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On the Concept of Finitism

Synthese 192 (8):2413-2436 (2015)

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Ontology and Arbitrariness.David Builes - 2022 - Australasian Journal of Philosophy 100 (3):485-495.details In many different ontological debates, anti-arbitrariness considerations push one towards two opposing extremes. For example, in debates about mereology, one may be pushed towards a maximal ontology (mereological universalism) or a minimal ontology (mereological nihilism), because any intermediate view seems objectionably arbitrary. However, it is usually thought that anti-arbitrariness considerations on their own cannot decide between these maximal or minimal views. I will argue that this is a mistake. Anti-arbitrariness arguments may be used to motivate a certain popular thesis in (...) Download Export citation Bookmark 4 citations
(1 other version)Takeuti's Well-Ordering Proof: Finitistically Fine?Eamon Darnell & Aaron Thomas-Bolduc - 2018 - In Maria Zack & Dirk Schlimm (eds.), Research in History and Philosophy of Mathematics The CSHPM 2017 Annual Meeting in Toronto, Ontario. New York: Birkhäuser.details If it could be shown that one of Gentzen's consistency proofs for pure number theory could be shown to be finitistically acceptable, an important part of Hilbert's program would be vindicated. This paper focuses on whether the transfinite induction on ordinal notations needed for Gentzen's second proof can be finitistically justified. In particular, the focus is on Takeuti's purportedly finitistically acceptable proof of the well-ordering of ordinal notations in Cantor normal form. The paper begins with a historically informed discussion of (...) Download Export citation Bookmark
In Defense of the Implicit Commitment Thesis.Ethan Brauer - 2022 - Ergo: An Open Access Journal of Philosophy 9.details The implicit commitment thesis is the claim that believing in a mathematical theory S carries an implicit commitment to further sentences not deductively entailed by the theory, such as the consistency sentence Con(S). I provide a new argument for this thesis based on the notion of mathematical certainty. I also reply to a recent argument by Walter Dean against the implicit commitment thesis, showing that my formulation of the thesis avoids the difficulties he raises. Download Export citation Bookmark

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