A Comment on Equality




          This all began when someone on a list devoted to mathematics

education objected to the sentence "pi = 22/7", which is the sort of thing

one sees on examinations in college calculus.  Apparently the schools are

so devoted to the use of calculators that students these days avoid the

use of symbols like the Greek letter for "pi" and the symbol for

sqrt(2), replacing them with decimal equivalents as soon as possible in

their calculations.  Approximate decimal equivalents, of course, but --

such is today's style -- written as if exact.  The question arose, what do

these students mean to say when they use the "=" sign this way?  Perhaps

they are not really wrong, given the interpretation they have in mind.


          I disputed this, and claimed that to these students equality was

identity rather than approximation, something that seems to be troubling

the schools these days in those places where the calculator is regarded as

the repository of truth.  My point was a psychological one, of course.  If

asked, the student saying "pi = 22/7" would probably admit this was not

quite so, if you wanted to get picky about it; but for daily life this was

his idea of equality.  He certainly was not talking about equivalence

classes of some other sort.


          Somehow this debate segued into semantic questions quite remote

from the misfortunes brought upon us by the use of calculators in school

mathematics instructon.  For example, is it true that "=" means identity

when Euclid says the square on the hypotenuse is "equal" to the sum of the

others?  And does "=" mean identity even in the apparently ambiguous

statement (x^2-1)/(x-1)  = x+1 ?  This last equation is ambiguous because

it is clear the two sides are not the same when considered as rational

functions (where the convention about implied domain gives us different

domains for them), and yet are the same as algebraic objects, i.e. as

members of the field of quotients of P[x].  (Or, more precisely, they are

the same if regarded as names of suitable equivalence classes.)


      I think that to answer the question of equality we do have to make

some decision about what the nature is, of those objects about which we

wish to make mathematical statements.


      Actually, one of my email pen pals had already signalled a common

attitude when he spoke of two "expressions" as being equal, thus finessing

(he thought) the distinction between a rational function and a member of

the field of quotients just mentioned.  At one point he objected to my

having written (tongue in cheek, imitating a straw man in a philosophy

department) that Lincoln was a seven-letter word, asking didn't I mean

"Lincoln" was a seven-letter word. I could ask him, I think, if he didn't

mean to ask if rather ""Lincoln"" was a seven-letter word, since Lincoln

is a man, "Lincoln" his name, and ""Lincoln"" the word expressing that

name.  Goodness.  Such philosophic niceties are the stuff of Logic 101

(for non-math students) as given in the department of philosophy, where

one can spend a month on the mortality of Socrates; but they will not

serve the mathematician or the general cultured modern citizen, except as

material for sophomoric levities.


      We will never get to the bottom of mathematics without a Platonic

stance.  In the days of the newmath they tried to get the kiddies to

distinguish between numbers and numerals, and got themselves in hot water

with exam questions such as "Write down the numeral that names the number

that solves the ...", which they soon discovered that wasn't for kids.  It

only got worse when it was discovered that the letter x was now the name

of a numeral, and the word "pronumeral" had to be invented for this new

idea. In most contexts this is all unnecessary, and in fact leads to

infinite regression.  We should be reasonable in our language, and make

distinctions only when needed, but since we do have to understand the

distinctions all the same, we must somewhere agree on a rock-bottom

reality upon which to begin our building of names and names-of-names, etc.


      When I say "13 = XIII" I am already Platonic.  I am saying that the

number symbolized by the left symbol is THE SAME AS the number symbolized

by the right.  Same number?  What's a number?  A number is an idea. 

It is not inherent in the ink that is spilled on paper to form the numeral "13". 

Nothing we write on paper is a mathematical object; mathematical objects

are ideals and dwell in Plato's heaven.  As soon as they are something else

we have trouble.


      All the stuff we write in ink, chalk or electrons, or speak viva

voce, are not mathematical objects, but are expressions we use to jog our

senses into recognizing the ideal objects that are the real subject of our

discourse.  I remember one of my professors once trying to deny this, when

defining an adjunction to a field.  He said, "If F is a field, F[X] is

defined as follows: let X be any object (an apple, a cousin, C#, etc.) and

form all formal expressions ..."  But this was fraudulent.  Apples and C#

do not live in Plato's heaven, whereas the field F did, and so combining

them in this way into "expressions" which are half inside and half outside

has no meaning for us.  (Well, maybe C# is up there somewhere, but among

the members of the chromatic scale, not among the fields ripe for



      Nobody ever told me that these permanent abstractions unrelated to

their manner of expression are what mathematical statements are, and that

Plato's heaven is where mathematics lives, but it is the only way I can

think about it.  F and F[X] inhabit Plato's heaven; otherwise F[X] has no

existence once I'm dead, or once the universe collapses; and this is

impossible to believe.  Fields cannot depend on me and my friends; we are

too temporary.  The fifth degree polynomial was unsolvable long before

Abel; can anyone deny it?  Then how can it ever cease to be unsolvable?

