[FOM] Re: FOM posting-Wittgenstein?

[Torkel Franzen]

Juliet Floyd says:

 >I am inclined to think that Wittgenstein's thought will likely prove
 >fruitful precisely through the negative reactions to his remarks that
 >logicians and mathematicians are bound to have, reactions such as Harvey
 >Friedman reports of himself.

  However, in the paper with Putnam you also emphasize "the valuable and
still relatively unappreciated point that Wittgenstein makes in §8 (and
also in §10) of Appendix I of RFM ... [namely that] if one assumes
(and, a fortiori if one actually finds out) that ~P is provable in Russell's
system one should (or, as Wittgenstein actually writes, one "will now
presumably") give up the 'translation' of P by the English sentence
'P is not provable'".

  The justification you give for this is that if ~P is provable in PM,
PM is w-inconsistent, and there is no model of PM in which the formula
NaturalNo.(x) that we have used in P to express "x is a natural
number" actually specifies (a part of the domain of the model
isomorphic to) the natural numbers.

  To this, the obvious rejoinder is that the interpretation of P as
equivalent to "P is not provable in PM" does not depend on PM being
w-consistent. If we define T as the theory PA+"PA is inconsistent",
the Gödel sentence P for T still expresses "P is not provable in T",
in the same sense as the Gödel sentence P' for PA expresses "P' is
not provable in PA".

  You take this rejoinder into account later on in the
paper. Essentially, if I understand you correctly, your argument is
that the relevant concept of truth (or "expresses") wasn't available
to Russell because of his foundational ambitions. Be this as it may, I
wonder if you could clarify how Wittgenstein's point is valid
or valuable or both today, from the point of view of a philosopher or
logician who uses standard logical concepts in his reasoning. In
particular, is there anything wrong with the observation
made above, that the Gödel sentence P for the w-inconsistent theory
PA+"PA is inconsistent" still expresses "P is not provable in T", in
the same sense as the Gödel sentence P' for the w-consistent theory PA
expresses "P' is not provable in PA"?

  You remark toward the end of the paper:

     That the Gödel theorem *shows* that (1) there is a well defined
     notion of "mathematical truth" applicable to every formula of
     PM; and (2) that if PM is consistent, then some "mathematical
     truths" in *that* sense are undecidable in PM, is *not* a
     mathematical result but a metaphysical claim. But that if P is
     provable in PM then PM is inconsistent and if ~P is provable
     in PM then PM is w-consistent is precisely the mathematical
     claim that Gödel proved. What Wittgenstein is criticizing
     is the philosophical naivete involved in confusing the two, or
     thinking that the former follows from the latter.

  To think that Gödel's theorem shows that there is a well-defined
notion of "mathematical truth" applicable to every formula of PM, or
every formula of the system P related to PM actually used by Gödel, or
to every formula of ZF, would indeed be a mistake. I haven't come
across this mistake myself in the philosophical or logical literature.
However, given a notion of mathematical truth applicable to the
language of PA, or of P, or of ZF, (and a soundness assumption) we can
conclude that there are true sentences undecidable in these
theories. Furthermore, this is not necessarily a metaphysical claim,
any more than mathematical theorems in general express metaphysical
claims.

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