On Frege’s Works on Philosophical Problems of Mathematics

Abstract

The struggle of the fundamental philosophical tendencies — materialism and idealism — in contemporary mathematics concentrates mostly on its foundations. And this is not fortuitous. Precisely in the domain of the foundations of mathematics appeared difficulties related to the most general and universal notions such as “infinity”, “set”, “function”, “object” and others. At the end of the nineteenth century it seemed to the mathematicians that the problem of the foundations of their science was completely solved. In 1900, H. Poincaré wrote: “Il n’y a plus aujourd’hui en Analyse que des nombres entiers ou des systèmes finis ou infinis de nombres entiers, reliés entre eux par un réseau de relations d’égalité ou d’inégalité. Les mathématiques, comme on l’a dit, se sont arithmétisées…. On peut dire qu‘aujourd’hui la rigueur absolue est atteinte.”² But a bit later contradictions were discovered in the theory of sets. It became clear that the problem of the foundations of mathematics could not be considered as solved. Idealist thinkers on mathematics began to speak of the “crisis” of the foundations of their science (Weyl, Brouwer, etc.). But this was in fact not a “crisis” but the discovering of the deep dialectical nature of the most primitive notions of mathematics and logic.

The author expresses his great gratitude to Sofija Aleksandrov Janovskaja for the scientific guidance and help given in the preparation of this article.