Shadows of the Truth: Metamathematics of Elementary Mathematics potx

Dividing Apples between People It is important not to separate mathematics from life.. When, as a child, I was told by my teacher that I had to be careful with “named” numbers and not to

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

Pina-coteca Vaticana Source: Wikipedia Commons Public domain.

Guido Reni was one of the first artists in history of visual arts whopaid attention to psychology of children Notice how the little angel counts

on his fingers the points he is sent to communicate to St Matthew

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 newand somewhat unusual point of view I have started to systemat-

ically record and analyze from a mathematical point of view

vari-ous difficulties experiencing by children in their early learning ofmathematics I hope that my approach will eventually allow me

to gain a better understanding of how we—not only children, butadults, 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 hiddenstructures of elementary mathematics which may intrigue, puzzle,and—like shadows in the night—sometimes scare an inquisitivechild

The real life material in my research is limited to stories that

my fellow mathematicians have chosen to tell me; they representtiny but personally significant episodes from their childhood I di-rected my inquiries to mathematicians for an obvious reason: onlymathematicians possess an adequate language which allows them

to describe in some depths their experiences of learning ics So far my approach is justified by the warm welcome it foundamong my mathematician friends, and I am most grateful to themfor their support For some reason (and the reason deserves a study

mathemat-on its own) my colleagues know what I am talking about!

The book was born from a chance conversation with my league Elizabeth Kimber I analyze her story, in great detail, inChapter 5 Little Lizzie, aged 6, could easily solve “put a number inthe box” problems of the type

col-7 +  = 12,

v

by counting how many 1’s she had to add to 7 in order to get 12 butstruggled with

+ 6 = 11,because she did not know where to start Much worse, little Lizziewas frustrated by the attitude of adults around her—they could notcomprehend her difficulty, which remained with her for the rest ofher life

When I heard that story, I instantly realized that I had hadsimilar experiences myself, and that I heard stories of challengeand frustration from many my fellow mathematicians I started toask around—and now offer to the reader a selection of responsesarranged around several mathematical themes

A few caveats are due The stories told in the book cannot beindependently corroborated or authenticated—they are memoriesthat my colleagues have chosen to remember I believe that thestories are of serious interest for the deeper understanding of theinternal and hidden mechanisms of mathematical practice becausethe memories told have deeply personal meaning for mathemati-cians who told the stories to me The nature of this deep emotionalbond between a mathematician and his or her first mathematicalexperiences 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 mostintimate layer of mathematical thinking

This bond with the “former child” (or the “inner child”?) is bestdescribed by Michael Gromov:

I have a few recollections, but they are not structural

I remember my feeling of excitement upon hitting on somemathematical ideas such as a straight line tangent to a curve andrepresenting infinite velocity (I was about 5, watching freely mov-ing thrown objects) Also at this age I was fascinated by the com-plexity of the inside of a car 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 strongerthan 100 whales and how to be stronger than all of them in theuniverse)

Also I recall many instances of acute feeling of frustration at

my stupidity of being unable to solve very simple problems atschool later on

My personal evaluation of myself is that as a child till 8–9, Iwas intellectually better off than at 14 At 14–15 I became inter-ested in math It took me about 20 years to regain my 7 year oldchild perceptiveness

I repeat Michael Gromov’s words:

It took me about 20 years to regain my 7 year old child ness.

I am confident that this sentiment is shared by many my

math-ematician colleagues This is why I concentrate on the childhood

of mathematicians, and this is why I expect that my notes will be

useful to specialists in mathematical education and in psychology

of education But I wish to make it absolutely clear: I am not

mak-ing any recommendations on mathematics teachmak-ing 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

math-ematics 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

sim-ply suggest to take a step further (or a step aside, or a step back in

life) and to take a look back in time, at one’s childhood years

A philosophically inclined reader will immediately see a

paral-lel with Plato’s Allegory of the Cave: children in my book see

shad-ows of the Truth and sometimes find themselves in a psychological

trap because their teachers and other adults around them see

nei-ther Truth, nor its shadows But I am not doing philosophy; I am

a mathematician and I stick to a concise mathematical

reconstruc-tion of what the child had actually seen

My book is also an attempt to trigger the chain of memories in

my readers: even the most minute recollection of difficulties and

paradoxes of their early mathematical experiences is most

wel-come Please write to me at

borovik@manchester.ac.uk

BIBLIOGRAPHY At the end of each chapter I place some

bibli-ographic references Here are some (very different) books most

closely related to themes touched on in this introduction: Aharoni

[610], Carruthers and Worthington [642, 644], Freudenthal [667],

Gromov [30], and Krutetskii [826]

