Among actual set theorists Platonism is quite popular as it gives them the feeling that they are discovering some kind of pre-existing eternal truth. Of course many other philosophies of mathematics would support the idea that mathematicians were discovering eternal truths about what are logical consequences of a particular axiom system, i.e., ‘ZFC entails Ramsey’s theorem.’ However, the special affection mathematicians have for Platonism seems to stem from it’s apparent ability to make the choice of axioms themselves more than a matter of convention or convenience. Most Platonists believe that the axioms are either true or false of actual sets in the same way the earth is either round or flat. Thus in this Platonistic framework math is no longer merely discovering conditional truths but truly investigating the nature of sets.

Unfortunately, the belief that if true Platonism would regard certain axiomitizations as true and others as false is simple incorrect. At least without significant additions to the theory. In fact as I shall show Platonism, just like the other philosophies of mathematics, in fact regards all consistent axiomitizations of set theory as equal with only our preferences to choose between them. As Platonism is essentially the extension of our normal notion of truth to mathematical objects it behooves us to first consider when this standard notion. Importantly, even for normal objects it is not just the external world but also our choice of language which determines what sentences we accept as true or false. For instance the fact that tables are flat surfaces one sets plates upon rather than objects designed for sitting upon has nothing to do with the external world and everything to do with our choice to use the symbol ‘table’ to mean the former and ‘chair’ to mean the latter.

This point has little import for most statements about the physical world as one of the implied components of meaning for most words is that they refer. Thus even though we might only stipulate that an electron means an indivisible object of negative electric charge found in hydrogen atoms this leaves no linguistic ambiguity as to whether electrons interact with the strong force. Since it is understood that we mean electron to refer to all existent objects satisfying the definition, all we need to do is go out and test whether those objects in the world meeting our definition of electron are affected by the strong force to determine the truth of this claim. In other words the use of the word electron is inessential in the claim “Electrons are affected by the strong force.” We can remove the word electron and translate the claim into, “There exists an indivisible object found in hydrogen atoms and affected by the strong force.”

The same tactic will not work for set theory. According to Godel’s completeness theorem, which is provable from ZFC, any consistent theory has a model. Thus if it is not possible to deduce a contradiction from some set of sentences there is an actual set of objects which satisfy those sentences. Since all Platonists about ZFC agree about what (recursive) sets of sentences are consistent they all agree about what sets of sentences have a model. In particular the both agree that the axioms of ZFC+~CH and ZFC+CH are consistent since ZFC proves CH is independent of ZFC. The upshot of this is that both Platonists who believes sets actually satisfy ZFC+CH (Continuum Hypothesis) and those who believe sets actually satisfy ZFC+~CH both agree that the sentences, “Objects exist which satisfy ZFC but not CH” and “Objects exist which satisfy ZFC and CH” are true.

Thus even assuming Platonism there is still no external matter of fact about what the right axioms are for set theory. The question is still entirely linguistic: which objects that all Platonists agree exist should we call sets.? While this result can probably be interpreted several ways at the very least everyone must agree on the following. If there is a sense in which one of “Sets satisfy CH” or “Sets don’t satisfy CH” than we must mean more by set than merely “an object which satisfies ZFC.” This means that any Platonist about sets who wants to argue that there is a determinate answer to CH has to meet the challenge of explaining what else he believes we mean by set beyond ZFC which entails his result.

Some (Kenny) might object that this shows no more particular difficulty for a realist about mathematics than does the existence of observationally indistinguishable theories about the physical world. For instance a theory asserting the reality of a electrons and another theory only asserting certain patterns in sense data. However, the difference here is significant. On a empiricist theory of meaning the choice between these two theories might be only a matter of meaning but once you accept ‘physically real’ as a real valid predicate one of the statements becomes true and the other false. In this mathematical case even granting the Platonist their ontological categories the difficulty still remains.

Moreover, Quine’s theory of indeterminacy of translation does nothing to diminish this result either. It only drives home the stake that the choice between sets which satisfy CH and those that do not is purely linguistic. Whether or not other statements occupy this category as well is unimportant. If you try to claim that set couldn’t just mean object that satisfies ZFC on Quine’s theory since this would be an unacceptable analytic truth you are just biting the bullet and admitting you mean more by set. Moreover, retreating to Quine seems self-defeating for a mathematical Platonist who wants to believe there is an object matter of fact about statements like CH.

Besides the entire excursion into Quine is really quite pointless as his arguments are just wrong. The inability to coordinate language through purely experimental methods is simply irrelevant to whether or not creatures with a similar physical basis might have more coordination of meaning. The entire idea that we can talk about meaning as if it was just some fact purely determined by an arrangement of atoms is just downright absurd. There simply have to be brute level facts about meaning somewhere if you want to admit anyALL meaning facts or answer the famous argument that this is a necessary consequence of indeterminacy in translation.