 There is considerable evidence, however, that the later Wittgenstein maintains his intermediate position that an expression is a meaningful mathematical proposition only within a given calculus and iff we knowingly have in hand an applicable and effective decision procedure by means of which we can decide it. For example, though Wittgenstein vacillates between “provable in PM” and “proved in PM” at (RFM App. III, §6, §8), he does so in order to use the former to consider the alleged conclusion of Gödel’s proof (i.e., that there exist true but unprovable mathematical propositions), which he then rebuts with his own identification of “true in calculus \(\Gamma\)” with “proved in calculus \(\Gamma\)” (i.e., not with “provable in calculus \(\Gamma\)”) (Wang 1991: 253; Rodych 1999a: 177). This construal is corroborated by numerous passages in which Wittgenstein rejects the received view that a provable but unproved proposition is true, as he does when he asserts that (RFM III, §31, 1939) a proof “makes new connexions”, “[i]t does not establish that they are there” because “they do not exist until it makes them”, and when he says (RFM VII, §10, 1941) that “[a] new proof gives the proposition a place in a new system”. Furthermore, as we have just seen, Wittgenstein rejects PIC as a non-proposition on the grounds that it is not algorithmically decidable, while admitting finitistic versions of PIC because they are algorithmically decidable.