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

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6.2 A formal treatment of subtra ction 65

of x, which w e would denote −x, was wr itten as opp(x), and, after

skipping a few intermediate applications of the commutative and

associative laws for addition, the example abov e would look like

(40 + 20) − (12 + 5) = (40 + 20) + opp(12 + 5)

= (40 + 20) + (opp(12) + opp(5))

= (40 + opp(12)) + (20 + opp(5))

= (40 − 12) + (20 − 5).

A very clean but excessively detailed calculation.

Tuna Altınel wrote to me recently:

Just in passing, you may mention that it took some time before

comfortably replacing the use of opp wi th that of the − sign. Cer-

tainly, it is another question whether this is related to the way

teaching was done, or a lack of personal mathematical vocation.

I have just found among my books my sixth grade math book

[650] and also the famous deﬁnition:

a − b = a + opp(b)

This is written in a pink colored rectangle as a rule f or integers

and followed in the same rectan gle by an explanation:

“soustraire un entier, c’est ajouter son opposé”.

The word soustraire is bold face, and above the ﬁrst part of

the sentence there is an upward a rrow pointing in the direction of

the “−” sign and an obliq ue double arrow from “entier” towards “b”.

The verb “c’est” is above th e equality si gn. Then there is another

upward vertical arrow from “ajouter” to “+” and an oblique d ouble

arrow from “opposé” to “opp”.

I must say I found this deﬁnition very easy to grasp. I have

a vague recollection that this was not the case for many of my

classmates. Now that I am looking at the pages I also see th at

there is another notation used for n egative nu mb ers: 5

−

meaning

minus ﬁve.

By the way, this appears to be a common theme in childhood

memories of mathematicians: those who were affected by the “new

maths” easily survived it—and frequently en joyed the experience.

Tuna Altınel continues:

I don’t know if th is is relevant to what I wrote above but the book I

quoted from has had an enormous place in my p ath to mathemati-

cianship. We had very bad teachers in the 6th and 7th grades.

The ﬁrst teacher I had in the sixth grade had to quit f or admin-

istrative reasons, then for a long time we didn’t have anyone, an d

ﬁnally someone who knew s ome math but not teaching, and whose

French was worse than the lowest level student i n my class.

My brother pushed me to studying the book myself and ﬁn-

ishing in summer the chapters left undone during the school

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66 6 What is a Minus Sign Anyway?

year. Luckily, my brother’s

and sister’s

notes were available also.

Themselves as well, but as a rule my brother would force me to

“fry myself in my own oil”.

6.3 A formal treatment of negative numbers

The next two stories are also from “new maths” era, French style;

The ﬁrst story is told by Olivier Gerard

You have already a few examples of new math courses in your

book. In my case it was in 1979–1980 with a very motherly female

teacher that I remember very vividly. In hindsight, it is evident sh e

mastered the principl es as well as the way to i mpose it on young

pupils. But many of her colleagues did not. Here is an account of a

point when I was ten about negative numbers. It doesn’t add much

if anything to what you already recount in your book, but let’s say

it is some corroborative evidence.

As I have recently explained this to an English speaki ng friend

in response to some inquiries and also as a counterpoint to some

theoretical discussions between us about concept formation in the

history of mathematics,

Concerning this problem of introducing negative numbers, a

little personal recollection.

When I was in junior high school (French College 6e and 5e,

for 11 and 12 years old students) 30 years ago, our mathematics

teacher took a lot of care to distinguish for us during two weeks

negative signs as afﬁ xed to numbers to make them negative, and

on the other hand n egative an d plus signs as operations between

numbers, givi ng us precis e rules of operations and developments

in which minus ⊖ and plus ⊕ where circled for operations and

simple +, − when attached to n umbers. Thi s was with out any ge-

ometric imagery ( the ﬁrst i ntroduction I remember of such was in

4e and 3e for introducing coordinate sys tems and vectors, distin-

guishing direction and sign, ﬁrst only in one dimension for weeks

then in two di mensions).

