Wittgenstein and concept‐extension in mathematics

• Published in: Philosophical investigations Vol. 48; no. 3; pp. 333 - 348
• Main Author: Wheeler, Samuel J.
• Format: Journal Article
• Language: English
• Published: Oxford Wiley Subscription Services, Inc 01.07.2025
• ISSN: 0190-0536; 1467-9205; 1467-9205
• DOI: 10.1111/phin.12457

I begin by attempting to get a perspicuous overview of what Wittgenstein means by saying that a mathematical proof forms concepts. I then distinguish these sorts of cases from those we might call concept‐extending proofs, which, rather than introducing new concepts, function to enrich those concepts that have already been given a home in our mathematical practice. At the same time, I also want to argue that the line between these two sorts of proofs is not always clear and will sometimes be blurred. I go on to compare paradigmatic examples of each, concluding with a case in which it is not immediately clear whether we ought to all the proof concept‐forming or extending. This last section includes a discussion of Sorin Bangu's recent account of Wittgenstein on mathematical proof.