The Multiple Complete Systems Conception as Fil Conducteur of Wittgenstein’s Philosophy of Mathematics

Abstract

The multiple complete systems conception is perhaps Wittgenstein’s most important
insight concerning the philosophy of mathematics after the Tractatus. According to
this conception, it is the invention of new systems complete in themselves, and not
the accumulation of truths, that characterizes mathematics. I explain, first, that
the multiple complete systems conception solidifies Wittgenstein’s Tractarian
Intensionalism (against Platonism and Intuitionism). Second, I present some of the
difficulties that it implies (its ‘paradoxical’ consequences) and show how they might
have determined some of the developments in Wittgenstein’s philosophy of mathematics.
These developments, I suggest, culminate in the idea that the conception is merely an
“equally arbitrary” alternative to other conceptions.