Weierstrass, Luzin, and Intuition

Abstract

Introductory note by the column editors. Three related essays, one in two parts, have appeared recently in Evolution of... columns: (1) Function: Part I (MONTHLY 105 (1998) 59-67) and Function: Part II (MONTHLY 105 (1998) 263270), by N. Luzin; (2) letters by N. N. Luzin to M. Ya. Vygodskil; (MONTHLY 107 (2000) 64-82), by N. Luzin,; (3) Comments on the paper Two letters by N. N. Luzin to M. Ya. VygodskiYl, by D. Laugwitz (MONTHLY 107 (2000) 267276). A large part of Luzin's paper (1) is an account of reasonably well-known topics in the history of the function concept, such as the vibrating string controversy, the Fourier discovery, the Baire system of classes of functions, and the Weierstrass notion of a (complex) analytic function. The less-well-known part of (1) tells us that many eminent mathematicians at the turn of the 20th century felt that Dirichlet's definition of a function was too broad and Weierstrass' definition of an analytic function was too restrictive, and that the fight against the restrictiveness of Weierstrass' definition, begun by Borel, eventually led to the development of the theory of quasi-analytic functions. Luzin's paper (2) involves two related themes. One is a fascinating account of his growth as a mathematician and the other is an impassioned critique of the preoccupation with rigor and with the logical buildup of the real numbers-tendencies associated with Weierstrass-to the detriment of intuition and of infinitesimals. Laugwitz's paper (3) is a modern echo of the themes in (2). Accounts of the evolution of mathematics seldom mention the word passions. They are much in evidence in the papers (1)-(3). Gray's account of these papers is aware of these passions but is itself usefully dispassionate. The distinction of this quartet of papers is that it not only presents a collection of crucial mathematical concerns and results but also conveys the strong feelings of some of the protagonists.