Intuition in Mathematics: from Racism to Pluralism

In the nineteenth and twentieth centuries many mathematicians referred to intuition as the indispensable research tool for obtaining new results. In this essay we will analyse a group of mathematicians (Felix Klein, Henri Poincaré, Ludwig Bieberbach, Arend Heyting) who interacted with Luitzen Egbertus Jan Brouwer (the father of the intuitionist foundational school) in order to compare their conceptions of intuition. We will see how to the same word “intuition” (in German Anschauung ) very different meanings corresponded: they varied from geometrical vision, to a unitary view of a demonstration, to the perception of time, to the faculty (shared by everybody) of considering concepts that habitually occur in our thinking separately. Furthermore, we will discover that these different meanings had a philosophical, very relevant counterside: they passed from a racial characterization of mathematics to a pluralistic view of it.