Alexandre Borovik

Didsbury

16 July 2011

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 cozzi, Pierre Arnoux, Autodidact, Bernhard Baumgartner, FrancesBell, SB, Mikhail Belolipetsky, AB, Alexander Bogomolny, RB,Anna Borovik (my wife, actually), Lawrence Braden, Michael Breen,

Ar-TB, BB, Dmitri Burago, LB, CB, LC, David Cariolaro, SC, EmilyCliff, Alex Cook, BC, V ˇC, Jonathan Crabtree, Iain Currie, RTC,

ix

PD, Ya˘gmur Denizhan, Antonio Jose Di Scala, SD, DD, Ted berg, Theresia Eisenkölbl, RE, ¸SUE, David Epstein, Gwen Fisher,Ritchie Flick, Jo French; Michael N Fried, Swiatoslaw G., IG,Herbert Gangl, Solomon Garfunkel, Dan Garry, Olivier Gerard,John Gibbon, Anthony David Gilbert, Jakub Gismatullin, VG,Alex Grad, IGG, Rostislav Grigorchuk, Michael Gromov, IH, LeoHarrington, EH, Robin Harte, Toby Howard, RH, Jens Høyrup,Alan Hutchinson, BH, David Jefferies, Mikhail Katz, Tanya Kho-vanova, Hovik Khudaverdyan, Elizabeth Kimber, EMK, JonathanKirby, SK, Ekaterina Komendantskaya, Ulrich Kortenkamp, CharlesLeedham-Green, AL, EL, RL, DMK, JM, Victor Maltcev, MM,Archie McKerrell, Jonathan McLaughlin, Alexey Muranov, AzadehNeman, Ali Nesin, John W Neuberger, Joachim Neubüser, An-thony O’Farrell, Alexander Olshansky one man and a dog, TeresaPatten, Karen Petrie, NP, Eckhard Pflügel, Richard Porter, HillaryPovey MP, Alison Price, Mihai Putinar, VR, Roy Stewart Roberts,

Eisen-FR, PR, AS, John Shackell, Simon J Shepherd, GCS, VS, pher Stephenson, Jerry Swan, Johan Swanljung, BS, Tim Swift,

Christo-RT, Günter Törner, Vadim Tropashko, Viktor Verbovskiy, RW, PW,

JW, RW, MW, Jürgen Wolfart, CW, Maria Zaturska, WZ and LoganZoellner

for sharing with me their childhood memories and/or their ucational and pedagogical experiences;

ed-to parents of DW for allowing me ed-to write about the boy;

and to my colleagues and friends for contributing their expertise

on history of arithmetic and history of infinitesimals, French andTurkish languages, artificial intelligence, turbulence, dimensionalanalysis, subtraction, cohomology, p-adic integers, programming,pedagogy — in effect, on everything — and for sharing with metheir blog posts, papers, photographs, pictures, problems, proofs,translations:

Santo D’Agostino, Paul Andrews, John Baez, John Baldwin, OlegBelegradek, Marc Bezem, Adrien Deloro, Ya˘gmur Denizhan , MurielFraser, Michael N Fried, Alexander Givental, AH, Mitchell Har-ris, Albrecht Heeffer, Roger Howe, Jens Høyrup, Jodie HunterMikael Johansson, Jean-Michel Kantor, H Turgay Kaptanoglu,Serguei Karakozov, Mikhail Katz, Alexander Kheyfits Hovik Khu-daverdyan, Eren Mehmet Kıral, David H Kirshner, Semen Sam-sonovich Kutateladze, Vishal Lama, Joseph Lauri, Michael Livshits,Dennis Lomas, Dan MacKinnon, John Mason, Gábor Megyesi,Javier Moreno, Ali Nesin, Sevan Ni¸sanyan, Windell H Oskay,David Pierce, Donald A Preece, Thomas Riepe, Jane-Lola Seban,Ashna Sen, Alexander Shen, Aaron Sloman, Kevin Souza, ChrisStephenson, Vadim Tropashko, Sergei Utyuzhnikov, Roy Wagner,Thomas Ward, David Wells, and Dean Wyles;

