On Monday, September 23, 2002, at 11:34 AM, Hal Finney wrote:

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> 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 goals. In this he has succeeded. --Tim (.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