The Indispensability Argument and Impure Set Theory

Question

Assume for the sake of this question that mathematics is reducible to set theory in such a way that the only mathematical objects there really are, are sets.

Suppose further that the Indispensability Argument is successful and shows us that we need to believe in sets.

My question is would this show us that impure set theory is indispensable? Does physics require impure sets? In other words, are there primitive relations of physics (where a primitive relation of physics is something unique to physics--- not something like equality/identity) that hold between mathematical entities (like numbers) and physical entities?

Are there any good papers that discuss indispensability and impure set theory?

Answer 1

I'm afraid I don't know of any papers that discuss indispensability and impure set theory directly. In a sense there's a reason to think that impure set theory is nice to have for physics. As you know, it's natural to represent functions in set theory as sets of ordered pairs, hence if you want to talk about functions that map spacetime points to other spacetime points, you'll need impure sets (if those functions don't float your boat, pick your favourite physical objects that you want to talk about functions from/to).

However, to say that impure set theory is indispensable is a much stronger claim. It's also going to be tough to establish, given (as I'm sure you're aware) that the universe(s) of pure sets is pretty big, far bigger than the physical universe (on any plausible theory of the physical universe; the biggest I think you can reasonably get it is if you allow mereological sums of spacetime points then you might be able to argue for the universe having cardinality $2^{2^{\aleph_0}}$, which is diddly squat in set theoretic terms). Given then the sheer abundance of sets, we can always represent new an interesting physical phenomena by pure sets, in such a way that the impure set theory is dispensable.

However, one should be mindful of the dialectic into which an indispensability argument is often inserted. Usually there is some sort of Quinean holism in the background providing the necessary oomph to think that indispensability to science matters. Given this, one's question really should not be "what is indispensable to science?", but rather, "what is indispensable to our best theory of the world?".

If the latter is the question, and if one thinks that categoricity is important for a mathematical theory (say for worries about first-order theories being unable to pin down their intended model up to isomorphism), one might be interested in the following paper by McGee:

McGee, Vann; `How We Learn Mathematical Language', The Philosophical Review Vol. 106, No. 1 (Jan., 1997), pp. 35-68

There he gives a full categoricity proof for $ZFC$ (on the assumption of unrestricted first-order quantification), by first adding urelemente and proving the categoricity of the pure sets from the impure universe. Thus if it turned out that urelemente were indispensable for this task, one might think that that impure set theory is an indispensable part of our best theory of the world after all.

[As a footnote, it should be noted that there are plenty of other ways to get categoricity given unrestricted first-order quantification. See, for example:

Martin, Donald A. (2001). Multiple universes of sets and indeterminate truth values. Topoi 20 (1)

who argues that any two universes of sets (satisfying certain criteria) can be combined,

and

McGee, Vann (1992) Two Problems with Tarski's Theory of Consequence Proceedings of the Aristotelian Society New Series, Vol. 92, (1992), pp. 273-292

where he argues for categoricity through the introduction of a satisfaction predicate.]

Answer 2

One generally only does mathematics 'up to isomorphism', so that when you talk about the elements of the cyclic group on 12 elements you are really talking generically about tons of other things, including the groups of roots of all kinds of equations, the factor groups of all sorts of larger groups, various constructions in the surreal numbers...

No one of these representations is 'more real' than any other. So the idea that you have to map the actors in some physical system onto the points in some constructed geometry in order to work with them mentally does not mean that they are actually being handled in the mathematics itself.

From an intuitionistic point of view, the mathematics is all psychological idealization of the intuitions that allow us to communicate about the outside world. From that point of view, as an idealization, all of the abstract representations that are isomorphic are 'really' what the underlying mathematics is made of.

Any of them is all of them. In that sense mathematics never handles real objects and 'impure' sets are an idealization of the notion of naming. So you are perfectly safe with the simplest model of space as a product of geometry-imbued continuua made of convergent sequences of ratios of enumerations of multiple copies of the empty set.

Science is always using approximations and idealizing them. The electron, treated mathematically, is not a real thing, it is a collection of behaviors mapped to a point. From a lower perspective, the Schroedinger wave that gives the distribution of the apparent behavior over space is more real, and so on down the rabbit hole. The approximations involve potentially infinitely many named anchor points, but we have no problem creating infinite names since we have models of the reals and the integers.