and to Tony Gardiner, Yordanka Gorcheva, Dan Garry, Olivier

Gerard, Stephen Gould, Mikhail Katz, Michael Livshits, Alison

Pease and Frederick Ross for sending me detailed comments on,

and corrections to, the on-line version of the book and /or

associ-ated papers

This text would not appear had I not received a kind invitation

to give a talk at “Is Mathematics Special” conference in Vienna

in May 2008, and without an invitation from Ali Nesin to give a

lecture course “Elementary mathematics from the point of view of

“higher” mathematics” at the Nesin Mathematics Village in

¸Sir-ince, Turkey, in July 2008 and in August 2009 Section 10.1 was

first published in a [106] in the proceedings volume of the Vienna

conference edited by Benedikt Löwe and Thomas Müller Parts of

the text first appeared in Matematik Dünyası, a popular

mathe-matical 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

de-scribes 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

au-thor 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

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 19

3.1 Fantasy units of measurement 19

3.2 Discussion 21

3.3 History 23

4 History of Dimensional Analysis 27

4.1 Galileo Galilei 28

4.2 Froude’s Law of Steamship Comparisons 30

4.2.1 Difficulty of making physical models 30

4.2.2 Deduction of Froude’s Law 31

4.3 Kolmogorov’s “5/3” Law 32

4.3.1 Turbulent flows: basic setup 32

4.3.2 Subtler analysis 34

4.3.3 Discussion 35

4.4 Dimension of Lagrange multipliers 36

4.5 Length and area 38

xiii

xiv Contents

5 Adding One by One 45

5.1 Adding one by one 45

5.2 Dedekind-Peano axioms 47

5.3 A brief digression: is 1 a number? 48

5.4 How much mathematics can a child see at the level of basic counting? 49

5.5 Properties of addition 52

5.5.1 Associativity of addition 53

5.5.2 Commutativity of addition 54

5.6 Dark clouds 55

5.7 Induction and recursion 57

5.8 Digression into infinite descent 59

5.9 Landau’s proof of the existence of addition 61

6 What is a Minus Sign Anyway? 65

6.1 Fuzziness of the rules 65

6.2 A formal treatment of subtraction 67

6.3 A formal treatment of negative numbers 68

6.4 Testimonies 71

6.5 Multivalued groups 73

7 Counting Sheep 77

7.1 Numbers in computer science 77

7.2 Counting sheep 79

7.3 Abstract nonsense 81

7.3.1 Existence and uniqueness 81

7.3.2 Unary algebras 81

7.3.3 Proofs 82

7.4 Induction on systems other than N 82

7.5 Categories 84

7.6 Digression: Natural numbers in Ancient Greece 85

8 Fractions 87

8.1 Fractions as “named” numbers 87

8.2 Inductive limit 89

8.3 Field of fractions of an integral domain 92

8.4 Back to commutativity of multiplication 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

Contents xv

10 Carrying: Cinderella of Arithmetic 109

10.1 Palindromic decimals and palindromic polynomials 109 10.2 DW: a discussion 113

10.3 Decimals and polynomials: an epiphany 114

10.4 Carrying: Cinderella of arithmetic 115

10.4.1 Cohomology 115

10.4.2 A few formal definitions 117

10.4.3 Limits and series 118

10.4.4 Euler’s sum 119

10.5 Unary number system 120

11 Pedagogical Intermission: Nomination and Definition 125

11.1 Names 125

11.2 Nomination 128

12 The Towers of Hanoi and Binary Trees 133

13 Mathematics of Finger-Pointing 135

13.1 John Baez: a taste of lambda calculus 135

13.2 Here it is 137

13.3 A dialogue with Peter McBride 139

14 Numbers and Functions 141

14.1 Chinese Remainder Theorem 141

14.1.1 History 141

14.1.2 Simultaneous Congruences 142

14.1.3 Algorithm 143

14.1.4 Example 144

14.2 The Lagrange Interpolation Formula 144

14.3 Numbers as functions 146

15 Graph Paper and the Arithmetic of Complex Numbers 149

15.1 Graph paper 149

15.2 Pizza, logarithms and graph paper 151

15.3 Multiplication of squares 153

15.4 Pythagorean triples 155

16 Uniqueness of Factorization 159

16.1 Uniqueness of factorization 159

16.2 Dialog with AL 161

16.3 Generalizations 162

16.4 The Fermat Theorem for polynomials 163

17 Pedagogical Intermission: Factorization 165

xvi Contents

18 Being in Control 167

18.1 Leo Harrington: Who is in control? 167

18.2 The quest for truth 171

18.3 The quest for logic 172

18.4 The quest for understanding 172

18.5 The quest for power 176

18.6 The quest for rigour 177

18.7 Suspicion of easy options 182

18.8 “Everything had to be proven” 185

18.9 Raw emotions 186

18.10 David Epstein: Give students problems that interest them 190

18.11 Autodidact 192

18.12 Blocking it out 193

19 Controlling Infinity 195

19.1 Fear of infinity 195

