Re: Raatikainen's critique of Chaitin

From: Eray Ozkural exa (erayo_at_bilkent.edu.tr)
Date: 09/04/04 

Date: 4 Sep 2004 13:09:06 -0700

Torkel Franzen <torkel@sm.luth.se> wrote in message news:<vcbllfq6vtd.fsf@beta19.sm.ltu.se>...
> erayo@bilkent.edu.tr (Eray Ozkural exa) writes:
>
> > Did Godel show that randomness had
> > something to do with incompleteness?
>
> Indeed not! But what in the particular example of incompleteness
> presented by Chaitin "shakes the foundations of mathematics" in a way
> that other examples of incompleteness do not?

I didn't say it "shakes the foundations of mathematics". Neither did
Chaitin. But he says that we must take incompleteness more seriously:

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Just look at Bertrand Russell's three-volume magnum opus (with
Whitehead) Principia Mathematica, which I keep in my office at IBM.
One entire formula-filled volume to prove that 1 + 1 = 2! And by
modern standards, the formal axiomatic system used there is
inadequate; it's not formal enough! No wonder that Poincaré decried
this as a kind of madness!

But I think that it was an interesting philosophical/intellectual
exercise to refute this madness as convincingly as possible; somehow I
ended up spending my life on that!

Of course, formalism fits the 20th century zeitgeist so well:
Everything is meaningless, technical papers should never discuss
ideas, only present the facts! A rule that I've done my best to
ignore! As Vladimir Tasic says in his book Mathematics and the Roots
of Postmodern Thought, a great deal of 20th century philosophy seems
enamored with formalism and then senses that Gödel has pulled the rug
out from under it, and therefore truth is relative. --- Or, as he puts
it, it could have happened this way. --- Tasic presents 20th century
thought as, in effect, a dialogue between Hilbert and Poincaré... But
truth is relative is not the correct conclusion. The correct
conclusion is that Hilbert was wrong and that Poincaré was right:
intuition cannot be eliminated from mathematics, or from human thought
in general. And it's not all the same, all intuitions are not equally
valid. Truth is not reinvented by each culture or each generation,
it's not merely a question of fashion.

Let me repeat: formal axiomatic systems are a failure! Theorem proving
algorithms do not work. One can publish papers about them, but they
only prove trivial theorems. And in the case-histories in this book,
we've seen that the essence of math resides in its creativity, in
imagining new concepts, in changing viewpoints, not in mindlessly and
mechanically grinding away deducing all the possible consequences of a
fixed set of rules and ideas.

Similarly, proving correctness of software using formal methods is
hopeless. Debugging is done experimentally, by trial and error. And
cautious managers insist on running a new system in parallel with the
old one until they believe that the new system works.

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I think he may have wiped away a good deal of nonsense.

> > Did he show the existence of a
> > maximally unknowable number?
>
> What, if anything, do you mean by "maximally unknowable"?

Numbers like Omega. We can talk about it, but we can never know it. Of
course, the existence of random reals was shown before, but Omega is
special, because it is not only random but also contains an infinite
amount of mathematical information.

Allow me to make it crystal clear for those who cannot see it. Here is
the argument, in very precise philosophical terms.
1. Omega acts as an oracle for the halting problem, the program to
solve any instance of the halting problem for a program p has constant
program-size complexity, given Omega_|p|.
2. Omega is a random real, its bits are completely independent, ie.
the outcomes of a hypothetical fair coin. (Which could not physically
exist by the way!)
3. Therefore, we cannot reason about this number which packs all the
information about the halting problem: it is beyond our cognitive
capacity.

Hence the adjective "unknowable". (If you need more explanation here,
I can point to you some online courses on epistemology)

The extra claim is that
4. There are no numbers which can pack more mathematical information
than Omega.

And hence the adverb "maximally", but this might actually be relative
to a theory of mind, or a theory of physics. I am not very sure of it,
actually.

Regards, 

--
Eray Ozkural

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