 Where, during the middle period, Wittgenstein speaks of “arithmetic [as] a kind of geometry” at (PR §109 &amp; §111), the later Wittgenstein similarly speaks of “the geometry of proofs” (RFM I, App. III, §14), the “geometrical cogency” of proofs (RFM III, §43), and a “geometrical application” according to which the “transformation of signs” in accordance with “transformation-rules” (RFM VI, §2, 1941) shows that “when mathematics is divested of all content, it would remain that certain signs can be constructed from others according to certain rules” (RFM III, §38). Hence, the question whether a concatenation of signs is a proposition of a given mathematical calculus (i.e., a calculus with an extra-mathematical application) is still an internal, syntactical question, which we can answer with knowledge of the proofs and decision procedures of the calculus.