19.2 Counting on and on 196

19.3 Controlling infinity 201

19.4 Edge of an abyss 205

20 Pattern Hunting 209

21 Visual Thinking vs Formal Logical Thinking 213

21.1 213

21.2 EH: Visualisation 216

21.3 Lego 217

22 Telling Left from Right 221

22.1 Why does the mirror change left and right but does not change up and down? 221

22.2 Pons Asinorum 223

22.3 TB 224

22.4 Maria Zaturska 225

22.5 MP 226

22.6 Digression into ethnography 226

22.7 BB 228

22.8 PD 228

22.9 Digression into Estonian language 230

22.10 Standing arches, hanging chains 230

22.11 Orientation of surfaces 231

References 233

Index 281

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 isrelevant 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:

Even worse: when we distribute 10 apples giving 2 apples to a son, we have

Where do “people” on the right hand side of the equation comefrom? Why do “people” appear and not, say, “kids”? There were no

“people” on the left hand side of the operation! How do numbers onthe left hand side know the name of the number on the right handside?

1Call me AVB; I am Russian, male, have a PhD in Mathematics, teachmathematics 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 nobad feelings about dividing 10 apples among 5 people, but I some-how felt that the problem of deciding how many people would getapples if each was given 2 apples from the total of 10, was com-pletely different I tried to visualize the problem as an orderly dis-tribution of apples to a queue of people, two apples to each person.The result was deeply disturbing: in horror I saw an endless line

of poor wretches, each stretching out his hand, begging for his twoapples (I discuss these my childhood fears in more detail in Sec-tion 19.1.)

Indeed, my childhood experience is confirmed by experimentalstudies, 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

1.2 Digression into Turkish grammar 3

sharing is not easy and may lead to mathematical discoveries! If

you do not believe, read a testimony from David Cariolaro:2

When I was 3 years old I was trying to divide evenly the LEGO

pieces 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

subtrac-tion; it is worth quoting from Romulo Lins [714]:

I once had a very interesting conversation with Alan Bell, at the

time when he was my 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

dispens-ing is reflected in the grammar of the Turkish language by the

pres-ence of a special form of numerals, distributive numerals.

What follows was told to me by David Pierce, Eren Mehmet

Kıral and Sevan Ni¸sanyan

First David Pierce:

Turkish has several systems of numerals, all based on the

cardi-nals; as well as a few numerical peculiarities

The cardinals begin:

bir, iki, üç, dört, be¸s, altı, (one, two, three, )

These answer the question

Kaç? (How many?)

The ordinals take the suffix -inci, adjusted for vowel harmony:

2DC is male, Italian, has a PhD in mathematics, holds a research

po-sition In this episode, the language of communication was his mother

tongue, Italian

4 1 Dividing Apples between People

birinci, ikinci, üçüncü, dördüncü, be¸sinci, altıncı, (first, second, third, )

These answer the questionKaçıncı?

The distributives take the suffix -(¸s)er:

birer, iki¸ser, üçr, Used singly, these mean “one each, two each” and so on, as in “Iwant two fruits from each of these baskets”; they answer the ques-tion

Kaçar?

Then Eren Mehmet Kıral continues:

When somebody is distributing some goods s/he might sayBe¸ser be¸ser alın (Each one of you take five) or

˙I ki¸ser elma alın (Take two apples each)

I do not know if it is a grammatical rule (or if it is important)but when the name of the object being distributed is not mentionedthen the distributive numeral is repeated as in the first example.The numeral may also be used in a non distributive problem Ifsomebody is asking students (or soldiers) to make rows consisting

of 7 people each then s/he might sayYedi¸ser yedi¸ser dizilin (Get into rows of seven)

In that context, a story told to me by one of my colleagues, ¸SUE3

is very interesting His experience of arithmetic in his (Turkish) ementary school, when he was about 8 or 9 years old, had a peculiartrouble spot: he could factorize numbers up to 100 before he learntthe times table, so he could instantly say that 42 factors as 6×7, but

el-if asked, on a del-ifferent occasion, what is 6 × 7, he could not answer.Also, he could not accept the concept of division with remainder: if

a teacher asked him how many 3s go into 19 (expecting an answer:

