Can someone explain how abstraction in mathematics works?

Question

I always hear that math is based on abstract thoughts and ideas. My question is: what is abstract thinking in the context of math? Can anyone provide an example of it?

Answer 1

Many mathematical objects have no independent concrete existence but exist only as a consequence of an axiomatic system.  Families of elliptic tori arise in the study of the congruent number problem of algebraic number theory, but I have never encountered one in my daily travels; however, I have visited them often in my late night musings...