-10: 3844322272. ISBN-13: 978-3844322279. 663. W. V. Fabricius, C. Sophian, and H. M. Wellman, Young children’s sensitivity to logical necessity in their inferential search behaviour, Child Development 58 (1987), 409–423. 664. L. M. Flevares and M. Perry, How many do you see? The use of nonspoken representations in first-grade mathematics lessons. J. Ed. Psych. 93 (2001), 330–345. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 268 References 665. H. Freudenthal, Geometry between the devil and the deep sea. Educational Studies in Mathematics 3 (1971), 413–435. 666. H. Freudenthal, Mathematics as an Educational Task. Reidel, Dordrecht, 1973. 667. H. Freudenthal, Didactical Phenomenology of Mathematical Structures. Springer, 1986. ISBN-13: 978-9027722614. 668. H. Freudenthal, Revisiting Mathematics Education. Kluwer, 1991. 669. K. C. Fuson, D. Wearne, J. C. Hiebert, H. G. Murray, P. G. Human, A. I. Olivier, T. P. Carpenter, and E. Fennema, Children’s conceptual structures for multidigit numbers and methods of multidigit addition and subtraction, J. Research Math. Ed. 28, no. 2 (1997), 130–162. Stable URL: http://www.jstor.org/stable/749759. 670. Équipe Galion, Mathématiques 6e. Programme 1977. OcdlHatier, 1980. 671. A. Gardiner, Mathematical Puzzling, UK Mathematics Foundation, Birmingham, 1996. ISBN 0-7044-17545. 672. A. Gardiner, Discovering Mathematics. The Art of Investigation. Dover Publications, Inc., Mineola, NY, 2007. xiv+206 pp. ISBN: 0-486-45299-9. MR2351062 673. A. Gardiner, Understanding Infinity. The Mathematics of Infinite Processes. Dover Publications, Inc., Mineola, NY, 2002. x+308 pp. ISBN: 0-486-42538-X. 674. R. Gelman, E. Meck, and S. Merkin, Young children’s numerical competence, Cognitive Development 1 (1986), 1–29. 675. H. P. Ginsburg, Children’s Arithmetic: How They Learn It and How You Teach It. Pro-Ed, 1989. ISBN-10: 0890791813. ISBN-13: 978-0890791813. 676. I. Grattan-Guinness, Not from nowhere. History and philosophy behind mathematical education. Int. J. Math. Ed. Sci. Tech. 4 (1973), 421–453. 677. E. Gray and D. Tall, Duality, ambiguity and flexibility: A proceptual view of simple arithmetic. J. Research Math. Ed. 26, no. 2 (1994), 115–141. 678. E. Gray, D. Pitta, and D Tall. Objects, actions, and images: A perspective on early number development. J. Mathematical Behavior, 18 (2000), 401–413. 679. H.B. Griffiths, Mathematics: Society and Curricula. Cambridge University Press, 1974. ISBN-10: 0521202876. ISBN-13: 9780521202879. 680. G. J. Groen and L. B. Resnick, Can preschool children invent addition algorithms? J. Educational Psychology 69 (1977), 645–652. 681. D. F. Halpern, C. P. Benbow, D. C. Geary, R. C. Gur, J. S. Hyde, and M. A. Gernbacher, The science of sex differences in science and mathematics, Psychological Sci. in the Public Interest 8 (2007), 1–51. 682. G. Hanna, H. N. Jahnke, Y. DeBruyna, and D. Lomas. Teaching mathematical proofs that rely on ideas from physics. Canadian J. Science, Mathematics and Technology Education 1 no. 2 (2001) 183–192. DOI: 10.1080/14926150109556460. 683. G. Harel and D. Tall, The general, the abstract and the generic in advanced mathematics, For the Learning of Math. 11, no. 1 (1991), 38–42. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK References 269 684. M. Hegarty and M. Kozhevnikov, Types of visual-spatial representations and mathematical problem solving, J. Ed. Psych. 91, no. 4 (1999), 684–689. 685. R. Hersh, Proof is convincing and explaining, Ed. Stud. Math. 24 (1993), 389–399. 686. R. Hersh, Independent thinking, College Math. J. 34, no. 2 (2003), 112–115. 687. R. Hershkowitz, Visualization in geometry: Two sides of the coin. Focus on Learning Problems in Mathematics 11 (1989) 61–76. 688. J. Hiebert, The position of the unknown set and children’s solutions of verbal arithmetic problems, J. Res. Math. Ed. 13, no. 5 (1982), 341– 349. Stable URL: http://www.jstor.org/stable/749008. 689. J. Hiebert and D. Wearne, Instruction, understanding and skill in multidigit addition and subtraction. Cognition and Instruction, 14 (1996), 251–283. 690. J. Hoffman, C. Johnson, A. Logg, Dreams of Calculus: Perspectives on Mathematics Education. 2004. http://www. bodysoulmath.org/books/dreams-of-calculus.pdf. 691. D. Holton, A. Ahmed, H. Williams, and C. Hill, On the importance of mathematical play, Internat. J. Math. Ed. Sci. Technology 32, no. 3 (2001), 43–56. 692. R. Howe and S. S. Epp, Taking place value seriously: arithmetic, estimation, and algebra. http://www.maa.org/pmet/resources/ PVHoweEpp-Nov2008.pdf. Accessed 17 July 2011. 693. R. Howe, Editorial: The greatest calamity in the history of science, ICMI News 18 (June 2011), http://www.mathunion.org/ pipermail/icmi-news/2011-July.txt. Accessed 17 July 2011. 694. C. Hoyles, The curricular shaping of students’ approaches to proof. For the Learning of Mathematics 17 no. 1 (1997) 7–16. 695. C. Hoyles and R. Noss (eds.), Learning Mathematics and Logo. MIT Press, 1992. ISBN-10: 0262082071. ISBN-13: 978-0262082075. 696. M. Hughes, Can preschool children add and subtract?, Educational Psychology 1 (1981), 207–220. 697. M. Hughes, Children and Number: Difficulties in Learning Mathematics. Wiley-Blackwell, 1986. ISBN-10: 0631135812. ISBN13: 978-0631135814. 698. F. Ilg and L. B. Ames, Developmental trends in arithmetic, Journal of Genetic Psychology 79 (1951), 3–28. 699. L. S. Kanevsky, Pursuing qualitative differences in the flexible use of a problem solving strategy by young children. J. Ed. Gifted 13 (1990), 115–140. 700. J. J. Kaput, D. W. Carraher, and M L. Blanton (eds.), Algebra in the Early Grades (Studies in Mathematical Thinking and Learning Series). Routledge, London, 2007. 552 pp. ISBN-10: 0805854738. ISBN-13: 978-0805854732. 701. A. Khait, The definition of mathematics: philosophical and pedagogical aspects, Sci. Ed. 14 (2005), 137–159. 702. C. Kieran, Concepts associated with the equality symbol, Ed. Stud. Math. 12 (1981), 317–326. 