Re: Godel's Dualism (Re: Reflections Godel Wang)

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Date: 12 May 2004 05:48:15 -0700

Torkel Franzen <torkel@sm.luth.se> wrote in message news:<vcbekptv901.fsf@beta19.sm.ltu.se>...
> erayo@bilkent.edu.tr (Eray Ozkural exa) writes:
>
> > That is in my opinion, a good example for the kind of absolutely
> > unsolvable (means unknowable by any finite discrete mechanism)
> > mathematical proposition of the kind Godel should have sought as far
> > as it can be inferred from the Gibbs lecture, but apparently could not
> > find.
>
> It is unclear what proposition you are referring to - Omega is not a
> proposition, but a real number. Do you have in mind statements of the
> form "the i:th digit of Omega is j"? There are several standard
> arguments to the effect that Chaitin's grandiose philosophical claims
> for Omega and for "the" constant c associated with a theory in his
> incompleteness theorem are completely unfounded. I'd be interested to
> hear any counter-arguments you or anybody else might have. A suitable
> starting point might be the discussion of Chaitin on the
> FOM list:
> http://www.cs.nyu.edu/pipermail/fom/2001-March/thread.html#4862
>
> and the references given there to papers by Raatikainen and others.

Well, as far as I can tell from Raatikainen's posts, he does not have
the required level of formal training in mathematics [*], and in
particular he has not understood Chaitin's Algorithmic Information
Theory (AIT) monograph, a piece of knowledge without which it is at
best unnecessary, and at worst philosophical fraud, to comment on his
work. Therefore, I won't devote much attention to his posts on FOM
list, for I don't find him an accomplished student of mathematics. But
I am curious of his papers on his website, I am sure there are
substantial philosophical arguments/claims in them, and the titles
suggest that he has interesting thoughts on Godel as well.

Please tell us how the constant "c" is unfounded. I'm very interested
to know about it, and so would Chaitin I guess.

It suffices for now to mention that Chaitin's information theoretic
proof of Godel's incompleteness is much more natural than number
theoretic joggling. [+] For what is done in arithmetization of logical
statements is nothing but low-level programming. One need not be a
great programmer to appreciate this simple fact, however, I suspect a
moderate amount of programming experience, (to acquire a feel of how a
program in a functional language will be translated to machine code)
should build the necessary intuition.

There are, on the other hand, interesting comments by other people.
Joe Shipman, for instance, has made sound remarks on Chaitin's work,
and Harvey Friedman and Hrant Marandjian have clarified things for
Raatikainen who does not seem to have grasped the basics of AIT.
Again, I am not surprised, in 21st century philosophers barely
understand any of 20th century mathematics.

An intelligible objection to Chaitin's "quasi-empirical mathematics"
stance is not given by Raatikainen, for he seems to be merrily lost in
Finnish forests. Nevertheless, Don Fallis raises the point that due to
the lack of compelling evidence for a quasi-empirical mathematical
program, Chaitin's statements could be seen as hanging from skyhooks,
a formidable objection that I could seriously consider.
http://www.cs.nyu.edu/pipermail/fom/2001-March/004889.html

I myself would have a hard time thinking that accepting P!=NP as an
axiom would yield great benefits in its consequences. I am skeptical
about that issue, but if Chaitin is right, we shall eventually see the
consequences.

Regards,

--
Eray Ozkural
[*] But he has taken enough social sciences courses to suspend his
arguments by means of linguistic attack. I have consequently checked
from his pretty web page, and saw that he has only degrees in
philosophy, not generally sufficient to comment on mathematical
research the way he attempts.... On a similar note, an immensely
popular philosopher, Hintikka, referred to Chaitin in his book as "a
computer scientist who claimed to have formalized randomness", and
then went on to suggest that his claims are ungrounded, etc. What a
foolishness! Chaitin doesn't merely claim things, he proves them! That
is what separates some philosopher in an ivory tower of hallucination,
from a mathematician who must construct watertight proofs, I think.
[+] And if one does not understand the information theoretic proof,
which is quite elementary using Kolmogorov complexity, one must not
bother looking further into the subject.

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