Borovik A. Shadows of the Truth: Metamathematics of Elementary Mathematics
Подождите немного. Документ загружается.
xi
This text would not appear had I n ot received a kind invitation
to give a talk at “Is Mathematics Special” confe r ence in Vienna
in May 2008, and without an invitation from Ali Nesin to giv e 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 [102] in the procee dings volume of the Vienna
conference edited by Benedikt Löwe and Thomas Müller. Parts of
the text first appeared in Matematik Dünyası, a po pular mathe-
matical journal edited by Ali Nesin [611].
My work on this book was partially supported by a grant from
the John Temple ton Foundation, a charitable institution which de-
scribes itself as a
“philanthropic catalyst for discovery in areas engaging in life’s
biggest questions.”
Howeve r, the opinions expressed in the book are those of the au-
thor and do n ot nece ssarily r eflect the views of the John Templeton
Foundation.
Finally, m y thanks go to the blogging commun ity—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 December 2010
SHADOWS OF THE TRUTH VER. 0.813 23-DEC-2010/7:19
c
ALEXANDRE V. BOROVIK
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
1 Dividing Apples between People . . . . . . . . . . . . . . . . . . . . 1
1.1 Sharing and dispensing . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Digression into Turkish grammar . . . . . . . . . . . . . . . . 3
1.3 Dividing apples by apples: a correct answer . . . . . . . 5
1.4 What are the numbers children are w orking 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 Intermissio n: Human Languages . . . . . . . 13
3 Units of measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1 Fantasy units of measurement . . . . . . . . . . . . . . . . . . . 19
3.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
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
xiii
xiv Contents
5 Adding One by One . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.1 Adding one by one . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.2 Dedekind-Peano axioms . . . . . . . . . . . . . . . . . . . . . . . . . 45
5.3 A brief digression: is 1 a number? . . . . . . . . . . . . . . . . . 46
5.4 How much mathematics can a child see at the
level of basic counting? . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.5 Properties of addition . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.5.1 Associativity of addition . . . . . . . . . . . . . . . . . . . 51
5.5.2 Commutativity of addition . . . . . . . . . . . . . . . . . 52
5.6 Dark clouds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.7 Induction and recursion . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.8 Digression into infinite d escent . . . . . . . . . . . . . . . . . . . 57
5.9 Landau’s proof of the existence of addition . . . . . . . . 59
6 What is a Minus Sign Anyway? . . . . . . . . . . . . . . . . . . . . . . 63
6.1 Fuzziness of the rules . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6.2 A formal treatment of subtraction . . . . . . . . . . . . . . . . 64
6.3 A formal treatment of negative numbers . . . . . . . . . . 66
6.4 Testimonies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.5 Multivalued groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
7 Counting Sheep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
7.1 Numbers in computer science . . . . . . . . . . . . . . . . . . . . 75
7.2 Counting sheep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
7.3 Abstract nonsense . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
7.3.1 Existence and uniqueness . . . . . . . . . . . . . . . . . 79
7.3.2 Unary algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
7.3.3 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
7.4 Induction on systems other than N . . . . . . . . . . . . . . . 80
7.5 Categories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
7.6 Digression:
Natural numbers in Ancient Greece . . . . . . . . . . . . . . 83
8 Fractions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
8.1 Fractions as “named” numbers . . . . . . . . . . . . . . . . . . . 85
8.2 Inductive limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
8.3 Field of fractions of an integral do main . . . . . . . . . . . 90
8.4 Back to commutativity of multiplication. . . . . . . . . . . 91
9 Pedagogical Intermissio n:
Didactic Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
9.1 Didactic transformation . . . . . . . . . . . . . . . . . . . . . . . . . 95
9.2 Continuity, limit, derivatives . . . . . . . . . . . . . . . . . . . . . 98
9.3 Continuity, limit, derivatives:
the Zoo of alternative approaches . . . . . . . . . . . . . . . . . 99
9.4 Some practical issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
SHADOWS OF THE TRUTH VER. 0.813 23-DEC-2010/7:19
c
ALEXANDRE V. BOROVIK
Contents xv
10 Carrying: Cinderella of Ari thmetic . . . . . . . . . . . . . . . . . . 107
10.1 Palind romic decimals and palindromic polynomials 107
10.2 DW: a discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
10.3 Decimals and polynomials: an epiphany . . . . . . . . . . . 112
10.4 Carrying: Cinderella of arithmetic . . . . . . . . . . . . . . . . 113
10.4.1 Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
10.4.2 A few formal definitions . . . . . . . . . . . . . . . . . . . 115
10.4.3 Limits and series . . . . . . . . . . . . . . . . . . . . . . . . . 116
10.4.4 Euler’s sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
10.5 Unary number system . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
11 Pedagogical Intermissio n:
