Mathematics: Problem Solving or Theorem Proving?

Abstract

The view that mathematics is problem solving has been challenged by the claim that, in the twentieth century, mathematics has been reduced to theorem proving. This raises the question: Is mathematics theorem proving, or problem solving? The purpose of the present chapter is to answer this question, which is a philosophical question about the nature of the method of mathematics. Indeed, since antiquity, saying that mathematics is problem solving has been an expression of the view that the method of mathematics is the analytic method, while saying that mathematics is theorem proving has been an expression of the view that the method of mathematics is the axiomatic method. The chapter argues that only the view that mathematics is problem solving is defensible, and the claim that, in the twentieth century, mathematics has been reduced to theorem proving is unjustified.