[feoren]

> "exactly like ours, except the truth of the continuum hypothesis is flipped"

We can and do create two alternate models of math with CH and ~CH as axioms, in this universe, right now. No need for alternate universes. There's no reason to think the CH is either true or false in the natural laws of our universe -- what would that even mean?

I suppose it's distantly possible that models where CH is true happen to represent our own universe much better than models where CH is false, and that there are other universes that are better represented by models where CH is false. Even if that were true, all the math is still the same, we're just preferring some models over others.

[LegionMammal978]

> what would that even mean?

Presumably something like "you can/cannot collect an uncountable group of points in physical space and still not have enough to fill a physical volume".

Anyway, the idea is that properties of 'ordinary' numbers and logical constructs could similarly just be models specifically useful for our own universe. E.g., propositional logic only works because our universe allows us to write truth tables that are causally valid, natural numbers only work because our universe allows us to count over discrete objects, etc.

There'd be no big gap between 'physics' and 'math': all 'math' that we can talk about would just be the 'physics' of things that work on paper in our universe. And in particular, 'the physics of math-on-paper' could conceivably work differently in an alternate universe, and our own ideas and discoveries would be inapplicable.

[feoren]

It's pretty hard to imagine what an "uncountable group of points" could possibly be, or how anyone could ever test for the existence of such a thing, but we're talking about any possible universe so I can't exactly refute what you're saying here. The very fact that we can even ask questions like "what is the cardinality of a 'set of points' that occupies physical volume?" shows that our math is not at all bound by the constraints of our own universe.

> propositional logic only works because our universe allows us to write truth tables that are causally valid, natural numbers only work because our universe allows us to count over discrete objects, etc.

No, none of this is true. Our universe also allows us to write truth tables that are not valid. We do not dematerialize upon writing down a logical fallacy. Our universe does not seem to contain any infinities at all, and if it does, they're almost certainly countable; yet we can still reason about uncountable infinities without ever having observed them. Our universe seems to exist in only 4 dimensions, yet we can still reason about high dimensional spaces. Why should the constraints of our universe matter to our math at all, other than making some things more obvious than others?

> all 'math' that we can talk about would just be the 'physics' of things that work on paper in our universe

That is just patently obviously not what math is. We have tons of math that is not describing the physics of our universe as we know it.