6, and 1 is left over), little ¸SUE was very uncomfortable—he knewthat 3 did not go into 19 ¸SUE added:

But I did not pay attention to 19 being prime I had the same lem when I was asked how many 3s go into 16 It is the samething: no 3s go in 16 Simply because 3 is not a factor of 16 This isperhaps because of distributive numerals I somehow built up anintuition of factorizing, but perhaps for the same reason (because

prob-of the intuition that distributive gave) I could not understand vision with remainder

di-As we can see, ¸SUE does not dismiss the suggestion that tributive numerals of his mother tongue could have made it easierfor him to form concept of divisibility and prime numbers (although

dis-he did not know tdis-he term “prime number”) before dis-he learned tiplication

mul-3 ¸SUE is Turkish, male, recent mathematics graduate

1.3 Dividing apples by apples: a correct answer 5

¸SUE only made his peace with remainders during his first year

at university, when the process of division with remainder was

in-troduced as a formal technique He is not alone in waiting years

be-fore finally being told that division with remainder is not a binary

operation because it produces two outputs, not one, as a binary

op-eration 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

rela-tion between “division” and “remainders” of integers until I went

to a lecture on the division algorithm in my first year of university,

which conveniently took place a few hours after a lecture in

com-puter science about how the JAVAprogramming 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

ap-ple to a person at a time, until no apap-ples 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

po-tentially unlimited number of recipients

I asked my teacher for help, but did not get a satisfactory

an-swer Only much later did I realize that the correct naming of the

numbers should be

10 apples : 5 people = 2apples

applespeople= 5 people.

(1.3)

I was not alone in my discomfort with “named numbers” and

“units” Here is a testimony from John Gibbon5:

4DG 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

5JDG is male, British, a professor of applied mathematics

6 1 Dividing Apples between People

Fig 1.2 Paul Cézanne Still Life with Basket of Apples 1890–94 The Art

Institute of Chicago Source: Wikimedia Commons Public domain.

At the age of 6 years I was asked the question “How many orangesmake 5?” I recall that I refused to answer This indicated to herthat I was unintelligent, which had been her worry Later in life

I realized why my 6 year-old mind had felt there was somethingwrong with the question The issue was one of units: “How manyoranges make 5 what?” was the problem turning round in my 6year-old mind On the one hand one cannot change oranges intosomething else so I rejected “How many oranges make 5 apples?”

On the other hand, if the answer was “How many oranges make 5oranges?” then we had a tautology I did not know what a tauto-logical argument was but I knew I felt uncomfortable with it

Therefore let us look into equations (1.3) with some attention

1.4 What are the numbers children are working with?

It is a commonplace wisdom that the development of cal skills in a student goes alongside the gradual expansion of therealm of numbers with which he or she works, from natural num-bers to integers, then to rational, real, complex numbers:

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, x−11, , xn, x−1n ]

of Laurent polynomials6in 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

confi-dently claim that the operations (1.1) and (1.2) on Page 1 have little

in common [264]—indeed, operation (1.2) involves an operand

“ap-ple/people” of a much more complex nature than basic “apples” and

“people” in operation (1.1): “apple/people” could appear only as a

result of some previous division

This difficulty was identified already by François Viéte who in

1591 wrote in his Introduction to the Analytic Art [237] that

If one magnitude is divided by another, [the quotient] is

heteroge-neous to the former 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

lan-guage that addresses the subtleties without adding unnecessary

6Laurent polynomials and Laurent series are named after French

mili-tary engineer Pierre Alphonse Laurent (1813–1854) who was the first

to introduce them Another his major achievement was construction of

the port of Le Havre

7IG is female, a PhD student in a leading British university She went to

school in Russia and was educated in Russian, her mother tongue

8 1 Dividing Apples between People

sophistication This is why I devote Chapters 3 and 4 to

discus-sion of dimendiscus-sional analysis, that is, the use of “named” numbers

in physics To my taste, it provides a number of interesting tary examples that may be used if not at school but then at least inteachers’ training

elemen-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 basicand important: that of forming a unit Taking a part of the worldand declaring it to be the “whole” This operation is at the base ofmuch of the mathematics of primary school First of all, in count-ing: when you have another such unit you say you have “two”, and