For two weeks we made only progressive exercises in this way,

reducing in small steps an expression with a lot of p arentheses

and visually clashin g signs to a ﬁnal result where we were at least

TA’s big brother Kuban Altınel i s a professor of computer science.

Evren Altınel is a medical doctor, but, in her own words, she enjoyed

mathematics at school.

OG tells about himself: “To match the kind of characterization of peo-

ple you use, I am a French-speaking male of 40 years, educated only in

French during the ﬁrst seventeen years of his life, now a mathematical

researcher and computer scientist in the private industry. I have been

interested in mathematics for an early age, at least 7 by my own recol-

lections and papers and if I take my parents word, at least 4 years old,

asking questions and reasoning about quantiti es and counts of th ings.”

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6.3 A formal treatment of negative numbers 67

authorized to make the ﬁnal arithmetical operation that we ha d

practiced in primary school. At ﬁrst we were told exactly what

kind of simpliﬁca tion to do from one line to the next, then we could

do it in the order we found the most convenient.

It was a lot of effort and demanded good focus and systematic

application of rules b efore they began to be internalized.

And then after this time, she told us to consider, “as a con-

venient abbreviation”, a number without a sign + as a positive

number, and to treat an y minus sign in front of a parenthesis as

the “subtraction” or “minus” operation.

She then rewrote the rules in a more d irect and compact form

as the projected shadow of a more rigorous and explicit “real ity”

we had been experiencing for days.

We were (at least the few boys and girls I used to chat with)

so happy to write and treat now the same exercises quickly and

efﬁciently (and a little mindlessly or automatically ) that we felt

a little superior to our former selves and began to look a t compli-

cated expressions with a sense of familiarity.

When discussing this episode with a former schoolmate, say

ten years later whil e a t the University, I was shocked that the only

thing h e remembered about this is th at in math, they were chang-

ing rul es all the time. I told him that it was on the contrary a very

close simile to what was done in natural languages all the time:

you lea rned quick but not very correct or not very precise ways to

say things when speaking orally with your parents, friends and

radio, but you learned also in school a full and scholarly way of

saying the same thing in much longer form which could be ana-

lyzed with grammar and could be used for more varied purposes

than the quick and dirty way. We had learned years before to sub-

tract numbers as a converse of add ition and we had then learn in

high school a fuller story where the origin and quality of things

were more explicit, we learned of new distinctions that we did not

imagine before.

The se cond “new maths” story comes from Pierre Arnoux

My s tory goes back to the last year of high school (“terminale”).

We were studying modern maths, and the lecture of the day was

the construction of the relative integers (Z) from na tural (p ositive)

integers (N) by taking the cartesian product and factoring it by

an equi valence relation; that is, the classical construction of the

symmetrization of the commutative monoid.

We could follow, and understand, all the s teps in the construc-

tion, but the goal of the whol e exercise eluded us completely; what

were we doing? I think nobody in the class understood what was

going on. After the class, during the 10 minutes break, I went to

see the tea cher, and asked her what was the point of the construc-

tion: after all, we had known negative integers for years, and there

PA is male, French, a professor of mathematics in a French university.

Another story from him i s on Page 22.

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68 6 What is a Minus Sign Anyway?

was ab solutely nothing new in this. Fortunately, the teacher un-

derstood very well the subject (she had participated in writing the

book), and was able to answer. She explained that we should con-

sider this as a ga me, and this proved that, to obtain negative inte-

gers, we did not need to invent something completely new coming

from outside: just playing with what we knew, the positive inte-

gers, and us ing a few tool, we coul d build negative in tegers, frac-

tions, real numbers, complex numbers . . .

