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

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19.3 Controlli ng inﬁnity 195

to a clearly described (even if inﬁnite) proced ure. And we have al-

ready familiar themes, like no mination and inventing names.

Fig. 19.1. Vision to the Youth Bartholomew. Mikhail Vasilyevich Nesterov,

1890. The T retyakov Ga llery, Moscow. Source: Wikipedia Commons. Public

domain.

Bartholomew was to become St. Sergius of Radonezh, the most ven-

erated spiritual leader of medieval Russi a.

David Jefferies

I can vivid ly remember grasping the concept of inﬁnity. I was

walking along a road, with walls and hedges coming in periodi-

cally 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. ,

I can remember also being told about decimals by my Dad, af-

ter 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 combinati on of th e inﬁnite and the rep-

resentation of irrational numbers.ixnumber!irrational It is clear

DJ is male, En glish, has a PhD in physics.

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196 19 Controlling Inﬁnity

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 imag-

inable memory device.

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 num-

bers 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 i t deﬁnitely made me feel really s at-

isﬁed, though I did n ot know why or what I would do with this

information. Also, no one of my friends seemed the least b it inter-

ested: 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 bed-

room, q uietly, 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 i mportant than the 9s were numbers like 100, 1010, 110, 111

then 1000, 1001, 1010 and so on, an d I played out these in l onger

and longer columns in my head until I was absolutely clear how

it worked, and I just knew that what I would now cal l the place-

integer system ﬁtted together in a completely satisfactory way. It

was sti ll early so I continued thinking about these numbers, and

remembered that we used to argue over whether or not there was a

largest number. We would make up peculiar names in thes e argu-

ments (the boys in the nursery class, that is): so somebody would

say that a zil lion was largest, and someone else might say, no, a

squil lion was, and so on, nonsense on nonsense. But if these dis-

cussions meant anything, I thought, it should all clear itself up in

the column pictures I n ow had in my mind. I then tried to pictu re

the 0/1 arrangement of the largest number, and was tickled at the

thought that if I th en cranked everything up by rolling the small-

est 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 d o not know where it

came from, but I suddenly realized that there was no largest num-

ber, a nd 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 i t, and recited what I had thought out. Poor fellow: it was

the overtime season and he worked more than 8 hours a day six

DD is male, American, mathematical physicist, works in a British uni-

versity.

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19.3 Controlli ng inﬁnity 197

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 h ad

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 nu mbers.

Although this incident is ﬁltered through my decades of doing

mathematical physics, it remains clear in my mind and a lways

has.

John W. Neuberger

At about the age of about seven, I lived with my family in a n

old two story wooden farm house in Iowa. On some very cold and

windy nights (the house would seem to lea n with the wind and

then relax wh en 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 wh ile, 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 mathemati cal instruction was always English.

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 deligh ted at university to begin to make a connec-

tion. 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.

I do not know if this is of interest to you or not but here i s a

thought-experiment of mine, probably around the a ge of 7-8, for

sure after 6 a nd before 10.

I started thin king about death and wanted to convince myself

I would never die, instead of thinki ng 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

JWN is male, American, a professor of mathematics.

MM is male, French, a professional philosopher with research interests

in philosophy of mathematics. Th e episode took pl ace at a ge 7 or 8.

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198 19 Controlling Inﬁnity

that a full life goes on in each dream (an interesting point about

personal i dentity, I guess). Now, that other life would be ﬁnite and

have only a ﬁnite n umber of nights. So, I thought further that in

each night there must be a ﬁ nite number of dreams, encapsulating

a ﬁnite number of lives. This was still short of inﬁnity, so I started

thinking that in each of these ﬁ nitely many dreams of the ﬁ nitely

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 s ure tha t I was safe that way (i.e. that I would go

on living forever), but I convinced myself that thes e were enough

lives to live, so that even if the process would end, I would still

have lived enough, and stopped thinking a bout it.

commented on M M’s post in my blog:

This is beautiful. I had the s ame “grand thought” at about

the same age and I remember that I h ad the feeling of

a true realization upon fully grasping its implication. The

difference being that I did not look for a solution to a prob-

lem (death) rather I had been thinking about the nature

of dreams and wh y 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 s tarted to think about what I wished

to do with the freedom I had ga ined knowing that I was

now in dream state, only to wake up. What a disappoint-

ment. 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 l ucid dreaming to normal dreaming. I had

tricked myself out of the lucid state. Am I still dreaming?

A reader of my blog, who signed his com ment only as JT, re-

marked that little Mathiew was safe because of König’s Lemma:

Every inﬁnite ﬁnitely branching tree ha s an inﬁn ite path (with no

repeated vertices).

19.4 Edge of an abyss

I ke ep 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 m ean—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

LB is male, Italian, an electronics engineer.

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19.4 Edge of an abyss 199

I have to come out from somewhere and c atch 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 wor ld of inﬁn ity is dead dangerous.

Here is a story from Alexander Olshansky

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 E ngels. Every Sunday on the sport grounds

of the base there were some sport competitions. A relay race of

800 meters + 400 meters + 20 0 meters + 100 meters

was quite popular; it was called Swedish relay. After two or three

races I h ave come to an ob vious 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 para d oxes: if the race continues the same way,

50 meters + 2 5 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).

