 Though mathematics and mathematical activity are purely formal and syntactical, in the Tractatus Wittgenstein tacitly distinguishes between purely formal games with signs, which have no application in contingent propositions, and mathematical propositions, which are used to make inferences from contingent proposition(s) to contingent proposition(s). Wittgenstein does not explicitly say, however, how mathematical equations, which are not genuine propositions, are used in inferences from genuine proposition(s) to genuine proposition(s) (Floyd 2002: 309; Kremer 2002: 293–94). As we shall see in  §3.5,  the later Wittgenstein returns to the importance of extra-mathematical application and uses it to distinguish a mere “sign-game” from a genuine, mathematical language-game.