A Few Remarks about Mathematical Intuition

Abstract

This short note is a summary of the talk given at the Adam Mickiewicz University in Poznań during the conference on Philosophy of Mathematics, III, on October 17, 2011. We suggest to understand mathematical intuition as a collection of (verbalized) judgments constituting the core of the context of discovery in mathematics. Mathematical intuition is present not only in the axioms: it can be observed in mathematical research practice as well. A few examples of mathematical intuition in action are given. We discuss the dynamic character of mathematical intuition, trying to reveal the sources of the corresponding changes of intuitive views in mathematics. There are several such sources, e.g.: paradoxes, new research programs (for instance, the program of arithmetization of analysis), explicit definitons of concepts understood so far in an intuitive way, and — of course — the development of the mathematical knowledge itself. We devote special attention to the distinction between standard, exception, and pathology. Finally, we say a few words about conflicts of intuitions in mathematics.