703. A. P. Kiselev. Elementary Geometry for Secondary Education Establishments. Moscow, 1892. (In Russian.) S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 270 References 704. S. S. Kutateladze, Nomination and definition, http://www.math. nsc.ru/LBRT/g2/english/ssk/nomination_e.html. 705. P. E. Langford, A longitudinal study of children’s understanding of logical laws in arithmetic and Boolean algebra, Educational Psychology 1 (1981), 119–140. 706. C. Lean and M. A. Clements, Spatial ability, visual imagery, and mathematical performance, Ed. Stud. Math. 12 (1981), 267–299. 707. D. Laugwitz, Motivations and linear algebra, Ed. Stud. Math. 5 (1974), 243–254. 708. S. Lerman, Problem-solving or knowledge centered: The influence of philosophy on mathematics teaching. International J. Mathematics Education, 14 no. 1 (1983), 59–66. 709. S. Lerman, Intersubjectivity in mathematics learning: a challenge to the radical constructivist paradigm? J. Research Math. Ed. 27, no. 2 (1996), 133–150. 710. G. Leversha, Crossing the Bridge. United Kingdom Mathematics Trust, 2008. ISBN 978-1-906001-06-3. 711. P. Liebeck, How Children Learn Mathematics. A Guide for Parents and Teachers. Penguin Books, 1990. ISBN 0-14-013488-3. 712. P. Liebeck, Scores and forfeits: an intuitive model for integer arithmetic. Ed. Stud. Math. 21 no. 3 (1990), 221–239. 713. P. Liljedahl and B. Sriraman, Musings on mathematical creativity, For the Learning of Math. 26, no. 1 (2006), 20–23. 714. R. Lins, Characterising the mathematics of the mathematics teacher from the point of view of meaning production, International Congress on Mathematical Education, Copenhagen, Denmark. July 4–11, 2004. 715. C. A. Maher and A. M. Martino, The development of the idea of mathematical proof: A 5-year case study. J. Research in Mathematics Education 27 no. 2 (1996), 194–214. 716. E. M. Markman, Realizing that you don’t understand: Elementary school children’s awareness of inconsistencies, Child Development 50 (1979), 643–645. 717. J. Martínez-Torregrosa, R. López-Gay, and A. Gras-Martí, Mathematics in physics education: scanning historical evolution of the differential to find a more appropriate model for teaching differential calculus in physics, Sci. Ed. 15 (2006), 447–462. DOI 10.1007/s11191-0050258-y. 718. J. Mason and D. Pimm, Generic examples: Seeing the general in the particular. Ed. Stud. Math. 15 (1984), 227–289. 719. G. McColm, A metaphor for mathematical education, Notices Amer. Math. Soc. 54, no. 4 (2007), 499–502. 720. H. Mendick, Masculinities in Mathematics. Open University Press, 2006. ISBN-10: 033521827X. ISBN-13: 978-0335218271. 721. J. Monaghan, Young peoples’ ideas of infinity. Ed. Stud. Math. 48, no. 2–3 (2001), 239–257. 722. M. Montessori, The Montessori Method (Anne Everett George, tranls.). Frederick A. Stokes Company, New York, 1912. 723. J. T. Mulligan and M. C. Mitchelmore, Young children’s intuitive models of multiplication and division. J. Research Math. Education, 28 (1997), 309–331. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK References 271 724. P. Nesher, Three determinants of difficulty in verbal arithmetic problems, Ed. Stud. Math. 7 (1976), 369–388. 725. T. Nunes, D. Desli, and D. Bell, The development of children’s understanding of intensive quantities, Internat. J. Ed. Research 39 (2003), 651–675. 726. T. Nunes, P. Bryant, D. Hallett, and D. Evans, Teaching children about the inverse relation between addition and subtraction, Math. Thinking and Learning 11, no. 1 & 2 (2009), 61–78. 727. L. Outhred and M. C. Mitchelmore, Young children’s intuitive understanding of rectangular area measurement. J. Research Math. Education 31 (2000), 144–168. 728. S. A. Papert, Mindstorms: Children, Computers, and Powerful Ideas. Perseus Books, 1993. ISBN-10: 0465046746. ISBN-13: 9780465046744. 729. N. C. Presmeg, Visualization in high school mathematics, For the Learning of Math. 63 (1986), 42–46. 730. N. C. Presmeg, Prototypes, metaphors, metonymies, and imaginative rationality in high school mathematics, Ed. Stud. Math. 23 (1992), 595–610. 731. R. T. Putnam, L. U. de Bettencourt, and G. Leinhardt, Understanding of derived-fact strategies in addition and subtraction. Cognition and Instruction 7 (1990), 245–285. 732. H. Radatz, Error analysis in mathematics education, J. Research Math. Ed. 10, no. 3 (1979), 163–172. Stable URL: http://www. jstor.org/stable/748804. 733. L. Radford, The seen, the spoken and the written. A semiotic approach to the problem of objectification of mathematical knowledge. For the Learning of Mathematics, 22 no. 2 (2002), 14–23. 734. L. Radford, Algebra as tekhne. Artefacts, symbols and equations in the classroom. Mediterranean J. Research in Math. Education 1 no. 1 (2002), 31–56. 735. L. Radford, Gestures, speech and the sprouting of signs, Mathematical Thinking and Learning, 5 no. 1 (2003), 37–70. 736. L. Radford, The semiotics of the schema. Kant, Piaget, and the calculator, in Activity and Sign—Grounding Mathematics Education. Festschrift for Michael Otte (M. H. G. Hoffmann, J. Lenhard, and F. Seeger, eds.). Springer, 2005, pp. 137–152. 737. L. Radford, The anthropology of meaning, Ed. Stud. Math. 61 (2006), 39–65. 738. A. Ralston, Research mathematicians and mathematics education: a critique, Notices Amer. Math. Soc. 51, no. 4 (2004), 403–411. 739. S. Restivo and W. K. Bauchspies, The will to mathematics: Minds, morals, and numbers. Foundations Sci. 11, nos. 1–2 (2006), 197–215. ISSN 1233-1821. 740. B. Rittle Johnson and M. W. Alibali, Conceptual and procedural knowledge of mathematics: Does one lead to the other? J. Ed. Psych. 91 (1999), 175–189. 741. B. Rittle Johnson, R. S. Siegler, and M. W. Alibali, Developing conceptual understanding and procedural skill in mathematics: An iterative process. J. Ed. Psych. 93 (2001), 346–362. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 272 References 742. S. A. Rose and M. Blank, The potency of context in children’s cognition: an illustration through conservation, Child Development 45 (1974), 499–502. 743. D. J. Rosenthal and L. B. Resnick, Children’s solution processes in arithmetic word problems, J. Ed. Psych. 66 (1974), 817–825. 