Brouwer’s Intuitionism Vis a` Vis Kant’s Intuition and Imagination

摘要

Within the last two decades there has been considerable interest in the life and work of L. E. J. Brouwer, one of the twentieth century’s most colorful and controversial figures (Stigt 1990; Dalen 1999, 2005; Kuiper 2004). Brouwer’s name has long been associated with two spheres of activity. As a pioneering actor in topology, he published a series of papers between 1909 and 1913 that were to exert a major impact on this fledgling field of research. Two famous results from this period were his proof of the invariance of dimension and his elegant fixed-point theorem, both of which were nonconstructive in nature (these results can be found in the classic text of Hurewicz and Wallman 1948). Soon afterward, however, Brouwer began an earnest program for a truly constructive mathematics. He took up a critical analysis of set theory and the concept of the continuum, activities that led him to pursue a radically new research orientation that came to be known as intuitionism. With it Brouwer aimed to sweep away standard methods that had long been used for proving existence theorems in analysis, techniques which had become increasingly controversial in the wake of the new foundational research undertaken by Cantor, Hilbert, Zermelo, Schoenflies and others.