The explanation was very convincing, and put a new light on

the course; I was very happy to und erstand that, and very sur-

prised that it had never been mentioned anywhere: we were, in

effect, giving complicated answers to tricky, a nd qui te philosoph-

ical, questions that had never been asked! I thin k that this short

conversation had a decisive impact in turning me towards mathe-

matics.

What surprised me also is that, a few years later, after my Ph.

D., I met a gain this high school teacher, and told her this story;

she had absol utely no memory of this di scussion!

When I remember this moment, It evokes two ideas for me:

• the ﬁrst one is that fu ndamental understanding often comes

in ﬂash; a ten-minutes di scussion all ows you to make a big

progress, while you were stuck for hours of hard work with-

out going anywhere (but, you should not forget that, proba bly,

without the hours of hard work, the ﬂash would not come . . . ).

• the second one is that it can happen that the same discussion

seems completely mundane and without particular interest to

one participant, and illuminating to the other one; in partic-

ular, as a teacher, we tend to forget that, because most of the

time, we know quite well what we are sp ea king about.

And ﬁnally, Ted Eisenbe r g

expresses ( perhaps with the beneﬁt

of hindsight) a wish for a more formal approach:

I recall that in the classroom we were tal king ab out removing

nested parentheses that were preceded by a negative sign. T hings

like:

−(3 − 5), −(5 − (3 − 5)), −4(−2(5 − 2(3 − 5))).

The class was having trouble and Mr. Pribnow asked if we could

somehow reverse the order of (a −b). I reali z ed that we could look

at it as:

a − b = a + (

−

1)b

= (

−

1)(

−

1)a + (

−

1)b

= (−1)[(

−

1)a + b]

= (−1)[b + (−1)a]

= (−1)[b − a]

= −(b − a).

TE is male, American, a professor of mathematics education.

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6.4 Testimonies 69

By doing this the whole notion of differentiating between the nat-

ural numbers N, the integers Z as being “signed” numbers, and

that the operation of subtracti on can be written as a sum of signed

numbers

a − b = a + (−b),

became crystal clear to me. I remember being terribly excited

about this observation and I went on to explain it to several

friends. A large door in algebra had opened for us, and to this day I

think of the subtraction of natural numbers in its equivalent form

of adding two signed nu mbers. It amazes me that this equivalence

is not drilled into children. I am convinced that so many of their

problems in beginning a lgebra would disappear if it were.

6.4 Testimonies

At this poin t, a few more childhood testimonies could be instruc-

tive.

BS, aged 6

Each morning in class we spent about 90 minutes on arithmetic,

and were usually issued with postcards containing three additions

(of 3 two or three digit n umbers) and three subtractions (of three

digit numbers). I never got to the subtractions, being too slow: but

ﬁnally worked out that they must be easier, as they involved only

two nu mb ers. So I sta rted doing the three subtractions ﬁrst; of

course I got them all wrong because I jus t added the two numbers.

I hadn’t realis ed it was a different sort of “sum”.

About three months after the above, I’d mastered the algo-

rithms for addition and subtraction in simple cases. We progressed

to subtractions of the sort 72−55. The teacher asked “Now if we try

to take 5 f rom 2 we can’t, so what can we do? Does anyone know?”

I offered “Take the 2 from the 5”, and was not pleased to be told

that wasn’t right. I was very indignant, even after being told the

“correct” process; mine was shorter, easier and gave an answer of

the required sort!

I had no idea that “Addition” and “Subtraction” were other

than formal algorithmic processes to write down an answer that

was “right”, reall y no idea at all that there was “applied arith-

metic” so as to sp ea k.

I don’t think that I got over this difﬁculty until very late in

my maths education. I am pretty sure that as a third year stu d ent

at u niversity I still found it difﬁcult to disentangle conceptually

numbers and numerals, addition/multiplication from the usual al-

gorithms to compute them. Clearly that’s got a lot to do with the

way I learned ari thmetic.

BS is male, English, h as a d octorate in mathematics and a lifetime

teaching mathematics in university setting.

