Borovik A. Shadows of the Truth: Metamathematics of Elementary Mathematics
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2 Pedagogical Intermission: Human Languages 15
The language of my mathematical instruction was Romanian,
which was also my mother tongue. The word for fraction in Roma-
nian is fractie, and that was the terminology used b y my teacher
and in textbooks. The word i s used frequently in common language
to express a pa rt 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 actual ly that was how fractions were introduced to us—p upils
(like parts of an object) and only later the notion was extended,
and so maybe I ha d problems accommodating to the new notion.
Or maybe both reasons . . .
Tuna Altınel
5
values 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. An other p otentially 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 p atriotic
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 Swift
6
:
By the way, related to the i ssue of language, I was taught to read
and write (in a Yorkshire primary school in the mid 1960s) using
an ‘experimental langua ge’ called ITA (Initial Teaching A lphabet);
I don’t know if this hindered or accelerated my development of
communication skills, b ut I do remember that I seemed to be the
only p erson in the class who, when we finally arrived at sta ndard
English when I was six or seven, had to translate everything back
to ITA before I was hap p y with its meaning. An echo of that ‘urge
5
TA is male, Turkish, holds a PhD in Mathematics from an American
univerity, teaches in a French university.
6
TS is male, English. He wrote about hi msel f:
My PhD was from an En glish university (namely the University of
Southampton). Most of my teaching and research talks have been
conducted in English, although I h ave, on occasions, lectured in
French. My university positions have been at English and Scottish
institutions.
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16 2 Pedagogical Intermission: Human Languages
to translate’ remains nowadays inasmuch as when I want to un-
derstand a piece of Mathematics properly, I usually have to trans-
late it into my own preferred notati on, etc., in order that I might
feel happy with its meaning.
Fig. 2.1. Initial Teaching Alphabet (Sir James Pitman, 1901–1985) . This
alphabet plays a key role in a story told by Tim Swift, Page 16.
Tim Swift added further:
My main mathematical interest has been that of differential ge-
ometry and its applications, and the trans lation issue is very rele-
vant in this part of Mathematics. There are so many different ‘for-
malisms’, and everyone has their own particular favorites: vector
bundles or principal bundles?; covariant derivatives or connection
1-forms?; ‘index-free notation’ or use of indices?; the framework of
jet bundles?; category theory langu age or not? . . . There is a joke
about dif ferential geometry being the s tudy of concepts invaria nt
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 d efined 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 Mic roscope [103]. Here I reproduce only a very illuminating
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2 Pedagogical Intermission: Human Languages 17
quote from the late Israel Gelf an d, one of the greatest mathemati-
cians of the 20th century. I had the privilege to work with him
and closely observe h is idiosyncratic ways of doing mathematics.
Once he told me something r emarkably c onsonant 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 s aying to me. To understand
a mathematical fact, you have to tran slate it into a mathematical
language which you know. Most mathematicians use three, four
languages. But I am a n old man and know too many langua ges.
When you tell me something from combinatorics, I have to trans-
late wh at you say in the languages of representati on theory, in-
tegral geometry, hypergeometric functions, cohomology, a nd so on,
in too many languages. This takes time.
It was amusing to watch how fellow mathematicians, not ac-
customed to the pecu liarities of Gelfand’s style, spoke to him the
first time. Very soon they became bewildered at why he insisted
on their giving him re ally basic, everyone-always-knew-it kinds of
definitions; then they were taken aback when he became furiou s 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 mo r ning, their second conversation u sually
was even more entertaining, because Gelfand started it with the
demand to repeat all the definitions; then he proceeded by que s-
tioning everything which was agreed upon yesterday, and eventu-
ally settled for a de finition given in a com pletely 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 man y different languages
and made possible the translation of much more complex mathe-
matics.
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3
Units of measurement
In this chapter, I wish to explain my personal obsession with di-
mensional analysis. It serves as a justification for the next chapter,
where I turn to some examples f r om the history of physics and the
natural sciences.