It is bloody eternal, and cannot be anything we write on paper, which we

know is not eternal.


      Now back to equality.  When I say "A=B" I say nothing other than

that the "two" Platonic objects whose (temporary) names are A and B are

in fact the same object, having for some reason been named twice.  Or even

only once, as in "2=2".  The symbols A and B don't have to look like each

other, but the things they name are identical. 


          Are there any cases in mathematics when we have to assign any

other meaning to "="?  It has been suggested that in the sentence

casually written 4/6=6/9 we don't really mean "is identical with", but

rather, "is in the same equivalence class with" according to the

well-known construction.  Or "represents the same rational number as", to

the same effect.  Now I must insist on the phrase "casually written"  as I

put it above.  By itself, without some conventions agreed among us, the

printed sentence is incomplete.  If we take, as we hope 5th grade children

do, the symbol "4/6" for the name of a certain rational number,

recognizing that this rational number has many names, then our sentence is

as I said, i.e. "4/6 = 6/9"  means that the two printed fractional

expressions are symbols for the same Platonic object.  "They" are equal in

that they are only an illusory "they", being in fact one object.  Any 5th

grader will understand that, and will accept 6/9 of a pizza as readily as

he accepts 2/3 of it.  He knows the fractions are the same.  He wouldn't

spend a nickel more for one than for the other.


      Having emerged from the 5th grade, and learned about equivalence

relations, and how to construct rationals from pairs chosen from N, we run

into the same casual sentence.  At first, of course, we learn carefully to

write, "[2/3] = [6/9]", the brackets indicating that the equation concerns

the equivalence classes of which {2,3} and {6,9} are representatives,

except of course that instead of {2,3} we are here employing the more

conventional notation (e.g. "2/3") for the ordered pair in question.  What

does the sentence "[2,3]=[6,9]" say?  It says the equivalence classes are

THE SAME.  They are up there with the field extensions, and they have been

there since the beginning of time, and those classes are not going to

die with you and me, either.  Of course, in daily life we don't keep the

brackets, though it is convenient to use the fractional notation because

it reminds us of the applicaton of rational numbers to daily measurement

and pizzas.  (And what to do on a calculator if we want an appproximate

decimal equivalent.)  So we say 2/3=6/9 but we mean the rational numbers

this ink or magnetic tape represents, and they are equal in the sense of

identical.  The same as -- and if you try to tell a fifth grader they

are not the same he will get *really* confused about that chapter on ratio

and proportion, which I wish the schools wouldn't go on so about.


      Sometimes we talk problems.  We say, find the dimensions of a

rectangle if the perimeter is P and the area is A.  We call x and y the

sides of the putative rectangle and posit that xy=A and x+y = P/2.  These

are rather incomplete sentences; they need context, and for kids in the

8th grade they need a lot of context if they aren't to turn into a ritual

abhorred by NCTM and mathematicians both.  First off, we are saying, "If

this problem has a solution, there must exist real (up there, of course)

numbers I choose to name x and y for the moment, such that these relations

hold between them. (The relations are also inscribed up there, and have

been there for a very long time).  So, [we continue] x and y, and the

given real positive numbers A and P, satisfy the relationships xy=A and

x+y = P/2."  What means the "=" in all this?  It means "the same real

number as", i.e.  identity of the objects (if any) that can make the

statements true.  The objects are no different in nature, by being as yet

unknown, from real numbers that are known, like 3; they are Platonic

abstractions if anything, and it is our duty to go forward to see if what

we have said about them can make sense.  If not, we have deceived

ourselves about their existence, but we haven't changed the meaning of



      I shall go no further, but I invite objection, and a sample

mathematical use of "=" which I cannot interpret in this manner.  It is

the way I have interpreted it all my life without running into trouble.

But since philosophy has never been a strong point with me, and I have

never understood a word of Immanuel Kant, so it is possible that I am

overlooking an important point.


          One common experience that has confused the question of the

meaning of equality has been the use of the word by Euclid, and his

followers in the schools in the past two hundred years.  For two thousand

years and more, children have been lisping, "Things equal to the same

thing are equal to each other."  We don't have any urtexts of Plato, but

the earliest known transcriptions use a Greek word meaning something very

like our English "equals" in Euclid's famous axioms, yet it is perfectly

plain that Euclid did not mean "equal" in the sense of identity in this

case.  Modern translations might have better used the word "equivalent"

instead, with a footnote explaining that the equivalence relation Euclid

had in mind was different in different parts of his treatise. 


          The well known properties of symmetry, identity and transitivity

are actually definitive of an equivalence relationship, while to mention

them for equality in the sense of identity is actually supererogatory.