so on The operation of multiplication is based on taking a set,declaring that this is the unit, and repeating it The concept of afraction 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 ture for a particular arithmetic problem Later we shall see that,once we set up a structure, it inevitably comes into interaction withother structures, thus leading to some (very elementary and there-fore very important) category theory coming into play (see Chap-ter 7)

struc-1.5 The lunch bag arithmetic, or addition of heterogeneous quantities

Usually, only Laurent monomials are interpreted as having cal (or real life) meaning But the addition of heterogeneous quan-tities still makes sense and is done componentwise: if you have

physi-a lunch bphysi-ag with (2 physi-apples + 1 orphysi-ange), physi-and physi-another bphysi-ag, 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 andstraightforward approach to vectors: a lunch bag is a vector (atleast this is how vectors are used in econometrics and mathemati-cal economics)

1.6 Duality and pairing

The “lunch bag” approach to vectors allows a natural tion of duality and tensors: the total cost of a purchase of amounts

introduc-1.6 Duality and pairing 9

g1, g2, g3of some goods at prices p1, p2, p3is a “scalar product”-type8

expression

X

gipi

We see that the quantities giand picould be of completely different

nature In physics, as a rule, the dot product involves

heteroge-neous 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

un-dergraduate linear algebra usually conceals the fact that dot

prod-uct is a manifestation of duality or pairing of vector spaces, thus

creating immense difficulties in the subsequent study of tensor

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

teach-ers 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

pair-ing of two vector spaces U and V of different nature; assumpair-ing 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· vand 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

undis-tinguishable Paradoxically, this provides another source of

diffi-culty for those students who are sensitive to formal logical aspects

of mathematical concepts

8Scalar product is also called dot product or inner product.

9CB is female, holds a PhD, works as an editorial director in a

math-ematics publishing house Her mother tongue is French, but she was

educated in English The episode described happened at age 12

10 1 Dividing Apples between People

Here is a testimony from BB10:

From the time I learned matrices (age 16 or so) I cannot rememberwhich are the columns and which are the rows Given that the ar-rangement of coefficients in a linear transformation can be writtenequally well in a matrix in two ways, it is something that alwaystakes me 10–15 seconds to recall even now

Of course, BB has reasons to be confused: for a mathematician,

dimensional vector space V and its dual V∗ Since the dual of thedual of a finite dimensional space is the same as the original space,that is,

(V∗)∗≃ Vcanonically, there is no intrinsic reason to distinguish between Vand V∗, that is, between the rows and the columns of a square ma-trix

Does the “transition matrix” transform the basis or the nates? (Actually, many books hide the appearance of the inverse

coordi-of the transpose by suitably defining the transition matrix.) Given

a matrix of a linear map, am I writing the map between the vectorspaces or between their duals?

I discuss these and other confusing issues of logical symmetryand 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 quantitieshighlighted by my correspondent Alex Grad12:

I remember that I was taught too that you can’t add apples andoranges, but I “resolved” the problem saying that

3 apples + 2 oranges = 5 fruits,including both categories in one more general, I am curious if thatreflects more interesting notions like in your case Laurent polyno-mials

10BB is male, Russian, has a PhD from an American university and holds

a research position in a British university

11PD is male, Bulgarian, has a PhD in Mathematics, teaches in an ican university

Amer-12AG is male, Romanian, a student of computer science His stories canalso be found on Pages 14 and 87

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 combina-

tions (called “dimensions”) of the three basic units: for length, time

and mass But then even this ring becomes too small since

physi-cists 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

mass1/2length3/2

Indeed, it would be natural to choose our units in such a way

that the permittivity ǫ0 of free space is dimensionless, then from

12 1 Dividing Apples between People

It pays to be attentive to the dimensions of quantities involved

in a physical formula: the balance of names of units (dimensions)

on the left and right hand sides may suggest the shape of the

for-mula Such dimensional analysis quickly leads to immensely deep

results, like, for example, Kolmogorov’s celebrated “5/3 Law” forthe 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 dition of neglect In 1999 NASA lost a $125 million Mars orbiterbecause a Lockheed Martin engineering team used English units

tra-of measurement (inches and feet) while the agency’s team used themore conventional metric system (meters and millimeters) for akey spacecraft operation Only very recently a programming lan-guage was created, F#, which automatically keeps control of units

of measurement and dimensions of quantities generated in the cess of computation

pro-Exercises

Exercise 1.1 To scare the reader into acceptance of the intrinsic

diffi-culty 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 × Aand 3 × B then there is a one-to-one correspondence between Aand B The first such proof, due to Lindenbaum, was announced

by Lindenbaum and Tarski in 1926, and subsequently ‘lost’; Tarskipublished 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 surements based on fundamental units:

mea-• speed of light c = 299, 792, 458 meters per second,

• gravitational constant G = (6.67428 ± 0.00067) × 10− 11m3kg−1s− 2and

• Planck’s constant h = 6.62606896 × 10−34m2kg · 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.