Little Sasha was walking on the edge of an abyss; being a

trained mathematician, his father had a false sense of security be-

cause pe rhaps he believed that Zeno’s “arrow” paradox (of which

Swedish relay is an obvio us version) is resolv ed in elementary cal-

culus by summation of the geometric progression

800 + 400 + 200 + 100 + 50 + 25 + ··· = 1600

= 4 × 400.

This is true; the runners will ind eed reach the end of the 4th circle,

and fairly quickly.

But if yo u think that Zeno’s p aradox 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 ba-

ton? Indee d, 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?

AO is male, Russian, holds professorships in mathematics in Moscow

and the USA.

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200 19 Controlling Inﬁnity

The baton paradox is a version of a supertask inve nted by Jon

Pérez Laraudogoitia [139] and is of ser ious importance for d iscus-

sion of foundations of statistical mechanics. I borrow its compact

description f rom Zurab Silagadze’s survey [149] of Zeno type para-

doxes as they appear in modern science:

In [139] Pérez Laraudogoitia constructed a beautifully simple su-

pertask 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 i nterval

of unit length. Some other particle of the same mass approaches

the system from the right with unit velocity. In elas tic collision

with identical particles the velocities are exchanged after the colli-

sion. 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 colli ding its left closest neigh-

bor. Therefore all particles are at rest after the collision supertask

is over and we are left with paradoxical conclusion [139] that the

total initial energy (and momentum) of the system of particles ca n

disappear by means of an inﬁnitely denumerable number of elas-

tic collisions, in each one of which the energy (and momentum) is

conserved!

Fig. 19 .2. Energy and momentum are disappearing in Pérez Laraudo-

goitia’s inﬁnite sequence of collisions.

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Patt ern Hunting

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 a n algebra book and

found out how and wh y

(x − 1)(x + 1) = (x

− 1).

That was enough to make me fal l in l ove with algebra.

But no-one, n o-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?

That could have led me on to

(x − y)(x + y) = x

− y

and also the differences between ascending squares being

1, 3, 5, 7 . . .

I ﬁgured that out much later, maybe even at university.

201

202 20 Pattern Hunting

Richard Porter

It was probably the ﬁrst ti me 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. al ternate in the

last digit between 5 and zero. Other examples given were the 2,

10 and 12 times tables. The teacher su ggested that there was no

such pa ttern for the 7 times table. I put up my hand and pointed

out that there was: (7, 14, 21 , 2 8, 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 Ger ard:

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 fascin ated by multipl ication 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 p ossible. 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 re-

served f or older pupi ls. I a sked 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 ali gnments. I ended a sking my mother to

help me paste these large sheets together and make neat f olds 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

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 p os session 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 knowl-

edge of addi tion. In the 3× column you add 3 to the previous resu lt.

You can also transpose rows to columns if you do not last track of

your copy.

RP is male, British, has a PhD in mathematics, teaches in a British

university.

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20 Pattern Hunting 203

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 multipli-

cation. I did this because, I would have had to ﬁll h alf a column

or half a row with additions to ﬁl l the number in the d iagonal by

addition only. I had a simple rul e 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 squ ares. 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 diagona l, make the difference of the previous two

terms, add two and th en a dd 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 u sed 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 pra ctical i.e.

something easy to start again when I was interrupted b ut I was

not able when I was 7 to formulate the relation between the dif-

ference and the number being squared so I always started from

10 × 10 and 11 × 11 and then mentally add ed 2 each time I went

one step further on the dia gonal, until I had the correct difference

to add for the next mis sing 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 certai n sense it would have b een more difﬁcult for me to

apply a rule like this when maki ng my multiplicati on tables than

to keep track of th e current difference to add.

I did not discu ss spontaneously mathematics with others when

I was i n primary school. I was mai nly satisﬁed to have my large

multiplica tion table, as a trophy.

In cinquième (s econd year of junior high school, I was 11 year

old) we were learning “produits remarquables”

When I saw

(a + b) × (a + b) = a

+ 2ab + b

on the blackboard in class, I remember saying to myself “thi s can-

not be right. I know really what is going on for years.”

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204 20 Pattern Hunting

I had no time to check immediately but when I came back from

school, I went to my room and I compared the f ormula from the

lesson, by rewriting it for b = 1 and reworked my old rule for

numbers on the diagonal and was really s hocked and depressed to

see they gave the same differences.

It was clear to me tha t this formula was really superior to my

rule, even though it was not more useful for ﬁlling a diagonal be-

cause usually you do not do that by large gaps. It was superior

technology, I remember thinking this was “silent” technology be-

cause you did not n eed to make a sentence in your h ea d to un-

derstand 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 und erstanding of multiplication tables

which was just a tiny part of the formula, that now all my class

would take for granted, wi thout having worked so hard with mul-

tiplicati on tables.

I slowly realized more vividly that what others were comput-

ing an d discovering a bout numbers applied to the numbers I was

using for my private purpose, tha t i t was the same business. That

it was not like the stories I could invent and th at nobody could in-

vent 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 bu t I believe it was the way I was thinking. The

fact of having been taught about numb ers 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 a s paradoxical.

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