744. W.-M. Roth and M. G. Bowen, Professionals Read Graphs: A Semiotic Analysis. J. Research in Mathematics Education 32 no. 2 (2001), 159– 194. 745. W.-M. Roth and J. S. Thom, Bodily experience and mathematical conceptions: from classical views to a phenomenological reconceptualization. Educational Studies in Mathematics 70 (2009), 175–189. 746. A. H. Schoenfeld, When good teaching leads to bad results: The disaster of well taught mathematics courses, Educational Psychologist 23 no. 2 (1988), 145–166. 747. A. Sfard, On the dual nature of mathematical conceptions: reflections on processes and objects as different sides of the same coin, Ed. Stud. Math. 22 (1991), 1–36. 748. A. Sfard, Reification as the birth of metaphor. For the learning of Mathematics 14 no. 1 (1994), 44–55. 749. A. Sfard, The gains and the pitfalls of reification—The case of algebra. Ed. Stud. Math. 26, nos. 2–3 (1994), 191–228. DOI: 10.1007/BF01273663. 750. A. Sfard, The many faces of mathematics: do mathematicians and researchers in mathematics education speak about the same thing? Mathematics Education as a Research Domain: a Search for Identity, Book 1, 2, 491–511, New ICMI Stud. Ser., 4, Kluwer Acad. Publ., Dordrecht, 1998, pp. 491–511. 751. A. Sfard, A mathematician’s view of research in mathematics education: an interview with Shimshon A. Amitsur. Mathematics Education as a Research Domain: a Search for Identity, Book 1, 2, New ICMI Stud. Ser., 4, Kluwer Acad. Publ., Dordrecht, 1998, pp. 445–458. 752. A. Sfard, On two metaphors for learning and the dangers of choosing just one, Ed. Researcher 27, no. 2 (1998), 4–13. DOI: 10.3102/0013189X027002004. 753. A. Sfard, Thinking as Communicating: Human Development, the Growth of Discourses, and Mathematizing. Cambridge University Press, 2010. ISBN-10: 0521161541. ISBN-13: 9780521161541. 754. S. Sheffield and K. Gallagher, Math and Nonfiction. Grades 3–5. Math Solutions Publications, 2004. ISBN 0-941355-62-4. 755. C. Shen and H. P. Tam, The paradoxical relationship between student achievement and self-perception: a cross-national analysis based on three waves of TIMSS data. Educational Research and Evaluation, 14 no. 1 (2008), 87–100. 756. R. S. Siegler, Individual differences in strategy choices: good students, not-so-good students, and perfectionists, Child Development 59 (1988), 833–851. 757. R. S. Siegler, Hazards of mental chronometry: An example from children’s subtraction. J. Ed. Psych. 18 (1989), 497–506. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK References 273 758. B. Sriraman, Can mathematical discovery fill the existential void? The use of conjecture, proof and refutation in a high school classroom. Math. in School 32, no. 2 (2003), 2–6. 759. B. Sriraman, Discovering a mathematical principle: The case of Matt. Mathematics in School (UK) 3 no. 2 (2004), 25–31. 760. B. Sriraman, Reflective abstraction, uniframes and the formulation of generalizations, J. Math. Behavior 23, no. 2 (2004), 205–222. 761. B. Sriraman, The characteristics of mathematical creativity, ZDM Math. Ed. 41, nos. 1–2 (2009), 13–27. DOI: 10.1007/s11858-008-0114z. 762. L. P. Steffe and P. W. Thompson, Interaction or intersubjectivity? A reply to Lerman. J. Research Math. Ed. 31, no. 2 (2000), 191–209. 763. R. M. Steinberg, Instruction of derived facts strategies in addition and subtraction. J. Research Math. Ed. 16 (1985), 337–358. 764. P. Suppes and F. Binford, Experimental teaching of mathematical logic in the elementary school, The Arithmetic Teacher 12 (1965), 187–195. 765. D. Tall, The notion of infinite measuring number and its relevance in the intuition of infinity, Ed. Stud. Math. 11, no. 3 (1980), 271–284. 766. D. Tall, Intuitions of infinity, Math. in School 10 (1981), 30–33. 767. D. Tall, Infinitesimals constructed algebraically and interpreted geometrically, Mathematical Education for Teaching 4 no. 1 (1981), 34– 53. 768. D. Tall, Comments on the difficulty of and validity of various approaches to the calculus. For the Learning of Mathematics 2 no. 2 (1981), 16–21. 769. D. Tall, Elementary axioms and pictures for infinitesimal calculus, Bull. Inst. Math. Appl. 18, nos. 3–4 (1982), 43–48. 770. D. Tall, Understanding the processes of advanced mathematical thinking, Enseign. Math. (2) 42, nos. 3–4 (1996), 395–415. 771. D. Tall, Cognitive development in advanced mathematics using technology. Math. Ed. Res. J. 12 no. 3 (2000), 210–230. 772. D. Tall, Natural and formal infinities, Ed. Stud. Math. 48, no. 2 & 3 (2001), 199–238. 773. D. Tall, A child thinking about infinity, J. Math. Behavior 20, no. 1 (2001), 7–19. DOI: 10.1016/S0732-3123(01)00058-X. 774. D. Tall, A theory of mathematical growth through embodiment, symbolism and proof. Annales de Didactique et de Sciences Cognitives, IREM de Strasbourg. 11 (2006), 195–215. 775. D. Tall, How Humans Learn to Think Mathematically. Princeton University Press, 2010. 776. D. Tall, E. Gray, M. Bin Ali, L. Crowley, P. DeMarois, M. McGowen, D. Pitta, M. Pinto, M. Thomas, and Yudariah Yusof, Symbols and the bifurcation between procedural and conceptual thinking, Canadian J. Science, Mathematics and Technology Education 1 (2001), 81–104. 777. D. O. Tall, M. O. J. Thomas, G. Davis, E. M. Gray, and A. P. Simpson, What is the object of the encapsulation of a process? J. Math. Behavior 18, no. 2 (2000), 1–19. 778. D. Tall and R. L. E. Schwarzenberger, Conflicts in the learning of real numbers and limits, Math. Teaching 82 (1978), 44–49. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 274 References 779. D. Tall and S. Vinner, Concept image and concept definition with particular references to limits and continuity, Ed. Stud. Math. 12 (1981), 151–169. 780. I. Thompson (ed.), Teaching and Learning Early Number. Open University Press, 1997. ISBN-10: 0335198511. ISBN-13: 9780335198511. 781. W. P. Thurston, Mathematical education. Notices Amer. Math. Soc. 37 (1990), 844–850. 782. P. Tsamir and D. Tirosh, Consistency and representations: The case of actual infinity. J. Res. Math. Ed. 30 (1999), 213–219. 