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BOLZANO’S MATHEMATICAL INFINITE

Part of: General and miscellaneous specific topics History of mathematics and mathematicians Philosophical aspects of logic and foundations

Published online by Cambridge University Press:  22 February 2021

ANNA BELLOMO  and
GUILLAUME MASSAS
ANNA BELLOMO*
Affiliation:
INSTITUTE FOR LOGIC, LANGUAGE, AND COMPUTATION UNIVERSITY OF AMSTERDAM AMSTERDAM, THE NETHERLANDS
GUILLAUME MASSAS
Affiliation:
GROUP IN LOGIC AND THE METHODOLOGY OF SCIENCE UNIVERSITY OF CALIFORNIA, BERKELEY BERKELEY, CA, USA E-mail: [email protected]
*
E-mail: [email protected]
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Abstract

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Bernard Bolzano (1781–1848) is commonly thought to have attempted to develop a theory of size for infinite collections that follows the so-called part–whole principle, according to which the whole is always greater than any of its proper parts. In this paper, we develop a novel interpretation of Bolzano’s mature theory of the infinite and show that, contrary to mainstream interpretations, it is best understood as a theory of infinite sums. Our formal results show that Bolzano’s infinite sums can be equipped with the rich and original structure of a non-commutative ordered ring, and that Bolzano’s views on the mathematical infinite are, after all, consistent.

Keywords

BolzanoCantorinfinitytransfinite arithmetic19th centuryinfinite sums

MSC classification

Primary: 01A55: 19th century
Secondary: 00A30: Philosophy of mathematics 03A05: Philosophical and critical
Type
Research Article
Information
The Review of Symbolic Logic , Volume 16 , Issue 1 , March 2023 , pp. 59 - 113
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

References

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