I study electronic engineering at university, 3rd course. I had to use mathematics a lot, from basic algebra to analysis. Yesterday, after watching some mathematics-related videos and reading some posts I made a conclusion that I don't have a general understanding of this beautiful field of science. I don't know if this kind of question is appropriate here but I want to ask a basic but nontrivial (for me) question:
What is mathematics?
That is, what branches it has, what is the foundation, the starting point, etc. This question emerged when I decided to get a more profound insight of the matter and start learning mathematics from scratch to form a systematized knowledge and after all I love using and studying it.
P.S.: I apologize for my English: it isn't my first language.
One encyclopaedic definition I've seen, which I think is very good, is that mathematics is the study of the logical consequences of axioms. (Technically, the consequences depend also on the rules of inference.) Such study cares not for what mathematical objects are, but what they do. For example, any construction of the real numbers will prove the same facts about them, so it's meaningless to ask whether they "are" Dedekind cuts or something else isomorphic. See also this David Hilbert comment.