The basis of mathematics is just a mental construct which has nothing to do with the outside world.
I'm waiting for Vector to swoop in on her giant eagle and bean you with a math textbook, but in the meantime... Math has literally everything to do with the outside world. Here's an example.
[loads of physics stuff]
That's physics, not mathematics. Physics uses mathematics, but physics isn't the basis of mathematics. In a sense, math is the basis of physics, not the other way around. The fact that physics uses math doesn't really matter when you're talking about how you could understand math by watching physical processes.
Ehm, no. Mathematics is not a natural, empiric science. You could learn nothing about mathematics from looking at anything because the basis of mathematics is just a mental construct which has nothing to do with the outside world.
With fear of derailing this thread, and leaving it forever in my updated replies list:
How on earth did we develop it, then?
That's a different story altogether. We have an intuitive, empiric or learnt understanding of things like sets, natural numbers and rational numbers. Based on that "understanding" (which is not really an understanding, just a "This is how it should be." kind of thing), we build a model which most closely resembles it. And then, we define stuff, make some theorems, propositions and stuff, prove them using our model and hope that it still works when we try to apply it.
That's the thing: It's all a model. That model didn't directly come from the outside world, but was carefully constructed so that it somewhat fits our ideas. There's no guarantee that it does, though.
No, ein, our formulas, theorems and axioms aren't built around base ten, they're built around real numbers and those are defined without even talking about base something-or-other.
And no, Max White, that doesn't change anything. We're still working with our known numbers, the natural numbers, rational numbers and shit like that are still the same.
Now, if you didn't work with the real numbers, but with a completely different field, all formulas would be different and it would be extremely difficult to understand.
Okay, let me clarify: Yes, mathematics as we use and have them are based on our vague understanding of numbers which is based on our vague understanding of the outside world which is vaguely based on the outside world. That's a lot of vagueness and furthermore, it's not guaranteed that everyone makes the same connections. Just because we see the same outside world doesn't mean that our understanding is the same. Just because our understanding of the outside world is similar doesn't mean that our understanding of numbers is the same and just because our understanding of numbers is similar doesn't mean that our model of mathematics is the same.
So really, if we're talking about an intelligent non-human being it's highly unlikely that it'll coincidentally develop the same understanding of mathematics.
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