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.
Can There Be an Alternative Mathematics, Really?