What kind of alternative mathematics systems exist? 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?
https://link.springer.com/chapter/10.1007/0-387-24270-8_30
 A: 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. 
A: I stumbled onto this question after hearing a story about a student using a different set of equations to arrive at all the correct results in class - echoing what Math Model said about Bohemian and Quantum mechanics. My interest in that story comes from exposure I had to "Godel, Escher, Bach: An Eternal Golden Braid" by Douglas Hofstadter. I don't have an answer but thought I'd try to connect the two ideas. 
Godel Escher Bach (GEB) provides a lot of insight into "strange loops" and in the beginning (I haven't had time to complete the book yet actually) it provides a good explanation of axioms in a system. My thoughts on your question are this: mathematics as we know it is based on a set of axioms that the other people in this thread have called 'logic.' That is fair, I cannot imagine off the top of my head any other axioms that make sense besides addition and multiplication etc.. 
However if a different set of axioms were used then that would effectively be a 'different' kind of mathematics. Who is to say what the results would be - they might simply not make sense in terms of "our" mathematics, but as long as they were consistent then they could be trusted and built upon. 
Good question!
