You can very fruitfully study the real numbers as has been done for centuries; that’s not up for debate.

Says who? Math is the study of finding all possible logical conclusions from a given set of assumptions. Reality has nothing to do with it.

I don’t think this is accurate. All logical conclusions would be a horrible mess of symbols because we don’t care about most of them (because they’re considered trivial) and the list would be infinite and filled with statements that are very similar to each other. The subject isn’t about brute force in that way.

It is more true to say that mathematics is about finding all meaningful / interesting logical conclusions from a given set of assumptions. What those two words mean is then the interesting question (and, of course, a matter of personal taste). Reality certainly has something to do with that choice, which is why I don’t think you can escape it. It’s fundamental to the subject, no matter how seemingly abstract you get.

But that depends on what particular people find meaningful or interesting. Lots of people--or at least lots of mathematicians :)--appear to find many things interesting that have nothing at all to do with "reality".

Also, it's important not to confuse "reality" with our models of it. Real numbers are unquestionably very useful in constructing models of reality. That will still be true even if it turns out that actual "reality" is not a continuum.

Yes. That’s why I said it’s a matter of taste.

My claim is that what makes things meaningful is how they relate to reality, and by reality I mean things like the three dimensional world and the passing of time we perceive. Whilst most of pure mathematics seems ridiculously abstract on first glance, I don’t believe there is a single topic that truly has no connection to reality.

> it's important not to confuse "reality" with our models of it

This is a frequently repeated point and I’m inclined to agree, but any such model also exists in reality and so it’s hard to know where to draw the line.

Yes, but there's still a distinction between the model and the thing modeled. A map of your city is not the same as the city, even though the map is a real object in the same reality as the city.

Similarly, real numbers are a model that is instantiated in the brains of mathematicians, papers in the literature, etc. Those things are of course real objects. But real numbers are used to model lots of other real things.

My original point was really a comment on the way that the subject is often taught: the real number system is not introduced as some arbitrary set of rules that is as good as any other (though of course it is); it is introduced as if it were as incontrovertible an idea as, say… integer addition. I don’t think it is. It can perfectly well be argued that it makes sense and is useful, but its naturality is another matter. I personally find the leap from R to C less controversial than that from Q to R.

You don't have to change the laws of logic that you use to do unitary QM (which, taken to extremes, is what gives you a "many worlds universe"). You can describe it just fine using mathematical equations based on the same laws of logic you're familiar with.

In other words, what you are calling "standard logic" actually contains many more possibilities than just the simple Boolean "rains or doesn't rain".

Can you give an example?

If you can't, doesn't that in itself answer your question?

Sure humans can; Greg Egan built a whole sci-fi universe that way:

https://www.gregegan.net/DICHRONAUTS/00/DPDM.html

But two time dimensions doesn't contradict our logic. (It contradicts our best current theories of physics, but that's not the same thing as contradicting our logic. Egan's model is logically consistent.) So this is not an example of what you were claiming.

> Anything I could suggest would by definition not make any sense to us

In other words, you are saying humans can't imagine a universe that contradicts our logic. I agree. And that is a "no" answer to the question you posed in the GGP post to this one.

Whether universes that contradict our logic could "exist", even if we can't imagine them, is a different question from the one you asked in that post. As for that:

> How would you even think about something like this if you throw out the foundation of our thinking?

Indeed. And if we can't even consistently think about such a thing, what's the point?

One interesting fact is that there is a school of mathematics called constructive mathematics. This branch of mathematics proves things without using the law of excluded middle.

In other words, you are saying humans can't imagine a universe that contradicts our logic. I agree. And that is a "no" answer to the question you posed in the GGP post to this one. Whether universes that contradict our logic could "exist", even if we can't imagine them, is a different question from the one you asked in that post.

I mean I could just say let us deny ∀x(x=x) and run with it, see what happens. It really depends on how one understands the word imagine in this context, I could postulate this but I can not really make sense of a thing not being equal to itself.

And if we can't even consistently think about such a thing, what's the point?

I agree with you that mathematics is pretty much divorced from reality, that mathematics makes up axioms and inspects to consequences of them, and also that some mathematical structures are useful to model aspects of the universe. Some were even specifically invented to model certain aspects of the universe, they were inspired by reality.

But at the bottom of mathematics there is somewhere the logic that underpins everything and one can ask the same questions about its relation with reality. Did we arbitrarily pick the axioms of logic or did we pick them because they make sense in our universe? Or are they universal and there can not be any universe that defies our logic? The question is essentially the following. While we - or at least some people - think that mathematics is quite independent of the real world in general, might this be less true than we think because we accidentally introduced aspects of our universe into it by building on top of a logic that reflects how our universe works? Or the other way around, are there other versions of mathematics build on top of other versions of logic that seem illogical to us but only because our universe instilled us with a certain logic.

Of course they do. That is perfectly consistent mathematically with there being two time dimensions. Two time dimensions is a property of the spacetime geometry, not of individual worldlines.

> whether you would want particles to trace out world surfaces

That would be a different mathematical structure, where "particles" would have something like "two time evolutions" instead of one. What that would imply about the spacetime geometry and its dimensions is a different question.

> I could just say let us deny ∀x(x=x) and run with it

How would you even do that? How would you even make a consistent mathematical structure with that as an axiom? And if you can't, doesn't that mean you can't deny that proposition?

> I could postulate this but I can not really make sense of a thing not being equal to itself.

You're contradicting yourself. If you can't make sense of it, you can't postulate it. "Postulate" doesn't mean "throw some words together". It means "take as an axiom and develop a consistent mathematical structure based on it". If you can't make sense of it, you can't develop a consistent mathematical structure based on it.

> Did we arbitrarily pick the axioms of logic or did we pick them because they make sense in our universe?

If we can't develop a consistent mathematical structure based on any other set of axioms, what choice do we have?

Are you sure that it is impossible to build something consistent on top of denying the law of identity?

Modern physics is anything but intuitive in my opinion. Maybe reality isn't well described by real numbers, or maybe it is; but my sense is the reason why physicists have used real numbers is because they've worked pretty well up until a certain relatively recent point, when other intuitions about reality have started to fall apart as well.

[1] https://plato.stanford.edu/entries/philosophy-mathematics/

This is also a lovely, very readable, book on the subject: https://mitpress.mit.edu/9780262542234/lectures-on-the-philo...

They may also be motivated by prestige, to varying degrees, though I think mathematicians are fairly low on the prestige-valuation scale (viz. Grigori Perelman).

Though I enjoy PoM myself, it actually feels a bit weird to wonder why mathematicians don’t care about it. To me, it’s like wondering why chess grandmasters don’t seem to care that chess is a fictional game and not real warfare.

I won’t argue with that. :-)

> it’s like wondering why chess grandmasters don’t seem to care that chess is a fictional game and not real warfare

You’re exactly right. That is why I will never be a chess grandmaster!