What do we mean when we say that something exists? The obvious answer is that it exists physically, that it is part of the physical world and that we can interact with it. But what about numbers, do they exist? It seems sensible to say that they don’t exist in this way, but given that we talk about them as if they did exist in mathematics it seems possible to defend the idea that they must exist, in order to make sense of talk about mathematics. Such a proposal would expand the universe to include abstract entities in addition to the familiar physical entities. But that is not a satisfactory solution either; it opens the door to the existence of all sorts of things, like an infinite number of possible objects. And when you admit that possible objects exist you have become confused about what existence means, or at least a very poor grasp of Occam’s razor. Possible objects are only possible because they don’t exist, because they aren’t part of the physical universe. And moreover a universe that contains all possible objects is simply metaphysically overburdened; possible objects bear no explanatory burden except to explain certain forms of discourse, and it would be a simpler conception to explain such discourse in terms of false ideas about what the universe contains.
So let us go back to the argument that the numbers must exist in some fashion. Certainly we say many things, supposedly true, about the numbers as if they existed. For example we say that “5 is a number”. In addition we believe that there is a fact of the matter about certain statements, such as “there are infinitely many twin primes” that are either determinately true or determinately false depending on the structure of the natural numbers, even though we don’t know yet whether they are true or false. Which implies that somehow the natural numbers are “out there”, beyond our conception of them, making these statements true or false.
But this argument turns, I think, on an equivocation between two senses of existence. Existence, in one sense, says only that something is part of the physical universe, nothing more. In another sense existence means something that can be reasoned about or talked about, logical or linguistic existence. The numbers, I would say, have a logical or linguistic existence. And this doesn’t mean that they are extra non-physical entities that are also part of the universe. We don’t interact with “the numbers” or have access to them in any special way. Rather these logical entities are created by our discourse and its rules. For example, I could define space as being divided into cubic regions one meter on each side. I could talk about the contents of these regions, their arrangement, ect. So in a sense these regions “exist”. And there is a fact of the matter about which region any given object is in, even if you don’t know which one that is. But that doesn’t mean that these arbitrary divisions of space are real, they simply reflect a way of talking about space.
And so it seems reasonable to say that numbers, and possible objects, have a logical or linguistic existence as well, since they are no more independent of the rules we have made about them then our arbitrary divisions of space were. They are entities that we think of, talk about, and reason about as if they exist. They are defined by certain rules, and the rules may imply certain facts that we aren’t aware of (because we can’t immediately understand all the logical consequences of those rules). And if they are useful they probably seem to reflect certain facts about the physical world. For example, the numbers seem to reflect certain facts about the cardinality of sets of physical objects and certain operations on those objects.
Of course none of this disproves the idea that these abstract objects exist as a kind of extension to the physical world. But it does show that we don’t need to add them to the world in this manner in order to explain how we can reason about them, and how there can seem to be facts about them independent of our conception of them. And thus, since it explains all the same facts, we should accept that simpler hypothesis that they exist only linguistically.