Abstract

This essay intends to identify intentionalism (infinity given by rules, not by extensions) and the idea of multiple complete mathematical systems (several “mathematics”) as the central characteristics of Wittgenstein’s philosophy of mathematics. We intend to roughly show how these ideas come up, interact to each other, how they develop and, in the end, how they are abandoned in the late period. According to the Tractatus Logico-Philosophicus, infinities can only be given by rules and there is a single numerical system (the number’s essence is the general idea of ordering). Intentionalism is up to at least 1933, but the idea of a single system is abandoned in 1929-30 (already in the Philosophische Bemerkungen). In its place one finds the idea of multiple, independent and complete numerical systems. This idea will engender some key moves in Wittgenstein’s philosophy of Mathematics. The notion of “seeing an aspect” from the Big Typescript, of instance, comes up so as to explain such systems. From 1934 onwards, Wittgenstein gradually abandons intentionalism and the idea of multiple, independent and complete systems. In his late philosophy, both ideas are used only as instruments to dissolve philosophical prose regarding mathematics