Let me preface this question with the following: I have a background in physics and thereby some knowledge on mathematics, but little knowledge about philosophy itself. The question in the title arose from a discussion I had with a friend the other day, which started with this thought: in physics, there is the so-called fine structure constant. This constant has the interesting property that it is just a number, it does not have any units. In other words, how we define a length, a time interval etc, i.e. our choice of e.g. meters and seconds as our units in which we measure the world (which is sort of arbitrary), does not influence the value of this constant.
Moreover, due to its status as a physical constant, i.e. a property of nature, it is universal (or rather, this is one of the fundamental assumptions of modern physics), meaning if we have a measuring apparatus for this constant, it does not matter whether we measure the constant's value here or in a galaxy billions of lightyears away. Let's assume that this universality is true.
The two factors above combined lead to statements such as "if we wanted to communicate with aliens, we should tell them about the fine structure constant". The thought process is that a sufficiently advanced alien civilization would have discovered the fine structure constant as we have, and found it to have the same value due to the above reasons. The question that we have arrived at is whether this is actually true. Is mathematics universal - or natural - in such a way that, even if such aliens were to operate mathematics under completely different axioms or in a completely different way from us (whatever that may look like), could we still "reduce" their and our maths to be "equivalent"? In other words, if they followed a completely different form of mathematics that we have not yet discovered, but their form is consistent and produces the same results, is it really "different" mathematics? (This goes somewhat into the direction of "If I have physical theory A and B, which operate differently but produce the same prediction of observations, are A and B really different?")
I realize that this is a very broad question, but maybe an answer could be a starting point for me to go deeper into this topic.