On 07 Aug 2010, at 00:05, Brian Tenneson wrote:

Bruno Marchal wrote:
Tegmark argues that reality is a mathematical structure and states that an open problem is finding a mathematical structure which is isomorphic to reality. This might or might not be clear: the mathematical structure with the property that all mathematical structures can be embedded within it is precisely the mathematical structure we are looking for.

The problem is in defining "embedded". I am not sure it makes set theoretical sense, unless you believe in Quine's New foundation (NF). I am neutral on the consistency of NF. With a large sense of "embedded" I may argue that the mathematical structure you are looking for is just the (mathematical) universal machine. In which case Robinson arithmetic (a tiny fragment of arithmetical truth, on which both platonist and non platonist (intuitionist) is enough. Indeed, I argue with comp that Robinson arithmetic, or any first order specification of a (Turing) universal theory is enough to derive the appearance of quanta and qualia.
Actually, I'm using what's called NF with urelements (NFU) which according to what I've read is consistent. http://plato.stanford.edu/archives/sum2009/entries/quine-nf/ (section 7. Coda).

I know my late colleague Boffa proved the consistency of variant of NF, like Crabbe (there is belgium school on NF!). But can we have a universal set in those variants? Don't we lose extensionnality with NFU? I should revise my NF! I think that I remember you are using NF motivated by such a universal set, am I right?

Where would I go about finding out a survey of concepts including "universal machine"? Are they known to exist?

Yes, and 'real' computers provide concrete examples. They are the pillar of recursion theory and theoretical computer science. Of course, mathematically we can debate on their best definition. Martin Davis(*) gave the "old" definition (similar to Turing, Post, ...) in 1956, and corrected it in a 1957 paper(*). Usually recursion theorist use the new one, because it leads to a mathematically clean notion of recursive equivalence (see the book by Rogers(**)). But in the context of applying this to biology, or to theoretical artificial intelligence, or to "machine theology", the old, larger definition, is better, because those applications are more intensional in nature (coding play a role). The old definition is also equivalent with Emil Post notion of creative set (a recursively enumerable set with a productive complement, and a set is productive if for all Wi included in it, you can find effectively a counterexample, that is a k in the set but not in Wi (Wi is the domain of Phi_i, the ith partial recursive function in some universal programming language). The notion of creative set is the set-theoretical notion of "universal machine". This is not obvious and has been proved by some people like John Myhill. The set of (gödel numbers) of provable sentences of a sigma_1 complete theory is creative, for example, and you can use that for making them emulating any universal machine. The best book is the book by Rogers(**), but Cutland wrote a nice introduction(***).

DAVIS, M., 1956, A note on universal Turing machines, Automata Studies, Annals of
mathematics studies, no 34, pp. 167-175, Princeton, N.Y.
DAVIS, M., 1957, The definition of universal Turing machines, Proceedings of the
American Mathematical Society, Vol 8, pp. 1125-1126.
ROGERS H.,1967, Theory of Recursive Functions and Effective Computability, McGraw-
Hill, 1967. (2ed, MIT Press, Cambridge, Massachusetts 1987).
CUTLAND N. J., 1980, Computability An introduction to recursive function theory,
Cambridge University Press.

How are they defined? It would be much easier if I didn't have to reinvent the wheel. The last sentence in the quote excites me: The leap from mathematics to things such as quanta and qualia is something I haven't really understood.

Well, alas, for almost precise historical reasons(:), you will not find many logicians interested in qualia. Thanks to quantum computer science, slowly but surely a growing number of logicians begin to see the interest of learning quantum mechanics. It is mainly my own work which shows that quanta can be a particular case of "sharable" qualia. I obtained this by using the work in (arithmetical, set-theoretical, analytical) self-reference logics (build on Gödel and Löb's results).

(:) for historical reasons, logicians have fought to be recognized as pure mathematicians, and most really dislike we remind them of the theo/philosophical origin/motivation of logic.

Digital mechanism (the tiny arithmetic TOE) entails already a large part of Quantum Mechanics, and then group or category theoretic considerations (and knot theory) might explain the 'illusions' of time, space, particle, and (symmetrical) hamiltonians, and why indeed physical reality should appear as an indeterminate state of a physical vacuum. But the logic-math problems remaining are not easy to solve. That is normal in a such top down, mind-body problem driven, approach to physics (and psychology/theology/biology).


Thanks. The problem, or methodological difficulty, is that it transforms a problem which has still no interest for most scientists (the mind-body problem) into a problem in mathematical logic (virtually known by nobody, except logicians). And then it put a doubt on the actual paradigmatic dogma: physicalism. We have to backtrack on Plato and Plotinus to see scientist approaching with a cold head deep issues (like what is life, consciousness, etc.). After the closure of the Platonist academy of Plato in Athene (in 523), the subject is still, for many people (both in science and philosophy, and religion) rather taboo today. Bah ... that's history ;)

I wish you progress on your NF approach, but keep in mind that if we assume the digital mechanist hypothesis, it is hard not to take advantage of the existence of the universal machine/numbers/set, in the sense made general by Church thesis. See the book above(****). What is cute with NF is the possibility of a universal object, as an extensional whole, in the universe, like if God could come at dinner tonight after all. But this is what makes me doubt of the consistency of NF. Do you think that NFU can tolerate such universal object?


(****) Other references have been archived by Gunther in the everything-list archive:


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