Ordinary discourse doesn’t include many appeals to the notion of identity, and certainly not any precise ones, even though truths about identity often surface in the form of claims that two apparently distinct things are actually the same. Thus we might turn to logic for a precise definition of identity. There it is treated as just one two place relation among many, one that holds between every object and itself and which never holds between two different objects. Such a description removes all the mystery surrounding identity, but it is also relatively uninformative. For example, it would be nonsensical to ask why two objects are identical, because that would be equivalent to asking why one object is identical to itself. And clearly there is no reason why an object is identical to itself, that is just a brute fact. Problems also arise with the logical definition of identity when considering possibility and necessity, and in other contexts, which suggest that there is more to identity than the logical relation captures, such that logical identity only reflects the nature of identity in certain limited circumstances.
Given that the logical definition of identity has failed up perhaps we are forced to turn to our ordinary use of identity. But when it comes to our ordinary use of terms such as “the same”, which seem to be those from which the idea of identity is extracted, it doesn’t seem much like a relation at all. Whenever we say that two things are the same there is an implicit understanding that they are the same something. For example, we might say that two people drive the same kind of car, uniting the two distinct objects under the umbrella of a single kind. Now it might be objected that this is simply too broad a conception of identity, that all we really seek to understand is the way in which a brick viewed at one moment is the same as it viewed in another moment. But, even here, we must stipulate that they are the same brick where being the same brick brings with it the idea of a temporally extended object. Otherwise we could object that what we saw in those two moments were really distinct objects, distinct brick-moments. And the same applies when we try to talk about the same brick seen by two different people at a single moment. It could be claimed that there are two non-identical brick-presentations under consideration, and thus to say that they are the same we need to say that they are the same brick, where this time brick brings with it the idea of an observer independent object that can be presented in a number of different ways. Under this conception identity seems to be a concept that is essentially tied to a way of dividing the world into a number of equivalence classes, such that saying two things are the same is really just to say that they belong to the same equivalence class under some division. And thus which things are identical and which aren’t seems completely a product of the way we divide the world, which is itself arbitrary, and thus it is hard to see how claims about identity could be significant, which is the real problem.
That seems odd, because there certainly appear to be important facts about identity, but we can’t easily get back to such facts and something like the logical conception from this starting point. We can’t for example, simply try to redefine identity without any such equivalence classes. Well, we could, but identity defined without such equivalence classes to bring together things that we might otherwise distinguish between leaves us with virtually no cases where identity holds at all. Under identity, so defined, the only time we might legitimately say that two things are identical is when we are referring to the same particular experience of something and saying that it is identical to itself. This would make identity simultaneously trivial and useless. Nor can we recapture the logical notion by simply picking one way of dividing the world into equivalence classes as “right”. Admittedly such a division would fix things to some extent, but the decision about which division is “right” seems itself completely arbitrary. Some might even make the case that the universe is best conceived of as a single object, and that what we think of as different objects, such as different particles at different moments of time, are really just properties of this universe-object, the property of it containing a particular particle in a particular place at a particular time. Certainly we can’t say that this way of logically describing the universe disagrees with our experience, it just makes talking about it a little difficult, and it completely does away with the possibility of any identity relation, since there is only one object that it might apply to, and is thus rendered meaningless simply because it doesn’t distinguish between objects.
A third way to bridge this gap, which we haven’t considered previously, is to consider equality as tied up with language, something that hasn’t come into play yet. The interesting claims involving identity, we might note, are those where we say that two things designated by different names are identical. For example, we might assert that the richest man in Europe is the tallest man in Europe, which could be to claim that those individuals are identical. And thus it might be supposed that identity is really a claim about what two terms refer to, namely the same thing, which couldn’t be properly captured by the suggestions above simply because they had no way to talk about referring. There does seem to be a kernel of truth in this suggestion, but it too has its failings. First of all it is still based on a kind of ordinary understanding about identity, and so still seems vulnerable to what it means for two things to be “the same thing”, which plays a key role in the definition. Again it seems that a division of the world into equivalences classes is necessarily involved. And in this case the division seems even more complicated. Certainly, for example, the richest man in Europe hasn’t been the richest for exactly as long as he has been the tallest, thus we run into problems in saying what exactly “the tallest person in Europe”. Does it mean the time extended person who is the tallest now, does it refer to the person-moment that is tallest now, or does it refer to all person-moments in all times that are tallest, regardless of whether they belong to the same time extended person? To make it true and meaningful we must pick some division, but, again, how to divide the world seems arbitrary. Secondly, it seems to fail to capture some of the ineffable essence of claims about identity. When we claim that the morning star is identical to the evening star we mean to say something to the effect that, roughly, the two objects under consideration are really one object. But so far none of the possibilities entertained have let us make this kind of assertion, because they don’t allow us to talk about two objects in one breath and a single object in the next.
To solve some of these problems I think we need to turn to why we talk about objects in the first place. The idea of objects, I claim, is a device that exists simply to make conceptualizing and theorizing about the world simpler. We divide the world up into objects and assign them properties, and then on the basis of this division and its laws we make predictions and check how well our model matches up to observations. Thus objects are from the beginning our invention, they don’t exist outside of us to find, even though talk about objects can be considered in the domain of objective fact. Thus we can make a distinction between talk about the model and talk about relationships between the model and the world as observed. To say that an object is red or at a particular location is to make a claim about the world, expressed through a relationship between statements about our model and the world. On the other hand, to say that a certain law holds of objects is strictly talk about the model. Of course we can make observations that contradict the proposed law, but we can never observe the law itself. And thus it is more accurate to say that our observations have revealed that the proposed model, including the law, doesn’t accurately reflect the world, rather than saying that they show the law doesn’t exist or is false. We can put this distinction to a number of purposes. For example we might point out that world-model correspondences can be given explanations in terms of the model, we can say why an object has a particular color or location by appealing to the structure of the model and its laws; but we can’t explain brute facts about the model itself, such as its laws, in the same way. But obviously this is tangential to the matter at hand.
So, to return to identity, allow me to simply assert that identity is a claim about the model, not about the world. It is saying that our model contains only one object that will be used to explain what might be thought of as two distinct phenomena, or what were explained using two objects in another model. Because identity fixes the model this explains why we have talking about identity from within the model; once you have decided what your objects are it is fairly useless to go over those facts again. Of course even under this understanding of identity there is still some arbitrariness about what your objects are. For example, we could double the number of objects by modeling the world with two objects wherever previously we had one, with one object accounting for all observations made from galactic north and another for the observations from galactic south. However I would point out that there is nothing “wrong” with such models, they are just a more complicated way of expressing essentially the same facts. But, for the sake of convenience, we tend to go for the fewest number of objects, and thus the maximum amount of identity, arguing that we should use a single object whenever possible, so long as contradictory properties (A and ~A) aren’t assigned to that object. Most importantly, of course, is simply that, regardless of whether there is arbitrariness here, claims about identity are significant, not as facts about the world, but as facts about the model, which makes arbitrariness somewhat irrelevant. I would elaborate further, but I fear I have gone on too long for one day already, so I will leave the rest to the reader (unless some especially interesting complications occur to me later).