783. B. Van de Rijt, R. Godfrey, C. Aubrey, J. E. H. v. Luit, P. Ghesquiere, J. Torbeyns, et al. The development of early numeracy in Europe. J. Early Childhood Research 1 no. 2 (2003), 155–180. 784. J. Vincent and B. McCrae, Mechanical linkages, dynamic geometry software and mathematical proof. Australian Senior Mathematics Journal 15 no. 1 (2001), 56–63. 785. J. Vlassis, Making sense of the minus sign or becoming flexible in ‘negativity’, Learning and Instruction 14 (2004), 469–484. 786. K. Weller, A. Brown, E. Dubinsky, M. McDonald, and C. Stenger, Intimations of infinity. Notices Amer. Math. Soc. 51, no. 7 (2004), 741– 750. 787. H.-H. Wu, The role of open-ended problems in mathematics education, J. Math. Behavior 13 (1994), 115–128. 788. H.-H. Wu, Review of three books by Gelfand et al. Math. Intelligencer 17 (1995), 68–75. 789. H.-H. Wu, The mathematician and the mathematics education reform, Notices Amer. Math. Soc. 43 (1996), 1531–1537. 790. H.-H. Wu, The role of Euclidean geometry in high school, J. Math. Behavior 15 (1996), 221–237. 791. H.-H. Wu, The Mathematics Education Reform: Why you should be concerned and what you can do, Amer. Math. Monthly 10, no. 4 (1997), 946–954. 792. H.-H. Wu, Basic skills versus conceptual understanding: A bogus dichotomy in mathematics education, Amer. Educator 23, no. 3 (1999), 14–19, 50–52. http://www.aft.org/pubs-reports/american_ educator/fall99/index.html. 793. H.-H. Wu, What’s sophisticated about elementary mathematics? Amer. Educator 33, no. 3 (2009), 4–14. http: //www.aft.org/pubs-reports/american_educator/issues/ fall2009/wu.pdf. 794. R. Zazkis and S. R. Campbell, Prime decomposition: Understanding uniqueness. J. Math. Behavior 15 (2) (1996), 207–218. 795. R. Zazkis and E. J. Chernoff, What makes a counterexample exemplary? Educ. Stud. Math. 68 (2008), 195–208. DOI 10.1007/s10649007-9110-4. 796. R. Zazkis and P. Liljedahl, Understanding primes: The role of representation. J. Res. in Math. Ed. 35 no. 3 (2004), 164–186. 797. A. Zvonkin, Mathematics for little ones, J. Math. Behavior 11, no. 2 (1992), 207–219.
798. A. Zvonkin, Children and 52 , J. Math. Behavior 12, no. 2 (1993), 141– 152. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK References 275 “Mathematicaly Gifted” Children 799. T. Andreescu, J. A. Gallian, J. M. Kane, and J. E. Mertz, Crosscultural analysis of students with exceptional talent in mathematical problem solving, Notices Amer. Math. Soc. 55, no. 10 (2008), 1249– 1260. 800. C. P. Benbow, Sex differences in mathematical reasoning ability among the intellectually talented: Their characterization, consequences, and possible causes. Behavioral and Brain Sci. 11 (1988), 169–232. 801. C. P. Benbow, Academic achievement in mathematics and science of students between ages 13 and 23: Are there differences among students in the top one percent of mathematical ability? J. Ed. Psych. 84 (1992), 51–61. 802. C. P. Benbow and O. Arjmand, Predictors of high academic achievement in mathematics and science by mathematically talented students: A longitudinal study. J. Ed. Psych. 82, no. 3 (1990), 430–441. 803. C. P. Benbow and R. M. Benbow, Biological correlates of high mathematical reasoning ability, Progress in Brain Research 61 (1984), 469– 490. 804. L. L. Chang, Who are the mathematically gifted elementary school children? Roeper Review 8 (1985), 76–79. 805. M. A. Clements, Terence Tao, Ed. Stud. Math. 15 (1984), 213–238. 806. S. Damarin, The mathematically able as a marked category, Gender and Ed. 12, no. 1 (2000), 69–85. 807. V. J. Dark and C. P. Benbow, Enhanced problem translation and short-term memory: Components of mathematical talent. J. Ed. Psych. 82, no. 3 (1990), 420–429. 808. C. M. Diezmann, Capitalising on the zeitgeist for mathematically gifted students, Australasian J. Gifted Ed. 11, no. 2 (2002), 5–10. 809. C. M. Diezmann and J. J. Watters, Two faces of mathematical giftedness, Teaching Math. 21, no. 2 (1996), 22–25. 810. C. M. Diezmann and J. J. Watters, Catering for mathematically gifted elementary students: Learning from challenging tasks, Gifted Child Today 23, no. 4 (2000), 14–19, 52. 811. C. Diezmann and J. Watters, The collaboration of mathematically gifted students on challenging tasks, J. Ed. Gifted 25, no. 1 (2001), 7–31. 812. C. Diezmann and J. Watters, The importance of challenging tasks for mathematically gifted students, Gifted and Talented International 17, no. 2 (2003), 76–84. 813. N. F. Ellerton, Children’s made-up mathematics problems—a new perspective on talented mathematicians, Ed. Stud. Math. 17 (1986), 261–271. 814. G. H. Gear, Accuracy of teacher judgement in identifying intellectually gifted children: A review of the literature, Gifted Child Quart. 20, no. 4 (1976), 478–489. 815. E. L. Grigorenko, L. Jarvin, Mei Tan, and R. J. Sternberg, Something new in the garden: Assessing creativity in academic domains (2008). Scientific Commons, http://www.psychologie-aktuell. com/fileadmin/download/PschologyScience/2-2008/14_ Grigorenko.pdf. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 276 References 816. C. Greenes, Identifying the gifted student in mathematics, Arithmetic Teacher, 28 no. 6 (1981), 14–17. 817. N. Hall, Teaching primary school students talented in mathematics, Australasian J. Gifted Ed. 6, no. 1 (1997), 21–26. 818. M. K. Heid, Characteristics and special needs of the gifted student in mathematics, Math. Teacher 76 (1983), 221–226. 819. B. Hermelin and N. O’Connor, Spatial representations in mathematically and in artistically gifted children, British J. Ed. Psych. 56, no. 2 (1986), 150–157. 820. E. Hong and Y. Aqui, Cognitive and motivational characteristics of adolescents gifted in mathematics: Comparisons among students with different types of giftedness, Gifted Child Quart. 48, no. 3 (2004), 191–201. 821. M. L. Johnson, Identifying and teaching mathematically gifted elementary school children, Arithmetic Teacher 30, no. 5 (1983), 25–26, 55–56. 822. D. P. Keating, The study of mathematically precocious youth, J. Special Ed. 9, no. 1 (1975), 45–62. 