Platonists believe that abstract objects such as numbers, colours, sets, ideas and so on are actually existing things (often referred to as the “Platonic realm”). So, if you’re a Platonist and a mathematician, you can take the axioms of maths as descriptions of the kind of objects, found in the Platonic realm, that you want to be working with. So, by “set” we mean those objects described by the Zermelo-Frankel axioms (for example). Then say we have some mathematical proposition that says “There exists some set S such that P(S)” where P is some property S could satisfy. If you’re a Platonist, you can take this statement at face value: there really is a set S, such that P(S). I wanted to think about how some other schools of thought could make sense of this claim (or any truth claim).

Another way of interpreting mathematical claims is formalism. For formalists, mathematical claims don’t have any meaning: they are merely strings of symbols (of course we don’t always write them as symbols, but there’s an unspoken assumption that anything we say without symbols can be translated into symbols). The axioms represent the base strings, and there are certain “translation rules” that tell us how to translate valid strings into other valid strings (this is a slightly simplified understanding of formalism but I think it’s sufficient). Clearly, on this interpretation, a true mathematical claim is one for which there exists a sequence of valid transformations from the axioms[1]. For the formalist, then, to say that S exists, is to say that “(∃S)P(S)” is a true claim.

I was thinking about another way a person might interpret mathematical claims (I don’t buy it, but it’s interesting anyway). Of course there seems to some sort of niceness about Platonism, since it allows one to take things at face value. On the other hand, you might (like me) be disinclined to think that there are really these abstract objects in existence. So why not take a sort-of “counterfactual Platonism”? So, we interpret the axioms as statements that would describe specific objects were abstract objects real[2]. Then we can just take the claim that S exists like the Platonists, just with the extra counterfactual bits: S is that object that would exist such that P(S), were abstract objects real.

Notes

1. I suspect I should be slightly more specific with my definition of “true” here. After all, according to Godels incompleteness theorems, there are true statements that can’t be proved or disproved. But, on formalism, a proof is just a valid sequence of translations of symbols starting from the axioms, so if we took what I said here, then truth and provability would be the same thing. What we would actually need to say is that a string is “true” if it can be validly arrived through string translations (as we said above) but then add (because of the principle of excluded middle) that every syntactically correct string is either true of false. There’s probably a better way to say this.
2. I suspect we would need to interpret this counterfactual using strict logical possibilities. This is because, plausibly the Brouwer axiom for alethic possibility is true and abstract objects are taken as necessarily existing entities, if they exist (and we would want them to exist in at least one of the possible worlds we’re quantifying over with our counterfactuals, lest the claims be vacuously true; of course this might not be a requirement, but let’s assume for generality that it is). But these two things would imply that they actually exist (let A be the proposition that some abstract object B exists. Then, by assumption, ◊□A. By Brouwer, ~A→□◊~A, and so by contraposition, ◊□A→A), which is what we wanted to avoid in the first place.