3.1 Fantasy units of meas urement
L’eau, qui valait au début du sièg e
deux késitah le bât, se vendait maintenant
un shekel d’argent ; les provisions de viande
et de blé s’épuisa ient aussi ; on avait
peur de la faim ; quelques-uns même parlaient
des bouches inuti les, ce qui effray ait tout le m onde.
Gustave Flaubert, Salammbô
As a bo y I was a voracious reader with, unsurprisingly, a strong
bent towards all kinds of fantasy, ad venture and exoticism. I re-
member being enchanted by Flaubert’s Salammbô (in Russian
translation) and being amused by the translator’s comment about
the line in the novel that I used as the epigraph to this chapter:
The water which was worth two kesitahs per bath at the opening
of the siege was now sold f or a shekel of silver . . .
The translator explained that Flaubert was much criticized f or us-
ing ancie nt u nits of value and measure with a blatant disregard
to their actual meaning and value. “So wh at?”—thou ght I—“it does
not matter what le bât, kesitah, or shekel were, it just sounds great.”
Reading Jules Verne’s Vingt mille lieues sous l es mers and
Alexandre Dumas’ Les Trois Mousquetaires introduced into my lan-
guage an exotic unit of distance, lieue. I had a rather vague u nder-
standing of how long a lieue was; apparently, it was sufficiently
19
20 3 Uni ts of measurement
long since D ’ Artagnan’s ability to ride eight lieues on horseback de -
served a special and ver y respe c tf ul mention in the 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 sud denly called to take part in a district math-
ematics competition. I had to solve some kind of a problem about
a river boat going upstream and downstream—I d o not remember
the proble m but I am pretty confident that it was close in spirit to
the follo wing one, which I use here for illustrative purposes.
It takes five days for a steamboat to get from St Louis to New
Orleans, and seven days to return from New Orleans to St Louis.
How long will it take for a raft to drift from St Louis to New 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 up stream can be easily calcu-
lated. This can be achieved by choosing the distance fro m St Louis
to New Orleans equal to 5 × 7 = 35 lieue. Then the speed of the
steamboat downstream is
35
5
= 7
lieue
day
,
while the speed upstream is
35
7
= 5
lieue
day
.
Since the speed of the current gets added to, or subtracted from,
the speed of the steamship in still water, the speed of the current is
7 − 5
2
= 1
lieue
day
and a raft will drift from St Louis to New Orleans for 35/1 = 35
days.
Alas, my solution was instantly dismissed as meaningless by a
teacher in charge of the compe tition and I was sternly reprimand ed
for my use of a silly word: lieue.
3.2 Discussion
Obviously, an introduction and use of an artificial unit of measure-
ment is logically e qu ivalent to an introduction of an (intermediate)
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3.2 Discussion 21
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 c annot 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, w e 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 suffi ciently interested to follow the mathematical en-
tertainments of the book with a pencil on paper; ve ry soon, I fo rgot
about the bo oks.
Howeve r, at the same mathematics competition where I intro-
duced the fantasy lieue I discovered that another problem looked
like being open to a Kar likania treatment. It was something about
money: two thing s together cost that much, etc. I do no t remembe r
the proble m; but I re member that I looked at it and decided to try
Karlikan ia’s approach. So, I denoted some quantity ap pearing in
the problem by a—I chose the quantity more or less at rando m—
and started to rewrite the problem as a balance d 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 princip le because, at the time of reading, it struck me as
slightly paradoxical.
To my surprise, ev erything in my first attempt at doing algebra
worked out in the smoothest possible way, and I f airly quickly got
the answer.
Alas, this my solution was also dismissed as goin g ahead of the
curriculum, and I was ag ain re primanded for wrongly labeling the
unknown by a, when eve r yone knew that it had to be x. I did n ot
argue w ith the teacher an d did n ot 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 w ere 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
time I had already rece ived from her and deeply interiorized her
lesson:
“Teachers are human beings; do you really think that they have
no righ t to be stupid? If you know that your teacher is thick, live
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22 3 Uni ts of measurement
with it, keep your mouth shut and treat her wi th respect, as you
would treat any other p erson”.
