Thursday, February 11, 2016

Is spacetime countable?

Plausibly there is such a thing as a true physics, an ideal physics that we are striving towards, a physics that includes both particular statements as well as laws. That true physics is true, and hence consistent. It is also the sort of theory we can produce, so it has countably many statements (maybe finitely, but perhaps we could continually add to it). Finally, it is very likely to be a first-order theory, since it looks like all of science involves first-order theories.

Suppose there is a spacetime. Then the true physics posits it. Imagine we now have the true physics. By downward Löwenheim-Skolem, the true physics is consistent with spacetime being countable. So, by Ockham's Razor, wouldn't we have good reason to think that spacetime is countable, since that's more parsimonious than its being uncountable? And, now returning to the early 21st century of the actual world, doesn't the fact that if we had the true physics we would have good reason to think spacetime is countable, give us good reason to accept the counterintuitive conclusion that spacetime is actually countable?

I am not sure. For even if a countable spacetime is ontologically simpler, its description in our mathematical language is more complex than that of an uncountable spacetime manifold. Does that matter? Yes, but only if our mathematical language actually cuts mathematical reality at the joints or if we are created to get science right. (I suspect a naturalistic physicist could have a hard time resisting the argument for a countable spacetime. But this is all very, very speculative, and my mind fogs up when I think about the relativity of uncountability, intended models and the like.)


Heath White said...

I don't think I understand what "spacetime is countable" means. If it's anything, 'spacetime' is a mass term, not a sortal. I know what "oranges are countable" means but not "air is countable." Help?

Alexander R Pruss said...

The set of points in spacetime is countable is what I meant.

Heath White said...

OK, another stupid question: I can pretty easily produce a theory or model (say, in geometry or math) which identifies points by three axes, which axes are real number lines. The set of points so defined is not countable. QED?

Alexander R Pruss said...

Any countable set of axioms for real number arithmetic can be satisfied by a countable structure. These things are paradoxical, which is why the Lowenheim-Skolem Theorem can also be called the "Skolem Paradox". (As background, think of Putnam's use of Lowenheim-Skolem to make trouble for theories of reference.)

I think what is at issue here is may be something like this. Suppose a physicist has a theory that uses the phrase "real number". As long as the theory is first-order, one can reinterpret the whole theory in such a way that phrases like "real number" doesn't mean things like real numbers, but members of some particular countable set which has the same first-order structure as the reals.

The theory under the aberrant interpretation will be just as empirically adequate as the theory under the intended interpretation (I am assuming, pace Putnam, that it makes sense to talk of an intended interpretation), BUT it will have a smaller ontology (countable rather than uncountable). So it SEEMS that the aberrant interpretation is to be preferred to the intended one by Ockham.

This argument can be resisted if we think there is something particularly natural about real numbers as opposed to the members of the corresponding set in the aberrant interpretation. But the question of how naturalness works within the mathematical domain seems difficult...

Alexander R Pruss said...

I should say that I was imprecise when I said "the true physics is consistent with spacetime being countable". The correct way to state this is that the true physics *can be interpreted* in a way that makes "spacetime" refer to a countable structure.

Moreover, when I think about how the Skolem proof works, the aberrant interpretation that makes "spacetime" refer to a countable structure can keep fixed the interpretation of fundamental physical terms like "photon", "electron", etc. as well as keeping fixed the interpretation of bridge terms like "dial on galvanometer" and "sensation as of seeing a red patch", as long as there are only countably many photons, electrons, galvanometers, distinguishable sensations, etc.

Basically, the issue is that only countably many of the points of spacetime are needed to make true the statements of a first-order physics. So why suppose there are more?