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Comment by LegionMammal978 | Hacker News Reader

> That's what's so cool about math: it could not possibly be any different.

Why not? There's not much tethering our axioms-on-paper to what is necessarily true, past what we can empirically observe. For instance, a universe that is "exactly like ours, except the truth of the continuum hypothesis is flipped" seems no less conceivable than our own universe, given that we don't even have any solid evidence for its truth or falsehood in the first place.

If we're willing to treat mathematical and logical ideas as physically contingent, then it's only a few further steps to "the concepts of identity and discreteness and measure in this universe are different than ours, so all our mathematical axioms are not applicable". Though it would be very difficult to translate any stories from such a universe into our own ideas.

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> "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.

> 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.

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.

Why not? Exactly because there is nothing tethering our axioms on paper to what is necessarily true. You could formulate something wildly different from ZF±C/Peano/whatever normal axiom system, but we wouldn't call it "math", and what we currently call "math" will work under any conditions

Our 'math' will work under any of our conditions (as far as we can observe), but who's to say they can't have 'math' in another universe that will work under any of their conditions, yet still be different from ours?

That's what GP was saying ("there are no conceivable alternate universes with different math", and none with a different derivative in particular), but I see no reason why math-as-we-know-it couldn't just be inapplicable to different 'conceivable' universes.

> For instance, a universe that is "exactly like ours, except the truth of the continuum hypothesis is flipped" seems no less conceivable than our own universe

Really? For that to be possible, the continuum hypothesis would have to be either true or false in our universe, which does not appear to be the case.

That's fair, and perhaps my example wasn't the best, but my point is that just as the continuum hypothesis is an artifact of the models we use to describe our universe (we use continuum-sized sets to describe physical space), more basic properties like "how numbers ought to work" could also be artifacts of our models. In particular, it wouldn't be inconceivable for an alternate universe to be better described by entirely different models from the ground up, which could be fairly described as "different math".

> we use continuum-sized sets to describe physical space

But... we don't. We use integers to describe physical space. We have real numbers as a mathematical construct, but we have never applied them to even a single physical problem. That's impossible to do, because specifying a real number takes an infinite amount of information.

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