I'd like to introduce you all to something called mathematicism -- the view that physical stuff is literally made out of mathematical stuff. And I'd like to discuss it as an alternative position on the topic of the existence or reality of abstract objects, opposite of both platonism and nominalism.
In the same way that when we can construct a series of sets that behave exactly like the natural numbers and so are indistinguishable and thus identical to them, so too can we construct complicated mathematical objects that behave indistinguishably from the fundamental constituents of reality and so are, for all intents and purposes, identical to them.
It's possible to build up complicated mathematical objects such as special unitary groups out of bare, empty sets. Special unitary groups are considered by contemporary theories of physics to be the fundamental kind of thing that the most elementary physical objects, quantum fields, are literally made of. Excitations of those quantum fields, which is to say particular states of those special unitary groups, constitute the fundamental particles of physics, which combine to make atoms, molecules, stars, planets, living cells, and organisms, including us.
This illustrates how, in a very distant way, we ourselves can be said to be made of empty sets. And as all of the truth functions, and so all the set operations, and all the other functions built out of set operations, can be built out of just conegation, and the objects they act upon are built up out of empty sets, everything can in a sense be said to be made out of negations of nothing.
It is not a special feature of contemporary physics that says reality is made of mathematical objects; rather, it is a general feature of mathematics that whatever we find things in reality to be doing, we can always invent a mathematical structure that behaves exactly, indistinguishably like that, and so say that the things in reality are identical to that mathematical structure. If we should find tomorrow that our contemporary theories of physics are wrong, it could not possibly prove that those features of reality are not identical to some mathematical structure or another; only that they are not identical to the structures we thought they were identical to, and we need to better figure out which of the infinite possible structures we could come up with it is identical to. We just need to identify the rules that reality is obeying, and then define mathematical objects by their obedience to those same rules. It may be hard to identify what those rules are, but we can never conclusively say that reality simply does not obey rules, only that we have not figured out what rules it obeys, yet.
The mathematics is essentially just describing reality, and whatever reality should be like, we can always come up with some way of describing it. One may be tempted to say that that does not make the description identical to reality itself, as in the adage "the map is not the territory". In general that adage is true, and we should not arrogantly hold our current descriptions of reality to be certainly identical to reality itself. But a perfectly detailed, perfectly accurate map of any territory at 1:1 scale is just an exact replica of that territory, and so is itself a territory in its own right, indistinguishable from the original; and likewise, whatever the perfectly detailed, perfectly accurate mathematical of reality should turn out to be, that mathematical model is a reality: the features of it that are perfectly detailed, perfectly accurate models of people like us would find themselves experiencing it as their reality exactly like we experience our reality. Mathematics "merely models" reality in that we don't know exactly what reality is like and we're trying to make a map of it. But whatever model it is that would perfectly map reality in every detail, that would be identical to reality itself. We just don't know what model that is.
There necessarily must be some rigorous formal (i.e. mathematical) system or another that would be a perfect description of reality. The alternative to reality being describable by a formal language would be either that some phenomenon occurs, and we are somehow unable to even speak about it; or that we can speak about it, but only in vague poetic language using words and grammar that are not well-defined. I struggle to imagine any possible phenomenon that could cause either of those problems. In fact, it seems to me that such a phenomenon is, in principle, literally unimaginable: I cannot picture in my head some definite image of something happening, yet at the same time not be able to describe it, as rigorously as I should feel like, not even by inventing new terminology if I need to. At best, I can just kind of... not really definitely imagine anything in particular.
All of this is building up to me addressing the central question in the philosophy of mathematics, which is about the existence of abstract objects, like numbers and such. There are two main answers to that question, and some positions intermediate to the two, but I want to offer a position that I consider to be off of that spectrum entirely.
One of the usual two positions is Platonism, sometimes called either Platonic realism or Platonic idealism, which holds that abstract objects, or as Plato called them "forms" or "ideas", are real in the same sense that concrete objects, like rocks and trees and tables and chairs, are real; but that they don't exist in our space and time, and instead live in some separate, spaceless, timeless realm, from which they somehow interact with the things in our realm that "partake" of them, in the way that a triangular rock "partakes of the form of the triangle". It is held by Platonists that the existence, in some way, of these abstract objects is necessary in order for mathematical and other abstract statements that seem nominally to be about them to be true: for instance, the Pythagorean theorem which describes the relations of the legs of a right triangle to the length of its hypotenuse is not made true by the existence of any particular triangular objects, but rather by facts about the form of triangles generally, even if no concrete triangular objects existed at all.
