Wittgenstein’s Philosophies of Mathematics: Systemic Intentionalism and the Employment of a New Method (Regarding Wittgenstein’s Philosophy of Mathematics Development)

Abstract

This essay intends to identify intentionalism (infinity given by rules, notby extensions) and the idea of multiple complete mathematical systems (several“mathematics”) as the central characteristics of Wittgenstein’s philosophy ofmathematics. We intend to roughly show how these ideas come up, interact toeach other, how they develop and, in the end, how they are abandoned in the lateperiod. According to the Tractatus Logico-Philosophicus, infinities can only begiven by rules and there is a single numerical system (the number’s essence is thegeneral idea of ordering). Intentionalism is up to at least 1933, but the idea of asingle system is abandoned in 1929-30 (already in the Philosophische Bemerkungen).In its place one finds the idea of multiple, independent and completenumerical systems. This idea will engender some key moves in Wittgenstein’sphilosophy 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, independentand complete systems. In his late philosophy, both ideas are used onlyas instruments to dissolve philosophical prose regarding mathematics.