823. D. P. Keating and J. C. Stanley, Extreme measures for the exceptionally gifted in mathematics and science, Ed. Researcher 1, no. 9 (1972), 3–7. DOI: 10.3102/0013189X001009003. 824. B. V. Kissane, The nature, development and nurture of mathematical talent, Ed. Research and Perspectives 10, no. 1 (1983), 40–58. 825. B. V. Kissane, Selection of mathematically talented students, Ed. Stud. Math. 17, no. 3 (1986), 221–241. Stable URL: http://www. jstor.org/stable/3482226. 826. V. A. Krutetskii, The Psychology of Mathematical Abilities in Schoolchildren. (Translated from Russian by J. Teller, edited by J. Kilpatrick and I. Wirszup.) The University of Chicago Press, 1976. ISBN 0-226-45485-1. 827. T. Lowrie, Visual and spatial problem-solving tasks to extend talented mathematicians in the regular classroom. Australasian J. Gifted Ed. 4, no. 1 (1995), 12–15. 828. D. Lubinski and C. P. Benbow, Study of mathematically precocious youth after 35 years: Uncovering antecedents for the development of math-science expertise. Perspectives Psych. Sci. 1 (2006), 316–345. 829. D. Lubinski and L. G. Humphreys, A broadly based analysis of mathematical giftedness, Intelligence 14 (1990), 327–355. 830. D. Lubinski and L. G. Humphreys, Some bodily and medical correlates of mathematical giftedness and commensurable levels of socioeconomic status, Intelligence 16 (1992), 99–115. 831. D. Lubinski, D. B. Schmidt, and C. P. Benbow, A 20-year stability analysis of the Study of Values for intellectually gifted individuals from adolescence to adulthood, J. Appl. Psych. 81 (1996), 443–451. 832. D. Lubinski, R. M. Webb, M. J. Morelock, and C. P. Benbow, Top 1 in 10,000: A 10-year follow-up of the profoundly gifted. J. Appl. Psych. 86 (2001), 718–729. 833. A. E. Lupkowski-Shoplik and S. G. Assouline, Evidence of extreme mathematical precocity: Case studies of talented youths. Roeper Review 16, no. 3 (1994), 144–151. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK References 277 834. D. T. E. Majoram, Teaching able mathematicians, Gifted Ed. Internat. 8, no. 10 (1992), 40–43. 835. M. C. Muratori, J. C. Stanley, M. U. M. Gross, L. Ng, T. Tao, J. Ng, and B. Tao, Insights from SMPY’s greatest former prodigies: Drs. Terence (“Terry”) Tao and Lenhard (“Lenny”) Ng reflect on their talent development. Gifted Child Quart. 50 (2006), 307–324. 836. M. W. O’Boyle, H. S. Gill, C. P. Benbow, and J. E. Alexander, Concurrent finger-tapping in mathematically gifted males: Evidence for enhanced right hemisphere involvement during linguistic processing. Cortex 30, no. 3 (1994), 519–526. 837. F. Pajares, Self-efficacy beliefs and mathematical problem-solving of gifted students, Contemp. Ed. Psych. 21 (1996), 325–344. 838. W. D. Parker, Psychological adjustment in mathematically gifted students, Gifted Child Quart. 40, no. 3 (1996), 154–157. DOI: 10.1177/001698629604000306. 839. N. C. Presmeg, Visualization and mathematical giftedness, Ed. Stud. Math. 17 (1986), 297–311. 840. T. M. Richardson and C. P. Benbow, Long-term effects of acceleration on the social-emotional adjustment of mathematically precocious youths, J. Ed. Psych. 82, no. 3 (1990), 464–470. 841. N. M. Robinson, R. D. Abbott, V. W. Berninger, and J. Busse, The structure of abilities in mathematically precocious young children: Gender similarities and differences. J. Ed. Psych. 88 (1996), 341–352. 842. N. M. Robinson, R. D. Abbott, V. W. Berninger, J. Busse, and S. Mukhopadhyah, Developmental changes in mathematically precocious young children, Gifted Child Quart. 41 (1997), 145–158. 843. D. L. Shea, D. Lubinski, and C. P. Benbow, Importance of assessing spatial ability in intellectually talented young adolescents: A 20-year longitudinal study. J. Ed. Psych. 93 (2001), 604–614. 844. E. J. Sowell, A. J. Zeigler, L. Bergwall, and R. M. Cartwright, Identification and description of mathematically gifted students: a review of empirical research, Gifted Child Quart. 34, no. 4 (1990), 147–154. DOI: 10.1177/001698629003400404. 845. P. Span and R. Overtoom-Corsmit, Information processing by intellectually gifted pupils solving mathematical problems, Ed. Stud. Math. 17, no. 3 (1986), 273–295. Stable URL: http://www.jstor. org/stable/3482229. 846. B. Sriraman, How do mathematically gifted students abstract and generalize mathematical concepts? NAGC 2002 Research Briefs, 16 (2002), 83–87. 847. B. Sriraman, The characteristics of mathematical creativity, Math. Educator 14, no. 1 (2003), 19–34. 848. B. Sriraman, Mathematical giftedness, problem solving, and the ability to formulate generalizations, J. Secondary Gifted Ed. 14 (2003), 151–165. 849. B. Sriraman, Gifted ninth graders’ notions of proof: investigating parallels in approaches of mathematically gifted students and professional mathematicians, J. Ed. of the Gifted 27, no. 4 (2004), 267–292. DOI: 10.4219/jeg-2004-317. 850. B. Sriraman, Are giftedness and creativity synonyms in mathematics? J. Secondary Gifted Ed. 17, no. 1 (2005), 20–36. DOI: 10.4219/jsge-2005-389. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 278 References 851. R. J. Sternberg, M. Ferrari, P. Clinkenbeard, and E. L. Grigorenko, Identification, instruction, and assessment of gifted children: A construct validation of a triarchic model. Gifted Child Quart. 40, no. 3 (1996), 129–137. DOI: 10.1177/001698629604000303. 852. R. J. Sternberg and E. L. Grigorenko, Are cognitive styles still in style?. Amer. Psychologist 52, no. 7 (1997), 700–712. 853. B. Tao, Parental involvement in gifted education, Ed. Stud. Math. 17, no. 3 (1986), 313–321. 854. D. A. Treffert, Extraordinary People: Understanding ‘Idiot Savants’. Harper and Row, New York, 1989. 855. H. Wagner and B. Zimmermann, Identification and fostering of mathematically gifted students, Ed. Stud. Math. 17, no. 3 (1986), 243–260. Stable URL: http://www.jstor.org/stable/3482227. 856. R. M. Webb, D. Lubinski, and C. P. Benbow, Mathematically facile adolescents with math-science aspirations: New perspectives on their educational and vocational development. J. Ed. Psych. 94 (2002), 785–794. 