Dr. Anna Bellomo

March 1^st -  August 31^st 2022

Affiliation: University Amsterdam

Research for a study about:

Research Project: Structures in Bolzano’s Mathematics

In philosophy of mathematics, ‘structuralism' can be divided into a ‘methodological’ or ‘mathematical’ structuralism, and a ‘philosophical’ structuralism. Methodological structuralism is taken to emerge primarily from within mathematical practice, and it is a preference for a certain set of attitudes towards the traditional definition of mathematics as a science of quantity, the role of axiomatics, the role of intuition in mathematical reasoning and argumentation,and the role of structure in mathematics. Philosophical structuralism, by contrast, focuses on questions related to the metaphysical and epistemological (access) problems raised by the thesis that mathematics is primarily about structures – the main question being, what the identity criteria for a structure are, and how there can be singular reference in mathematics if we can only individuate mathematical objects of discourse up to isomorphism.


In this research project, I  intend to further investigate Bolzano's understanding of mathematical objects and collections and analyse it according to the 'conceptual grid' of methodological and philosophical structuralism as presented and developed in recent literature. I intend to defend two theses: first, that from the late 1820s onwards, Bolzano's philosophy of mathematics becomes less structuralist in the sense of philosophical structuralism, whereas his early work suggests a more structuralist understanding of mathematics in some respects. Second, that this anti-structuralist approach of his does indeed have repercussions for his theory of collections as foundations for mathematics.

Lecture

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

Date: June 9, 2022

Time: 3-5pm (CEST)

Platform: Online APSE-CEU-IVC Talks (https://wienerkreis.univie.ac.at/news-events/apse-ceu-ivc-talks-summerterm-2022-overview/)https://wienerkreis.univie.ac.at/news-events/einzelansicht/news/online-apse-ceu-ivc-talks-hans-joachim-dahms-current-fellow-at-ivc-es-liegt-in-der-luft-eine-s/?tx_news_pi1%5Bcontroller%5D=News&tx_news_pi1%5Baction%5D=detail&cHash=ae9f62ad861ecc14aafe5ea97995b190

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.

Report

During my visit at the Institute Vienna Circle as an IVC fellow I had the opportunity to further develop some of the work started during my PhD studies. I have worked on the notion of collection in Bernard Bolzano’s work and its relationship to the idea of structure, motivated by an interest in comparing Bolzano’s use of collections as foundational entities in his mathematics to the idea of a mathematical foundation developed by Penelope Maddy for Zermelo-Fraenkel set theory. That research has now formed the basis of a paper in which I defend the idea that, unlike later theories of sets, Bolzano’s theory of collection is not abstract enough to serve as the basis for a mathematical structure in the modern sense. As a consequence, we cannot describe Bolzano’s project as trying to build a foundation for mathematics in the sense Maddy develops, but there are some deep affinities with the project of being able to define precisely the basic concepts of mathematics.

Partial results of my research have been presented in the APSE-CEU-IVC talk series of summer 2022. I have also taken advantage of my time in Vienna to present at the “Perspectives on Categoricity” workshop organised by Georg Schiemer (May 27, 2022).