Shadows of the Truth: Metamathematics of Elementary Mathematics potx

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. Gui d o Reni, 1630 –1640. Pina-
coteca Vaticana. Source: Wikipedia Commons. Pu blic 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 h is 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 systemat-
ically record and analyze from a mathematic al point of view vari-
ous difficulties experiencing by children in their early learnin g 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 hid den
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 di-
rected my inquiries to mathematicians for an obvious reason: only
mathematicians po ssess an adequate language which allows them
to describe in some depths their experiences of learning mathemat-
ics. So far my approach is justified by the warm welcome it found
among my mathematician friends, and I am most gr ate ful to them
for their suppor t. 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 col-
league 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 difficu lty, which remained with he r 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 f rom 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 corrobor ate d or authenticated—they are memor ies

that my colle agues 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 per sonal meaning for mathemati-
cians 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 gr an ted and suggest that it be u sed as a key to the most
intimate layer of mathematical thinking.
This bo nd 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 mov-
ing thrown objects). Also at this age I was fascinated by the com-
plexity of the inside of a car wi th the hood lifted.
Later I had a similar feeling by imagining first infinite ordinal s
(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 a t 14. At 14–15 I became inter-
ested 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 perceptive-

ness.
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vii
I am confident that this sentiment is shared by many my math-
ematician 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 mak-
ing 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 o f mathematical structures in math-
ematics. Mathematics has an astonishing power of reflection, and a
self-referential study of mathematics by mathematical means plays
an inc reasingly important role within mathematical culture. I sim-
ply 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 paral-
lel with Plato’s Allegory of the Cave: children in my boo k see shad-
ows of the Truth and sometimes find themselves in a psychological
trap becau se their teachers and other adults around them see nei-
ther Truth, nor its shadows. But I am not doing philosophy; I am
a mathematician and I stick to a concise mathematical reconstruc-

tion 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 wel-
come. Please write to me at

BIBLIOGRAPHY. At the end of each chapter I place some bibli-
ographic 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
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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 Ar-
cozzi, Pierre Arnoux, Autodidact, Bernha rd Baumgartner, Frances
Bell, SB, Mikhail Belolipetsky, AB, Alexander Bogomolny, RB,
Anna Borovik (my wife, actually), Lawrence Braden, Michael Breen,
TB, BB, Dmitri Burago, L B, CB, LC, David Cariolaro, SC, E mily
Cliff, Alex Cook, BC, V
ˇ
C, Jonathan Crabtree, Iain Currie, RTC,

ix
x
PD, Ya
˘
gmur Denizhan, Antonio Jose Di Scala, SD, DD, Ted Eisen-
berg, 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 Kho-
vanova, Hovik Khudaverdyan, Elizabeth Kimber, EMK, Jonathan
Kirby, SK, Ekaterina Komendan tskaya, Ul rich 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 N eubüser, An -
thony O’Farrell, Alexander Ols hansky one man and a dog, Teresa
Patten, Karen Petrie, NP, Eckhard Pflügel, R ichard Porter, Hillary
Povey MP, Alison Price, Mihai Putinar, VR, Roy Stewart Roberts,
FR, PR, AS, John Shackell, Simon J. Shepherd, GCS, VS, Christo-
pher Stephenson, Jerry Swan, Johan Swanl jung, 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 sh aring with me their childhood memories and/or their ed-
ucational a nd 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 infinitesimal s, 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, Ya
˘
gmur Denizhan , Muriel
Fraser, Michael N. Fried, Alexander Givental, AH, Mitchell Har-
ris, Albrecht Heeffer, Roger Howe, Jens Høyrup, Jodie Hunter
Mikael Johansson, Jean-Michel Kantor, H. Turgay Kaptanoglu,
Serguei Karakozov, Mikhail Katz, Alexander Kheyfits Hovik Khu-
daverdyan, Eren Mehmet Kıral, David H. Kirshner, Semen Sam-
sonovich 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;
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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 associ-
ated papers.

