Identity As Difference

According to Bertrand Russell, a number is a class of similar classes, which consists in any classes containing the same number of elements. However, to avoid circularity, we must redefine that similarity without resorting to the concept of a number at all---even a one-to-one correspondence between the elements of any two classes will not suffice, as it still depends on the numbers two and one. Ultimately, the similarity of classes consists in each element of each class relating to any and no other element of each other class: according to this number-free definition, there is no difference between any elements of the same class, as anyone of them relates to the same element of each other class, despite no other doing so. Within similar classes, the only quality any elements of the same class have is their belonging to that class, by which the only difference between them is their not being each other. So similar classes reduce the identities of each one’s elements to pure otherness. Likewise, the only difference between those classes themselves is their not being each other, since their number of elements is by definition the same, and any other quality any of them could have would necessarily belong to its elements---which have no quality other than belonging to that same class. This reduces the identities of similar classes themselves to pure otherness, hence to the otherness between each one’s elements. Finally, by being a class of any classes containing the same number of elements, that number of (any) elements is not only the similarity between those classes, but also each one of them: it is each one’s identity---the pure difference of otherness---as much as their common identity---the pure identity of sameness. Then, a number is the same as the otherness between similar classes---another class: theirs---and also the same as the otherness between each one’s elements---another element: its own class---or is absolute otherness. And because numeric sameness is absolute otherness, any number is another number, which in turn is another number, so any number is any other number:[1] numbers are both different from and identical to each other,[2] just like similar classes or similar-class elements are. Hence the quotient indeterminacy of dividing zero by zero, by which all numbers are the same: the identity of numbers is difference.[3]

[1] This is partially apparent in square roots of positive real numbers, like the square root of nine, which is both three and minus three.

[2] Or any number is infinity, and indeed, since infinity plus one is both of a higher order than infinity and identical to it, order is the hierarchical difference between a number and itself.

[3] The identity of numeric difference makes numbers false, by preventing them from being different from each other, although the difference of numeric identity makes numbers true, by preventing them from being identical to each other: mathematics is the realm not of numbers, but of numeric truth.