I am not someone who is deep into the field of mathematics (the most advanced I reached to learn in school were limits and I have forgotten a good part of basic mathematical procedures because I have not even practiced linear equations for a long time) but the axiomatic way in which mathematics appears to be structured has always seemed interesting to me

But, given that, from school mathematics to mathematics "per se" there is an abysmal stretch in which one begins to wonder: " non-linear algebra? non-Euclidean geometry? Zermelo-Frankel sets? WTF? " I want to ask: how to even have a basic idea of "what is mathematics and how is it organized" in its extension? Is there anything that serves as the backbone for the other mathematical branches? If so, how transversal is it?

If not, what really allows all these branches, say arithmetic, geometry, topology etc to be encompassed under the term "mathematics"?