After talking about the infinite and Frege Arithmetic, it is time to turn to potentialism about set-theory. In this session we will turn to potentiality about the height of the cumulative hierarchy.
Main Readings:
For our final meeting on structuralism, we cover the modal set-theoretic structuralism from Parsons and Linnebø. This rounds out the book we have been following and also our time with structuralism (for now).
We are now covering the theory of structuralism in mathematics. This first meeting will be an overview of the general position and its various parts together with specific focus on set-theoretic structuralism. Over the course of the coming meetings, we will spend some time on various conceptions of structuralism and their proponents.
Next reading will be Solomon Feferman’s Conceptions of the Continuum, available here.
Marta will present the paper, followed by a discussion.
Slides used in the presentation are available here.
Tibo will give a short presentation supporting the (in)famous Quine thesis that SOL is in reality ST, followed by a debate. The rules will be uploaded in due time.
Slides used during the presentation are available here (link now unfortunately broken).
Suggested Bibliography (They are freely accessible from your UvA student account)
Quine W.V.O., Philosophy of Logic (2nd Edition), Chapter 5: ‘The Scope of Logic’, Harvard University Press, 1986. [PDF accessible from the UvA Library]
Anna Bellomo presented her ongoing PhD research on Bolzano’s conceptions of infinity, as well as her involvement on the e-Ideas framework.
‘In the philosophy of mathematics circles, Bernard Bolzano (1871-1848) is mostly known for two things: his contributions to the so-called rigorisation of analysis, and his proto-Cantorian theory of size for infinite sets. In this talk, I will focus on the latter and summarise some recent findings suggesting that, contrary to what has been so far the default interpretation of Bolzano’s treatment of the countable infinite, his focus was not a theory of size for infinite sets, but solving some problems relating to the treatment of (non-convergent) infinite sequences.’
Luca Incurvati will be presenting his recently published book Conceptions of Set and the Foundations of Mathematics. The book is accessible through the UvA Library.
Book summary:
Sets are central to mathematics and its foundations, but what are they? In this book Luca Incurvati provides a detailed examination of all the major conceptions of set and discusses their virtues and shortcomings, as well as introducing the fundamentals of the alternative set theories with which these conceptions are associated. He shows that the conceptual landscape includes not only the naïve and iterative conceptions but also the limitation of size conception, the definite conception, the stratified conception and the graph conception. In addition, he presents a novel, minimalist account of the iterative conception which does not require the existence of a relation of metaphysical dependence between a set and its members. His book will be of interest to researchers and advanced students in logic and the philosophy of mathematics.