Dan Fabulich writes:
> Chaitin's paper is at the same time much better and much worse than I
> was expecting.
I think you are talking about
http://www.cs.auckland.ac.nz/CDMTCS/chaitin/vienna.html?
> Chaitin's paper relies heavily on an equation he's cooked up which
> takes parameters.
I felt that Chaitin's larger point was that axioms have a certain
information content, and they simply can't prove theorems that have a
larger information content. As he put it, with five pounds of axioms
you can't prove a ten pound theorem. This suggests that mathematical
truth may be much more complex than we imagined.
Prior to Godel it was suspected that with the right axioms you could prove
everything that was true. We now know this to be false. Chaitin has
taken Godel's result and made it quantitative.
> The relationship to epistemology, though, looks rather weak to me.
I'm not sure that this has much relevance outside of mathematics (or
perhaps meta-mathematics). It challenges the fundamental axiomatic
approach of mathematics and logic which goes back to the ancient Greeks.
However Chaitin's suggestion that mathematics should become more of an
engineering discipline has not been well received. If mathematicians
had wanted to be engineers they wouldn't have gone into math.
We use the phrase "mathematical certainty" as the highest degree
of assurance possible. Chaitin would challenge that by opening up
mathematics to areas where no proofs are possible. But math without
proofs is arguably something different than math.
Chaitin has some hand-waving about possible connections between Godel's
results and the uncertainty principle, or between his discovery of the
essentially random nature of mathematics and the randomness in QM.
However, the paradoxes of QM take on an entirely different hue when
considered in light of many-worlds interpretations, where there is
no randomness.
Interestingly, Chaitin's theories can take on a different role there;
Algorithmic Information Theory plays a key part in many-universe models
that go beyond QM. Wei Dai has a mailing list for exploring the notion
that all universes exist, and that the "measure" or contribution of
a universe is proportional to the algorithmic information content of
that universe, by Chaitin's measure (others have also developed similar
metrics).
There is an interesting paradox here; you'd think a universe starting
with a certain information content could not generate anything which had
a larger information content (a la Chaitin's results). However, I don't
think this is true; our own universe may have had almost zero information
content but it generated us. My explanation is that to single out a
human being in the multiverse takes a huge amount of information, in the
form of its "address". The actual rule seems to be that the information
in a subsystem is limited by the sum of the initial information in the
universe, plus the information needed to locate and specify the extent
of the subregion. I wonder what if any connections this might have to
Chaitin's limitations.
Hal
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