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]


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 goals. In this he has succeeded.


(.sig for Everything list background)
Corralitos, CA. Born in 1951. Retired from Intel in 1986.
Current main interest: category and topos theory, math, quantum 
reality, cosmology.
Background: physics, Intel, crypto, Cypherpunks

--Tim May
"Dogs can't conceive of a group of cats without an alpha cat." --David 
Honig, on the Cypherpunks list, 2001-11

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