MoL-2024-31: Koolstra, Jori (2024) Conceptualization as Explanatory Coherence: an Application to Set Theory’s Potentialism vs. Minimalism Debate. [Report]
Preview |
Text
MoL-2024-31.text.pdf - Accepted Version Download (698kB) | Preview |
Abstract
This thesis is concerned with the philosophy of set theory, and therefore with the foundations of mathematics. Our standard theory of set, ZFC, is rooted in the iterative conception, according to which sets arise as collections of some things. Boolos [12] (1971) inaugurated the modern study of conceptions of set, emphasizing their justificatory role in deriving axioms and avoiding the paradoxes of naive set theory. More recently, Incurvati [54] (2020) conducted an extensive comparative analysis of several conceptions of set, advocating for the correctness of the iterative conception based on certain virtues it possesses over the other conceptions discussed.
Different accounts of the iterative conception have been proposed in the literature. Of particular interest are Linnebo’s 2010 and 2013 papers [69] [68], which introduce a potentialist perspective on iteration formulated in modal plural logic. According to Linnebo, the conventional language employed in set theory lacks the necessary granularity to properly address certain foundational concerns, such as the paradoxes. This view has, however, recently been challenged by Soysal [100] (2020). She argues that e.g. V is not a set simply because its existence conflicts with the axioms supported by the iterative conception (and this conception is correct). If this conception-based explanation goes through, Linnebo’s modal apparatus may turn out to be an ‘idle wheel’, as Soysal’s argument applies even to the minimalist account of the iterative conception, according to which its content is exhausted by saying that the sets are those objects that occur in the cumulative hierarchy. Therefore, to evaluate Linnebo’s conception of set, we need a general theory of conceptualization in mathematics on which we can compare such different accounts.
I introduce a Lakatosian account of conceptualization, drawn from Incurvati [54], on which conceptions arise when open-texture in a foundational concept is sharpened. Due to this foundationality, we must exercise careful control on stipulation in conception formation, in particular to handle circular justification, for instance regarding the height of the cumulative hierarchy. The novel solution I propose is to derive conception-admissibility from explanatory coherence. This can be achieved by answering as many (why-)questions as possible. Moreover, I will argue that explanatory richness in this sense is also a virtue that should weigh on a conception’s correctness. Finally, I show that Soysal’s minimalist answer to why V is not a set is explanatorily unstable, unlike the solution derived from Linnebo’s potentialism. I will therefore conclude that his modal apparatus does genuine explanatory work, and thus the potentialist account must be preferred as more correct.
| Item Type: | Report |
|---|---|
| Report Nr: | MoL-2024-31 |
| Series Name: | Master of Logic Thesis (MoL) Series |
| Year: | 2024 |
| Subjects: | Logic Mathematics |
| Depositing User: | Dr Marco Vervoort |
| Date Deposited: | 10 Nov 2026 14:29 |
| Last Modified: | 10 Nov 2026 14:29 |
| URI: | https://eprints.illc.uva.nl/id/eprint/2462 |
Actions (login required)
 |
View Item |