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70 6 What is a Minus Sign Anyway?

BS’s story is echoed in a testimony from AH

AH, aged 9

My clearest memory of learning mathematics was when I was nin e

years of age. We had a blackboard-full of subtractions using de-

composition to do. Because I had always previously subtracted the

smaller ‘number’ from the larger ‘number’ (actually, digits but no-

one ever told me that) in each column, when I had to borrow ‘1’

from the ‘number’ in the next column and then continue with the

subtraction, I remember feeling that this was time consu mi ng and

although I knew it was going to produce a page of wrong answers, I

didn’t feel inclined to do it. Therefore I remember still subtracting

the smallest digit from the largest, which was incorrect.

The next day I think we moved on to the next aspects of math-

ematics. I d on’t recall ever doing them again in that class, but I

suppose we di d .

RE, a ged 6

When I was—I sup pose—about 6, my mother told me about nega-

tive nu mbers. We had not covered them at s chool, and she is not

very mathematical. (I think she even expressed some scepticism

that these were “proper numbers” rather than something that sci-

entists had “made u p”.)

All the same, I got the basi c idea , and was very excited about

these strange new numbers. Next time we were studying subtrac-

tion in school, I got all the questions wrong! Probably I wanted to

show off my new knowledge, so for “7 −4 =?” I answered “−3”, and

so on.

Looking back, I had not understood that while a d dition is com-

mutative, subtraction is not!

Karen Petrie, a ged 7

My confusion came at about the age of 7. We were sent home with

a worksheet with many p roblems of the form a−b = c, where a and

b were given integers and we had to provide c. One of the problems

was something like 6 −7 =? The rest had all ha d positive answers

so this stumped me. My father, took some time to explain to me

why th e answer was −1. I went back into school to tell th e teacher

the next day, she was horriﬁed that my father had explained this

‘mistake’ to me. She did so by saying you do not get a negative

number of apples do you there are only 0 or more than 0. I took

AH is female, a New Zealander living in England . She has a PhD in

Mathematics Education. She is a mathematics teacher educator. This

episode occurred during her schooling in New Zealand.

RE is male, English, holds a PhD in mathematics.

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6.5 Mul tivalued groups 71

this to heat in such a manner that later on in my schooling when

negative numbers were introduced, I was convinced th ey were al-

ways the resu lt of mista kes. It took me a long time to realize that

they were valid mathematical concepts, that could be useful, they

just do not relate very well to apples.

6.5 Multivalued g roups

The stories told by BS (Page 69) and AH (Page 70) are ye t another

conﬁrmation of what increasingly ap pears to be a g eneral principle:

almost every error made by children because the y thoug ht that “it

was natural to do it that way” is developed in the “grow n up” math-

ematics into a serious theory.

In BS’ and AH’s cases, a child did not care much about what is

addition and what is subtraction, and, moreover, which is which.

A systematic application of this approach means that non-n egative

integers N ∪ {0 } can be endowed with a 2-valued binary operation

(x, y) 7→ [x + y, |x − y|].

This turns out to be one of the basic examples of the so-called n-

valued groups as introduce d by Sergei Novikov and Victor Buch-

staber; see [9] an d Exercise 6.4 below for a more detailed discus-

sion. To express the confusion which operation is which, [ ] denotes

a multiset, that is, an unordered set with possibly repeated ele-

ments. Multiset is a very n atural concep t; fo r example, the roots of

the equations

(x − 1)

= 0

and

(x − 1)

= 0

form multisets [1, 1] and [1, 1, 1].

The origin s of the theory of n-valued groups lie in algebraic

topology.

I do not know why this is happening; maybe mathematics is rich

enough to contain an analogue of everything—or maybe there are

some intrinsic reasons when e very “simple and natural” alternative

to “canonical” mathematics generates a rich mathematical theory.

