Thoughts On Identity

We might look upon identity in a strict and technical sense as a relationship; however, the common view is that identity is not a “real” relationship. Sometimes it is cited that this is because a “real” relation must be between two or more objects. This seems to be only a way to skirt the issues that are involved with the relationship of identity, and not because identity doesn’t share the same structure as other so-called “real” relationships.

If we take a look at some two-place relation, say, ‘is taller than’, we can clearly see that in order for it to convey any sense, there must be two objects that are being compared: a is taller than b. Now, it certainly seems as if the relationship of identity is also a two-place relation—it shares the same structure as the relation ‘is taller than’. Clearly, if we were to assert that ‘a is identical to’ we would not have said anything meaningful. We must say that ‘a is identical to b’: it just so happens that the nature of this two-place relation requires (in order for it’s statement to be true) that a and b be the same object. But this doesn’t seem much different from the requirement that for a to be taller than b, then a really does have to be taller than b. In other words, there does not seem any structural reason why identity should not be viewed as a “real” relation.

There is something peculiar about the relationship of identity however. There is a way to examine identity that brings out a contradiction or paradox in its nature. Upon a certain inspection, we can come to see that the relationship of identity is both empty and absolutely full. Put differently, we are able to see how identity is both nothing and everything.

On the one hand, a = b is tautological, and so, it is, in a sense, vacuous. On the surface, it tells us nothing about a or b other than that a is b: it reveals nothing beyond the fact that it is. We can see this similar to Hegel’s take on mathematics—it is existence without essence. Thus, we see how the relationship of identity is empty.

On the other hand, any identity statement is, in a sense, full of itself. To understand this we can move to a mathematical example. To begin, we can look at a particular case, say 4 = 4. Like we’ve already seen, this doesn’t seem to tell us anything about 4 beyond the fact that it is. It is 4. But, if we know four as it is within the structure N, then we can delve into this and see that it is something like //// = ////. Thus, if we know that 4 = 4, we know with certainty that there are four objects, and only four objects, and that the structure of the relations between these four objects form the whole of the identity of 4 with itself. Therefore, in a phenomenological sense—in the actual occurrence of a grouping of four and only four objects, the identity relation conveys the whole of everything: it is absolutely full. We can also see that this will hold for any n.

So perhaps now we can see that the relationship of an object with itself is a rather peculiar relation; however, this does not seem to be a reason to discount it as a “real” relation. Moreover, with respect to different representations of numbers, the particular example of 4 = 4 is not only empty & full, but might also reveal to us identity amongst alternative representations, ex., //// = …. or abcd = 1234 or o’’’ = x₀, x₁, x₂, x₃ or etc..

OK. So this ties into how structures, singularities, and objects are really ‘pseudo-concepts’, as we saw in Russell explaining Wittgenstein, insofar as they have no existence outside their relations with other structures, singularities, and objects. However, the flip side of this is that there can be no relations without instantiations in systems, and systems are, by their very nature, structured. Here we can turn to what Shapiro had said about properties being difficult to differentiate and in turn rephrase this with respect to structures, singularities, and objects—in short, places as positions in structure or as place-holders in systems. This is to say that there can’t really ever be pure differentiation, i.e., there are no objects that can be singled out as an object—a particular—without also considering every other relation of that “object” to the rest of the structure—the universal—it is embedded within. The particular is in the universal & the universal is in the particular.

To put this all a little differently, we can see that there can be no singularity that exists if it does not at least bear a relation to itself—identity. But identity has a paradoxical nature that makes one into two: a is identical to b. It is from this paradox that the rest of, for example, N is generated. Let’s take a look:

1)   If we have a singularity that bears no relation to itself, according to the results of the thought experiment, then we have no singularity at all. 0 = 1.

2)   If we have a singularity that does bear a relationship to itself, then we have ‘1 is identical to 1’. However, as we have seen, Rxx—1 = 1, appears to give the impression of two objects. So, in a sense, we have 1 = 2. Leaving that aside, since we have seemed to established that relations are objects, we have from Rxx the singularity ‘1’ and the next singularity ‘2’ as R.

3)   Of course, if we now have 2, then either 2 bears no relation to itself (or anything else) and then it is nothing, or we have 2 = 2. If we have 2 = 2, then we have, like above, the manifestation of the next singularity 3.

4)   Etc. (repeat until bored or dead)

Thus, whatever is the structure that is created from this process, we can see that there is no way to divorce any of the particulars from its relations with the rest of the structure & there would be no universal structure without the relationships amongst the particulars, which themselves are either structures dependent upon relations—and vice versa—or they do not exist.

To sum up perhaps cryptically: an atomic either has to relate to itself (before it can ever relate to any other atomic) or it doesn’t exist, but in its relation to itself it generates difference. This difference itself is the manifestation of any other differentiated atomic to which any given atomic must bear a relation to for either one to exist. Get it? It’s similar to the Buddhist Pratītyasamutpāda—interdependent co-arising.