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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.
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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
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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 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
Cliff, Alex Cook, BC, VČ, Jonathan Crabtree, Iain Currie, RTC,
ix
x
PD, Yağmur Denizhan, Antonio Jose Di Scala, SD, DD, Ted Eisenberg, Theresia Eisenkölbl, RE, ŞUE, 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, Yağmur 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 Nişanyan, 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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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 Şirince, 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
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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 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
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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
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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
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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
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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 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 Nişanyan.
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, beş, 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.
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birinci, ikinci, üçüncü, dördüncü, beşinci, altıncı, . . .
(first, second, third, . . . )
These answer the question
Kaçıncı?
The distributives take the suffix -(ş)er:
birer, ikişer, 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
Beşer beşer alın. (Each one of you take five) or
İ kişer 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
Yedişer yedişer dizilin. (Get into rows of seven)
In that context, a story told to me by one of my colleagues, ŞUE3
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 ŞUE was very uncomfortable—he knew
that 3 did not go into 19. ŞUE 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, ŞUE 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
ŞUE is Turkish, male, recent mathematics graduate.
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1.3 Dividing apples by apples: a correct answer
5
ŞUE 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.
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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.
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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[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.
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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
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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.
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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.
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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.
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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.
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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
correspondent VČ2 :
[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.
VČ is male, Croatian, a lecturer in mathematical logic and computer
science.
13
14
2 Pedagogical Intermission: Human Languages
particularly suitable word to serve as definite article (except the
adjective odred̄eni = 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.
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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.
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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
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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.
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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.
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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
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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.
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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.
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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.
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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.
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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.
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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:
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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].
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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.
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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
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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
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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
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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].
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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.
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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
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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
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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.
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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.
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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]:
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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:
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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.
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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.
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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.
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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.
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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.
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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.
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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
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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.
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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.
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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′ ).
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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.
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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
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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,
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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.
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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.
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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.
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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
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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.
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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.”
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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
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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.)
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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.
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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.”
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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
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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.
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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.
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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.
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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.
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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/.
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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
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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.
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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
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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
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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;
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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.
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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
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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 .
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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?
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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 Şirince 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, Şirince, 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.
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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 . . .
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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
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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.
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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].
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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.
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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.
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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
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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
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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.
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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:
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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:
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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”.
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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).
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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.
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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.
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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.
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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
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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 . . .
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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
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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.
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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.
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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?
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10.5 Unary number system
123
Fig. 10.1. A page from Leibnitz’s paper about binary system [216].
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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: ŞUE 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.
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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.
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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.
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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].
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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-
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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).
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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
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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.
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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?
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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:
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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.
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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
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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
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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
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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).
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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),
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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.
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148
14 Numbers and Functions
Fig. 14.1. Make this diagram into a proof of the Chinese Remainder Theorem (Exercise 14.5).
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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.
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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.
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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.
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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:
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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.
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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.
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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.
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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”.
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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.
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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.
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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.
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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.
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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.
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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
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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.
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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
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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:
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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.
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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 Yağmur 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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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
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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
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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.
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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.
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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.
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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.
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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.
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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.
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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”,
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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
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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!
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19.4 Edge of an abyss
207
Fig. 19.2. Energy and momentum are disappearing in Pérez Laraudogoitia’s infinite sequence of collisions.
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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.
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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.
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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.
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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.
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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:
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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
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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.
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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.
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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.
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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
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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
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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.
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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.
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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
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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.
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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.
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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
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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
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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 .
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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
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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, Yağmur, 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
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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
Nişanyan, 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
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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
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c A LEXANDRE V. B OROVIK