50 CHAPTER 6

The Logic of Equality

An equation is a statement, or assertion, that one thing is identical

to another. For example, “The ﬁrst president of the United States

was George Washington.” Equality is timeless: Although we wrote

“was,” because both sides of the equation existed in the past,

logically speaking, “is” or “will be” have the same force as “was.”

Philosophically, equality can get pretty complicated. Consider

the deﬁnition: “A unicorn is a horse with a single horn. ” A uni-

corn and a single-horned horse are identical—by deﬁnition—even

though they (it) do not exist. Yet “is” might imply existence.

To rid ourselves of this problem, we drop the word “is” and its var-

ious forms, and replace it with the symbol =.Now,wedonothaveto

take an ontological stand on the terms on either side of the symbol.

Whatever their mode of being, we are asserting their identity.

We may also use symbols, usually single letters, to stand for the

terms. For example, if F stands for the ﬁrst president of the United

States and G stands for George Washington, then the assertion in

the ﬁrst paragraph can be symbolized as follows: F = G.

Before algebra was invented, this kind of symbolism was unavail-

able, and people had to use ordinary language, which was often

cumbersome. Nevertheless, equations were important even then,

and have continued to become more and more useful as science has

advanced.

Almost anything we do can be phrased as an equation. For

example, solving X = D for the unknown X,whereD stands for

“what we are having for dinner tonight,” is equivalent to ﬁnding

out what we are having for dinner tonight. From now on, though,

we will restrict our discussion to mathematical equations, where

the terms denote mathematical entities.

The History of Equations

Long before algebra as we know it, ancient peoples were working

with equations:

1. The base angles of an isosceles triangle are equal.

EQUATIONS AND VARIETIES 51

2. If a circular bath has diameter 10 cubits, then the

circumference is 30 cubits.

3. A triangle whose three sides have lengths 3, 4, and 5 is a

right triangle.

The ﬁrst equation is a proposition from Euclid. The assertion is

claimed to hold for all isosceles triangles. So it is a general theorem.

The second equation is from the Bible, describing a large basin in

the Temple in Jerusalem. It is not exactly true, but no doubt it was

meant to be only approximate. In this book, we will be concerned

only with exact equations.

The third equation is an example of a solution to what is called

a Diophantine equation because it has unknowns (the three side-

lengths) which are constrained to be integers (whole numbers). The

equation we are thinking of here is x

2

+ y

2

= z

2

, which expresses the

Pythagorean Theorem about the side lengths x, y ,andz of a right

triangle. As a mere equation about any old right triangles, we can

suppose x, y,andz are any positive (real) numbers. What makes

it a Diophantine equation is our self-imposed desire to restrict the

solutions we are interested in to whole numbers.

Diophantus was a Greek mathematician from postclassical

times whose writings inﬂuenced the founders of modern European

number theory. Because Diophantus lived before the invention of

symbolic algebra, his work can be hard to read. It is amazing what

ancient mathematicians accomplished without the aid of the slick

symbolic language we use now.

When symbolic algebra was ﬁrst set down in Europe in a treatise

by Franc¸ois Vi

`

ete in the late 1500s, he called it “the art of ﬁnding

the solutions to all problems.”

1

In short order, Ren

´

e Descartes added

the connection between algebra and geometry, and he also thought

that now all geometrical problems could be ﬁrst quantiﬁed and

then solved. Because he also thought that the physical universe

was governed entirely by geometrical laws, he felt that “analytic

geometry” or “algebraic geometry” could be used to discover all of

science.

1

You can ﬁnd an English translation of this treatise at the end of (Klein, 1992).

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