Ideal Concepts, Intuitions, and Mathematical Knowledge Acquisitions in Husserl and Hilbert

Abstract

We analyze Husserl’s and Hilbert’s studies on the role of “ideal concepts” and that of the notion of “mathematical intuition” in the context of mathematical knowledge acquisitions. We note a striking similarity between Husserl’s analysis in 1901 (presented at Hilbert’s seminar) and Hilbert’s analysis in the 1920’s on the problem justifying the ideal concepts. We also analyze the similarity and difference on Husserl’s standpoint and Hilbert’s standpoint on mathematical objectivities and on mathematical intuitions. In the course of this analysis we compare these with Gödel’s and some Hilbert scool members’ standpoints. We also propose a view on mathematical knowledge acquisitions along a “dynamic” interpretation of the Husserlian philosophy of mathematics, which provides a view different from the traditional views such as realist, nominalist, constructivist, conventionalist, empiricist views.

The original version of the paper was presented at the “Epistemology and Sciences” Colloquium at the Poincaré Institute in Paris on Dec. 5th 2000. Parts of this paper were also presented in some meetings, including colloques “Categorial Themes of Husserl’s Logic” organized by the Husserl Archives of Paris in April 2001, “Husserl et Wittgenstein” organized by Université de Picardie and l’Institut Universitaire de France in April 2001, “Logic and Philosophy of Science” organized by Academie de Paris and Université Paris-I Sorbonne in March 2000.

This work was partly done when the author stayed at Université Paris-I as an invited professor in 2000/2001. The author would like to express his thanks for their support on this work.