Exercises

Exercise 6.1 Gwen Fisher

contributed this old chestnut:

KP is female, British, holds a PhD, is researcher is computer science.

GF is female, American; she h as a MA in mathematics and a PhD

in mathematics education, teaches mathemati cs and creates mathe-

matically inspired beadwoven art—see http://www.beadinfinitum.

com/ and http://www.gwenbeads.etsy.com/.

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72 6 What is a Minus Sign Anyway?

There was a problem that stumped me as a child. I th ink my 3rd

grade teacher told it to me, and I didn’t fully understand it until I

was in gra d school. It goes like this. . .

3 men go to stay at a hotel. The cost of 1 room is $30, so they

decide to split the room 3 ways, each paying $10. Later, the hotel

clerk realiz es that he overcharged the men by $5 total, so he gives

the porter 5 one dollar bills to give back to th e men. The porter

steals $ 2 and gives the men back the other $3. Thus, each man

paid $9 for the room. So, 3 men, times $9 per man is $27, plus the

$2 stolen by the porter is $29. But the room was originally $30, s o

where did the other $1 go?

Exercise 6.2 Expand the system of natural numbers N by taking a dis-

joint union of two isomorphic copies of N (denoted N and −N) and a set {0 }

containing of a new symbol 0:

Z = N ⊔ {0 } ⊔ −N.

Deﬁne on Z an operation of addition “+”, such that:

• its restriction to N and −N coincides with the earlier deﬁned operation

+ on N and −N.

• 0 is the identity element for +.

• hZ; 0, +i is a commutative group,

thus making Z into the familiar additive group of integers.

Exercise 6.3 After that deﬁne on Z an operation of subtraction “−” an d

prove the identity discussed by John Baldwin:

(a + b) − (c + d) = (a − c) + (b − d).

You may be interested to know that this exercise is Theorem 237

of Landau’s Grundlagen der Analysis [ 53, p. 101]. But Landau was

not in hurry to introduc e subtraction; he ﬁrst developed the the-

ory of fractions, real and complex numbers, and only then proved

the properties of subtraction, simultaneou sly for all these number

systems.

Exercise 6.4 [9] For a set X, we denote b y (X)

be the set of unordered

n-tuples of elements of X, possibly with repetitions. We shall use square

brackets [ ] to write unordered n-tuples. For example, [1, 2, 1] and [1, 1, 2]

represent the same element of (N)

, while [1, 1, 2] a nd [1, 2, 2] are different

elements of (N)

. We shall call (X)

the n-multiset of X.

An n-valued multiplication on X is a map

∗ : X × X → (X)

x ∗ y = [z

, z

, . . . , z

], z

= (x ∗ y)

The fol lowing three axioms deﬁne n-valued group structure.

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6.5 Mul tivalued groups 73

• A ssociativity. The n

-sets:

[x ∗ (y ∗ z)

, x ∗ (y ∗ z)

, . . . , x ∗ (y ∗ z)

[(x ∗ y)

∗ z, (x ∗ y)

∗ z, . . . , (x ∗ y)

∗ z]

are equ al for all x, y, z ∈ X.

• Un it. An element e ∈ X such that

e ∗ x = x ∗ e = [x, x, . . . , x]

for all x ∈ X.

• Inverse. A map inv: X → X such that

e ∈ inv(x) ∗ x and e ∈ x ∗ inv(x)

for all x ∈ X.

Check that a 2-valued operation

x ∗ y = [x + y, |x − y|]

deﬁnes a 2-valued group structure on R

= {x ∈ R | x > 0} because it

satisﬁes the axi oms of the unit, inverse and associativity:

• Un it: e = 0.

• Inverse: inv(x) = x.

• A ssociativity: 4-subsets of R

[x + y + z, |x − y − z|, x + |y − z|, |x − |y − z||]

and

[x + y + z, |x + y − z|, |x − y| + z, ||x − y| − z|]

are equ al for all nonnegative x, y, z.

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