@article {4443783, title = {S\lupecki{\textquoteright}s generalized mereology and its flaws}, journal = {History and Philosophy of Logic}, volume = {35}, number = {3}, year = {2014}, pages = {289{\textendash}300}, abstract = {

One of the streams in the early development of set theory was an attempt to use mereology, a formal theory of parthood, as a foundational tool. The first such attempt is due to a Polish logician, Stanis\unmatched{0142}aw Le{\'s}niewski (1886{\textendash}1939). The attempt failed, but there is another, prima facie more promising attempt by Jerzy S\unmatched{0142}upecki (1904{\textendash}1987), who employed his generalized mereology to build mereological foundations for type theory. In this paper I (1) situate Le{\'s}niewski{\textquoteright}s attempt in the development of set theory, (2) describe and evaluate Le{\'s}niewski{\textquoteright}s approach, (3) describe S\unmatched{0142}upecki{\textquoteright}s strategy without unnecessary technical details, and (4) evaluate it with a rather negative outcome. The issues discussed go beyond merely historical interests due to the current popularity of mereology and because they are related to nominalistic attempts to understand mathematics in general. The introduction describes very briefly the situation in which mereology entered the scene of foundations of mathematics {\textendash}- it can be safely skipped by anyone familiar with the early development of set theory. Section 2 describes and evaluates Le{\'s}niewski{\textquoteright}s attempt to use mereology as a foundational tool. In Section 3, I describe an attempt by S\unmatched{0142}upecki to improve on Le{\'s}niewski{\textquoteright}s work, which resulted in a system called generalized mereology. In Section 4, I point out the reasons why this attempt is still not successful. Section 5 contains an explanation of why Le{\'s}niewski{\textquoteright}s use of Ontology in developing arithmetic also is not nominalistically satisfactory.

}, doi = {http://dx.doi.org/10.1080/01445340.2014.917837}, author = {Urbaniak, Rafal} } @article {4443772, title = {Stanislaw {L}e{\'s}niewski: rethinking the philosophy of mathematics}, journal = {European review}, volume = {23}, number = {1}, year = {2014}, pages = {125{\textendash}138}, abstract = {

Near the end of the XIXth century part of mathematical research was focused on unification: the goal was to find {\textacutedbl}one sort of thing{\textacutedbl} that mathematics is (or could be taken to be) about. Quite quickly sets became the main candidate for this position. While the enterpize hit a rough patch with Frege{\textquoteright}s failure and set-theoretic paradoxes, by the 1920s mathematicians (roughly speaking) settled on a promising axiomatization of set theory and considered it foundational. Quite parallel to this development was the work of Stanislaw Le{\textbackslash}{\textquoteright}sniewski (1886-1939), a Polish logician who did not accept the existence of abstract (aspatial, atemporal and acausal) objects such as sets. Lesniewski 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 Lesniewski{\textquoteright}s project and to briefly comment on its contemporary relevance.

}, author = {Urbaniak, Rafal} } @article {Urbaniak2009, title = {Slingshot arguments: two versions}, journal = {The Reasoner}, volume = {3}, number = {4}, year = {2009}, pages = {4{\textendash}5}, author = {Urbaniak, Rafal} } @article {urbaniak2006some, title = {Some non-standard interpretations of the axiomatic basis of Le{\'s} niewski’s Ontology}, journal = {The Australasian Journal of Logic}, volume = {4}, year = {2006}, pages = {13{\textendash}46}, abstract = {

We propose an intuitive understanding of the statement: ‘an ax-iom (or: an axiomatic basis) determines the meaning of the only specific constant occurring in it.’ We introduce some basic semantics for functors of the category s{\textfractionsolidus}n,n of Le{\textasciiacute}sniewski’s Ontology. Using these results weprove that the popular claim that the axioms of Ontology determine themeaning of the primitive constants is false.

}, author = {Urbaniak, Rafal} }