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Fearless Symmetry by Robert Gross, Avner Ash

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REPRESENTATIONS 7
Our definition of one-to-one correspondence, however, does not
tell you the “essential nature” of a one-to-one correspondence. We
have given you no clue why you should care about one-to-one
correspondences, nor does our definition tell you how to make a
one-to-one correspondence.
Even when a mathematical definition technically has all of the
properties listed by the OED , it often strikes nonmathematicians as
unusual. A mathematical definition can redefine a commonly used
word to mean something else. For example, mathematicians refer
to “simple” groups, which are in fact not particularly simple. They
define the words “tree” and “quiver” in ways that have nothing to
do with oaks and arrows.
Sometimes a mathematician defines an object in terms of its
properties, and only then proves that an object with these prop-
erties exists. Here is an example: The greatest common divisor of
two positive integers a and b can be defined to be a positive number
d so that:
1. d divides a.
2. d divides b.
3. If c is any other number that divides both a and b,thenc
divides d.
With this definition, it is not obvious that the greatest common
divisor exists , because there might not be any number d that
satisfies all three properties. So right after making the definition,
it should be proved that a number with the properties outlined
actually exists.
Counting (Continued)
In our example, each pebble corresponded to one sheep in the
morning and one sheep in the afternoon. This sets up the rule
that associates to each morning sheep the afternoon sheep that
shared its pebble. This rule is a one-to-one correspondence under
the conditions of our story.
But we do not need to know any set theory, nor what a one-to-
one correspondence is, to count sheep in this way. In fact, we do

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