Re: Tegmark's TOE Cantor's Absolute Infinity
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. Hal Finney Of course, and I express this point as a footnote to my Occam's razor paper (something to the effect of remaining agnostic about whether recursively enumerable axiomatic systems are all that there is). Somewhere I speculated that these other systems require observers with infinitely powerful computational models relative to Turing machines and that these observers are of measure zero with respect to observers with the same computational power as Turing machines... Cheers A/Prof Russell Standish Director High Performance Computing Support Unit, Phone 9385 6967, 8308 3119 (mobile) UNSW SYDNEY 2052 Fax 9385 6965, 0425 253119 () Australia[EMAIL PROTECTED] Room 2075, Red Centrehttp://parallel.hpc.unsw.edu.au/rks International prefix +612, Interstate prefix 02
Re: Tegmark's TOE Cantor's Absolute Infinity
I'm not so sure that I do perceive positive integers directly. But regardless of that, I remain convinced that all properties of them that I can perceive can be written as a piece of ASCII text. The description doesn't need to be axiomatic, mind you. As I have mentioned, the Schmidhuber ensemble of descriptions is larger than the Tegmark ensemble of axiomatic systems. Cheers Hal Finney wrote: But as an example, how about the positive integers? That's a pretty simple description. Just start with 0 and keep adding 1. From what we understand of Godel's theorem, no axiom system can capture all the properties of this mathematical structure. Yet we have an intuitive understanding of the integers, which is where we came up with the axioms in the first place. Hence our understanding precedes and is more fundamental than the axioms. The axioms are the map; the integers are the territory. We shouldn't confuse them. We have a direct perception of this mathematical structure, which is why I am able to point to it for you without giving you an axiomatic description. Hal Finney A/Prof Russell Standish Director High Performance Computing Support Unit, Phone 9385 6967, 8308 3119 (mobile) UNSW SYDNEY 2052 Fax 9385 6965, 0425 253119 () Australia[EMAIL PROTECTED] Room 2075, Red Centrehttp://parallel.hpc.unsw.edu.au/rks International prefix +612, Interstate prefix 02
