Wittgenstein and concept-extension in mathematics

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.