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Stanisław Leśniewski: Rethinking the Philosophy of Mathematics

Published online by Cambridge University Press:  29 January 2015

Rafal Urbaniak
Rafal Urbaniak*
Affiliation:
Centre for Logic and Philosophy of Science, Ghent University, Belgium, and Department of Philosophy, Sociology and Journalism, Gdansk University, Poland. E-mail: [email protected]
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Abstract

Near the end of the nineteenth century, a part of mathematical research was focused on unification: the goal was to find ‘one sort of thing’ that mathematics is (or could be taken to be) about. Quite quickly sets became the main candidate for this position. While the enterprise hit a rough patch with Frege’s failure and set-theoretic paradoxes, by the 1920s mathematicians (roughly speaking) settled on a promising axiomatization of set theory and considered it foundational. In parallel to this development was the work of Stanislaw Leśniewski (1886–1939), a Polish logician who did not accept the existence of abstract (aspatial, atemporal and acausal) objects such as sets. Leśniewski attempted to find a nominalistically acceptable replacement for set theory in the foundations of mathematics. His candidate was Mereology – a theory which, instead of sets and elements, spoke of wholes and parts. The goal of this paper will be to present Mereology in this context, to evaluate the feasibility of Leśniewski’s project and to briefly comment on its contemporary relevance.

Type
Focus: Logic and Philosophy in Poland
Information
European Review , Volume 23 , Issue 1 , February 2015 , pp. 125 - 138
Copyright
© Academia Europaea 2015 

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References

Notes and References

1.‘In the late nineteenth century, it was a widespread idea that pure mathematics is nothing but an elaborate form of arithmetic. Thus it was usual to talk about the arithmetisation of mathematics, and how it had brought about the highest standards of rigor.’ See Ferreirós, J. (2012) The early development of set theory. In: E. N. Zalta (ed.), The Stanford Encyclopedia of Philosophy (http://plato.stanford.edu/cite.html), Winter 2012 edition, section 3.Google Scholar
2.That is, a sequence x 1,x 2,… such that for every rational number ϵ>0 there is an integer n such that for all integers numbers j, i, |x i−x j|<ϵ.0+there+is+an+integer+n+such+that+for+all+integers+numbers+j,+i,+|xi−xj|<ϵ.>Google Scholar
3.See Appendix A to remind yourself what these axioms are.Google Scholar
4.See Appendix B for details.Google Scholar
5.For a good anthology of Frege’s basic writings see Beaney, M. (1997) The Frege Reader (Malden, Oxford, Carlton: Wiley-Blackwell).Google Scholar
6.Very roughly speaking, this is a logic that handles reasoning employing not only quantification over objects (‘for all objects x’) but also over properties of objects (e.g. ‘for no x there is a P such that Px and not-Px.’).Google Scholar
7.Frege thought that sets, which he called ‘extensions’, were logical objects, whatever that might mean, so his aim was even more ambitious: to show that mathematics is ultimately just logic.Google Scholar
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11.This is a fairly natural way to go: Leśniewski first used mereological intuitions to handle Russell’s paradox in 1914, and developed a semi-formalized axiomatization of Mereology in 1916.Google Scholar
12.Nowadays, we rather call ingredients ‘parts’ and what Leśniewski called parts we call ‘proper parts’.Google Scholar
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