[FOM] FOM posting-Wittgenstein?
Mark Steiner
marksa at vms.huji.ac.il
Mon May 5 16:15:00 EDT 2003
For convenience I will reproduced the entire "notorious passage" which
angered Goedel:
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)
Following is my reading of the passage and remarks thereupon. I will not
presume to summarize Floyd's alternative reading and refer readers to her
reply.
Paragraph 1 of this passage presents an argument, A, whose conclusion is
(relativized to PA): P is a true but unprovable proposition of PA.
Paragraph 2 presents a refutation of this argument.
Argument A is not Goedel's proof, nor is the conclusion Goedel's theorem.
(Though Goedel did present something like argument A in the Introduction to
his theorem, so he has no gripe against Wittgenstein.)
Nevertheless, A can be viewed as an informal version of a MATHEMATICAL
THEOREM. To formalize the theorem one need only define "truth for PA" in ZF
using techniques it would be presumptuous for me to outline in this list.
These techniques were of course published by Tarski in his famous paper.
Wittgenstein would not accept that this concept is the same as "truth" as
used in ordinary language, so let's call the Tarski concept "shmuth." Shmuth
has some nice properties, such as:' "There are infinitely many primes" is
shmue iff there are infinitely many primes' IS A THEOREM OF ZFC and so for
the other "biconditionals." Also we have the theorem that: a sentence of the
form 'for all x, phi(x)' is shmue iff every sentence of the form phi(n), n a
numeral (or "shnumeral" if you prefer), is shmue. Furthermore, all the
axioms of PA are shmue (this is also a theorem of ZFC) and the rules of
inference preserve shmuth. Hence all theorems of PA are shmue (a theorem of
ZFC). I remark that the notion of provability in PA used here, which depends
on the notion of a sequence of formulas, is quite naturally given in the
language of ZFC, as core mathematicians agree to allow set theory to be the
basic language of modern mathematics, and sequences are thus set theoretical
objects by almost all core mathematicians.
Argument A can now be formalized by most members of this list as a formal
theorem in ZFC to the effect that P is shmue iff P is not provable in PA.
But now if P is provable in PA then it must be "shmue" by the above, which
is a contradiction. Hence P is not provable and therefore shmue. This
theorem does NOT rely on intepreting P as "P is not provable in PA." This
theorem is not refutable by any philosophical argument.
LW would of course not accept that shmuth is truth, because for him one
cannot disconnect truth of a sentence of PA from provability in PA. But
shmuth has some of the core properties of truth, and our theorem is
sufficient to persuade a right thinking Wittgensteinian that:
If we would like to expand the notion of truth in PA by expanding PA, it is
proposition P and not its negation which should end up provable. For since
it is incontestible that all substitution instances of P are in fact
provable in PA, we would do violence to our ordinary concept of truth to
allow not-P to be "true" (i.e. provable) in any expansion of PA. Thus, the
conclusion of argument A has real content even for LW. In fact he could have
used it to buttress his idea that "truth" is not a fixed concept at all. Of
course, in ZFC P is already provable since in ZFC we can prove every Tarski
biconditional and thus prove any statement which we can prove is shmue. But
we don't use in the proof anything like the strength of ZF so we are not
relying on the GENERAL rule that we can add to PA any statement provable in
ZF.
LW's remark that winning in chess might be losing in another game is not
relevant to a case, like this one, where the games are logically connected.
Take three dimensional chess-the moves are determined by the rules for two
dimensional chess mathematically. There is, further, no reason why LW should
not have recognized this fact and even used it to illustrate his point about
the variable nature of the truth concept. It is this reason that I called my
article "Wittgenstein as his own worst enemy" and speculated as to the
reasons why LW slipped into attacking a valid mathematical argument.
Concerning the second theorem of Goedel, I gave a very similar argument to
Harvey's in a review of a book by Detlefsen in The Journal of Philosophy,
Vol. 88, No. 6. (Jun., 1991), pp. 331-336.
Like Harvey, I argued that if we apply Goedel II to ZFC (rather than PA)
there is no serious argument that CON(ZFC) does not mean "ZFC is consistent"
. As I wrote above, core mathematicians, even if they deride "logic", are
more than willing to allow set theory to define the basic concepts of
classical mathematics: sequences, topologies, functions, everything is a
"set." (When I was an undergraduate, the Columbia math department refused
to hire a logician, but brought logicians--such as Haim Gaifman--as visiting
lecturers to teach "foundation" courses in which all basic notions of
mathematics were defined in set theoretic terms.) That is: the sentence
CON(ZFC) in its standard mathematical meaning, without any need for
"interpretation" simply says that ZFC is consistent. I think this is also
Harvey's view, though of course, he did much more with it than I did.
Mark Steiner
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