My first school teacher—and I dearly loved her—was a life long best
friend of my mother; half a dozen of my classmates were children
of teachers fr om our school. I grew up surrounded by teachers w ho
were also frien ds of my family, or our neighbo rs, or parents of my
friends, and I retained for the rest of my life a ve r y close affinity
with the teaching co mmunity.
So, I was inoculated—but usually an unexpecte d intellectual
conflict with a teacher could be quite difficult for a child. Here is
a testimony from Pierr e Arnoux
1
; because of it mathematical con-
tent, I quote it here, not in Chapter 18 where more sad stories are
assembled.
The story dates back from my first year in college (“sixième” in
French), I was ten years old. The teacher gave us as homework a
problem of the type “John is three times older th an Peter, and if
you subtract the age of Peter from twi ce the age of John, you get
40” (I do not remember the exact sta tement). We were supposed to
solve this by words, but I could not make it, and I asked my father.
He explained to me that we could give name to the ages, so it
came to J = 3P , 2J − P = 40, then replace everywhere J by 3P
and solve the problem, obtaining 5P = 40, P = 8, J = 24. This
seemed rather compli cated, and very fuz zy, but it clearly worked
since it gave the solution.
Next day, when I came back, th e teacher asked me to give my
solution on the blackboard; I was very unsure of myself, because
the solution seemed somewhat “illegal”, since it was clearly out-
side of what we had been taught. I said so, then explained my
solution (which seemed s lightly strange to the other students, as
far as I remember). When I had fini shed, the teacher said “well,
this is the solution, why do you see a problem? Why all th is fuss?”
and dismissed me to my place.
I remember very clearly my shock at feeling that I had found
something completely new (to me, at l east!) and rather difficult,
and that the teacher was completely una ble to perceive the differ-
ence between this (in fact, the beginning of algebra) and what he
had previously taught us. And I also remember that my respect
for that teacher was very much decreased after that.
3.3 Hi story
When writing this book, I sent to a few historians of mathematics
the follo wing questions:
1
PA is male, French, a professor of mathematics in a French univer-
sity. Another story from him is on page 67, and this one continues on
page 176 .
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3.3 History 23
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 co ncerning travel problems
that could be reconstructed as similar to the one that I considered,
and sent me his rough literal translation of Problem 108 fro m 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 foll owing: multiply 5 by 8, makes 40, and say thus: in
40 days one will make the trip 8 ti mes, and the other 5 times, so
both together will make it 13 times.
Now say: if 40 days equal 13 tri p s, 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 introd uction
of a convenient dimensio nless u nit, a trip. For abacists who liv ed in
a strictly regimented traditional society, it was psychologically dif-
ficult to move away from established units of m easurement. 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 wor d “mile” does not appear in the solution.
Albrecht Heeffe r wrote to me with further examples:
To answer your question, is this mechanism of an arti ficial unit
been done in the history of algebra? Yes ind eed, if have found it in
several abbaco manuscripts, and it functions as an intermediate
unknown is some sens e. 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 un kn own is used for, let us say, the
largest numb er, c. Then one supp os es th at the smaller, a is 1. This
allows to derive that c = b
2
and hence to d erive a value for b and
c. In the las t stage, the value of a is derived to meet the extra
conditions.
Again, it appe ars that the artificial unit is dimen sionless.
To freely use arbitrary, m ade on-the-fly units of measurement,
one has to be conditioned in a cultural relativism. The latter was a
relatively late phenomenon of the human civilization, and Flaubert
was one of its proponents in the literature.
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24 3 Uni ts of measurement
Fig. 3.1. This woodcut from the Margarita Philosophica of Gregorius
Reisch, pu blished 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 Matthi as Tom-
czak, [841].
This brie f excursion into history justifies devoting the whole of
the next chapter to history of the dimensional analysis.
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