In the first part of the talk, I shall explore the consequences of distinguishing the foundations of meaning and the foundations of truth in mathematical statements, or imagination and rigor as motors of mathematical development. The foundations of meaning can be sought in our largely unconscious perception of the world, which modern cognitive science is exploring.

In the second part, I shall illustrate this by comparing two approaches to understanding mathematical problems: creating appropriate abstract structures, which is exemplified by Galois, Hilbert, Bourbaki and Grothendieck, or creating geometric models where we can use our intuition of space, of which Riemann, Poincaré and Thom were masters. This part requires some basic mathematical knowledge.