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BOLZANO’S MATHEMATICAL INFINITE

Published online by Cambridge University Press:  22 February 2021

ANNA BELLOMO
Affiliation:
INSTITUTE FOR LOGIC, LANGUAGE, AND COMPUTATION UNIVERSITY OF AMSTERDAM AMSTERDAM, THE NETHERLANDS E-mail: a.bellomo@uva.nl
GUILLAUME MASSAS
Affiliation:
GROUP IN LOGIC AND THE METHODOLOGY OF SCIENCE UNIVERSITY OF CALIFORNIA, BERKELEY BERKELEY, CA, USA E-mail: gmassas@berkeley.edu
Corresponding

Abstract

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.

Type
Research Article
Copyright
© Association for Symbolic Logic, 2021

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