857. R. M. Webb, D. Lubinski, and C. P. Benbow, Spatial ability: a neglected dimension in talent searches for intellectually precocious youth, J. Ed. Psych. 99, No. 2 (2007), 397–420. DOI: 10.1037/00220663.99.2.397 858. E. Winner, Gifted Children: Myths and Realities. Basic Books, 1996. ISBN 0-465-01759-2. Miscellaneous 859. C. Aldred, Egyptian Art. Thames and Hudson, 1980. 860. Bill the Lizard, Six visual proofs, http://www.billthelizard. com/2009/07/six-visual-proofs_25.html. Accessed 22 July 2010. 861. A. Huxley, Views of Holland, in Essays in Modern Thought (Thomas R. Cook, ed.), Ayer Co Pub; Facsimile edition (June 1935), pp. 97–106. ISBN-10: 0836903323. ISBN-13: 978-0836903324. 862. G. W. Koch, S. C. Sillett, G. M. Jennings and S. D. Davis, The limits to tree height, Nature 428 (22 Apr 2004), 851–854. 863. B. Mellor, The Exhibition of Mathematical Art at the 2008 Joint Mathematics Meetings, J. Math. Arts 2 no. 1 (2008), 39–45. 864. E. Mcdowell, Motor car and galimoto: An intercultural lesson in pragmatism, creativity, and perseverance. http://www.learnnc.org/ lp/pages/4945, accessed 30 December 2009. 865. H. Neuhaus, The Art of Piano Playing. Kahn & Averill, 1998. ISBN-10: 1871082455. ISBN-13: 978-1871082456. 866. L. Pasteur, Sur les relations qui peuvent exister entre la forme cristalline, la composition chimique, et le sens de la polarisation rotatoire, Ann. Chim. Phys. 24, 442 (1848). 867. E. M. Purcell, Life at low Reynolds number, Amer. J. Physics 45 (1977) 3–11. Available at http://brodylab.eng.uci.edu/ $~jpbrody/reynolds/lowpurcell.html; accessed 7 September 2008. 868. T. Relph, Counting sheep. http://www.lakelanddialectsociety.org/sheep.htm. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK References 279 869. K. Schmidt-Nielsen, Scaling: Why Is Animal Size So Important? Cambridge University Press, 1984, 256 pp. ISBN-10: 0521319870. ISBN-13: 978-0521319874. 870. D’A. W. Thompson, On Growth and Form. Cambridge University Press, 1961. 871. M. Tomczak, Science, Civilization and Society. http://www. incois.gov.in/Tutor/science+society/index.html. Accesses 22 July 2010. 872. P. Valéry, Cahiers. Gallimard, 1974. 873. M. Wittlinger, R. Wehner, and H. Wolf, The ant odometer: stepping on stilts and stumps, Science 312, no. 5782 (30 June 2006), 1965–1967. DOI: 10.1126/science.1126912. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 280 References J’espère que nos neveux me sauront gré, non seulement des choses que j’ai ici expliquées, mais aussi de celles que j’ai omises volontairement, afin de leur laisser le plaisir de les inventer. Descartes, Géométrie. S HADOWS OF THE T RUTH V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK Index abacist, 23, 24 dell’Abbaco, Paolo, 23 abstraction, 77 addition, 72 associativity of, 65 commutativity of, 73 componentwise, 8 of heterogeneous quantities, 8 Aharoni, Ron, ix, 8 Akilov, Gleb Pavlovich, 57 Alechina, Natasha, ix, 173 algebra, 138, 172, 190, 191 linear, 9 Peano, 82, 85 tensor, 9 unary, 85 algorist, 24 algorithm division, 5 Allegory of the Cave, vii Altınel, Evren, 68 Altınel, Kuban, 68 Altınel, Tuna, ix, 15, 67, 68 analysis, 192 p-adic, 118 dimensional, 8, 12, 24, 28, 32, 41 functional, 105 real, 101 Andrews, Paul, x angle trisection of, 178 approximation, 104 Archimedes, 85 Arcimboldo, Giuseppe, 2 Arcozzi, Nicola, ix, 173 area, 38 arithmetic Dedekind-Peano, 85 floating point, 78 mental, 187 Peano, 46 Arnold, Vladimir Igorevich, 32, 34, 36 Arnoux, Pierre, ix, 22, 70, 181, 182 arrow, 84 article definite, 13 indefinite, 13 Axiom of Choice, 12 of Induction, 47, 81 axioms Dedekind-Peano, 47, 63, 79 Baez, John, x, 135 Baldwin, John, x, 65, 66, 75, 140 Ball, Rowena, 32 Baron-Cohen, Simon, 168 Bass, Hyman, 97 Baumgartner, Bernhard, ix, 93 Belegradek, Oleg, x, 163 Bell, Alan, 3 Bell, Frances, ix, 165 Belolipetsky, Mikhail, ix, 126 Bezem, Marc, x, 164 BODMAS, 65 Bogomolny, Alexander, ix, 188 Boroditsky, Lera, 227 Borovik, Anna, ix, 170, 222 bound 281 282 Index uniform Lipschitz, 104 Braden, Lawrence, ix, 179 Brahmagupta, 141 Breen, Michael, ix, 52, 188, 209 Buchstaber, Victor Matveevich, 73, 75 bundle jet, 16 principal, 16 vector, 16, 228 Burago, Dmitri, ix, 38 inclusive “or”, 13 parallel “and”, 13 constant dimensionless, 35 gravitational, 12 construction compasses and ruler, 178 elementary geometric, 192 continuity, 100, 102 Conway, John Horton, 12 Cook, Alex R., ix, 116 coset, 117 cosine, 102 counting, 195 sheep, 79 Crabtree, Jonathan, ix, 19 2-current, 38 Currie, Iain, ix, 45 curve closeed, 38 level, 37 cylinder, 29 C++ , 47, 79 calculator, 175 calculus, 99, 101, 102, 105, 138 lambda, 135, 137–139 Caramuel y Lobkowitz, Juan, 120 Cariolaro, David, ix, 3, 197 carry, 115, 116, 121 category, 84, 230 Cauchy, Augustin-Louis, 100 Cézanne, Paul, 6 Chevallard, Yves, 99 Chorin, Alexander, 35, 36 Church, Alonzo, 136 C INDERELLA, 177 circle, 29, 192 Cliff, Emily, ix, 175 2-coboundary, 117 2-cocycle, 116, 117, 121 code Gray, 58 coefficient binomial, 83 cofunctor, 230 cohomology, 17, 116 column, 10 combinatorics, 17 Commodore 64, 170 commutativity of addition, 196 compass, 177 compasses, 178 composition of morphisms, 84 Comte, August, 98, 99 cone, 29 congruence, 223 connective, 13 consequtive “and”, 13 exclusive “or”, 13 D’Agostino, Santo, x Da Vinci, Leonardo, 221 De Morgan, Augustus, 50, 51 decimal, 113, 115, 202 decimal place, 117 decimal point, 116 Dedekind, Richard, 47, 100 definition, 17, 192 formal, 129 recursive, 81 Deloro, Adrien, x, 126, 192 ˘ Denizhan, Yagmur, x, 170, 230 denominator, 91, 200 common, 89, 90 derivative, 103 covariant, 16 partial, 37 Descartes, 280 Di Scala, Antonio Jose, x, 170 dice, 175 differentiation, 180, 216 digit significant, 116 units, 178 Dikstra, Edsger, 139 dimension, 11, 12 direction cardinal, 226, 227 S HADOWS OF THE T RUTH 23-N OV-2012/7:23 V ER . 