[FOM] Wittgenstein?

[T P Uschanov]

Juliet Floyd wrote:

> I do not hold that Wittgenstein "understood Goedel better than
> Goedel"--that is a ridiculous assertion that I hope and expect
> readers of the FOM will not associate with my readings of him. 
> The situation--historically, mathematically, and philosophically--
> is more complex.

Well, I have reason to believe that it was I who was being alluded
to when the "ridiculous assertion" was mentioned earlier. It was an
off-hand remark I made in a non-scholarly forum, partly in order to
provoke, and I wouldn't have made it for the life of me if I had
known that it would wind up being resurrected years later and asso-
ciated with Wittgensteinian philosophy of mathematics as a whole --
let alone that an excellent philosopher of mathematics like Juliet
Floyd would feel the need specifically to disown it!

What I wish to replace the withdrawn "ridiculous assertion" with are
two separate assertions:

(1) That Wittgenstein's understanding of Gödel's theorem in terms of
mathematics -- as opposed to the philosophy of mathematics, which was
for him a separate discipline -- was not either remarkably eccentric,
vitiated by straightforward mistakes, or otherwise ridiculous. (This
was alleged by a number of the early critics of his writings on FOM,
e.g., Kreisel, Bernays and Dummett; Gödel himself thought the same in
the 1970s, and Mark Steiner does so today.) There is evidence from
Alister Watson and R. L. Goodstein, both students of Wittgenstein
and working mathematicians, that Wittgenstein's understanding of
Gödel's theorem as a piece of mathematics in the mid to late 1930s,
when he wrote his notorious remarks on it, was basically accurate
and that he fully accepted Gödel's proof.

(2) That Wittgenstein's understanding of Gödel's theorem in terms of
the philosophy of mathematics -- as opposed to mathematics, which was
for him a separate discipline -- has been misunderstood by most
commentators, this having the effect that they have mislocated his
dissatisfaction with Gödel. To put it very shortly, Wittgenstein's
disagreement with Gödel consists in his non-realist view that there
is no reason to suppose that truth is a system-independent notion any
more than provability is. He did not view PM as an attempt to codify
*the* truth about mathematics, or as a kind of organon for the benefit
of future mathematicians, but as one way of dealing with these parts
of mathematics among others. As Juliet Floyd and Hilary Putnam have
written, for Wittgenstein, "we should look on PM as a 'system' in
the sense in which a system of non-Euclidean geometry is a 'system'
of geometry -- a sense in which the same sentence (Satz) can be true
in one system and false in another. And [he] does *not* deny that a
proposition which cannot be proved in Russell's system ... can in
*some* sense be 'true' or 'false' (outside the system) -- he only
asserts that this is a 'different sense' from the sense in which it
is true or false *as* a 'proposition of Principia Mathematica'." (Cf.
Steiner's remark about Wittgenstein's "suspicion that Gödel was doing
FOM, rather than fom--or perhaps FOM in the guise of fom".)

In other words, when one wants to say "There are statements that are
true that cannot be proven within system S", one would be better off
saying "There are statements that are true within system S that
cannot be proven within system S". Wittgenstein's understanding of the
philosophical implications of Gödel's theorem was better than Gödel's,
not because he was the one to believe that truth is a non-system-
independent notion in this way, but because he was the one to raise
the very possibility that it might be seen as one. (Before discussions
of his understanding of Gödel's theorem locate his disagreement with
Gödel here, we are not in a position to argue whether Wittgenstein's
understanding was better than Gödel's *also* because he believed that
truth is a non-system-independent notion.)

I have read the recently mentioned Floyd/Steiner exchange in Philosophia
Mathematica over five times -- it is a truly excellent and entertaining
pair of papers -- and I have to say that I side with Floyd about
practically everything. Regrettably the exchange is not available
electronically (perhaps Floyd and Steiner will respond to Harvey
Friedman's request for summation?), but Floyd's earlier (2000) joint
paper with Putnam actually is available in PDF form at:

http://staff.washington.edu/dalexand/Putnam%20Readings/Notorious.pdf

I think it is the best brief paper on the issue published so far,
and I'd love to see discussion of it here. This also goes for:

http://www.cs.tut.fi/~jkorpela/mathem/numexp.html

which is a paper by Jukka Korpela on another sub-topic of this dis-
cussion: Wittgenstein's definition of numbers.

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