per-For me personally, this is a serious practical issue Every tumn, I teach a foundation year (that is, zero level) mathematicscourse to a large class of students which includes 70 foreign stu-dents from countries ranging from Afghanistan to Zambia Stu-dents in the course come from a wide variety of socioeconomic, cul-tural, educational and linguistic backgrounds But what matters

au-in the context of the this book are au-invisible differences au-in the cal structure of my students’ mother tongues which may have hugeimpact on their perception of mathematics For example, the con-

logi-nective “or” is strictly exclusive in Chinese: “one or another but not

both”, while in English “or” is mostly inclusive: “one or another orperhaps both” Meanwhile, in mathematics “or” is always inclusiveand 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 consecu-

tive1

But it is as soon as you approach definite and indefinite cles that you get in a real linguistic quagmire In the words of mycorrespondent V ˇC2:

arti-[In Croatian, there are] no articles There are many words that can

“serve” as the indefinite articles (neki=some, for example), but no

1Rudiments of a “consecutive and” can be found in my native Russianand traced to the same ancient Slavic origins

2V ˇC is male, Croatian, a lecturer in mathematical logic and computerscience

13

14 2 Pedagogical Intermission: Human Languages

particularly suitable word to serve as definite article (except the

adjective odre ¯ deni = 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 vanje problema tog tipa? = You mean you found a method or the

rješa-method for solving problems of that type?)

What is truly astonishing is that subtler linguistic aspects ofmathematics 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 inLondon I am a native English speaker The lesson was about frac-tions, 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 doingfractions We are being fooled here.” And in that moment I sawmathematics as a set of conventions for which this teacher at leastdid not have a coherent understanding I needed to know why theword ‘of’ and the operation of multiplication were linked, and theteacher could not tell me

On the other hand, the realization of the linguistic nature ofmathematical difficulties can come in later life This is a testimonyfrom Alex Grad4:

When I was about 9 years old, I first learned at school about tions, and understood them quite well, but I had difficulties in un-derstanding the concept of fractions that were bigger than 1, be-cause you see we were taught that fractions are part of something,

frac-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 takeone), 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

3JA 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.”

4AG is male, Romanian, a student at Computer Science faculty which

belongs to Engineering School He says about himself: “yes, my tion 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.

occupa-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

Roma-nian 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

En-glish (I think ), so I think that it’s very likely that my

diffi-culty 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 Altınel5values the fact that his school (the famous

Galata-saray Lisesi in Istanbul) taught him to see and feel cultural

differ-ences 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

5TA is male, Turkish, holds a PhD in Mathematics from an American

univerity, teaches in a French university

6TS 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

16 2 Pedagogical Intermission: Human Languages

to translate’ remains nowadays inasmuch as when I want to derstand a piece of Mathematics properly, I usually have to trans-late it into my own preferred notation, etc., in order that I mightfeel happy with its meaning

un-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 alphabetplays 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 ometry and its applications, and the translation issue is very rele-vant in this part of Mathematics There are so many different ‘for-malisms’, and everyone has their own particular favorites: vectorbundles or principal bundles?; covariant derivatives or connection1-forms?; ‘index-free notation’ or use of indices?; the framework ofjet bundles?; category theory language or not? There is a joke

ge-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

trans-late”, as defined by Tim Swift) matter so much in mathematics?

Because mathematics is itself a Babel of separate but closely

in-tertwined 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

mathemati-cians 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

trans-late what you say in the languages of representation theory,

in-tegral geometry, hypergeometric functions, cohomology, and so on,

in too many languages This takes time

It was amusing to watch how fellow mathematicians, not

ac-customed 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

ques-tioning everything which was agreed upon yesterday, and

eventu-ally 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

synchro-nization marker which aligned together many different languages

and made possible the translation of much more complex

mathe-matics

Units of measurement

In this chapter, I wish to explain my personal obsession with mensional analysis It serves as a justification for the next chapter,where I turn to some examples from the history of physics and thenatural sciences

di-I would like to start with a warning wisely formulated by

The definition of the unit is in the eye of the beholder My mumwould say “One more spoonful” yet to me, there were four morepeas!

