On Monday, September 23, 2002, at 11:34 AM, Hal Finney wrote:
I have gone back to Tegmark's paper, which is discussed informally
at http://www.hep.upenn.edu/~max/toe.html and linked from
http://arXiv.org/abs/gr-qc/9704009.
I see that Russell is right, and that Tegmark does identify
mathematical
structures with formal systems. His chart at the first link above
shows
Formal Systems as the foundation for all mathematical structures.
And the discussion in his paper is entirely in terms of formal systems
and their properties. He does not seem to consider the implications if
any of Godel's theorem.
I still think it is an interesting question whether this is the only
possible perspective, or whether one could meaningfully think of an
ensemble theory built on mathematical structures considered in a more
intuitionist and Platonic model, where they have existence that is more
fundamental than what we capture in our axioms. Even if this is not
what Tegmark had in mind, it is an alternative ensemble theory that is
worth considering.
I think this is exactly so, that Reality nearly certainly has more that
what we have captured (or perhaps can _ever_ capture) in our axioms.
Godel's results can be recast in algorithmic information theory terms,
as Greg Chaitin has done, and has Rudy Rucker has admirably explained
in Mind Tools.
For example, a few excerpts (out of a full chapter, so my excerpts
cannot do it justice):
It turns out that there's a real sense in which our logic cannot reach
out to anything more complicated than what it starts with. Logic can't
tell us anything interesting about objects that are much more complex
than the axioms we start with. [p. 286]
Now we may reasonably suppose that the world around us really does
contain phenomena that code up bit strings of complexity greater than
three billion [Tim note: Rucker had earlier estimated that the
complexity of all of modern math and science is reasonably explained
and axiomatized in a thousand or so books, or about 3 billion bits,
give or take]. Chaitin's theorem tells us that that our scientific
theories have very little to say about these phenomena. On the one
hand, our science cannot find a manageably short explanation for a
three-billion-bit complex phenomenon. On the other hand, our science
cannot definitively prove that such a phenomenon _doesn't_ appear to
have a short, magical explanation. [p. 289]
Discussion:
It seems plausible that we ourselves will eventually have a knowledge
base of more than 3 billion bits, perhaps hundreds of billions of bits
(I expect diminishing returns, in terms of basic theories, hence an
asymptotic approach to some number...just my hunch). Some
Jupiter-sized brain may have a much richer understanding of the
cosmos and may be able to understand and prove theorems about much more
complicated aspects of reality.
It seems likely that the current limits on our ability to axiomatize
mathematics are not actual limits on the actual universe!
(Unless one adopts a weird Distress-like model that future
hyperintelligent beings will bring more and more of the mathematical
structure of the universe into existence merely through their increased
ability to axiomatize.)
Personally, for what's it's worth, I vacillate/oscillate between a
Platonist point of view that Reality and the Multiverse/Universe/Cosmos
actually has some existence in the sense that there appears to be an
objective reality which we explore and discover things about and a
Constructivist/Intuitionist point of view that only things we can
actually construct with atoms and programs have meaningful existence.
(I believe in the continuum in the axiomatic sense, in the sense of the
reals as Dedekind cuts, in the ideas of Cauchy sequences, limits, and
open sets, but I don't necessarily always believe that in any
existential sense there are infinities. The real universe does not
appear to have an infinite number of anything, except via abstraction.)
The two views--Platonism vs. Constructivism--are not necessarily
irreconcilable, though. Paul Taylor's book, The Foundations of
Mathematics, discussed the reconciliation.
Lastly, the Schmidhuber approach, as I understand it, is closer to the
Chaitin/Rucker point above than the Tegmark approach is. By considering
all outputs of UTMs as string complexity increases, one is including
ever-richer axiom systems. (Chaitin talks about Omega, which Rucker
also discusses.)
I don't want to diss Tegmark, but as I said when I first started
posting to this list, Tegmark seems to have a fairly simple view of
mathematics. His famous chart showing the branches of mathematics and
then his hypothesis that perhaps the multiverse has variants of all of
of the axioms of these branches, isn't terribly useful except as a
stimulating idea (hence this list, of course). Naturally Tegmark is not
claiming his idea is _the_ theory, so stimulation is presumably one of
his
