 It should also be noted at the outset that commentators disagree about the continuity of Wittgenstein’s middle and later Philosophies of Mathematics. Some argue that the later views are significantly different from the intermediate views (Frascolla 1994; Gerrard 1991: 127, 131–32; Floyd 2005: 105–106), while others argue that, for the most part, Wittgenstein’s Philosophy of Mathematics evolves from the middle to the later period without significant changes or renunciations (Wrigley 1993; Marion 1998). The remainder of this article adopts the second interpretation, explicating Wittgenstein’s later Philosophy of Mathematics as largely continuous with his intermediate views, except for the important introduction of an extra-mathematical application criterion.