What Is Mathematics?

Abstract

This chapter has a reflective-philosophical nature. Usually people do not ask themselves what mathematics is. However, they have implicit ideas about it. The most common idea, in my opinion, is that mathematics is a collection of procedures (algorithms, formulas, etc.) intended to be used in solving various examinations during their mathematics studies.

I explain that mathematics is a collection of theories (number theory, group theory, game theory, for example). Each theory has the following structure: a set of abstract objects, relations on this set, as well as operations on the set. Each theory has a deductive system, including axioms and rules for proving theorems. To clarify this complicated definition, I use the example of the theory of school arithmetic. Most of this structure is already well known to pupils in the sixth grade. I describe its axioms, including the axiom of mathematical induction, and use the axioms to prove some of its theorems.