This has always bothered me: it would seem that the concept of Euclidean space a

Question

This has always bothered me: it would seem that the concept of Euclidean space and real numbers themselves arose out of necessity for describing the physical universe that we live in. Mathematics, on the other hand, has become more broad than this and has generalized itself to adapt to any field, of which real numbers are only one. It is unclear to me whether or not we would even have reason to conceptualize the idea of real numbers if we lived in a universe without them. Furthermore, geometry is probably a requisite for every physical theory in existence, although I could be wrong on that point. Stochastic (as opposed to physical) chemistry comes to mind as a weak counter-example. This also brings to mind the fact that I don't know what makes something a physical theory, as opposed to... something else. I think that ideally all physical theories and models are intended to approximate a real world system, but isn't that true for all geometry branches?!

What valid logical criteria exists (if any) to classify geometry in mathematics as opposed to physics?

Explanation / Answer

Mathematics does not need to bother itself with real-world observations. It exists independently of any and all real-world measurements. It exists in a mental space of axioms, operators and rules.

Geometry conforms to that description.

Physics depends on real-world observations. Any physics theory could be overturned by a real-world measurement.

None of maths can be overturned by a real-world measurement. None of geometry can be.

Physics starts from what could be described as a romantic or optimistic notion: that the universe can be usefully described in mathematical terms; and that humans have the mental ability to assemble, and even interpret, that mathematical description.

Maths need not concern itself with how the universe actually works. Perhaps there are no real numbers, one might think it is likely that there is only a countable number of possible measurements in this universe, and nothing can form a perfect triangle or point.

Maths, including geometry, is a perfect abstraction that need bear no relation to the universe as it is. Physics, to have any meaning, must bear some sort of correspondence to the universe as it is.

So although geometry appears to bear relation to the universe as it is, it need not do so in order to satisfy its own axioms: it has a consistency requirement, as opposed to a descriptive requirement.

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