I am not very amenable to this position at all, holding it to fall heavily afoul of the principles I've laid out extensively before against the position I call "transcendentalism".
The second of the usual two positions is called nominalism, which holds that abstract objects are merely empty names, that do not refer to real things that exist at all, and are just names for the similar properties of, and collections of, particular concrete objects. I am much more amenable to that position generally, but I think that a kind of existence can nevertheless be applied to abstract objects after all, a kind of existence abstracted away from the more familiar notion of concrete existence.
In the most restricted sense, one could say "only what I am experiencing right here right now exists". Everything else that we talk about existing is some degree of inference and abstraction away from that. There is a position in the philosophy of time, called presentism, that holds that only the present exists, neither the past nor the future. I agree with them to the extent that in a sense only the present exists: only the present presently exists, right now. But a part of what I'm experiencing right now in the present is memory, from which I infer (automatically, intuitively, without thinking about it) the existence of other times, having an experience of moving between different times, from those remembered past times and toward projected future times, and there is a perfectly serviceable sense in which I can say that those other times "exist" in a timeless sense of the word: they don't exist now, presently, for sure, but they still exist at other times.
And in that "movie", so to speak, of my past, present, and future experiences that I have now inferred, I have the experience of seeming to move around different places, so I further infer that other places exist too, besides just the here that I am experiencing now. Like with presentism, only the place I am at exists here, but those other places can still reasonably be said to exist elsewhere.
In this way, a spatiotemporal kind of existence is already abstracted away from the more primitive kind of existence relevant to my local, present experiences. But beyond that, some philosophers such as David Lewis hold, and I agree, that other possible worlds, like the kind that we use to make sense of talk of alethic modalities like necessity and possibility, really exist, and aren't just useful fictions, even though they don't actually exist, because "actual" is an indexical term like "present" or "local": it refers to things relative to the person using the word. Just as other times don't presently exist but are still real in a more abstract sense, so too, on this account, other possible words don't actually exist, because "actually" means "in the possible world I am a part of", but they are nevertheless still real in a still more abstract sense.
Likewise, to finally get on to my point about the existence of mathematical objects, since we can in principle equate our concrete universe with some mathematical structure or another, and that mathematical structure definitely concretely exists (because it just is the concrete universe), we can say that other mathematical structures, i.e. abstract objects, don't concretely exist — because "concretely" is indexical, like "actually", it means "as a part of the mathematical structure that is our universe" — but they can nevertheless be reasonably called "real" in some even broader sense, the most abstract sense possible: they abstractly exist. This position is held by physicist Max Tegmark, and he calls it the "ultimate ensemble"; it is more broadly called the mathematical universe hypothesis, or mathematicism, and it has precursors tracing back to the Pythagorean philosophers of ancient Greece.
This kind of existence for abstract objects does not run afoul of my position against transcendentalism the way that Platonism does, because the abstract objects don't exist in some wholly different kind of way separate from the kind of concrete objects that we can empirically observe. They are just the loosest part of the broader framework of explanation for our empirical observations. We cannot directly observe other times or places, only the local present, but postulating the existence of other times and places helps to explain the patterns in our local, present experiences. Those other times and places aren't held to be discontinuous or of a completely different nature than the local present, they are just postulated extensions of the here and now. Likewise, I hold, with postulating other possible worlds, continuous with the one we find ourselves in and of same nature as it; and also likewise with other abstract objects besides whichever one is identical with the concrete universe, continuous with it and of the same nature as it.
But still, that last step into abstract rather than concrete existence is a significant one. This view of the relation between the concrete and abstract bears a similarity to what Immanuel Kant called the phenomenal and the noumenal, where on his account we cannot ever have direct experiential contact with noumena, but instead only project our ideas about them behind the world of phenomena that we experience, much like how on my account the truly abstract has no direct influence on the concrete world we experience, and we can only project our ideas of abstract objects behind that concrete world in an attempt to understand and explain it.
In relation to platonism and nominalism, we might say that while the platonist affirms the reality of two kinds of existence, and the nominalist denies the reality of one of those kinds, mathematicism like mine outline here instead denies that there are two wholly separate kinds at all, holding the latter kind (the kind the nominalist affirms) to be merely a subset of the former kind (the kind the nominalist denies). Rather than there being no abstract objects, or both abstract objects and concrete objects, on this account all concrete objects are but parts of a single abstract object -- the abstract object of which we are a part, namely the concrete universe.
