Throwing Away the Baby with the Bath Water: Wittgenstein, Goodstein and the Equation Calculus

Abstract

Reuben Louis Goodstein (1912-1985) was a student of Wittgenstein at
Cambridge in 1931-1934. In this paper, I provide an brief overview of his work in
mathematical logic that shows the influence of Wittgenstein's ideas, including the
replacement in his equation calculus of mathematical induction by a rule of
uniqueness of a function defined by a recursive function. The latter is found in
Wittgenstein's Big Typescript. I also show that the fundamental ideas of the equation
calculus are found not only in the middle period but, in nuce, in the remarks
on mathematics in the Tractatus Logico-Philosophicus and I use this to develop an
argument against a fashionable reading of that book, the so-called 'New Wittgenstein'.
Another link with Wittgenstein is the rejection of quantification theory; in
the last part of the paper use Goodstein's critical remarks on the Law of Excluded
Middle, which also include a critique of Brouwer's half-way rejection, to shed light
on Wittgenstein's.