I’ve been distracted by various things (including preparing talks/lectures for my upcoming NZ trip): but at last it’s time to get back to thinking about Francesco Berto’s book.
The last chapter is called “Gödel versus Wittgenstein and the Paraconsistent Interpretation”. Which gives you an idea of how much Berto is taking on in this chapter. There’s the question of the interpretation of Wittgenstein’s prima facie point-missing remarks about the incompleteness theorem. Then what are we to make of Routley and Priest’s take on the message of the first theorem? And, ambitiously, there’s the claim that the “paraconsistent interpretation” throws light on, or can be seen as bring out strands in, Wittgenstein’s take on Gödel. Which is a lot to try to deal with in a bit over twenty pages.
And in fact I think I’ll pass over the discussion of Wittgenstein here. Berto himself acknowledges that his reading at best reflects some “intuitions at the core of Wittgenstein’s philosophy of mathematics”, and that it leads him to advocate “a strong revisionism with respect to classical logic and classical mathematics” — and such revisionism doesn’t sound too Wittgensteinian.
One quick remark though, which connects up with another line of discussion in this blog. Berto quotes some remarks of Wittgenstein’s on Hilbert, and endorses Wittgenstein’s supposed critique of Hilbert’s “metamathematics”. Berto summarizes Wittgenstein as emphasizing that “Hilbert’s metamathematics is, in fact, nothing but mathematics”. But that’s no critique of Hilbert — for it just reiterates what was surely Hilbert’s view too. The Hilbertian project is use finitarily “safe” mathematics to prove consistency results about theories considered as finite objects, and thereby remove the temptation to look for foundations for those theories outside mathematics (in logic, in intuition, or whatever): see here and here. Which rather suggests that Wittgenstein was as insensitive — shall we say? — in his readings (if any?) of Hilbert as he seems to have been in his readings of Gödel. And it suggests too that Berto, while leaning over backwards to try to find why he thinks is a charitable reading of Wittgenstein, is not extending the same courtesy to Hilbert.
But be that as it may: I’ll now set aside what else Berto says about Gödel, and concentrate on the remarks about his “paraconsistent interpretation”. To be continued