FOM: Wittgenstein and Goedel; Wittgenstein and Turing
Mark Steiner
marksa at vms.huji.ac.il
Mon Jan 31 11:50:38 EST 2000
In a way, what I am about to write below HAS to be misleading.
There
is no way to understand what W wrote about f.o.m. without understanding
his general positions on such matters as "rule-following." At this
point, I won't go into them; you can see what I have to say in the
volume, Benacerraf and his Critics.
Because of W's radical positions and his inability to write discursively
(his style is part of his philosophy, in the unusual sense that it often
displays what it says), what he said is often disputed by the
scholars--but this is true also for Kant. Saul Kripke's wonderful book
on Wittgenstein has been disputed by many, including I believe people
who are reading this now. His version of W is sometimes called
"Kripkenstein." I hope nobody calls what follows Kripkensteiner. I
have chosen W's views on f.o.m. topics to discuss, but not the more
fundamental issues like the applicability of mathematics, which W was
one of the FEW philosophers to worry about in this century. (Which is
why I'm interested in him.) I should say that I don't think that W
would have been NECESSARILY against f.o.m. in the sense that Steve
defined it recently. W was against Hilbertian "worries" about
consistency in mathematics; he thought it was absurd and "superstitious"
that mathematicians should be paranoid about "hidden contradictions" in
arithmetic (see below his debate with Turing). I have the feeling,
though, that Hilbert's stuff on this subject might have been
"political"--i.e. he wanted to destroy Brouwer's possible influence on
the mathematical community (and it was a real threat, as "core"
mathematicians like Weyl seem to be Intuitionist fellow-travelers) by
shoring up the epistemological foundations of classical mathematics. It
is true (probably) that W reacted, in class, to Hilbert's statement,
"Nobody will ever drive us out of Cantor's paradise" with the retort, "I
won't drive anybody out of Cantor's paradise--when I finish, you'll
leave by yourself." But W was under the impression that the theory of
infinite cardinals had no application to "core" mathematics or to
physics, and that its appeal was metaphysical (and you have to admit
that Cantor was opposed by Poincare on exactly the same grounds). In
light of recent postings, which W could not have predicted, W might well
have been wrong about this. Meanwhile, Harvey admits that, as of today,
there is no real reason for "core" mathematicians to learn the typical
proof procedures of ZFC--they just assume that what they are doing is
translatable into ZFC. Now to Goedel
1. Wittgenstein and Goedel
No words of Wittgenstein have occasioned more criticism by
logicians
than these on Goedel's theorem:
I imagine someone asking my advice; he says: “I have constructed
a proposition (I will use ‘P’ to designate it) in Russell’s symbolism,
and by means of certain definitions and transformations it can be so
interpreted (or clarified) that it says: ‘P is not provable in Russell’s
system’. Must I not say that this proposition on the one hand is true,
and on the other hand is unprovable? For suppose it were false; then it
is true that it is provable. And that surely cannot be! And if it is
proved, then it is proved that it is not provable. Thus it can only be
true but unprovable.
Just as we ask: “‘provable’ in what system?”, so we must also ask:
“‘true’ in what system?” ‘True in Russell’s system’ means, as was said:
proved in Russell’s system; and ‘false in Russell’s system’ means: the
opposite has been proved in Russell’s system. Now what does your
“suppose it is false” mean? In the Russell sense it means ‘suppose the
opposite is proved in Russell’s system’; if that is your assumption, you
will now presumably give up the interpretation that it is unprovable.
And by ‘this interpretation’ I understand the translation into this
English sentence.–if you assume that the proposition is provable in
Russell’s system, that means it is true in the Russell sense, and the
interpretation “P is not provable” again has to be given up. If you
assume that the proposition is true in the Russell sense, the same thing
follows. Further: if the proposition is supposed to be false in some
other than the Russell sense, then it does not contradict this for it to
be proved in Russell’s system. (What is called “losing” in chess may
constitute winning in another game.) (RFM, I, Appendix III, §8)
As it happens, Goedel himself read this and reacted with anger:
As far as my theorem about undecidable propositions is concerned
it is indeed clear...that Wittgenstein did not understand it (or
pretended not to understand it). He interprets it as a kind of logical
paradox, while in fact it is just the opposite, namely a mathematical
theorem within an absolutely uncontroversial part of mathematics
(finitary number theory or combinatorics).
Yet, Wittgenstein's account of Goedel's theorem, is obviously
based on Goedel's own Introduction to the theorem, where he says:
...we can find a formula F(v) of PM with one free variable
v...such
that F(v), interpreted according to the meaning of the terms of PM,
says: v is a provable formula. We now construct an undecidable
proposition of the system PM...
We now show that the proposition [R(q);q] is undecidable in PM.
For
let us suppose that that proposition [R(q);q] were provable; then it
would also be true. But in that case....which contradicts the
assumption. If, on the other hand, the negation of [R(q);q] were
provable, then ... Bew[R(q);q] would hold. But then [R(q);q], as well
as its negation would be provable, which again is impossible.