This text would not appear had I not received a kind invitation
to give a talk at “Is Mathematics Spec ial” conference in Vienna
in May 2008, and without an invitation from A li 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 o f the Vienna
conference edited by Benedikt Löwe and Thomas Müller. Parts of
the text fi rst appeared in Matematik Dünyası, a popular mathe-
matical journal edited by Ali Nesin [627].
My work on this book was partially suppo r ted by a g rant from
the John Templeton Foundation, a charitable institution which de-
scribes itself as a
“philanthropic catalyst for discovery in areas engaging in life’s
biggest questions.”
Howeve r, the opinions expressed in the book are those of the au-
thor and do not necessarily reflect the views of the John Te mpleton
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 O wl.
Alexandre Borovik
Didsbury
23 Novem ber 2012
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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 Intermis sion: Human Languages . . . . . . . 13
3 Units of measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1 Fantasy units of measurement . . . . . . . . . . . . . . . . . . . 19
3.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4 History of Dimensional Analysis . . . . . . . . . . . . . . . . . . . . 27
4.1 Galileo Galilei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.2 Froude’s Law of Steamship Comparisons . . . . . . . . . . 30
4.2.1 Difficulty of making physical models . . . . . . . . 30
4.2.2 Deduction of Froude’s Law . . . . . . . . . . . . . . . . . 31
4.3 Kolmogorov’s “5/3” Law . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.3.1 Turbulent flows: basic setup . . . . . . . . . . . . . . . 32
4.3.2 Subtler analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.4 Dimension of Lagrange multipliers . . . . . . . . . . . . . . . 36
4.5 Length and area . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
xiii
xiv Contents

5 Adding One by One . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.1 Adding one by one . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.2 Dedekind-Peano axioms . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.3 A brief digression: is 1 a number? . . . . . . . . . . . . . . . . . 48
5.4 How much mathematics can a child see at the
level of basic counting? . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.5 Properties of addition . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.5.1 Associativity of addition . . . . . . . . . . . . . . . . . . . 53
5.5.2 Commutativity of addition . . . . . . . . . . . . . . . . . 54
5.6 Dark clouds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.7 Induction and recursion . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.8 Digression into infinite descent . . . . . . . . . . . . . . . . . . . 59
5.9 Landau’s proof of the existence of add ition . . . . . . . . 61
6 What is a Minus Sign Anyway? . . . . . . . . . . . . . . . . . . . . . . 65
6.1 Fuzziness of the rules . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6.2 A formal treatment of subtraction . . . . . . . . . . . . . . . . 67
6.3 A formal treatment of negative numbers . . . . . . . . . . 68
6.4 Testimonies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.5 Multivalued groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
7 Counting Sheep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
7.1 Numbers in computer science . . . . . . . . . . . . . . . . . . . . 77
7.2 Counting sheep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
7.3 Abstract nonsense . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
7.3.1 Existence and uniqueness . . . . . . . . . . . . . . . . . 81
7.3.2 Unary algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
7.3.3 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
7.4 Induction on systems other than N . . . . . . . . . . . . . . . 82
7.5 Categories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
7.6 Digression:
Natural numbers in Ancient Greece . . . . . . . . . . . . . . 85

