My Memories of L. E. J. Brouwer

Abstract

In March of 1921 — I had just completed my first semester at the university, studying physics - H. Hahn joined the mathematics faculty. His first release was the announcement of a seminar on the curve concept (Neueres über den Kurvenbegriff), one two-hour meeting weekly during the spring semester. I hesitated, but finally mustered up my courage to audit the first session. There, without introduction, Hahn formulated the problem of making precise the idea of curve that everyone has but that no-one had been able to articulate. Various attempts had been made. G. Cantor thought of defining curves as one-to-one images of a segment; C. Jordan then defined them as its images by continuous mappings. But Cantor himself proved the first, and Peano showed the second, of these definitions to be unacceptable since both include squares and even cubes, which no one calls curves. Thereafter arcs were studied — topological images of a segment; that is, obtainable from it by mappings that are both one-to-one and continuous. Among the arcs one does not find squares or cubes, but neither does one find ellipses or lemniscates. Hence more complicated objects, called graphs, were built up by joining arcs in certain ways; but even this concept was too narrow. Other attempts were unsatisfactory for other reasons. So, Hahn said, the seminar would culminate with the realization that the problem was still unsolved and that a satisfactory definition of the curve concept still was unknown.¹ In conclusion of the first session, Hahn presented Cantor’s most elementary concepts of point set theory.