Wolfgang Schwarz

What is it Like to be a Set?

I've been invited to this year's German-Italian Colloquium in Analytic Philosophy, for which I've put together some remarks on the philosophy of mathematics: "Emperors, dragons and other mathematicalia" (PDF). I mainly argue that mathematical sentences should be interpreted as quantifications over possibilia. Technically, this isn't really new. Daniel Nolan in particular has made a very similar suggestion (PDF). What hasn't been emphasized enough, I believe, is that this interpretation not only works from a technical point of view, but is quite attractive for various philosophical reasons. (Unlike Nolan, I argue that it isn't a reform, but a faithful interpretation of mathematics.)

The basic idea is eliminative structuralism: To interpret a mathematical sentence S, find a nice axiomatization A of the relevant mathematical theory, then replace all mathematical terms in A->S by universally bound variables. The remaining question is whether something satisfies the axioms. But if the axioms are consistent, and the quantifiers are possibilist quantifiers, this is not a big deal.

As I've taken some material for the paper from my PhD thesis, I've illustrated this programme with Lewis's megethological axiomatisation for set theory. The main advantage of Lewis's axioms is that they are satisfied if only there exist sufficiently many things (as Lewis proves in "Mathematics is Megethology"). So we don't need any bold assumptions on possible structures, only a bold assumption about the number of possibilia.

On the other hand, it seems that Lewis's system faces problems with impure sets that a more traditional structuralism based on, say, the ZFC axioms, might avoid. For in Lewis's system, no fusion of a set with any part of the empty set is a member of a set. But on some realisations of Lewis's axioms, Julius Caesar (for example) will be such a fusion. So it isn't true on all realisations of the axioms that Julius Caesar has a unit set, which implies by the structuralist analysis that "Julius Caesar has a unit set" is not true.

If one uses the ZFCU axioms instead of Lewis's, Julius Caesar will be an individual on some selections of the membership relation, a set on others, and on some selections he will (in addition) be a mixed fusion of sets and individuals. But none of that implies that he doesn't have a unit set. On the contrary: in ZFCU, every set and every individual has a singleton. (Stronger systems like NBGU contain entities, proper classes, that are not members. If we use such a system, we must again insure that Julius Caesar is not one of those.)

On this interpretation, there is no good reason for adding urelemente to ZFC in the first place: even if we allow urelemente, Julius Caesar will not determinately be one of them. (That's the whole point of treating set theory as a quantification over possibilia.) If we use ZFC without urelemente, so that everything is a set on any selection, all ordinary uses of ZFCU will be validated, as far as I can see. It's just that they are ultimately only uses of ZFC. (Fundierung excludes that Julius Caesar is his own unit set.) ZFC may even turn out to be sort of categorical on this interpretation: since (I believe) any non-isomorphic models of ZFC differ in size, if the quantifiers in ZFC are absolutely unrestricted possibilist quantifiers, the size of modal space might determine a class of isomorphic models. That is, every set theoretical sentence will then be either true on all selections of the membership relation or false on all selections.