In the last two sections, we saw how Berto interprets Wittgenstein’s remarks in (RFM I, App. III) in the light Priest’s and Routley’s ‘single argument’, were
319RFM VII, §36. 320Berto 2009b, p. 209.
321For proof of the decidability and completeness results, see (Priest 1994). For proof of the
Gödel’s theorem recast as applying to naive proof concept. In this section, we will offer some criticisms of this idea.
In his interpretation, Berto makes some assumptions about Wittgenstein’s philosophy of mathematics which are inspired by the interpretations of Shanker and Rodych. These include the idea that Wittgenstein held a strong calculus conception of language, that he was a strong finitist and that he held that every mathematical question was decidable. In the sections on Shanker and Rodych we saw that there are reasons to doubt this. Before we go on to criticise Berto, we should note that this is not a severe problem for him, as his main claims, namely that Wittgenstein saw Gödel’s theorem leading to a paradox when put in the context of an absolute notion of ‘provability’, and that he was fine with this paradox, do not depend on these assumptions. At most he loses those aspects of his interpretation that are directly relevant to this, namely that Wittgenstein advocated an inconsistent arithmetic because it would be finite and complete, and it is very easy for him to change his interpretation in the light of similar criticisms as raised against Shanker and Rodych—that Wittgenstein had this in mind is very unlikely anyway, as the relevant work in paraconsistent models had not been done. Berto’s main claims, however, do have other serious problems.
First of all, just a superficial reading of the text makes it very implausible that Wittgenstein had anything like a naïve notion of proof in mind when he wrote §11. There, as in other remarks, Wittgenstein talks what would be the consequences of proving the unprovability of P in ‘Russell’s system’. Russell’s system, by which Wittgenstein of course means the system ofPrincipia Mathematica, is a formal system, and not at all comparable to Priest’s and Routley’s naïve notion of proof—and even if it were, Wittgenstein would have phrased it differently, had he meant anything of the sort. Wittgenstein is thus highly unlikely to have meant any such thing by his remark.
Furthermore, while it is true that Priest’s and Routley’s notion of naive proof excludes there being a language/metalanguage-distinction, and it is somewhat plausible that Wittgenstein rejected such a distinction as well, other elements of his philosophy make it difficult to subscribe to him a view similar to Priest and Routley. On the calculus conception of mathematics, Wittgenstein would have, much like Berto says, rejected there being any calculus that is considered a metacalculus of any other, but he still would have seen the two supposed levels at work in Gödel’s proof as being two different calculi, and therefore no contradiction would be derivable, as there would just be two different sentences in two different calculi.322Of course, as we said above, this view of Wittgenstein’s
philosophy of mathematics, is not essential for Berto.
On the language-game conception, however, things don’t fare much better.
Priest’s and Routley’s naïve conception of truth has a certain unity to it, and as Priest notes, the very possibility of it being formalised is essential for their argument. Wittgenstein’s view of mathematics in the late period was completely opposite to this picture, for him there was no essence to our notion of proof, and the idea that the whole of mathematics could be formalised in one formal system would surely be anathema to him. It is therefore very unlikely that he would have wanted to say anything like Berto attributes to him.
Lastly, the point of Priest’s and Routley’s argument is to show that mathemat- ics is in fact inconsistent. That is to say, it is their claim that mathematics (and logic) is or should be inconsistent. Despite Wittgenstein’s indifferent attitude towards contradiction, it is unlikely that he would have wanted to make such strong claims. The dialetheists champion contradiction, Wittgenstein merely wanted to show that it are not as important as many philosophers believe. He consistently argued that if a contradiction were to be found, this would not mean a that a calculus was useless:
I want to object to thebugbear of contradiction, the superstitious fear that takes the discovery of a contradiction to mean the destruction of the calculus.323
This is not the same as saying that there are in fact important, inconsistent calucli, and that this should be so. Consider for instance (RFM VII, §15):
‘Then are you in favour of contradiction?’ Not at all; any more than soft rulers324
Berto, however, is completely right in that the dialetheist program is very much in Wittgenstein’s spirit, and it is not unlikely that he would have been very sympathetic towards it.
This is however just speculation and is in no way enough to show that this is what Wittgenstein had in mind. Berto’s reading is however highly original and interesting, and he could of course argue that he never intended to argue that this was in fact what Wittgenstein had in mind, but rather that in the light of further developments in logic and philosophy, Wittgenstein’s remarks do not seem as crazy and foolish as the early commentators thought. This is not at all implausible, and if this is his aim, he is certainly correct, but it does not show that Wittgenstein’s remarks were particularly interesting, nor that he was right—at most Berto could claim Wittgenstein as some kind of authority in his dialetheism, but in light of the severe criticism Wittgenstein has suffered
323(WVC, p. 196) See as well Wittgenstein’s remarks in (RFM I, App. III) where he declares
contradictions to be harmless and only of interest because the bother people, or (RFM III, §82): “My aim is to alter theattitudeto contradiction and consistency”.
for such views (as witnessed by the reaction of the early commentators), it is unlikely that this would be of much help to him.