0.822 c A LEXANDRE V. B OROVIK Index 283 dispensing, 1–3 displacement, 9 Disquisitiones Arithmeticae, 83 distribution, 5, 195 dividend, 183 divisibility, 4 division, 3, 12, 187 long, 114 by zero, 200 long, 116, 121, 159, 175, 179, 188 with remainder, 4, 5, 159 without remainder, 183 divisor, 183 on a variety, 228 Dolan, James, 135 domain, 122 Euclidean, 162, 164 integral, 164 unique factorization, 162, 164 dominoes falling, 42 Doyle, Peter, 12 drag, 30 Froude’s, 31 quadratic, 31, 40 Droste effect, 59 duality, 8 of vector spaces, 9 Dudeney, Henry Ernest, 190 Dumas, Alexandre, 20 duodecilion, 127 Dyer-Bennet, John, 56 Euler, Leonhard, 119 exponentiation modular, 56, 57 extremum constrained, 37 econometrics, 8 economics mathematical, 8 Eisenberg, Theodore, x, 71 Eisenkölbl, Theresia, x, 189, 198 electric charge, 11 Elements, 39, 48, 59, 101, 129 energy flow, 32 kinetic, 33 spectrum, 33 Epstein, David, x, 190 equation, 172 quadratic, 190 simultaneous linear equations, 191 Euclid, 28, 39, 48, 59, 101, 129, 225 Euler, Leonard, 83, 102 F#, 12 factor highest common, 91 factorial, 218 factorisation prime, 4 factorization, 187 Farquharson, Joseph, 79, 83 Feynman, Richard, 80 figure solid, 29 Fisher, Gwen, x, 74, 189 Flaubert, Gustave, 19, 24 Flick, Ritchie, x, 133 fluid viscous, 31 flux, 180 folklore mathematical, 105 Fon-Der-Flaass, Dmitrii Germanovich, xi force, 9, 11 form quadratic, 106 formalism, 16 epsilon-delta, 101 Formula Lagrange Interpolation, 146 formula Moivre, 102 quadratic, 190 formulae Weierstrass substitution, 157 fraction, 1, 14, 70, 75, 200 in lowest terms, 91 frame, 228 Fraser, Muriel, x French, Jo, x, 65 french, Jo, 66 Freudenthal, Hans, 100 Fried, Michael N., x, 196, 213 Froude, William, 30 function analytic, 78 continuous, 99 S HADOWS OF THE T RUTH 23-N OV-2012/7:23 V ER . 0.822 c A LEXANDRE V. B OROVIK 284 Index definable, 101 Euclidean, 162, 164 Euler’s, 56 exponential, 99, 197 hypergeometric, 17 piecewise monotone, 101 successor, 47 transition, 228 uniformly Lipschitz differentiable, 104 functional, 136 functor, 230 G., Swiatoslaw, x, 127 Gaby, Alice, 227 Galatasaray Lisesi, 15 Galileo Galilei, 28 Gangl, Herbert, x, 102 Gardiner, Tony, xi Garfunkel, Solomon, x, 172, 178 Garry, Dan, x, xi, 5 Gaudi, Antoni, 231 Gauss, Carl Friedrich, 83 Gauss, Karl, 231 Gelfand, Israel Moiseevich, 1, 17 G EO G EBRA, 177 geometry, 29 differential, 16, 17 elementary, 192 Euclidean, 101, 223 integral, 17 non-euclidian, 192 origami, 178 geometry, synthetic differential, 103 Gerard, Olivier, x, xi, 68, 182, 183, 210 Gibbon, John D., x, 5 Gilbert, Anthony David, x Gilbert, Tony D., 91 Gintrac, Jean Louis, 41 Gismatullin, Jakub, x, 169 Givental, Alexander, x, 28, 39 Glaser, Anton, 120 Gorcheva, Yordanka, xi Gould, Stephen, xi Gowers, Timothy, 60 Grad, Alex, x, 10, 14, 87 gradient, 37 graph, 228 Gray code S HADOWS OF THE T RUTH V ER . 0.822 cyclic, 58 Grigorchuk, Rostislav Ivanovich, x Gromov, Michael, vi, x Grothendieck, Alexandre, 191 group n-valued, 73, 75 cohomology, 118 commutative, 74 forma, 38 multivalued, 73 of coboundaries, 118 renormalization, 28 Grundlagen der Analysis, 61, 62, 75 Günter Törner, x, 59 Guugu Yimithirr, 226 Hadamard, Jacques Salomon, 223 Harrington, Leo, x, 167 Harriot, Thomas, 120 Harris, Michael, 154 Harris, Mitchell, x Harte, Robin, x, 180 Heeffer, Albrecht, x, 23, 25 Henderson, David, 129 Henkin, Leon, 81, 82 Hersh, Reuben, 188 Hiebert, James, 46 highest common factor, 91 Hokusai, Katsushika, 32, 33 homomorphism, 146 augmentation, 11 Hooke, Robert, 231 Howard, Luke, 150 Howard, Toby, x, 116 Howe, Roger, x, 127 Høyrup, Jens, x, 174 Hunter, Jodie, x Hutchinson, Alan, x, 137, 188 Ibn Tufail, Abu Bakr, 11 identity Fibonacci, 153, 154 index, 16 induction, 51 mathematical, 50 successive, 51 infinitesimal, 102 infinity, 195, 200, 201, 204 inheritance, 79 Initial Teaching Alphabet, 15, 16 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK Index 285 Kortenkamp, Ulrich, x, 173 Krutetsky, Vadim Andreevich, 113, 115 Kutateladze, Semen Samsonovich, x, 128, 129, 138 integer 10-adic, 118, 121 2-adic, 122 p-adic, 118 Gaussian, 153 negative, 70 non-negative, 73 one-digit, 178 relative, 70 symbolic, 78 integral, 138, 157 surface, 31 integration, 99 interval nested intervals, 102 invariance scale, 29 invariant, 29 inverse, 75, 228 isometry, 181 Kıral, Eren Mehmet, x, 3, 4 Kahane, Jean-Pierre, 97 Kant, Immanuel, 221 Kantor, Jean-Michel, x Kaptanoglu, H. Turgay, x Karakozov, Serguei Dmitrievich, x Katz, Mikhail, x, xi, 103 Keisler, H. Jerome, 102 Kheyfits, Alexander, x Khovanova, Tanya, x, 180 Khrushchev, Nikita Sergeevich, 196 Khudaverdyan, Hovhannes, x, 30 Khudaverdyan, Hovik, x, 118, 155 Kimber, Elizabeth, v, x, 45, 82, 125 Kirby, Jonathan, x, 94 Kirshner, David H., x Kiselev, Andrei Petrovich, 223, 224 Knuth, Donald, 103, 105 Kock, Anders, 103 Kolmogorov, Andrei Nikolaevich, 32, 34–36 Komendantskaya, Ekaterina, x, 199 König, Dénis, 205 Lagrange, Joseph Louis, 36 Lakoff, George, 101 Lama, Vishal, x Landau, Edmund, 61, 63, 75, 81, 82 language mathematical, 17 programming, 5 Laurent, Pierre Alphonse, 7 Lauri, Joseph, x, 176 Law Froude’s, 39–41 Froude’s of Steamship Comparisons, 30, 32, 39 Kolmogorov’s “5/3 Law”, 12 law associative, 65 commutative, 67 Coulomb’s, 11 Kolmogorov’s, 35 Lawvere, William, 103 Leedham-Green, Charles, x, 186 L EGO , 3, 218 Leibniz, Gottfried, 120, 123 Lemma Lemma König’s, 205 length, 11 Levshin, Vladimir, 21 lieue, 20 limit, 100–102, 118, 200 epsilon-delta definition of, 101 inductive, 89, 91 Lindemann, Ferdinand von Lindemann, 179 Lindenbaum, Adolf, 12 Lindquist, Christer, 169 line contour, 37 real, 102 straight, 129 width of, 177 Lins, Romulo, 3 Livshits, Michael, xi, 104 Livshitz, Michael, x logarithm, 99 S HADOWS OF THE T RUTH 23-N OV-2012/7:23 J AVA, 5 Jefferies, David, x, 201 Johansson, Mikael, x Jones, Alexander, 129 V ER . 0.822 c A LEXANDRE V. B OROVIK 286 Index logic mathematical, 101 logic, intuitionist, 103 Lomas, Dennis, x Löwe, Benedict, xi lowest common multiple, 91 MacKinnon, Dan, x Maltcev, Victor, x, 92, 168 manifold Riemannian, 38 Manin, Yuri Ivanovich, 29 Mann, Heinrich, 27, 195 map bilinear, 9 Gauss, 231 linear, 10, 228 smooth, 231 Mason, John, x mass, 11 Matematik Dünyası, xi M ATLAB, 77–79, 166 matrix, 10, 228 orthogonal, 106 square, 10, 228 transition, 10, 228 Mazur, Barry, 160 McBride, Peter, 139, 140 McKerrell, Archie, x, 196 McLaughlin, Jonathan, x measure Lebesgue, 138 mechanics, 29 Megyesi, Gábor, x, 118 minus, 65, 140 Molière, Jean-Baptiste, 115 monoid symmetrization of, 70 monomial, 127 Laurent, 8 monoplacer, 127 Montessori Method, 129 Montessori, Maria, 128, 130 Moon, 42 Moreno, Javier, x morphism, 84 identity, 84 motion projectile, 28 Müller, Thomas, xi multiple S HADOWS OF THE T RUTH V ER . 