What Is Mathematics?

  • Shlomo Vinner
Part of the Mathematics in Mind book series (MATHMIN)


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.


  1. Courant, R., & Robbins, H. (1941). What is mathematics? New York: Oxford University Press.zbMATHGoogle Scholar

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Shlomo Vinner
    • 1
  1. 1.The Hebrew University of JerusalemJerusalemIsrael

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