The analogy of this argument with the Richard antinomy leaps to
the
eye. It is closely related to the "Liar" too; for the undecidable
proposition [R(q);q] states that q belongs to K, that is, by (1), that
[R(q);q] is not provable. We therefore have before us a proposition
that says about itself that it is not provable [in PM]...From the remark
that [R(q);q] says about itself that it is not provable it follows at
once that [R(q);q] is true, for [R(q);q] *is* indeed unprovable (being
undecidable).
I submit that this account of Goedel's theorem is precisely the
argument that W criticized. It does NOT follow from the theorem itself,
which says nothing about "truth," or "interpretations." Nor is there
any NECESSITY to interpret the Goedel sentence as SAYING "I am not
provable." Nor does Goedel's theorem force us to say that "P is true"
in mathematics is anything other than "P is provable in system ... " if
we don't want to (and recall I posted that Macpherson adopted W's
position exactly on this matter).
I happen to be one of those who believe that Wittgenstein DID
miss the
boat on Goedel's theorem. Recent evidence indicates that he was so
incensed at Goedel's Introduction, that he couldn't read the rest of the
theorem.
At the same time, had Wittgenstein not "lost his cool", he could
have
attacked Goedel's popular version of the proof as unwarranted by the
theorem and its real proof. Namely, again, recent research indicates
(what is anyhow obvious) that Goedel's theorem was motivated by
philosophical realism--as Prof. Davis says, a reaction against logical
positivism.
In fact, I believe strongly, that Goedel's theorem could just
have well
have been used by W to bolster his own ideas in the philosophy of
mathematics, which were in strong opposition to those of Goedel. One of
these, in my opinion, is that truth (including mathematical truth)
cannot be regarded as any kind of "correspondence" between a sentence
and "reality." Goedel's theorem (though psychologically perhaps could
not have been discovered except by a realist, an empirical proposition
which may or may not be true) simply "shows" that truth is a "family
resemblence" concept, which has no essential nature--similar to
"winning" in games. Goedel's argument (as W SHOULD have seen it) simply
shows how to extend "truth in PM" to truth in a stronger system.
It is true that W seems to say that truth is conventional, where
it
would seem that Goedel's proof shows that we have no real choice but to
call the sentence [R(q);q] "true" (i.e. assertible in a stronger system,
for Wittgenstein). But, contrary to what has been asserted in some of
the W literature, I see no evidence that W held that mathematics is
conventional, though he did say that mathematical theorems are rules.
The choice whether to adopt a rule R or whether to adopt the contrary
rule -R is NOT conventional in mathematics, says W. This is his general
position, but, in discussing Goedel's theorem, he "lost his cool,"
because Goedel's theorem had become an icon of mathematical realism.
The authority of Goedel himself had been invoked to show that
mathematical truth is logically independent of provability (and Goedel
himself intended his authority to be so invoked, it seems).
This is why I have entitled my forthcoming article in
Philosophia
Mathematica, "Wittgenstein as his own worst Enemy." It will be
published along with a reply by Juliet Floyd, who believes that what I
thought W should have said, he actually did. So the differences between
us should not make that much of a difference to this list.
2. Wittgenstein and Turing
Since I'm now on intersession, I'll continue. Turing was a
participant
in W's philosophy of mathematics class at Cambridge in 1939. He came
every week, and was one of the few who dared take on the Master. (I
doubt if there are any now reading this, but Harvey is correct in saying
that the W cults exist, meaning that there are are those who believe
fervently that, since W never meant to make a mistake, anybody who
thinks he did must have misunderstood him, to quote a joke I read on the
Internet concerning somebody else.) When, one week, Turing didn't show
up, W refused to lecture on anything important. It was crucial to W
that Turing agree that W was giving a correct "description" of
mathematics. Unfortunately, Turing would not agree that
"mathematicians" are superstitious when they are concerned about the
consistency of arithmetic. He argued that "bridges would fall" if there
were a hitherto unnoticed inconsistency in arithmetic, because from the
inconsistency you could derive anything about the bridge. W replied
that once we noticed an inconsistency, we simply would not use it to
derive "anything." But Turing replied that we could prove an arbitrary
proposition from a contradiction without actually going through the
contradiction p & -p. His idea was that -p implies p -> q and then from
p you can get q by MP. I doubt that W would accept that mathematicians
would accept this move [although in fact "core" mathematicians do accept
that the null set is a subset of every set, which is the same move], but
his deeper resistance to Turings argument was based on his idea that you
can't detach the application of a mathematical formalism from the
formalism itself. The mathematical formalism in fact gets its meaning
from praxis. A mathematical practice which has been successful in
building bridges will not be retroactively nullified just because
somebody in the future finds, somewhere in the calculus of the
formalism, a contradiction. Another example would be the infinitesimal
calculus, which as Prof. Davis points out, had no consistent formulation
at a time when it was in heavy use in physical calculation. In further
support of W's ideas, one could cite the practice of contemporary
physicists, whose mathematical apparatus, in the case of quantum
electrodynamics, is not known to be formalizable as a consistent
system. Nevertheless, physicists know perfectly well how to work with
it, and in fact are able to generate the most fantastically precise
results in the history of physics--namely, the calculation of the
magnetic moment of the electron, correct to 12 decimal places!! (Last
time I checked.)
I think both of these Wittgensteinian ideas deserve to be discussed
without cant or dogma.
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