The Conception of Bernard Bolzano About the Continuum and the Achievements of Contemporary Mathematics
The Conception of Bernard Bolzano About the Continuum and the Achievements of Contemporary Mathematics

Summary/Abstract: The present paper is devoted to the contribution of the eminent philosopher and mathematician Bernard Bolzano to the problem of continuum. The paper consists of two parts. In the first part it is compared the conception of Bolzano about the continuum with that of Aristotle. Bolzano’s view includes two essential moments: first, that the continuum is composed of noncontinual elements (points), and second, that two points which are at a certain distance one to other cannot compose a continuum. Aristotle has two definitions about continuity: in the first one he in fact develops the idea of continuity as a physical connectedness, and in the second one he emphasizes the infinite divisibility of the continuum. The second part of the paper gives an evaluation of Bolzano’s contribution to the problem of continuum in comparison with the achievements of the contemporary mathematics on the problem. There is a tendency to consider continuity in two senses: as an infinite divisibility and as a connectedness. The contemporary topology (as a branch of mathematics) develops the concept of continuity in the second sense. Bolzano’s conception bears in some respects the resemblance to the contemporary definitions in topology of the concepts of closeness, neighborhood, isolated point, etc. His definition about the continuum can be considered as a remote precursor of the contemporary topological definition of continuity as connectedness.

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