On 05 Aug 2010, at 01:18, Brian Tenneson wrote:

Hmm... Lawvere has tried to build an all encompassing universalmathematical structure, but he failed. It was an interesting failureas he discovered the notion of topos, (discovered also independentlyby Groethendieck) which is more a mathematical mathematician than amathematical universe.Also Tegmark is not aware that Digital Mechanism entails the nonlocality, the indeterminacy and the non cloning of matter, and thatDM makes the physical into a person-modality due to the presence ofthe mathematician in the arithmetical reality.Quanta are special case of first person plural sharable qualia. -------------I'm not looking for a truly all-encompassing mathematicalstructure. What I'm looking for is a mathematical structure inwhich all mathematical structures can be embedded. By mathematicalstructure, I mean there is a symbol set S consisting of constantsymbols, relation symbols, and function symbols, and the pairing ofa set with a list of rules that interpret the symbols. In Tegmark'spapers on "ultimate ensemble TOE" and "the mathematical universe,"he refers to what I call a mathematical structure as a "formalsystem" (and also mathematical structure).The structure I'm looking for wouldn't encompass anything that isn'ta mathematical structure, like a category with no objects/elements.

`You may encounter a problem with the notion of 1-person, and`

`'material' bodies.`

Tegmark argues that reality is a mathematical structure. What'scute about his argument is that while invoking the concept of a TOE,his argument is independent of what that TOE might be. He defines aTOE to be a complete description of reality. Whether or not thiscan be expressed in a finite string is an open problem as far as Iknow. (I doubt it can.) He argues that a complete description ofreality must be expressible in a form that has no human baggage andI would add to that is something that exists independent of humansin the sense that while the symbols used to provide that completedescription will depend on humans, what is pointed to by the symbolsis not.

`Computationalism entails something very near such view indeed. It`

`entails also that if such structure make sense, then its cardinality`

`is unknowable by the self-aware beings that could be generated inside.`

`The statement that the cardinality of the mathematical universe is`

`countable or not is absolutely undecidable, from 'inside'.`

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 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.`

I am confident that I have found such a structure but only over afixed symbol set; I need such a structure to be inclusive of allsymbol sets so as to cast away the need to refer to a symbol set.

This again follows from Church thesis, for the 'computationalist' TOE.

The technique I used was to use NFU, new foundations set theory withurelements--which is known to be a consistent set theory, to firstfind the set of all S-structures.

All right, then.

Then I take what I believe is called the reduced product of all S-structures. Then I show that all S-structures can be embeddedwithin the reduced product of all S-structures. Admittedly, thereis nothing at all deep about this; none of my arguments are deeperthan typical homework problems in a math logic course.

That may be already a lot for non mathematical logicians ...

My next move is to find justification for the existence of a mathstructure with the important property that all structures can beembedded within it --independent of the symbol set-- and thuseliminating the need to refer to it.One thing I wonder is how to define all your notions such as"mathematician," "n-brains," "n-minds," and "digital mechanism" interms of mathematical structures.

`This is done. Everything is defined in term of number and number`

`relation. But it is not asked that the relation is arithmeticaly`

`definable. For example, the ONE of Plotinus is Arithmetical truth, and`

`can be represented by the set of Gödel numbers of true arithmetical`

`proposition. Of course the "internal machine will have to build richer`

`epistemological tools (like analytical tools, set theoretical tools)`

`to talk about all this. The internal epistemology is richer than the`

`needed external ontology. It is made consistent by some use of Skolem`

`paradox.`

I'm particularly interested in defining something that modelsawareness and using it to find self-aware structures such as"mathematicians."

`Mathematician are just Löbian machine/theory/number. They are`

`(recursively) equivalent with Robinson Arithmetic + the induction`

`axiom, i.e. Peano arithmetic. They can be characterized by being`

`universal and knowing it. They can dream. More exactly, they cannot`

`not dream, and physical realities appears when collection of machine`

`glue well the dreams. This needs natural coherence property, and it is`

`suggested that those coherence properties are provided by the`

`intensional variant of the logic of self-reference (the Z and X logics).`

`This solves conceptually, I think, the mind body problem, in a way`

`compatible with the computationalist breakdown of the identity thesis.`

`I hope you see that to find (truing-emulable) self-aware structure is`

`not enough, given the inconsistency of the identity thesis with`

`mechanism (cf UDA). You have to recover the physical reality from the`

`breakdown of the identity thesis: physics has to be recovered by all`

`the infinite computations leading to the state of that self-aware`

`structure. This makes, strictly speaking, the physical reality out of`

`existence. Like in Plotinus theory of matter. Both God and Matter are`

`out of the intelligible Whole (the Noûs).`

`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).`

Bruno http://iridia.ulb.ac.be/~marchal/ -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-l...@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.