0.822 lowest common, 91 multiplication, 187 n-valued, 75 associativity of, 95 column, 114 long, 187 of fractions, 14 multiplier Lagrange, 36, 37 multiset, 74, 75 Muranov, Alexey, x, 93, 95, 120 Neman, Azadeh, x, 101, 185 Nesin, Ali, x, xi, 185 Nesin, Aziz, 185 Nesterov, Mikhail Vasilyevich, 201 Neuberger, John William, x, 203 Neubüser, Joachim, x Ni¸sanyan, Sevan, x, 3 Niven, Ivan Morton, 179 Noether, Emmy, 62 nomination, 128 nonillion, 127 norm, 38 Nosov, Nikolay, 173 notation, 16, 17 index-free, 16 novendecillion, 127 Novikov, Sergei Petrovich, 73 number “named”, 1, 8, 32 bisquare, 153 complex, 6, 70, 75 decimal, 168 final, 199 finite, 204 integer, 6 largest, 203 last, 199 monoplace, 127 named, 79 natural, 47, 48, 199 negative, 68 odd, 3 polygonal, 85 prime, 4 rational, 6 real, 6, 70, 75 signed, 71 transcendental, 122 triangular, 85 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK Index 287 wave, 33 number system base-n, 121 numeral, 13, 72, 81 cardinal, 3, 80 distributive, 3, 4, 80 ordinal, 3, 80 sheep, 80 Núnez, Raphael, 100, 101, 104 O’Farrell, Anthony, x, 189 object, 84 initial, 85 source, 84 target, 84 Obsessive Compulsive Disorder, 169 Ockley, Simon, 11 octillion, 127 Olshansky, Alexander Yurievich, x, 205 Olshansky, Yuri Nikolaevich, 205 operation binary, 5 operator, 136, 137 opposite, 67 optimization constrained, 36 Ore, Oystein, 141 Oskay, Windell H., x, 89 Osserman, Robert, 231 Owl (Otus Persapiens), xi pairing of vector spaces, 9 paradox Zeno’s, 205 parallelepiped, 95 partition, 2 Pascal, Blaise, 177, 221 Patten, Teresa, x, 174 Pease, Alison, xi pendulum, 41 permittivity, 11 permutation, 218 Perry, John, 150 Petrie, Karen, x Pflügel, Eckhard, x physics, 180, 187 π, 179 Piaget, Jean, 196 S HADOWS OF THE T RUTH Pierce, David, x, 3, 55, 57, 81, 82, 84, 85, 129 Pitman, Sir James, 16 place decimal, 176 place value, 127 Plato, vii, 85 point limit, 102 pointing, 130 pole, 228 Pólya, George, 28 polymorphism, 79 polynomial, 113 Laurent, 7, 10 Pons Asinorum, 223–225 Pormpuraaw, 227 Porter, Richard, x, 210 Povey, Hillary, x, 202 precalculus, 106 Preece, Donald A., x, 126 preoder, 90 price shadow, 38 Price, Alison, x, 171 probability, 175 Problem Lagrange Interpolation, 145 problem scaling, 28 word, 14 procedure decision, 101 product cartesian, 70 dot, 9 inner, 9 scalar, 9 tensor, 10, 228 progression geometric, 206 projection stereographic, 156, 157 prompt, 78 protractor, 178 Putinar, Mihai, x, 181 quadrillion, 127 quintillion, 127 quotient, 7 partial, 5 V ER . 0.822 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK 288 Index quotition, 2 Radzivilovsky, Vladimir, 129 Ramachandran, Vilayanur, 46, 167 Reisch, Gregorius, 24 relation equivalence, 70 remainder, 5, 141, 176 Reni, Guido, iv, ix, 98 representative coset, 117 Riepe, Thomas, x ring graded, 7 Laurent polynomial, 11 Noeterian, 131 unit of, 164 Robert, Alain M., 102 Roberts, Roy Stewart, x, 49, 50, 99, 126 Robinson, Abraham, 101, 103 Ross, Frederick, xi, 103 rounding, 78 row, 10 RSA, 166 ruler, 178, 181 Saint-Exupéry, Antoine de, v Salinger, Jerome David, 205 scaling, 27, 28 Seban, Jane-Lola, x section, 228 semantics, 139 Sen, Ashna, x St. Sergius of Radonezh, 201 series Laurent, 7 set directed, 90 empty, 86 set theory internal, 102 sextillion, 127 Shackell, John R., x, 197 sharing, 1–3 sheaf, 228 Shen, Alexander, x Shepherd, Simon J., x, 87 signified, 130 signifier, 130 Silagadze, Zurab, 206 S HADOWS OF THE T RUTH V ER . 0.822 Sloman, Aaron, x Souza, Kevin, x space dual, 228 finite dimensional vector, 228 vector, 228 speed relative, 40 square completion of the square, 190 magic, 209 perfect, 121 squillion, 203 Stephenson, Christopher, x, 209 S TICKLE B RICKS , 217, 218 Stokes, Sir George, 31 structure o-minimal, 101 substitution universal trigonometric, 157 subtraction, 3, 65, 66, 72, 75 as ‘change giving’, 3 repeated, 12 supertask, 206 surface orientable, 231 regualr, 231 Sutrop, Urmas, 230 Swan, Jerry, x, 217 Swanljung, Johan, x, 175 Swift, Tim, x, 15–17 symbol minus, 65, 67 opposite, 67 symmetry, 209 formal, 223 rotational, 216 system direct, 90 table multiplication, 4, 217 Taimina, Daina, 129 Tarski, Alfred, 12, 101 Tenejapan Tzeltal, 231 tensor, 8 Theorem Binomial, 83 Chinese Remainder, 141, 146 Fermat’s Little, 83 Lagrange’s, 57 23-N OV-2012/7:23 c A LEXANDRE V. B OROVIK Index 289 Pythagoras, 27, 28 Pythagorean, 39 Stokes, 31 theorem Euler’s, 57 theory category, 8, 16, 46 model, 105 relativity, 192 representation, 17 Thompson, D’Arcy, 30, 39 tiling Penrose, 216 time, 11 Tomczak, Matthias, 24 topolgy set-theoretical, 105 topology, 103 algebraic, 74 transformation didactic, 98, 104, 105 linear, 228 transpose, 228 transposition didactique, 99 tree, 205 triangle, 29 isosceles, 223 trillion, 127 Trimble, Todd, 135 Tropashko, Vadim, x, 157, 173 Trubachov, Anatoly Mikhailovich, 32 tuple unordered, 75 turbulence, 12, 35 Tyrrell, Malcolm, 217 Tzeltal, 226, 231 dimensionless, 23 fundamental, 12 Planck’s, 12 Utyuzhnikov, Sergei, x value intermediate, 101 place, 127 van Gogh, Vincent, 35 vector, 8 velocity, 11 Verbovskiy, Viktor, x, 198 Verne, Jules Gabriel, 20 Viéte, François, 7 viscosity, 31 visualization, 95, 193 vortex, 34 Vygotsky, Lev Semenovich, 125 Wagner, Roy, x, 23 Walker, Peter, 159 Ward, E. Honor, 221 Ward, Thomas, x, 221 wave number, 33 Weierstrass, Karl, 100 Wells, David, x Wigner, Eugene P., 93 Wolfart, Jürgen, x, 125 work, 9 writing mirror, 222 Wyles, Dean, x, 218 unit, 75, 122 of measurement, artificial, 21 Zagier, Don, 56 Zaturska, Maria, x, 225 Zeno, 205 zero, 228 zero divisor, 118 zillion, 203 Zoellner, Logan, x, 116 S HADOWS OF THE T RUTH 23-N OV-2012/7:23 V ER . 0.822 c A LEXANDRE V. B OROVIK