[FOM] wittgenstein and goedel

Wed May 7 12:00:50 EDT 2003

Mark Steiner wrote:
>
> For convenience I will reproduced the entire "notorious passage" which
> angered Goedel:
>
> Wittgenstein:
> "I imagine someone asking my advice; he says: "I have constructed a
> proposition ..." [snip: see previous message]

Jeffrey Ketland wrote:
>
> I'm curious to know what Goedel said or wrote about Wittgenstein's
> arguments. Can someone provide guidance?

You should have a look at Hao Wang's book "A logical journey: From Goedel to
philosophy".  Unfortunately I do not have it here with me now to quote...  The
comments by Goedel are *really* depreciative ---it would be fun if someone
could check there and post them to the list.

Harvey Friedman wrote:
>
> Is there anybody on the FOM list prepared to outline the LW attack or
> any other attack interactively?

mjmurphy wrote:
>
> I don't see it as an "attack".  That is, it is not a matter of LW's
> attempting to refute some piece of mathematics.  I would say vaguely that he
> was "attempting to change attitudes" (towards consistency, for example) or,
> if this were a different list, that he was engaged in a kind of "ideological
> critique". (See below)

This observation strikes me as going straight to the point.  The later
Wittgenstein was much more interested in a kind of "philosophical therapy" (of
mathematics, psychology, philosophy, etc) than anything else.  All that he
wanted with this therapy was to "describe the use of language, and leave
everything how it is" (Philosophical Investigations 124-127).

> Putting aside the issue of "hidden contradictions" for a moment, I think the
> point LW was trying to make here with respect to the "obsession" with
> consistency is something like the following.  He asks Turing: "Why should we
> care if our system is inconsistent?"  Turing's answer is "Because in the
> application of an inconsistant system we may cause a bridge might fall
> down."  Another possible answer is "Because if we interpret the sentences of
> the sysem as a set of instructions, then given the instruction "p & ~p", we
> will not  know how to behave."  But LW would say that we should not allow
> contingent matters of psychology (in the latter case) or physics (in the
> former) to determine the proper focus of the philosophy of mathematics.   If
> Turing's answer is convincing, then it would be okay to have an inconsistent
> system at some future date if we were all made immortal and didnt't care if
> our bridges collapsed.  But this too would be an irrelevant, external
> consideration.
>
> This kind of criticism doesn't refute anything.  LW is urging the FOM people
> to "throw off their chains"; to consider roads untaken (maybe leading to
> paraconsistent logic?) etc. etc.  It isn't like anyone has to do that, or
> even be moved by the plea.

   Indeed, paraconsistent logicians enjoy quoting Wittgenstein as a prophet:

"...even at this stage, I predict a time when there will be mathematical
investigations of calculi containing contradictions, and people will actually
be proud of having emancipated themselves from consistency." (Philosophical
Remarks, p.332; Ludwig Wittgenstein and the Vienna Circle, p.139)

* * *

There has been some discussion in the list as to whether Wittgenstein's work
would have (or would have had) some impact at the mathematical praxis.
Someone mentioned the Early Wittgenstein's proposal of truth-tables, and the
Intermediary Wittgenstein's flirting with intuitionism and his sympathy for
some finitistic and construtivistic doctrines, to which one could add perhaps
the positing of an anti-platonist attitude at about the same time (see
Hintikka & Hintikka, Investigating Wittgenstein, 1986).

   It is understandably difficult to administer one's teachings without at the
same time letting one's prejudices show off.  I find it reasonable to claim,
at any rate, that the Later Wittgenstein's influence can / could only have had
a *negative* effect in mathematical investigations.  Indeed, if on the one
hand Wittgenstein *did not stimulate* some projects, on the other hand he did
at least *discourage* certain research lines, qualifying them as
"uninteresting" ---among them the transfinite set theory and the elaboration
and developing of new formal systems (see Glock's A Wittgenstein Dictionary,
1996).  Luckily, Turing and Ramsey did not take Wittgenstein too serious.
But other of his students did (check Ray Monk's biography of Wittgenstein).

   Take for instance Goedel's First Incompleteness Theorem, where G is the
sentence which says, intuitively, that "G is not provable".  The comments by
Wittgenstein on the theme are largely inconclusive.  One is more likely to
find Wittgenstein criticizing the interpretation of G than the correctness of
Goedel's proof.  One way of understanding that position is by recovering van
Heijenoort's distinction (in "Logic as language and logic as calculus", Revue
Internationale de Philosophie 17, 1967) between "language as a universal
medium" and "language as calculus", as reelaborated by Hintikka & Hintikka:
according to the former viewpoint, one cannot observe one's own language from
outside and describe it, as one does with other objects that can be specified,
referred, described, discussed, and about which one can theorize in one's
own language;  according to the latter viewpoint, though, one can do all that,
raising metatheoretical questions about one's given logic and even proposing a
modification on its interpretation, for instance, on what concerns the domain
of the quantifiers.  According to van Heijenoort, the adherence of Frege and
Russell to the former viewpoint explains the total absence of semantical
notions in their works, in particular the inattentiveness about the
distinction between the notion of provability and the notion of validity based
on the (proto-)set theory.  Seemingly, such a distinction would only start to
be taken for serious after the developments of Loewenheim in 1915 and Skolem
in 1920.  According to Hintikka & Hintikka, Wittgenstein's adherence to the
very same viewpoint explains why is it that Wittgenstein (and, to some extent,
Frege before him, and Quine after him) would consider the whole talk about
semantics to be "ineffable" (rather than "impossible").