Nomination and Definition . . . . . . . . . . . . . . . . . . . . . . . . . . 123
12 The To wers of Hanoi and Binary Trees. . . . . . . . . . . . . . 129
13 Mathematics of Finger-Pointing . . . . . . . . . . . . . . . . . . . . 131
13.1 John Baez: a taste of lambda calculus . . . . . . . . . . . . . 131
13.2 Here it is . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
13.3 A dialogue with Peter McBride . . . . . . . . . . . . . . . . . . . 135
14 Numbers and Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
14.1 Chinese Remainder Theorem . . . . . . . . . . . . . . . . . . . . 137
14.1.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
14.1.2 Simultaneous Congruences . . . . . . . . . . . . . . . . 138
14.1.3 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
14.1.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
14.2 The Lagrange In terpolation Formula . . . . . . . . . . . . . 140
14.3 Numbers as functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
15 Graph Paper and the Arithmetic of Complex
Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
15.1 Graph paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
15.2 Pizza, logarithms and graph paper . . . . . . . . . . . . . . . 147
15.3 Multiplication of squares . . . . . . . . . . . . . . . . . . . . . . . . 149
15.4 Pythagorean triples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
16 Uniqueness of Factorization . . . . . . . . . . . . . . . . . . . . . . . . 155
16.1 Uniqueness of factorization . . . . . . . . . . . . . . . . . . . . . . 155
16.2 Dialog with AL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
16.3 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
16.4 The Fermat Theorem for polynomials . . . . . . . . . . . . . 159
17 Pedagogical Intermissio n: Factorizatio n . . . . . . . . . . . . 161
SHADOWS OF THE TRUTH VER. 0.813 23-DEC-2010/7:19
c
ALEXANDRE V. BOROVIK
xvi Contents
18 Being in Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
18.1 Leo Harrington: Who is in con trol? . . . . . . . . . . . . . . . 163
18.2 The quest for logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
18.3 The quest for understanding . . . . . . . . . . . . . . . . . . . . . 167
18.4 The quest for power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
18.5 The quest for rigour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
18.6 Suspicion of easy options . . . . . . . . . . . . . . . . . . . . . . . . 176
18.7 “Everything had to be proven”. . . . . . . . . . . . . . . . . . . . 179
18.8 Raw emotions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
18.9 David Epstein: Give students problems that
interest them . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
18.10 Autodidact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
18.11 Blocking it out . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
19 Controlling Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
19.1 Fear of infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
19.2 Counting on and on . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
19.3 Controlling infinity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
19.4 Edge of an abyss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
20 Pattern Hunting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
21 Visual Thi n kin g vs Formal Logical Thinking . . . . . . . . 205
21.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
21.2 EH: Visualisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
21.3 Lego . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
22 Telling Left from Right . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
22.1 Why does the mirror change left and right but
does not change up and down? . . . . . . . . . . . . . . . . . . . 213
22.2 Pons Asinorum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
22.3 TB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
22.4 Maria Zaturska . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
22.5 MP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
22.6 Digression into ethnography . . . . . . . . . . . . . . . . . . . . . 217
22.7 BB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
22.8 PD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
22.9 Digression into Estonian language . . . . . . . . . . . . . . . 221
22.10 Standing arches, hangin g chains . . . . . . . . . . . . . . . . . 222
22.11 Orientation of surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 222
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
SHADOWS OF THE TRUTH VER. 0.813 23-DEC-2010/7:19
c
ALEXANDRE V. BOROVIK
1
Dividing Apples between People
It is important n ot to separate mathematics from life.
You can explain fractions even to h e avy 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 ans wer you immed iately.
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 life
1
; I believe it is
relevant for the principal theme of this book.
When, as a child, I was told by my teacher that I had to be
careful with “named” numbers and not to add apples and people,
I remember asking her why in that case we can divide apples by
people:
10 ap ples : 5 people = 2 ap ples. (1.1)
Even worse: when we distribute 10 apples giving 2 apples to a per-
son, we have
10 ap ples : 2 apples = 5 people (1.2)
Where do “people” on the rig ht hand side of the equation come
from? Why do “people” app ear and not, say, “kids”? There were no
“people” on the left hand side of the operation! How do numbers on
the left hand side know the name of the number on the right hand
side?
1
Call me AVB; I am Russian, male, have a PhD in Mathematics, teach
mathematics in a British university.
