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If you put two things in a bag, and put three more things in there, there are now five things in the bag. I don't think there are other valid answers.

Math starts from that basic simple assumption. Then it can slowly dog-paddle into the weird end of the pool that blurs the line into philosophy or epistemology.

"If I have five things in the bag and I cut one in half, do I now have six things in the bag? What if cut one of them infinity times? Are there any exceptions in which two plus three may NOT equal five or fifteen times twenty may NOT equal three hundred? In either case, how do I prove it? Can I prove it in 24 dimensions?"

But yeah, even when exploring new math, it's still like a numerical story that's writing itself, as if the mathematician is taking dictation, as opposed to making stuff up, because then the math doesn't work.
If Newton and Leibnitz had not come up with calculus, somebody else would have, and sooner rather than later, conditions were ripe for a tool exactly like calculus to be useful. And it would have ended up looking exactly the same.

Is this ALWAYS the case? Is there any math that is purely creative? Can there be any math like this? Maybe the academic world is rife with this sort of thing, and I simply don't know about it.