   This is all to say that Wittgenstein's view of the language as a universal
medium might have led him to an identification of the notions of "being true"
and "being provable" in Russell's PM.  More specifically, this amounted not
only in assuming that "True(A)<->Provable(A)", but in fact assuming that
"True(A)=Provable(A)".  All that Wittgenstein can say about G, or any other
mathematical statement, is that any such proposition should extract its
meaning from its own proof, for "in mathematics, process and result are
equivalent" (Remarks on the Foundations of Mathematics I:82; check also
Tractatus 6.1261).  Wittgenstein's first commitment, then, was to a
clarification of the usage (thus, the meaning) of the expression "being
provable".  If one believes to have reasons to assert that "not-Provable(G)"
has been proved, than, for Wittgenstein, one seems to have been left with
one among the following options (Remarks on the Foundations of Mathematics
Ap.I:5-9, V:18-19):

(a) one is mistaken ---in that case, the least one should do is to change
one's interpretation of "being not provable";

(b) one has indeed proved G, but in another mathematical system, or in a
physical system, for instance ---in that case, Wittgenstein sees no problem,
for there certainly are true propositions from other systems which are not
provable in PM as much as there are true propositions of PM which are not
provable "out there";

(c) one has indeed proved G in one's initial system, and so is left with a
contradiction ---in that case, Wittgenstein says that this should not be
necessarily believed to cause any harm to the system, and he sees no reason
not to think that the "principle of non-contradiction is simply false in that
particular case".

Wittgenstein's critique to (c) starts with a comparison of G to the Liar
sentence, L ---a comparison which is not fully adequate, as we would, in
the same context, reconstruct L as saying, intuitively, that "L is not
true", rather than "L is not provable".  But then again, under the above
interpretation, such a distinction would not say anything to Wittgenstein.
(One should note, by the way, that what the Liar gives us is the reason why
the truths of arithmetic cannot be *defined* inside Peano Arithmetic, PA.)
Next, Wittgenstein's critique follows the strategy of disqualification,
asserting that propositions such as G are totally useless:

"It is as if one should extract from certain principles about natural forms
and architectonic style the idea that to Mount Everest, where nobody can live,
belonged a chalet in the baroque style." (Remarks on the Foundations of
Mathematics, Ap.I:19)

Now, this is clearly not a decisive observation: first of all because
Wittgenstein should not be criticizing something for *not* having a use:
*couldn't* it have one (perhaps an invented one), at another language game?;
second, because nowadays one is already aware of several other examples of
mathematical statements which are "more natural", that is, which do not
translate metamathematical assertions, but which are formally undecidable
in PA.

* * *

  Wittgenstein's misconstruction of Goedel's Second Incompleteness Theorem
seems a bit more embarrassing.  In the scope of a general negativist attitude
towards the metamathematical enterprise ("The mathematical problems of what is
called foundations are no more the foundations of mathematics for us than the
painted rock is the support of the painted tower", Remarks on the Foundation
of Mathematics, V:13), Wittgenstein is led to some very delicate positions.
An example of that is his view about proofs of relative consistency, or
equiconsistency, constituting just a sort of silly game (Philosophical
Remarks, p.335).  Now, to prove the consistency (or rather the non-triviality)
of classical propositional logic, CPL, for one, one can easily proceed through
a soundness result: all axioms are tautologies, and inference rules are
truth-preserving; thus, by induction on the length of proofs, all theorems
will also be tautologies; finally, given the classical meaning of negation,
two contradictory propositions cannot be simultaneously tautological, and so
they cannot be simultaneously theorems of CPL.  To use that consistency result
of CPL to prove the consistency of its proper extension, classical
quantificational logic, CQL, the canonical proof consists in constructing a
"forgetful functor" erasing all quantifiers of CQL, mapping thus all formulas
of CQL to formulas of CPL.  Now, if there would be contradictory theorems in
CQL, they would be taken by that functor to contradictory theorems of CPL, but
we already now that the latter cannot be found.  One can by similar means
prove the equiconsistency of different systems of geometry, such as euclidian
and riemannian geometry.

   About that theme, Wittgenstein's judgment is intriguing, as he roughly
maintains that "all that we have is a mapping, a function taking rules of a
game into rules of another game" (Philosophical Remarks, p.335).  Eager to
demonstrate the uselessness of Hilbert's Metamathematics, Wittgenstein seems
here to grossly misunderstand its aims, namely, prove (by finitary means) the
consistency of PA, inside PA.  Note that neither the consistency proof of CPL
nor the consistency proof of CQL are written entirely inside CPL or CQL
(induction, for instance, is not part of these systems).  Now, PA _does_
constitute the system in which PA's consistency proof is sought, and it is on
_that_ seeking that one is limited by the Second Incompleteness Theorem.

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