Set theory and physics

Assume the correct physics has precise particle positions (similar questions can be asked in other contexts, but the particle position context is the one I will choose). And suppose we can specify a time t precisely, e.g., in terms of the duration elapsed from the beginning of physical reality, in some precisely defined unit system. Consider two particles, a and b, that exist at t. Let d be the distance between a and b at t in some precisely definable unit system.

Here’s a question that is rarely asked: Is d a real number?

This seems a silly question. How could it not be? What else could it be? A complex number?

Well, there are at least two other things that d could be without any significant change to the equations of physics.

First, d could be a hyperreal number. It could be that particle positions are more fine-grained than the reals.

Second, d could be what I am now calling a “missing number”. A missing number is something that can intuitively be defined by an English (or other meta-language) specification of an approximating “sequence”, but does not correspond to a real number in set theory. For instance, we could suppose for simplicity that d lies between 0 and 1 and imagine a physical measurement procedure that can determine the nth binary digit of d. Then we would have an English predicate M_d(n) which is true just in case that procedure determined the n binary digit to be 1. But it could turn out that in set theory there is no set whose members are the natural numbers n such that M_d(n). For the axioms of set theory only guarantee the existence of a set defined using the predicates of set theory, while M_d is not a predicate of set theory. The idea of such “missing numbers” is coherent, at least if our set theory is coherent.

It seems reasonable to say that d is indeed a real number, and to say similar things about any other quantities that can be similarly physically specified. But what guarantees such a match between set theory and physics? I see four options:

1. Luck: it’s just a coincidence.

2. Our set theory governs physics.

3. Physics governs our set theory.

4. There is a common governor to our set theory and physics.

Option 1 is an unhappy one. Option 4 might be a Cartesian God who freely chooses both mathematics and physics.

Option 2 is interesting. On this story, there is a Platonically true set theory, and then the laws of physics make reference to it. So it’s then a law of physics that distances (say) always correspond to real numbers in the Platonically true set theory.

Option 3 comes in at least two versions. First, one could have an Aristotelian story on which mathematics, including some version of set theory, is an abstraction from the physical world, and any predicates that we can define physically are going to be usable for defining sets. So, physics makes sets. Second, one could have a Platonic multiverse of universes of sets: there are infinitely many universes of sets, and we simply choose to work within those that match our physics. On this view, physics doesn’t make sets, but it chooses between the universes of sets.