Logic and Intuition in Mathematics and Mathematical Education

Abstract

A good mathematics teacher is not only a good mathematician, but also a good teacher. In other words, a good mathematics teacher is not only able to solve mathematical problems, (s)he is also able to explain how mathematical problems are solved. Many mathematicans (and mathematics teachers) are, however, able to solve mathematical problems without knowing or understanding how they solve these problems: solving a mathematical problem often involves a multitude of unconscious or intuitive mental processes. And a person who solves certain problems without knowing or understanding how (s)he solves these problems is not able to explain to others how these problems can be solved. Consequently, many mathematics teachers would be better teachers if they knew more about the psychology of mathematicians and mathematical invention. In this article, the distinction between mathematical invention and mathematical discovery will be discussed from a psychological viewpoint. Some ideas about the psychology of mathematicians and mathematical invention will be formulated. These ideas fit in with so-called universal Darwinism and will be helpful in understanding the distinction between mathematical intuition on the one hand, and deduction or logic on the other.