In the first part of the article, Giuseppe Veronese’s concept of arithmetical constructivism is reconstructed from his dispersed remarks. It is pointed out that although for Veronese time is a necessary condition for the construction of natural numbers by an individual subject and the subject cognizes time in an a priori way, it is not a (proto-)intuition of the subject. This is a fundamental difference between the concept proposed by Veronese and the constructivism of Kant and Brouwer. Veronese’s justification of (...) the subject’s ability to cognize infinity, represented in his text by infinite sequences, is analyzed in the second part of the paper. The only condition for the cognition of the infinite sequence is to have a rule according to which its successive terms “follow one another.” The price of the ability to cognize the infinitive “whole” (sequence) may be the uncognizability of certain “parts” (terms of sequence). Infinite sequences, cognizable according to Veronese, are allowed as objects of mathematical research, although they do not meet the condition of constructability. (shrink)

In 1891, Hilbert came to the conclusion that geometry is not a science of the physical space but is an abstract science. Hilbert’s new position was repeated in the introduction to Grundlagen der Geometrie in 1899. The aim of this paper is to show the sources of change of Hilbert’s position in 1891. We showed that the di-rect source of the change was having heard a lecture by Wiener on projective geome-try. Wiener presented there a project of geometry as a (...) science of abstract objects i.e. objects as sets of minimal number of – only relational – properties. We showed that this idea was derived by Wiener from Dedekind’s arithmetic of natural numbers, pre-sented in 1888 in Was sind und was sollen die Zahlen? Thus – and this is the essential thesis of our paper – the original source of Hilbert’s idea of geometry as a abstract science is included in Dedekind’s concept of arithmetic. (shrink)

Book review: Lukas Benedikt Kraus, Der Begriff des Kontinuums bei Bernard Bolzano, Beiträge zur Bolzano-Forschung, vol. 25, Academia Verlag, Sankt Augustin 2014, pp. 112.

The main purpose of the present paper is to demonstrate that as early as 1904 pre-eminent American mathematician Maxime Bôcher was an adherent to the presently relevant argument of reasonableness, or even necessity of parallel development of two philosophical methods of reflection on mathematics, so that its essence could be more fully comprehended. The goal of the research gives rise to the question: what two types of philosophical deliberation on mathematics were proposed by Bôcher?