This essay originally appeared on Quora.com as an answer to the question, “When human civilization becomes extinct, where will all the mathematical truths that we have developed over millennia end up?”
After I spent a few years on philosophy as a youth, it began to seem like a body of irreconcilable differences. That is, philosophical positions seem to ultimately be founded on a simple difference of opinion, or difference in cast-of-mind, that just can’t be changed by argument. This doesn’t mean that philosophy is pointless, per se. I think it can be very instructive to uncover what those fundamental differences are and see how they play out in our thinking. But when philosophers act as if they’re going to win everyone over with some devastating argument and put one of these differences to rest, I can’t take them seriously. I think that these impasses are some of the most interesting aspects of human cognition.
This question is a great example of that. The question of whether or not math “exists” — why do people even have different opinions on this? I think it comes down to how you understand the relation between form and matter, and how real you want form to be. A nautilus shell grows in the shape of a spiral; what does that mean? Is a spiral a real thing apart from a nautilus shell, or just a convenient way to describe a nautilus shell? When we try to scrutinize the relationship between form and matter, we feel a certain confusion, and questions like this arise. It ultimately hangs on this question: are you enterprising, or are you contemplative?
Western philosophy has produced dozens of schools of thought regarding how real abstractions are: the Forms, hylomorphism, nominalism, fictionalism, formalism, and so on. Even though some of these theories (e.g. formalism) don’t involve “matter” at all, I think they’re still rooted in the same confusion that arises when we try to scrutinize the relation between form and matter.
I think there are, fundamentally, two answers. It’s a sliding scale, with many nuanced compromises, but there are two basic positions.
The first one: mathematics is just a game that humans play with symbols. In that case, mathematics vanishes with us. It ends up in the same “place” as chess or basketball or politics, i.e. it’s not there. If mathematical “truths” are truths about the way we play games, then they’re not relevant when we’re not here. I suppose you could say that they exist in some latent space of possibilities, but that’s just another way of saying that, hypothetically, a species could exist that plays certain games.
The second one: mathematical truths don’t need humans to exist. This is Platonism, or some variety of it. The idea is that math exists on its own, without us. That might sound kind of weird, but I wrote an answer a while ago explaining why someone would think this way.
This is my blog and I can ramble about whatever I want, so I’m gonna discuss Frege now. There was a guy named Gottlob Frege who argued that math is mind-independent in some sense because it’s not subjective; there is no your Pythagorean theorem and my Pythagorean theorem. You can’t change the Pythagorean theorem based on your feelings. So it must exist independently of us.
Now, let’s imagine a dialogue between a realist (math doesn’t need people) and an irrealist (math only exists with people), using the Fregean argument as a point of departure. I’m going to stop talking about math here and instead talk about abstraction. You can think of abstraction as the broader class of things that include math. We can define abstraction using very precise technical terms as “airy-fairy spooky stuff”. Numbers, language, ideas, patterns, colors, and money are all abstractions.
The irrealist can sidestep this argument by paraphrasing it in terms of modality; it is only possible for triangular objects to work a certain way. The realist asks a counterquestion: what does it mean for an object to be triangular? Is a triangular object objectively triangular? Or is that just some “filter” we put on it?
Notice how this is the same question at a deeper level. First, they were arguing over whether or not triangles were real. Now they’re arguing over whether or not triangularity describes an objective feature of an object. The irrealist can respond that nothing is objectively triangular; triangles are just a useful fiction (or name, or whatever) for describing certain objects. The realist can press the point and ask why it works on some objects and not others. If the irrealist says “Because some objects are more triangular than others!” then he’s ceding some ground to the realist. To be cagey, the irrealist should say that it doesn’t matter why it’s useful, only that is useful.
At this point in the history of ideas, this seems to be more or less where things stand. Irrealists about abstraction generally subscribe to some kind of pragmatism; they get around the point about the reality of abstraction by simply ignoring it, and they can do this because their whole philosophy is premised on only paying attention to useful things. Realists argue for the reality of math the same way they always have, and may try to attack the irrealist position by insisting that, pragmatically, we do have to acknowledge the reality of abstraction. This is Quine’s indispensability argument.
It goes deeper than that, though. Remember how I began this answer: there are two different casts of mind here, and ne’er the twain shall meet. I strongly suspect that the irrealists — medieval nominalists, modern fictionalists, and all the rest — have always thought the same way. The core tendency of irrealists is always a tendency to simply ignore certain lines of argument, in order to avoid the conclusion. And the core tendency of realists has always been to fixate on certain questions, in full knowledge of where the conclusion is.
(It may seem that I’m siding with the realists here, given that I’m saying that argumentation always goes a certain way if you allow it to happen. And indeed, I used to side with the realists for precisely that reason. In recent years, however, it has occurred to me that reasoning yourself into something doesn’t make it true. Logic establishes propositional relations only; it does not guarantee the truth of premises.)
In my usual pedantic fashion, I’m going to ask why. Why is it that realists want abstractions to be real? And why is it that irrealists would rather abstractions not be real? It may be that both are ultimately looking for confirmation of their base vision of reality. And this is the root issue I’ve been seeking this whole time.
Irrealists about abstraction see reality as the place where I can do things. The real world is the world where I can do stuff. The set of things I can do changes continuously, so reality is constantly in flux.
Realists about abstraction have a more quietistic, “primordial” view of reality. Reality is just a bare, unanalyzable that-which-is. The most important question about anything is whether or not it’s real, and “real” is something we all understand on a gut (“intuitive” if you wanna be fancy) level, even if we can’t articulate it.
And that, I think, is the fundamental difference in perspective that has generated the past two millennia or so of debate around the reality of abstraction. Is reality a place where you can do things? Or is reality just reality, full stop? Are you enterprising, or are you contemplative?