The identity of indiscernibles says that if two things can’t be distinguished from each other, or, in other words, if they have all the same properties, then they are really the same thing. The identity of indiscernibles seems to have some intuitive force, but there are philosophers who challenge the doctrine. For example Max Black asks us to consider a possible world in which there are two identical spheres a fixed distance from each other. He contends that if we don’t introduce an observer into this universe then the spheres have all the same properties, but at the same time are not identical (by virtue of being numerically distinct), thus showing that indiscernibles are not necessarily identical.
Here I am concerned not with indiscernible objects though, but indiscernible systems or universes. We can say that two systems or universes are indiscernible when all the same facts hold in both. The question then is: are indiscernible systems necessarily identical? For example, consider a first order logic axiomatization of the natural numbers. From this axiomatization follow all the things that we can say about the natural numbers themselves, although certain facts about the structure of the natural numbers cannot be talked about. And because of this there is more than one “model” that satisfies these facts. Brief digression: a model is simply a structured collection of objects which the properties and functions of a formal system are taken to be about. Thus the natural numbers as a single infinite sequence 0, 1, 2, … satisfy our first order axiomatization, because all the facts that follow from that system are true statements about our model, the sequence of numbers. But there are other models that also satisfy our axiomatization. For example the natural numbers 0, 1, 2, … followed by an infinite number of infinitely long ω chains ( … ω1, ω1 + 1, ω1 + 2, …, ω2, ω2 + 1, ω2 + 2, … ect). Each new ω value is an infinite distance from the previous sequence of objects, and from the natural numbers (the first sequence), and thus cannot be reached by the successor function (taking the next number) starting from one of those numbers. It might seem impossible that all the first order facts following from our axiomatization are true about this system as well, but they are. So here we have two seemingly very different models that are indiscernible form each other in first order logic, but are different, and so it would seem that it is false that indiscernible systems are necessarily identical.
But perhaps we shouldn’t be so hasty. To say that these models are different we had to approach them with a more powerful logic, in which it made sense to talk about what kind of infinite sequence we were dealing with. And with these more powerful facts in hand there was a difference between the two, and so they weren’t indiscernible after all, and we could say that there were in fact two of them. But perhaps that is a poor approach; let us consider what would happen if we had only the first order facts to work with, would they be different then? Well we certainly couldn’t tell them apart. If someone handed us two “models” of our system and we could probe them only with various first order facts we would be at a loss to tell one from the other. Now the assumption that there is a difference is based on the idea that there are certain facts, that can’t be captured by our system, that do distinguish the two. But the assumption that such facts exist is in a system that has a purely first order axiomatization is perhaps a questionable one. The system is not like some Platonic form, which we can appeal to answer the question of whether every possible fact is true or false. Indeed the appeal to models is somewhat questionable to begin with. Of course models are a very useful mathematical tool, and they provide a convenient way to decide whether a statement is true, by allowing us to appeal to the model. But what we were investigating to begin with was our first order axiomatization, not a particular model of it. If we were investigating models we would have begun with a system that captured all the facts about models, and thus would always be able to tell them apart. Thus I would argue that in our original axiomatization there is no fact of the matter at all about whether every natural number is some finite successor of 0.
Let us return to a moment to the universe containing two spheres. Is this universe the same as one with an unusual geometry in which one sphere is some distance from itself? All the physical facts, constructed from the basic physical entities and properties, will be the same. In both universes it is a fact that a sphere, defined with a certain complete physical description, is a certain distance in a certain direction from a sphere with that same complete physical description. Of course these universes could come apart, if for example there was a change in one of the spheres, but as things stand all the facts about them are the same.
Our idea that there could be a difference, in the cases of both the axiomatized system and the one sphere/two sphere universe, comes from the idea that there could be more facts, facts not found within the systems, that distinguish them. But what would it even mean to appeal to facts external to a universe? I think in both these cases our intuition that there could be a difference is rooted in our idea that when we have a fact that can be either true or false we assume that it is a determinate matter whether it is true or false (assuming that fact is well-formed), even if we don’t know it. In most cases this is true, but it doesn’t have to be. For example, in the quantum world there is really no matter of fact in some cases about what state a system is in (the cat is neither alive nor dead). And in first order logic there really is no fact of the matter whether the Gödel sentence is true or false (there are consistent models where it is true and consistent models where it is false). Thus I would argue that there is no fact of the matter at all about these supposed distinctions. And thus we can safely conclude that they are in fact identical, since there is no reason to distinguish them.
But this means, going back to the original spheres example, that there is no matter of fact about whether there is one sphere are two, as there are indistinguishable universes in which the same physical facts hold in which there is one sphere and in which there are two. Assuming that we don’t think there is some metaphysical property of “thisness” that does in fact distinguish them then the question as to whether they are identical is an empty one, there is simply no fact about the matter. And since it is perfectly consistent to treat them as identical we can, and maintain the identity of indiscernibles as a metaphysical fact.