Bolzano's theory of real numbers

Abstract

Recently K. Rychlik1 has published for the first time a manuscript by B. Bolzano containing a theory of real numbers. This publication is of great importance for the study of the history of mathematics in the nineteenth century, since it shows that as early as 1 83 0-1 83 5 an attempt was made at an arithmetical foundation of the theory of real numbers. Such a foundation was afterwards realised by K. Th. Weierstrass, H. Ch. R. Meray, G. Cantor and J. W. R. Dedekind (1858) in the second half of the nineteenth century. Though A. L. Cauchy2 had observed in 1827 that irrational numbers may be considered as limits of sequences of rational numbers, this is something quite different from Bolzano's attempt to develop a theory of real numbers based on a method of approximation . Rychlik's edition contains besides the text of the manuscript a collection of fifteen mathematical footnotes, a Vorwort and a Nachwort in which he tries to evaluate and justify Bolzano's theory. There is also a foreword by L. Rieger containing valuable observations concerning Bolzano's theory. In order to make clear the reason for this study I describe Rychlik's position concerning Bolzano's theory as it appears from the Vorwort and the Nachwort. I refer to Rychlik's edition by R, followed by the number of the page. It should be stressed that both Rychlik and Rieger are aware that Bol-