# The Principles of Mathematics (1903)

### § 26

Some important points arise in connection with the theory of identity. We have already defined two terms as identical when the second belongs to every class to which the first belongs. It is easy to show that this definition is symmetrical, and that identity is transitive and reflexive (i.e. if x and y, y and z are identical, so are x and z; and whatever x may be, x is identical with x). Diversity is defined as the negation of identity. If x be any term, it is necessary to distinguish from x the class whose only member is x: this may be defined as the class of terms which are identical with x. The necessity for this distinction, which results primarily from purely formal considerations, was discovered by Peano; I shall return to it at a later stage. Thus the class of even primes is not to be identified with the number 2, and the class of numbers which are the sum of 1 and 2 is not to be identified with 3. In what, philosophically speaking, the difference consists, is a point to be considered in Chapter VI.(§ 26 ¶ 1)