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?
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".