8 Fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
8.1 Fractions as “named” numbers . . . . . . . . . . . . . . . . . . . 87
8.2 Inductive limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
8.3 Field of fractions of an integral domain . . . . . . . . . . . 92
8.4 Back to commutativity of m ultiplication. . . . . . . . . . . 93
9 Pedagogical Intermis sion:
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
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10 Carrying: Cinderella of Arithmetic . . . . . . . . . . . . . . . . . . 109
10.1 Palind romic 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 Intermis sion:
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 Intermis sion: Factorization . . . . . . . . . . . . 165
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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
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1
Dividing Apples between People
It is important not to separate mathematics from life.
You can explain fractions even to he avy 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 life
1
; 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 app les : 5 people = 2 apples. (1.1)
Even worse: when we distribute 10 apples giving 2 apples to a per-
son, we have
10 app les : 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 rig ht 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 discomfor t. I had no
bad feelings about dividing 10 apples among 5 people, but I some-
how felt that the problem of deciding how many people would get
apples if each was give n 2 apples from the total of 10, was c om-
pletely different. I tried to visualize the problem as an orderly dis-
tribution 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 Sec-
tion 19.1.)
Indeed , my childhood e xperience 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 discovere d later that these
operation w ere called partition and quotition in [623]. But even
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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 LEGO
pieces th at I had at that time with my brother— and fa iled in that
respect and burst in tears. When I told my Mum that I could not
divide evenly the pieces she recognized th at I indeed discovered
odd numbers, and that was my first mathematical discovery.
Finally, notice that there are similar special cases of subtrac-
tion; 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 P h. 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 h ave to pay $35 and give a
$100 bill to the clerk. He gives me a $5 bi ll 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 an d doing a subtraction were
quite different thin gs, 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 lo gical d ifference betwee n the operations of sharing and dispens-
ing is reflected in the grammar of the Turkish language by the pres-
ence 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 h as several systems of numerals, all based on the cardi-
nals; a s 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 po-
sition. In this epi sode, the language of communication was his mother
tongue, Italian.
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4 1 Dividing Apples between People
birinci, ikinci, üçüncü, dördüncü, be¸sinci, altıncı, .
(first, second, third, . . .)
These answer the question
Kaçıncı?
The distributi ves take the suffix -(¸s)er:
birer, iki¸ser, üçr, . . .
Used singly, these mean “one each, two each” and so on, a s in “I

want two fruits from each of these baskets”; they answer the ques-
tion
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)
In that context, a story told to me by one of my colleagues, ¸SUE
3
is very interesting. His experience of arithmetic in his (Turkish) el-
ementary 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 d ifferent occasion, what is 6 ×7, h e could not answer.
Also, he could not accept the con cept of division with remainder: if
a teacher asked him how many 3s go into 19 (expecting an answer:
6, and 1 is left ov er), 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 h ad the same prob-
lem when I was asked how many 3s go into 1 6. It is th e 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 dis tributive gave) I could not understand di-
vision with remainder.
As we can see, ¸SUE does not dismiss the suggestion that dis-
tributive numer als of his mother tongue could have made it easier
for him to form concept of divisibility and pr im e numbers (although
he did not know the term “prime numbe r”) before he learned mul-
tiplication.
3
¸SUE is Turkish, male, recent mathematics graduate.
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1.3 Dividi ng apples by apples: a correct answer 5
¸SUE only made his peace with remainder s during his first year
at university, when the process of division with remainde r was in-
troduced as a formal technique. He is not alone in waiting years be-
fore finally being told that division with re mainder is not a binary
operation because it produces two outputs, n ot one, as a binary op-
eration should: partial quotient and remainder. Indeed here is a
story from Dan Garry
4
:
When I was seven, I had to take a week off because I was sick. We
were s tudying 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 rela-
tion 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 com-
puter science about how the JAVA programming language handles
integer division.
1.3 Divi ding 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 fix ed data set: 10 apples and 5 people, and you
can easily visualize giving apples to the people, in rounds, o ne ap-
ple to a person at a time, until no apples were left. But, as I have
already mentioned, an attempt to visualize the second p roblem in
a similar way, as an orderly distribution of apples to a queue of
people, two apples to each person, necessitates dealing with a po-
tentially unlimited number of recipients.
I asked my teacher for help, but did not get a satisfactory an-
swer. Only much later did I realize that the co r rect naming of the
numbers should be
10 app les : 5 people = 2
apples
people
, 10 app les : 2
apples
people
= 5 people.
(1.3)
I was not alone in my discomfort with “named nu mbers” and
“units”. Here is a testimony from John Gibbon
5