Even Euclid must have understood that much, and in fact he used these

axioms (which hardly required mention in the case of genuine, Platonic,

equality) for several sorts of equivalence. Other axioms, involving

"addition" and "subtraction" of "equals", are also non-trivial when the

objects being (e.g.) added are geometric entities.  Our high school

textbooks, taking Euclid's use of "equals" and "add" at face value, are

still repeating these axioms idiotically in the trivial case where the

objects are numbers and the equalities are identities, and add to them

equally unnecessary "axioms" about equals multiplied by equals and even

"equals taken to equal powers", where all that is needed is the

information contained in the definitions, that the operations involved

produce a unique answer, i.e. are well-defined.


          Plainly, Euclid did not mean "equals" in my sense, for he

used those axioms when he meant "congruent" or "scissors-congruent".  Thus

his own proof of the Pythagorean Theorem leans on the partitioning of the

three squares into triangles that "add up" -- though even here his

triangles aren't yet congruent, but are 1:1 scissors congruent themselves,

having equal bases and heights. (Euclid proves, stage by stage, how to

construct a single triangle scissors-congruent to any given polygon.)


      Things are even more sophisticated when Euclid, in Book V, which

school children certainly are never exposed to, defined "equality" of

ratios of like objects, and somehow presumed this sort of equality to be

an equivalence relationship, obeying the axioms for what he called

equality.  This "equality" is far from identity if the symbols are

looked upon, for the ratio of a large circle to its diameter is expressed

as C:D, while the "equal" ratio for a small circle might be expressed

"c:d", quite a different set of symbols.  But they are the same, just as

4/6 and 6/9 are the same, when the symbols are regarded as names and not

ink spots.  There is, from the Platonic point of view, more justice in

Euclid's use of "equals" than in the freshman's "equation", "pi=22/7".


      Euclid's axioms on 'equals added to equals", etc., have been

ingnorantly copied over the centuries, sometimes to govern areas, i.e.

numbers, rather than the geometric objects themselves, and sometimes to

refer to group elements, where no axiom at all is needed, group addition

being postulated as uniquely defined.  That is, to deduce x+a=y+a from x=y

you don't need Euclid, only the fact that "x+a" is well-defined in any

group, so that if x=y (i.e. the symbols represent the same group element)

how many forms of x+a can there be?  But Euclid's geometry needed some

such axiom because congruence and scissors-congruence were otherwise not

fully defined there, except by implication in the successive theorems by

which the examples of such "equality" were gradually extended to more and

more geometric objects (or their ratios, or other qualities).


          I should add here that Euclid's "equality" applied also to limits

of scissors-congruent pairs. The way in which what we call "limits"

appears in Euclid, however, takes much elucidation; it is not for this



      One of my correspondents mentioned his unease at hearing students

take Euclid's "equals" to mean "equal in area".  He was right.  As it

happens, two polygons are equal in Euclid's sense (i.e., they were

scissors-congruent)  if and only if their areas are equal, but the

corresponding conjecture for polyhedra is false.  It was Han Sah of Stony

Brook who pointed this out to me only a short time before his death, and

it was something I had been ignorant of all my life. In plain geometry for

the past one or two hundred years, school children have been given a

confusing story on congruence, and the most confusing part has been the

constant assignment of numbers to geometric entities (even where

"analytic" geometry isn't in question) as if those assignments were

obvious.  Yet the statement that two solid spheres are to each other as

the cubes on their diameters, while it can be constued as a relationship

among numbers describing these things, is not what Euclid proved, and by

Sah's observation, is not even equivalent to what Euclid proved, though it

is what today's school child is taught.  (It is true, too, but it is not

geometry in Euclid's sense.)


      Teaching Euclid's system with full rigor is impossible at the school

level.  I myself have never been through Hilbert's axiomatization, to

deduce from it the real number system by which the objects describable in

Hilbert's system can be represented using coordinates in the plane.  In

teaching it -- more or less -- to children, however, I would by all means

explain that Euclid doesn't mean "equal" when he says what in Greek

apparently sounds like a version of that word.  Even though I could not

offer all the proofs, I would try to explain the difference between

congruence and equality.  And the great mystery of how the Pythagorean

theorem can be understood with a real "equals", when interpreted as a

statement about areas (numbers) and addition in R, is worth elucidation;

it should come as a surprise, not as something swept under the rug as

obvious.  (The Pythagorean theorem is certainly not about sums in R, but

about set unions and rigid motions.)  To claim the verb "equals" has just

changed its meaning as between the two interpretations is to obscure an

important insight into geometry, not to make things simple for children.

In talking about the 3-4-5 triangle, I believe it is important to

distinguish between the numerical equality 9+16=25 and the geometric

relationship between the three literal squares on the diagram.  It is

unfortunate that it is popular these days to talk about "Pythagorean

triples", i.e. integral solutions of the Diophantine equation a2+b2=c2,

under the heading "geometry".