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 strongbent towards all kinds of fantasy, adventure and exoticism I re-

member being enchanted by Flaubert’s Salammbô (in Russian

translation) and being amused by the translator’s comment aboutthe 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

1JC is male, Australian

19

20 3 Units of measurement

The translator explained that Flaubert was much criticized for ing ancient units of value and measure with a blatant disregard

us-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 lan- guage an exotic unit of distance, lieue I had a rather vague under- standing of how long a lieue was; apparently, it was sufficiently long since D’Artagnan’s ability to ride eight lieues on horseback de-

served 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 ematics competition I had to solve some kind of a problem about

math-a river bomath-at going upstremath-am math-and downstremath-am—I do not rememberthe problem but I am pretty confident that it was close in spirit tothe following one, which I use here for illustrative purposes

It takes five days for a steamboat to get from St Louis to NewOrleans, 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 Or-leans?

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 lated This can be achieved by choosing the distance from St Louis

calcu-to New Orleans equal calcu-to 5 × 7 = 35 lieue Then the speed of thesteamboat downstream is

35

lieueday ,while the speed upstream is

35

lieueday .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

7 − 5

lieuedayand a raft will drift from St Louis to New Orleans for 35/1 = 35days

Alas, my solution was instantly dismissed as meaningless by ateacher in charge of the competition and I was sternly reprimanded

for my use of a silly word: lieue.

3.2 Discussion 21

3.2 Discussion

Obviously, an introduction and use of an artificial unit of

measure-ment 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

ele-mentary 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

en-tertainments of the book with a pencil on paper; very soon, I forgot

about the books

However, at the same mathematics competition where I

intro-duced 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

Cru-cially, I remembered Karlikania’s key idea: when you move

some-thing to another side of the equality, you change its sign I

remem-bered 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

in-cident 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

22 3 Units of measurement

time I had already received from her and deeply interiorized herlesson:

“Teachers are human beings; do you really think that they have

no right to be stupid? If you know that your teacher is thick, livewith it, keep your mouth shut and treat her with respect, as youwould treat any other person”

My first school teacher—and I dearly loved her—was a life long bestfriend of my mother; half a dozen of my classmates were children

of teachers from our school I grew up surrounded by teachers whowere also friends of my family, or our neighbors, or parents of myfriends, and I retained for the rest of my life a very close affinitywith the teaching community

So, I was inoculated—but more frequently an unexpected lectual conflict with a teacher could be quite difficult for a child.Here is a testimony from Pierre Arnoux2; because of it mathemat-ical content, I quote it here, not in Chapter 18 where more sadstories are assembled

intel-The story dates back from my first year in college (“sixième” inFrench), I was ten years old The teacher gave us as homework aproblem of the type “John is three times older than Peter, and ifyou subtract the age of Peter from twice the age of John, you get40” (I do not remember the exact statement) We were supposed tosolve 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 itcame to J = 3P , 2J − P = 40, then replace everywhere J by 3Pand solve the problem, obtaining 5P = 40, P = 8, J = 24 Thisseemed rather complicated, and very fuzzy, but it clearly workedsince it gave the solution

Next day, when I came back, the teacher asked me to give mysolution on the blackboard; I was very unsure of myself, becausethe solution seemed somewhat “illegal”, since it was clearly out-side of what we had been taught I said so, then explained mysolution (which seemed slightly strange to the other students, asfar 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 foundsomething completely new (to me, at least!) and rather difficult,and that the teacher was completely unable to perceive the differ-ence between this (in fact, the beginning of algebra) and what hehad previously taught us And I also remember that my respectfor that teacher was very much decreased after that

2PA is male, French, a professor of mathematics in a French sity Another story from him is on page 70, and this one continues onpage 181

univer-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

dif-ficult 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

24 3 Units of measurement

Fig 3.1 This woodcut from the Margarita Philosophica of Gregorius

Reisch, published in Freiburg in 1503, shows “Arithmetica” watching

a competition between an “abacist” and an “algorist” Image source:

Wikipedia Commons Public domain Text is quoted from Matthias

Tom-czak, [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 arelatively late phenomenon of the human civilization, and Flaubertwas one of its proponents in the literature

This brief excursion into history justifies devoting the whole ofthe next chapter to history of the dimensional analysis