Re: Panu Raatikainen's review of two of Chaitin's books.

From: Jesse F. Hughes (jesse_at_phiwumbda.org)
Date: 05/27/04

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    Date: Thu, 27 May 2004 20:14:15 +0200

    erayo@bilkent.edu.tr (Eray Ozkural exa) writes:

    > Torkel Franzen <torkel@sm.luth.se> wrote in message news:<vcbhdu29q5e.fsf@beta19.sm.ltu.se>...
    >> jesse@phiwumbda.org (Jesse F. Hughes) writes:
    >>
    >> > No, no, no. That the value of Omega is such and such can certainly be
    >> > a mathematical fact. I don't dispute that. It is that this fact is
    >> > true "for no reason" that is dubious. At the least it requires
    >> > explanation and argument. (Note: I'm not saying that all mathematical
    >> > facts are true "for some reason" either. I haven't a clue what either
    >> > claim would mean.)
    >>
    >> As emphasized by Raatikainen, to claim that statements about the
    >> bits of Omega are "true by accident" or that there is "no reason" for
    >> their truth is to claim that statements of the form "the Turing
    >> machine T eventually halts" are true "by accident" or "for no reason".
    >> It is entirely obscure what this claim is supposed to mean or on what
    >> it is based.
    >
    > Raatikainen's analysis seems wrong because the above substitution
    > takes the literal meaning of Chaitin's expression, while it should
    > have read, in my opinion:
    > "The _structure_ of the set of halting problems is the way it is for
    > no reason."
    >
    > rather than
    >
    > "The individual halting problems are true for no reason (other than
    > by definition)."

    And that helps how? What does it mean for a mathematical truth to be
    "true for no reason"?

    > My previous criticism was the way Raatikainen put it. The above
    > analytic argument of yours seems much clearer, and makes further
    > analysis possible.
    >
    > However, I must admit that Chaitin here is mistaken in his expression.
    > After reading his work, the impression I got was not that
    > "(individual) bits of Omega are true for no reason", rather that [I'm
    > writing in upper case to highlight it]
    >
    > ALL MATHEMATICAL KNOWLEDGE CAN BE EXPRESSED WITH A REAL NUMBER
    > CALLED OMEGA, AND IT IS RANDOM, THEREFORE THE WHOLE OF MATHEMATICS IS
    > TRUE FOR NO REASON, TO PUT IT INTUITIVELY.

    Gosh. That's much worse than anything Chaitin has written. Well, at
    least anything that I've read.

    All of mathematical knowledge can be expressed by Omega? In what
    sense is this true?

    What does this technical term "random" have to do with being true for
    no reason? How is that intuitive? How *could* it be intuitive, when
    Chaitin's notion of randomness seems to have nothing much to do with
    the intuitive notion of randomness (and when neither seem to have
    anything much to do with "reasons" for mathematical truths).

    > Needless to say, this statement presupposes usual discrete
    > assumptions, i.e. the universe is a discrete computer. That is, Omega
    > is a "god number" that prescribes all logical existence.

    Hoo-doggies. That "Omega prescribes all logical existence" is a
    *usual* assumption?

    > Whether the assumptions of digital philosophy are true is an
    > independent discussion, and possibly beyond the scope of comp.theory
    > and sci.math.

    I think that Chaitin may offer you a tidy reward if you'll agree not
    to publicly defend his claims any more. 

    -- 
But in our enthusiasm, we could not resist a radical overhaul of the
system, in which all of its major weaknesses have been exposed,
analyzed, and replaced with new weaknesses.
   -- Bruce Leverett (presumably with apologies to Ambrose Bierce)
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