 There seem to be two reasons why the later Wittgenstein reintroduces extra-mathematical application as a necessary condition of a mathematical language-game. First, the later Wittgenstein has an even greater interest in the use of natural and formal languages in diverse “forms of life” (PI §23), which prompts him to emphasize that, in many cases, a mathematical ‘proposition’ functions as if it were an empirical proposition “hardened into a rule” (RFM VI, §23) and that mathematics plays diverse applied roles in many forms of human activity (e.g., science, technology, predictions). Second, the extra-mathematical application criterion relieves the tension between Wittgenstein’s intermediate critique of set theory and his strong formalism according to which “one calculus is as good as another” (PG 334). By demarcating mathematical language-games from non-mathematical sign-games, Wittgenstein can now claim that, “for the time being”, set theory is merely a formal sign-game. 