Collections, structures and foundations in Bolzano's philosophy of mathematics

APSE-CEU-IVC Talks
The Philosophy Department of the Central European University, the Institute Vienna Circle and the Unit for Applied Philosophy of Science and Epistemology (of the Department of Philosophy of the University of Vienna) are jointly organizing a series of talks this term

Date: 09/06/2022

Time: 15h00

Online Plattform: The meeting will be online via Zoom | Talks in Philosophy of Science and Epistemology PSE

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Abstract:

Bernard Bolzano's contributions to mathematics and its philosophy often see him mentioned in the same breath as the grandfathers of axiomatic set theory, Georg Cantor and Richard Dedekind. This is because he also developed a theory of collections, which was used to account for  \complex (as opposed to atomic) objects in general, thus including mathematical objects. Bolzano's way of understanding the role of collections in mathematics is however very different from that of later mathematicians, essentially because of a different understanding of the relationship between extensionality, abstraction, and structure.

In this talk I will use Maddy's (2017) sketch of the foundational roles of Zermelo-Fraenkel set theory with Choice (ZFC) to clarify the conceptual distance between Bolzano's understanding of collections as a tool for mathematical foundations and the way we understand and use sets in the context of axiomatic foundational theories thereof. Bolzano's collections do not allow for the same understanding of mathematical structure as the one afforded by later set theories, and this I argue has far-reaching consequences for how direct a line we can draw between Bolzano and later set theorists.