Borovik A. Shadows of the Truth: Metamathematics of Elementary Mathematics

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18.9 David Epstein: Give students p roblems that interest them 185

Dudeney. Of course, there were many other types of Dudeney prob-

lems that I was unable to s olve, 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:

Perhaps it’s appropriate for me to tell you my own vi ews on the

pedagogical principles related to th is 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 a n

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 poss ibilities. 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 u sing the laborious methods. Then

the teacher introduces the method, a lgebra and simultaneous lin-

ear 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 s p ace

theory or topology. I restricted the theorems I proved in the course

to those that could be used to solve my initial problems. The syl-

labus was rather different from the s yllabus of the ofﬁcial course.

Many university mathematics courses are based not on helping

students to solve problems in which they may already be inter-

ested, but instead in terms of laying the foundations for some fu-

ture course. The future course also consists of laying down th e

foundations for a future course, and so on . . . sa tisfaction inﬁnitely

delayed, except for the small minority who become research math-

ematicians.

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, includi ng some research problems.

It is worth quoting from the autobiographic book Ré c oltes et Se-

mailles by the g reat mathematician Alexandre Grothendieck [123]:

Je passais pas mal de mon temps, même pendant les leçons (chut. . . ),

à faire des p roblèmes 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 ci el,

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 naturell es ne manquaient

pas. Ainsi, quand les longueurs a, b, c des trois cotés d’un triangle

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186 18 Bein g i n C ontrol

sont connues, ce triangle est connu (abstracti on faite de sa posi-

tion), 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 p ei ner, mais j’ai dû ﬁnir par

y arriver, à force. De toutes façons, quand une chose me “tenait”,

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 a ll, I th ink, 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, n ot my problems.

And yet there were plenty of truly natural questions. For in-

stance, 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, f or example, the area of

the triangle as a function of a, b, c. Same th ing for a tetrahe-

dron wh en you know the lengths of the six edges—what is the

volume? On this one I think I’ve suffered, but I must have even-

tually succeeded. Anyways when something really had “caught”

me, I wouldn’t count the hours nor days tha t I woul d spend on it,

should I forget all the rest! (And so it is still now. . . )

18.10 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 fascina ted by ele-

mentary geometry, in part because a popular TV series on as tron-

omy and relativity caught my imagination and provided some fas-

cinating sta tements, like the usual visualizations of strange non-

euclidian things. I found the possibility of proving “obvious” sta te-

ments by general principles absolutely fascinating, much more

fascinating than proving more complicated statements. However,

when I asked teachers about non-euclidian geometry, their neg-

ative reaction alienated me very much. Further I found the way

some geometric objects were d eﬁned too ugly to accept. For exam-

ple, 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 ev-

ery statement twice: ﬁrst by elementary geometric constructions

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18.11 Blocking it out 187

with the help of problem-speciﬁ c thought experiments, then by

analysis. That the later proofs were much shorter, more general

and l ooked 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 to-

wards deﬁnitions was “Well, that’s O.K. and obvious, but wha t are

these things real ly 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 in vited to do university entrance

exam. M y math education is completely autodidactic, and was

from German books, but shifted—from age ca. 16 on, after I ob-

tained 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, whi ch 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 in-

terests are still strongly guided by aesth etical impressions, esp. i f

concepts all ow visualizations. Some theories have very special aes-

thetical 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 d e-

rive the whole theory from a handful of basic ideas, which were

often helpf ul in seemingly different math contexts too. Now I read

more and faster, but restrict usually to understanding the tech-

niques of the proofs.

18.11 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 CC

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 rec-

ollections ab out 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.

CC is male, Italian, a professor of logic with research interests in logic,

philosophy of mathematics and epistemology.

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188 18 Bein g i n C ontrol

I’m af raid that I don’t have any recollections of my own to add to

your collection—not because my learni ng of mathematics has been

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:

+ 11

+ 12

+ 13

+ 14

365

see Figure 18.1.

JG is male, has a PhD in mathematics, teaches in a British university.

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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 num-

bers, 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 und er the Microscope [103], and me n-

tioned my own story in Chapter 1 of this book.

When I was 7 years old, the problem of distributing 10 ap ples

between some peo ple, 2 apples a per so n, was very disturbing to me.

I tried to visualize the problem as an o r derly distribution of ap-

ples to a queue of p eople, two apples to each person. Unfortunately,

this necessitated dealing with a potentially unlimited number of

recipients. In horro r I saw an endless line of poor wretches, each

stretching out his hand, begging f or 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 unfortun ate exp erience of the notorious bread queues of the

summer of 1962. There was no famine as such: neither I nor any-

one 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 big-

ger brother who had been standing in the queue fro m 5 in the morn-

ing. A couple of hours later, when I was ap proaching the counter,

another brother came to replace me. We were 6, 11, 15 years old,

respectively.

189

190 19 Controlling Inﬁnity

These bread queues we r e one of the causes of the ev entual fall

from power of Nikita Khrushchev , the supreme Soviet leader of the

time. His name was freque ntly mentioned in the queue, usually

accompanied by expletives. This was the most useful lesson in pol-

itics that I ever got. But it also had an impact on my perce ption of

mathematics.

