Description

Rodrigo Blanco Betes (ILCLI): "An alternative approach to conceptual structuralism"

Date: January 30, 2026. 15:00.
Venue: Carlos Santamaria Zentroa, room 4.

Abstract:
Solomon Feferman (2011) argued that the Continuum Hypothesis (CH) may not be a definite mathematical question, given his philosophical conception of mathematics, which he calls conceptual structuralism. This position has recently been criticized by Koellner (2022) as unfounded, obscure, and overly conventionalist. In this talk, I aim to revisit Feferman’s perspective, clarifying its more obscure aspects and extending his line of thought where it was left incomplete.
In my view, conceptual structuralism imposes two fundamental constraints on mathematical conceptions. First, it emphasizes that mathematical structures must be useful for scientific purposes. Second, it maintains that mathematical conceptions must be clear (in a sense to be discussed) to possess definite truth values. Feferman relies much more heavily on this latter requirement in formulating his case concerning CH—a strategy that invites many of Koellner’s criticisms. However, both authors pay comparatively little attention to the other constraint of conceptual structuralism: the one related to mathematical applicability.
My main aim is to develop a more fine-grained notion of mathematical legitimacy, one that remains faithful to the spirit of conceptual structuralism while aligning it more closely with mathematical practice. I argue, however, that adopting this refined notion would make it considerably more difficult to defend the illegitimacy of CH, as Feferman would do.

References:
Feferman, S. (2011). Is the Continuum Hypothesis a definite mathematical problem? Draft for Exploring the Frontiers of Incompleteness (EFI) Project, Harvard 2011-2012.
Koellner, P. (2022). Feferman on the Indefiniteness of CH. Retrieved from https://bpb-us-e1.wpmucdn.com/websites.harvard.edu/dist/f/94/files/2022/07/EFI_Feferman_comments.pdf