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> Without persuading other people of the “truths” that you discover, there is no real mathematics.

Why? If I sat around and studied math by myself and discovered something true yet not yet known but didn't share it, it's still true. Are you saying I didn't "do math" because I didn't share the result? Math exists on another plane and it has 'truths' that we haven't discovered, yet are still 'true', no?

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No, they're saying that what is true in mathematics is contingent, not absolute. It all depends on which set of axioms use, what assumptions you make.

The area of a triangle doesn't have 1 unique formula, it has many, depending on the system you use. A triangle in plane geometry has a different area than a triangle in spherical geometry, and different again in hyperbolic geometry.

When you get to studying the geometry of manifolds, you realize the area of a triangle can be any damn thing you want, depending on how you construct the manifold you embed it in.

This is a needless qualifier for the argument at hand.

A proof can just be "assuming these axioms.....the area of a triangle is X"

Sure, but just that some random statement is true isn't interesting. It's interesting if it helps understand some abstract structure better.

It could also be interesting by having a practical use.

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