Jogando o bebê junto com a água do banho: Wittgenstein, Goodstein e o cálculo equacional

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.