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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
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
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