Trends in the Foundations of Mathematics in the Early Twentieth Century by Maxime Bôcher

Jerzy Dadaczyński

The paper adopted the text of M. Bôcher Conceptions and Methods of Mathematics from 1904 as the source for the prehistoric development of the future trends in the foundations of mathematics. Th e paper of the American mathematician is not primarily devoted to this trends. Nevertheless, one can read from it, how the “map” of approaches to the foundations of mathematics in 1904 was developing. I n 1904 logicism was no longer a tendency, but a well developed direction in the foundations of mathematics. M. Bôcher saw its main difficulties associated with the discovery of antinomies, but he was sure that the corrections in the logic underlying mathematics will remove these diffi culties. In this way he announced “the second” logicism – by B. Russell – and the axiomatization of the set theory. T wo points, derived from the paper from 1904, predicted the future Hilbert’s formalism in 20s. Th is was, firstly, methodical nominalism and, secondly, posing the question whether it was possible to prove – after eliminating the known antinomies – the global consistency of mathematics. Th is question has become the central goal of the formalism and the first reason for building the metamathematics by Hilbert. Methodological nominalism, in turn, became in Hilbert’s formalism a method of treatment of the names of formal language of mathematics.  The paper from 1904 negatively treated Kantianism as a basis for the development of mathematics. However, it is a source confirming that Kantianism was still actual in many mathematical environments. On this ground, three years after the publication of the paper of the American mathematician, intuitionism was created. Generally it can be stated that already in 1904 one could discover in the foundations of mathematics trends that have led to the emergence of logicism, formalism and intuitionism.


Keywords: Maxime Bôcher · philosophy of mathematics · foundations of mathematics · logicism · intuitionism · formalism

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