1
2 1 Dividing Apples between People
Fig. 1.1. The First Law of Arithmetic: you do not add fruit and people.
Giuseppe Arcimboldo, Autumn. 15 73. Musée du Louvre, Paris. Source:
Wikipedia Commons. Public domain.
There were much deeper reasons for my discomfort. I had no
bad feelings about dividing 10 apples among 5 people, but I some-
how felt that the problem of deciding how many people would get
apples if each was given 2 apples from the total of 10, was com-
pletely d ifferent. I tried to visualize the pro blem 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 two
apples. (I discuss these my childhood fears in more detail in Sec -
tion 19.1.)
Indeed , my childhood experien c e is confirmed by exp erimental
studies, see Bryant and Squire [255]. To emphasize the difference
between the two operations, I started to call operation (1.1) sharing
and (1.2) dispensing or distribution. I discovere d later that these
operation were called partition and quotition in [607]. But ev en
SHADOWS OF THE TRUTH VER. 0.813 23-DEC-2010/7:19
c
ALEXANDRE V. BOROVIK
1.2 Digression into Turkish grammar 3
sharing is not easy and may le ad 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 i n 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 firs t mathematical discovery.
Finally, notice that there are similar special cases of subtrac-
tion; it is worth quoting from Romulo Lins [688]:
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 actuall y
doing a subtraction. For instance, I have to pay $35 and give a
$100 bill to th e clerk. He gives me a $5 bill and says ‘forty’, gives
me a $10 bill and s ays ‘fifty’, and finally gives me a $50 bill and
says ‘a hundred’. I argued that this and doing a subtraction were
quite dif ferent 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 tha t a ‘change gi ving’ operation? The same argument
applies to ‘sharing’ and ‘division’.
1.2 Digression into Turkish grammar
A logical differenc e betwe en the operations of sharing and d ispens-
ing is reflected in the grammar of the Turkish language by the pres-
ence of a special form of nu merals, 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 q uestion
Kaç? (How many?)
The ordinal s take the suffix -inci, adjusted for vowel harmony:
2
DC is male, Italian, has a PhD in mathematics, holds a research po-
sition. In this episode, the language of communication was his mother
tongue, Italia n.
SHADOWS OF THE TRUTH VER. 0.813 23-DEC-2010/7:19
c
ALEXANDRE V. BOROVIK
4 1 Dividing Apples between People
birinci, ikinci, üçüncü, dördüncü, be¸sinci, altıncı, . . .
(first, second, thi rd, . . . )
These answer the q uestion
Kaçıncı?
The distributives take the suffix -(¸s)er:
birer, iki¸ser, üçr, . . .
Used singly, these mean “one each, two each” and so on, as in “I
want two fru its from each of these baskets”; th ey answer the ques-
tion
Kaçar?
Then Eren Mehmet Kıral continues:
When somebody is distributing some goods s/he might say
Be¸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 mentioned
then the distributive numeral is repeated as in the first example.
The numeral may also be used in a n on distributive problem. If
somebody is asking students (or soldiers) to make rows consisting
of 7 people each th en s /he migh t say
Yedi¸ser yedi¸ser dizilin. (Get into rows of seven)
In that context, a story told to m e by one of my colleagues, ¸SUE
3
is ver y intere sting. His expe r ience of arithmetic in his (Turkish) el-
ementary school, when he was about 8 or 9 years old, had a peculiar
trouble spot: he could factorize numbers up to 100 before he learnt
the times table, so he could instantly say that 42 factors as 6×7, but
if asked, on a diff erent 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 (exp ecting an answer:
6, and 1 is left over), little ¸SUE was very unco mfortable—he knew
that 3 did not go into 19. ¸SUE added:
But I did not pay attention to 19 being prime. I had the same prob-
lem when I was asked how many 3s go into 16. It is the same
thing: no 3s go in 16. Simply because 3 is not a factor of 16. This is
perhaps because of distributi ve numerals I somehow built up an
intuition of factorizing, but perhaps for the same reason (because
of the intuition that distributive gave) I could not understand di-
vision with remainder.
As we can see, ¸SUE does not dismiss the suggestion that dis-
tributive n umerals of his mother tongu e could have made it easier
for him to form conce pt of divisibility and prime numbers (although
he did not know the term “prime number”) before he learned mul-
tiplication.
3
¸SUE is Turkish, male, recent mathematics graduate.
SHADOWS OF THE TRUTH VER. 0.813 23-DEC-2010/7:19
c
ALEXANDRE V. BOROVIK