Wittgenstein on Mathematical Progress

The objective of this article is to try to elucidate Wittgenstein’s extravagant thesis that each and every mathematical advancement involves some “semantical mutation”, i.e., some alteration of the very meanings of the terms involved. To do that we will argue in favor of the idea of a “modal incompatibility” between the concepts involved, as they were prior to the advancement, and what they become after the new result was obtained. We will also argue that the adoption of this thesis profoundly alters the traditional way of construing the idea of “progress” in mathematics.