Re: Raatikainen's critique of Chaitin
From: Eray Ozkural exa (erayo_at_bilkent.edu.tr)
Date: 09/05/04
- Next message: Isaac: "Radius of convergence, really confused"
- Previous message: Isaac: "Re: Contour integral question"
- In reply to: Eray Ozkural exa: "Re: Raatikainen's critique of Chaitin"
- Next in thread: Aatu Koskensilta: "Re: Raatikainen's critique of Chaitin"
- Messages sorted by: [ date ] [ thread ]
Date: 5 Sep 2004 10:28:57 -0700
erayo@bilkent.edu.tr (Eray Ozkural exa) wrote in message news:<fa69ae35.0409031752.648eb4e1@posting.google.com>...
> Why do you think, for instance, do some people who read Chaitin's work
> and understood it say that Raatikainen's argument about Chaitin's
> "true for no reason" is not informative? Or why do some people like me
> say that we can put an enormous amount of mathematical knowledge in "n
> = n" (for certain n) while this information is not contained in
> "forall x (x=x)", therefore Raatikainen was wrong, those are bit
> strings in the theory?... Can you see how trivial that is? These are
> only two of several flaws in that paper, which I have discussed on
> three separate threads.
The above is just a summary, I've just explained the situation in some
more detail to Daryl.
It boils down to this.
Raatikainen's criticism about strength of formal systems is not
completely wrong, but his formal argument is wrong. If he had not
tried to give formal arguments, his paper would have been much better.
You see once you are formal, you are not allowed any mistakes,
everything could crash with a single mistake.
A better approach would be to outline the shortcomings of Chaitin's
formalization, to show how loose the relation between the complexity
of a theorem and the complexity of the axiom schema of the theory can
get. (And not to try to discredit the mathematical theory)
Here, I briefly indicate where Raatikainen has gone wrong with respect
to his criticism of Chaitin's views on theorem proving power:
1. He is examining Theorem LB, which is the weakest incompleteness
theorem in Chaitin's monograph.
2. He does not seem to be aware of the invariance theorem, or what
asymptotic means.
3. He does not seem to understand that computers are supposed to be
physical machines.
4. He does not seem to understand that the inference rule in
formalization of axiomatic theory in Chaitin's work can be any
truth-preserving system of rules, e.g. it does not have to be
predicate calculus. To one set of rules quantifiers, connectives and
logical symbols have a meaning, and individual numbers have *no*
meaning. To another set of rules, logical symbols have absolutely *no*
meaning, while numbers have a meaning.
(The hardest to understand is 4.)
Almost all of these mistakes are *conceptual*, hence why I think
Raatikainen has not shown the proper effort to work through Chaitin's
proofs in the first place. In fact 1. and 2. suggest us that he is on
the wrong side of the planet, looking for New York City.
Even more depressing are my recent observations on some of the
loopholes in Chaitin's statements. It was sufficient that Raatikainen
had demonstrated the following trivial fact philosophically (without
diving into indexings and all that!):
Mere complexity does not necessarily entail being informative about
the world.
It's really so simple that I'm cracking as I write this. Let's have
this infinitesimal pin and pick a random number between 0 and 1. The
probability that the random real we picked is Omega is almost 0! (But
not 0!) That is, if we wanted to solve the halting problem, having a
truly truly random, very real number would be of absolutely no use!
Although the complexity of a random real is just as much as the number
Omega, it normally contains no information about the halting problem.
Unfortunately!
This is case 2 that Aatu was talking of, and I wholeheartedly agree
with him on this issue. I think this philosophical demonstration was
sufficient to show that Chaitin's theory does not give the ultimate
characterization of theorem-proving strength. (However, without giving
a better mechanical alternative, it is useless to show the
limitations, I believe). This can be said, because Chaitin does not
talk about lower bounds in his theory, while his informal statements
imply the existence of lower bounds.
Discussion of case 1 is philosophically more interesting, e.g. what
are the hardest kinds of mathematical problems we can imagine, what is
a golden standard of theorem strength or interestingness, of axiom
interestingness, of mathematical truth? But this post is not the place
for it, I've tried to explain that some of the relativity described in
4. may have something to do with the solution, in other posts. I do
think, however, that we might need to go above *syntax* to fully
capture how and when mathematical truth manifests itself as
irreducible axioms. IOW, we need to explain how humans can accomplish
it, we need to explain how this "second order data" (as Godel
suggested) is *ever* processed by mathematicians, to come up with,
say, new axioms!
Regards,
-- Eray Ozkural PS: "Humans are not computers, humans do it with divine methods. They download the data directly from God." is no explanation in my opinion! Please!
- Next message: Isaac: "Radius of convergence, really confused"
- Previous message: Isaac: "Re: Contour integral question"
- In reply to: Eray Ozkural exa: "Re: Raatikainen's critique of Chaitin"
- Next in thread: Aatu Koskensilta: "Re: Raatikainen's critique of Chaitin"
- Messages sorted by: [ date ] [ thread ]
Relevant Pages
|