:
4
DG is male, 21 years old, was born and raised in Englan d . He is a fin al
year undergraduate s tudying Computer Science and Mathematics in a
British university.
5
JDG is male, British, a professor of applied mathematics.
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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 th ere 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 tauto-
logical argument was but I knew I felt uncomfortable with it.
Therefore let us look into equations (1.3) with some attention.
1.4 What ar e the numbers children are working
with?
It is a commonplace wisdom that the development of mathemati-
cal skills in a student goes alon gside the gradual expansion of the
realm of numbers with which he or she works, from natural num-

bers to integers, then to rational, real, complex numbers:
N ⊂ Z ⊂ Q ⊂ R ⊂ C.
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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[x
1
, x
−1
1
, . . . , x
n
, x
−1
n
].
of Laurent polynomials
6
in n variables over Q, where symbols
x
1
, . . . , x
n
stand for the names of objects involved in the calculation: apples,
persons, etc. This explains why educational psychologists confi-
dently claim that the operations (1.1) and (1.2) on Page 1 have little

in common [264]—indeed, operation (1.2) involves an ope r and “ap-
ple/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 heteroge-
neous to the former . . . Mu ch 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 IG
7
:
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, a s 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 obtainin g 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 te ach Laurent polynomials
to kids (or even to teachers); but we n eed some simple c ommon lan-
guage that addresses the subtleties without adding unnecessary
6

Laurent polynomials and Laurent series are named after French mili-
tary engineer Pierre Alphonse Laurent (1813–1854) who was the first
to introduce them. Another his major a chievement was construction of
the port of Le Havre.
7
IG is femal e, a PhD student in a leading British university. She went to
school in Russia and was educated in Russian, her mother tongue.
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8 1 Dividing Apples between People
sophistication. This is w hy I devote Chapters 3 and 4 to discus-
sion of dimensional analysis, that is, the use of “named” num bers
in physics. To my taste, it provides a number of interesting elemen-
tary 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 classi c operations there i s a fi f th 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 count-
ing: 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” le vel, “forming a unit” may be viewed as setting
up an appropriate Laurent polynomial ring as an ambient struc-
ture 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 there-
fore very important) category theory coming into play (see Chap-
ter 7).
1.5 The lunch bag arithmetic, or addition of
heterogeneous quantities
Usually, only Laurent monomials are interpreted as having physi-
cal (or real life) meaning. But the addition of he terogeneous quan-
tities 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 mathem ati-
cal economics).
1.6 Duality and pairing
The “lunch bag” approach to vectors allows a natural introduc-
tion of duality and tensors: the total cost of a purchase of amou nts
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1.6 Duality and pairing 9
g
1
, g
2
, g
3
of some goods at prices p
1

, p
2
, p
3
is a “scalar product”-type
8
expression

g
i
p
i
.
We see that the quantities g
i
and p
i
could be of completely different
nature. In physics, as a rule, the dot produ ct involves heteroge-
neous magnitudes. In introductory physics courses, the do t product
usually makes its first appearance on the scen e as work done by
moving an objec t, 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 un-
dergraduate linear algebra usually conceals the fact that dot prod-
uct is a manifestation of duality or pairing of vector spaces, thu s
creating imme nse difficulties in the subsequent study of tensor al-
gebra. As the following testimony from CB
9
shows, the boredom

and confusion start even earlier:
I remember the very first conceptual difficulty I ever had: that was
the sca lar product of vectors. I could n ot figure why an operation
involving two vectors should yield a plain number, and my teach-
ers could not explain what that number meant in relation to the
two vectors. As a result I ha ted scalar products as all we did with
them was a meaningless if easy algebraic manipulation.
Indeed scalar (or dot) produc t as it appears in physics is a pair-
ing of two vector spaces U and V of different nature; assuming that
we are working over the real num bers R, pairing is a map
U × V → R
(u, v) → u ·v
which is bilinear, that is,
(au
1
+ bu
2
) ·v = au
1
· v + bu
2
· v
and similarly
u · (av
1
+ bv
2
) = au ·v
1
+ bu ·v

2
,
in both cases for all a, b ∈ R and all vectors u, u
i
∈ U and v, v
i
∈ V .
If it is possible to ignore physical (or financial) meanings of the
vector spaces U and V , then the two spaces become lo gically undis-
tinguishable. Paradoxically, this provides another source of diffi-
culty for those students who are sensitive to formal logical aspects
of mathematical concepts.
8
Scalar product is also cal led dot product or inner product.
9
CB is female, holds a PhD, works a s an editorial director in a math-
ematics publishing house. Her mother tongue is French, but she was
educated in English. The epi sode described happened at age 12.
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