My daughter was counting hooks in a line, but she did something interesting. As she counted hooks, she counted the spaces between hooks, so that the next space had the same number as the previous hook. Perhaps the ordinary way to think of the space between hooks is the same—as zero—that doesn’t change the count, but she felt, quite unprompted by me, that each space was unique in its place between 1 and 2, 2 and 3, etc—That it had an identity. Normally identity in math is denoted with “=”. Expressing the identity of 1 is done by writing “1=1” or . A space is the opposite of an object, but the difference between hook 1 and hook 2 could be thought to have an identity, because it can be identified with its context between hook 1 and hook 2, for example, and could be written  . It was in Plato’s The Sophist that it was offered that not-being be the spaces or differences between beings. The inequality, or difference, between 1 and 2, , or between 2 and 3 are such spaces.

I am contrasting a definition of number as a difference with a common belief that a number is a class or likeness of members. Normally the number three is essentially a set of three members, where the members of the set could be any three things we wish to collect or associate. However, we run into difficulty when five year old children count spaces instead of objects. What do the spaces have in common, how can they be identified, except as differences between objects? And in that case, how are we to deal with the idea that three “spaces” share a likeness, when all they are is purely a difference?

The point of this is to argue that different differences (or inequalities) are different from each other. Put simply

As argued in a previous post (https://questionsarepower.org/2015/01/31/degrees-of-difference/), the difference between 1 and 2 is different from the difference between 2 and 3, even though 2-1=1 and 3-2=1, we have to keep the context of the first and second subtractions with their results.

To return to that argument briefly, observe how the difference between age 1 and 6 is much greater than the difference between age 40 and 45, even though the distance between each set is 5. Further, one difference between and circle and a square is that the square can be triangulated while the circle cannot, but this difference can be turned into a similarity between a square and a strict rectangle. This would suggest that a difference “=” a likeness. Back to our inequality above, that we could denote the first difference “2-1” with ,and the second difference “3-2” with ,. If we were to lose these contexts and allow both to be reduced to the same “,” contradictions would follow. For example if we were to lose the contexts in the inequality

by writing simply

In English, the idea that something “does not equal ”, reduces to “=”, so we replace two of the “not-equals” with one “equals”:

Now, what is  equal to?

Since we already know , it cannot be that . That would be a contradiction, but the alternative: