[FOM] Wittgenstein?

[T P Uschanov]

Torkel Franzen wrote:

 > The Peano axioms are provable in PA, but this cannot be what you
> mean by "true in PA", since you hold that we should say "There are
> statements that are true within PA that cannot be proven within
> PA".

There was a reason why I chose just that example. Because I would argue 
that since the Peano axioms are part of what simply has to be taken as 
true if PA is to be PA at all (instead of some other arithmetic, however 
similar), it is therefore dubious to hold that they are true in PA in 
exactly the same sense of "true" as the average statement-which-is-true- 
in-PA. This is just an extension of Wittgenstein's general point in the 
so-called notorious Gödel paragraph (as summarized perfectly correctly 
even by enemies of it such as Steiner, and now seemingly Bays): that 
sentences featuring words such as "provable," "within," "true," "false," 
etc., are natural-language glosses on formal-language mathematical 
results, and that such glosses can never have more than a merely 
contingent relation to (a) these formal-language results themselves, (b) 
other natural-language glosses involving the use of the same words, even 
if these are glosses on results in the same formal language. 
(Wittgenstein's own aside: "What is called 'losing' in chess may 
constitute winning in another game.")

I did not intend to claim simply that "we should say" that "There are 
statements that are true within PA that cannot be proven within PA," 
only that if someone wants to convey the notion which "There are 
statements that are true that cannot be proven within PA" is ordinarily 
intended to convey, then that would be a better way of putting it. I am 
very sorry if I was unclear on this point.

And if all this sounds intolerably harebrained, eccentric or evasive to 
you, then there is really little else left for me to say that has not 
already been said, only better. I simply think that viewpoints like this 
follow naturally from Wittgenstein's extreme version of non-realism, if 
followed through completely (in cold blood, or so to say).

By the way, I found Bays's new paper very interesting, and I moreover 
think that it is impossible to predict just what Wittgenstein would have 
said if he had been confronted with Bays's argument. (Perhaps he would 
even have received enough of the schadenfreude he was pursuing with his 
remarks from the mere fact that realist mathematicians are compelled to 
choose to prioritize either axiomatization or interpretation, which 
choice Bays presents as a possible way out of the remarks.)

-- 
T. P. Uschanov, Research Assistant        | e-mail:
Department of Philosophy                  | <tuschano at cc.helsinki.fi>
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