Re: Raatikainen's critique of Chaitin
From: Eray Ozkural exa (erayo_at_bilkent.edu.tr)
Date: 31 Aug 2004 04:28:07 -0700
Torkel Franzen <email@example.com> wrote in message
> firstname.lastname@example.org (Eray Ozkural exa) writes:
> > Raatikainen does *not* understand
> Instead of going on about what Raatikainen does not understand, can
> you define or explain in what sense some mathematical statements are
> true for a reason and others are not?
Yes, you are right; that's a very fair request. I think Chaitin's
statements, too, are not quite precise sometimes, at least to a
philosophical audience. He might be assuming that the reader already
thinks like him, sidestepping some of the real issues which may occur
to the readership.
Chaitin's argument is, at a coarse-grain level of representation,
1. Define the halting probability of a program, Omega.
2. Prove that Omega is random by defining random real and showing its
equivalence to Solovay randomness.
3. Show by arithmetization that Omega occurs in number theory.
4. Therefore, there are mathematical constructs in foundations of
mathematics which are random...
>From this Chaitin immediately proceeds to the conclusion that "the
randomness of these constructs show that the bits of Omega are true
for no reason". In my opinion, this neglects apparently needed
rigorous arguments to justify the philosophical conclusion.
Hence, my own philosophical argument to show a way to get to the
metaphysical claims in Chaitin's statement:
1. The halting problem's structure is random, it cannot be attributed
to mechanical causes less complex than itself (by the very definition
of randomness), in this case infinite complexity.
2. It does not seem likely that the world is more complex than Omega
3. Therefore, the bits of Omega are true for no mechanical cause
(reason) in this universe.
I have spent some effort in furnishing a persuasive argument to reach
Chaitin's realist conclusions. Nevertheless, I do not endorse this
argument myself, because I do not consider myself a realist. There may
be some philosophical loopholes in the argument for the existence of
Omega; the whole argument has a somewhat disturbing theological
flavor: we seem to be presupposing what we wanted to prove.
This might not be the only possible argument to obtain similar
conclusions. For instance, there is an alternative line of
argumentation that assumes the equivalence of *minds* (and not
necessarily of physical universe as implicit in the above argument) to
*discrete computation*. Since mathematical thought is a kind of
thought, limits of minds apply to mathematics itself. Then, the random
structure of the halting problem (i.e. which programs halt), applies
to mathematics itself: mathematics at large is irreducible (because in
the idealist sense, Omega is part of number theory), and we cannot
reason about its axiomatic truth. (For us, some mathematical
statements are true for no reason!)
This second approach is less problematic, but might still be cured by
a constructivist interpretation. [*]
Although I have read some of Chaitin's online books, I have not read
all of them. I am unaware of a detailed argument in his works similar
to the two arguments which I presented.
-- Eray Ozkural [*] I think that a constructivist interpretation may be preferable.