 A second important aspect of the intermediate Wittgenstein’s strong formalism is his view that extra-mathematical application (and/or reference) is not a necessary condition of a mathematical calculus. Mathematical calculi do not require extra-mathematical applications, Wittgenstein argues, since we “can develop arithmetic completely autonomously and its application takes care of itself since wherever it’s applicable we may also apply it” (PR §109; cf. PG 308, WVC 104).