19.2 Co unting 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 keep ing it under control. I start with a short

story from Archie McKerrell

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 sti ll seems almost part of the

structure of the u niverse!

6 + 18 + 19 + 19 = 62

Michael Fried brough t to my attention to the follow ing passage

from Piaget’s Geneti c Ep istemology [366, 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 coordina-

tion 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 f riend who quoted it as the point of

departure of his interest i n 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 f un,

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 coun ted them, and once again there were ten. He

went around the circle in the other way an d got ten again. And no

matter how he put the pebbles down, when he counted them, the

number came to ten. He di scovered here what is known in math-

ematics as commutativi ty, that is, the sum is independent of the

order. But how did he discover this? Is this commutativity a prop-

erty 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

AMK is male, Scottish, h as a PhD. He teaches at a a British university

and is doing research in theoretical physics.

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19.2 Counti ng on and on 191

was he, the subject, who put the pebbles in a li ne and then in a cir-

cle. Moreover, the sum was not in th e pebbles themselves; it was

he who united them. The knowledge that this future mathemati-

cian discovered that day was drawn, then, n ot from the physical

properties of the pebbles, but from the actions that he carried out

on the pebb les. This knowledge is wha t I call logical mathematical

knowledge and not physical knowledge.

David Cariolaro

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 2

—and memorized them, th ereby getting

acquainted with the behavior of the exponential function . . .

I remember one mathematical event which took place

early in my life—I must have been 4 or 5 at the time.

At tha t age I had the ﬂexibility to keep a hula-hoop go-

ing 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 na me 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, an d 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 Shackell

I don’t remember the i ncident 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 ap p roximately as follows:

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

tion.

PW is male, American, a professor of mathematics.

JRS is male, English, a professor of mathematics.

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192 19 Controlling Inﬁnity

- “One thousand three hundred and twenty three,

one thousand three hundred and twenty four.” -

“John, s top that counting.” - “One thousand three

hundred and twenty ﬁve, one thousand three hun-

dred 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 Verbovskiy

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 num-

ber I could count at that time. Teaching was done in an

abbreviated form: . . . after a thous and goes two thousand,

then three th ousand, . . . , ten thousand, eleven thousand,

twenty thousand, thirty thousand, . . . , one hundred thou-

sand, two hundred thousand, three hundred thousa nd,

. . . , one million. I a ccepted that as a l iteral truth, and

came to conclusion that one thousand multiplied by 37

gives one million. That is, it was assumed that one thou-

sand is followed by one thousand an d one, bu t 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 cra z y, 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 han-

dled 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 brain-

teaser that required ingenuity.

Perhaps, it was when my love to logic was born, be-

cause in logic proofs are very consequentia l . . .

Theresia Eisenkölbl

My brother and I had learned (presumabl y from our parents) how

counting goes on and on without an end. We understood the con-

struction but we were left with some doubt that you could really

VV is male, Russian, has a PhD in mathematics, holds a research posi-

tion in mathematical logic.

TE is female, learned mathematics in German. She has a PhD in Mathe-

matics (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 Counti ng on and on 193

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 kinder-

garten 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 thou-

sand many times. We ended up bei ng convinced that it is possible

to count to a million but slightly unhappy tha t we coul d 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 a nd tried very h ard to explain to me that there is no

ﬁnal / biggest number when you count. I remember I kept asking

her “Where is the gate?”

In my mind I imagined a closed gate that

stops numbers from increasing! Su rely 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 woul d continue addi ng 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 natu ral

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 (Rus-

sian or English), I always answer using the language she chooses

for her ques tion. She was brought as bi lingual from birth; and

switches from language to language easily.

When I was quite young—maybe 5—I asked wha t the biggest

number was, and my father (a physician) told me there was none;

SK is Ukrainian, his mother tongue is Russian, he has a PhD in math-

ematics 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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194 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 q uestion: “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 h ave to wind up somewhere, and it could

not be before B. Later, when I fully understood what the p roblem

was about, it really bothered me that I had produced such a silly

response.

I don’t remember where I encountered the s tatement 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 di-

vision by zero is just prohib ited. I don’t remember precisely why

I got so excited about this statement to the extent that I immedi-

ately went to see the grandmother of my friend (who was retired

mathematics teacher) to discuss the issue. She told me in a di-

dactic manner that indeed it is possible to divide by zero and any

number divided by zero is i nﬁnity. My question “why” was not an-

swered to my satisfaction. I was told that “this i s just so, you will

learn more when you get older”. The only explana tion I’ve got was

through a kind of limit p rocess: 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 con-

cepts seem natural to me. The p roblem I couldn’t overcome was,

that if we, say, take two different numerators we have two differ-

ent limit sequences, their ratio doesn’t change, still we get seem-

ingly the same answer: inﬁnity. I remember that I found it very

weird that for inﬁ nity you immediately get very strange relati ons

which are considered to be correct.

19.3 Co ntrolling i nﬁnity

And I end this chapter with examples of a child’s victory over inﬁn-

ity. Again, contro l is the principal issue. We also see the the weap on

used by children is reduc tion of the inﬁnite to the ﬁnite or at least

VS is male, Russia, a professor of ma thematics in a British university.

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