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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