Bolzano’s Infinite Quantities

Foundations of Science 23 (4):681-704 (2018)
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Abstract

In his Foundations of a General Theory of Manifolds, Georg Cantor praised Bernard Bolzano as a clear defender of actual infinity who had the courage to work with infinite numbers. At the same time, he sharply criticized the way Bolzano dealt with them. Cantor’s concept was based on the existence of a one-to-one correspondence, while Bolzano insisted on Euclid’s Axiom of the whole being greater than a part. Cantor’s set theory has eventually prevailed, and became a formal basis of contemporary mathematics, while Bolzano’s approach is generally considered a step in the wrong direction. In the present paper, we demonstrate that a fragment of Bolzano’s theory of infinite quantities retaining the part-whole principle can be extended to a consistent mathematical structure. It can be interpreted in several possible ways. We obtain either a linearly ordered ring of finite and infinitely great quantities, or a partially ordered ring containing infinitely small, finite and infinitely great quantities. These structures can be used as a basis of the infinitesimal calculus similarly as in non-standard analysis, whether in its full version employing ultrafilters due to Abraham Robinson, or in the recent “cheap version” avoiding ultrafilters due to Terence Tao.

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Kateřina Trlifajová
Czech Technical University, Prague

Citations of this work

Bolzano’s Mathematical Infinite.Anna Bellomo & Guillaume Massas - 2021 - Review of Symbolic Logic:1-55.
Sizes of Countable Sets.Kateřina Trlifajová - 2024 - Philosophia Mathematica 32 (1):82-114.

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References found in this work

Non-standard Analysis.Gert Heinz Müller - 2016 - Princeton University Press.
Foundations of Set Theory.Abraham Adolf Fraenkel & Yehoshua Bar-Hillel - 1973 - Atlantic Highlands, NJ, USA: Elsevier.
Paradoxien des Unendlichen.Bernard Bolzano - 2012 - Hamburg: Meiner, F. Edited by Christian Tapp.
Wissenschaftslehre.Bernard Bolzano & Alois Höfler - 1837 - Revue de Métaphysique et de Morale 22 (4):15-16.

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