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

Bruno Marchal wrote:Tegmark argues that reality is a mathematical structure and statesthat an open problem is finding a mathematical structure which isisomorphic to reality. This might or might not be clear: themathematical structure with the property that all mathematicalstructures can be embedded within it is precisely the mathematicalstructure we are looking for.The problem is in defining "embedded". I am not sure it makes settheoretical 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 arelooking for is just the (mathematical) universal machine. In whichcase Robinson arithmetic (a tiny fragment of arithmetical truth, onwhich both platonist and non platonist (intuitionist) is enough.Indeed, I argue with comp that Robinson arithmetic, or any firstorder specification of a (Turing) universal theory is enough toderive the appearance of quanta and qualia.Actually, I'm using what's called NF with urelements (NFU) whichaccording 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 toreinvent the wheel.The last sentence in the quote excites me: The leap from mathematicsto things such as quanta and qualia is something I haven't reallyunderstood.

`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 largepart of Quantum Mechanics, and then group or category theoreticconsiderations (and knot theory) might explain the 'illusions' oftime, space, particle, and (symmetrical) hamiltonians, and whyindeed physical reality should appear as an indeterminate state ofa physical vacuum. But the logic-math problems remaining are noteasy to solve. That is normal in a such top down, mind-body problemdriven, approach to physics (and psychology/theology/biology).Interesting!

`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?`

Bruno

`(****) Other references have been archived by Gunther in the`

`everything-list archive:`

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