What kind of alternative mathematics systems exist?
What I mean is, mathematical systems that use a different sort of "basic premises" or e.g. logic(s) than the contemporary "mainstream" mathematics.
Is such possible even? I would think that yes, because one could perceive the mathematics that we know to e.g. be a bit "western". However, there's also at least some similarity between mathematics from different cultures, they seem to come to agree on the topics that they all studied.
However, this is not enough to make me confident about the "mainstream" mathematics being "the only mathematics there is". While e.g. logical consistency is useful for making progress, I don't think it's clear in philosophy of science, whether the "mainstream" mathematics is "complete". That is, whether it will stay without hitting into some fundamental problems further down the line. Since there are examples of this in the history which has also led to the development and refinement of mathematics.
Related:
Can There Be an Alternative Mathematics, Really?
Well, there's something called fuzzy logic if you're interested in checking that out. The thing about math is that it's not really accepted unless it's consistent, and it isn't called a universal language for no reason, so all of math is sort of grounded in logic and reasoning. It doesn't care how you feel about it. There's different theories about how to approach subjects though such as nonstandard analysis, and if you want to talk about physics, there's something called Bohemian mechanics that gets the same results as quantum mechanics with different assumptions, but what is important is that the results are consistent.