[yrui]

I recommend reading histories of mathematics including more recent histories like Morris Kline's "Mathematics: the loss of certainty." According to other books by Reuben Hersh, such as "the Mathematical Experience" and "What is mathematics really?", mathematics can be seen as a social activity where what gets studied and what gets accepted as proven has a definite relationship to human existence and human social life. If you agree with this, even in part, history of math becomes important.

A few words on your questions:

Mathematics is no different from any other field except that its objects of interest are all abstractions. As an engineer you use it, so it appears to be about number and symbol pushing, but the theory of how and where to push the numbers and symbols and why it works was discovered by a mathematician at some point. The proofs are not ultimate but hopefully convincing given certain usually acceptable assumptions or axioms. We like the axiomatic method because axioms are usually not controversial and everything proceeds from them using strict logical reasoning.