On 29 Dec 2012, at 20:51, Brian Tenneson wrote:

Why not take the categories of all categories (besides that Lawyere tried that without to much success, except rediscovering Grothendieck topoi).

I'm more interested in the smallest mathematical object in which all mathematical structures are embedded but the category of all categories will do.

Except that it is too big, and eventually lawvere extract the topi from this, which model well, not the mathematical reality, but the mathematician itself.

Also, we have already discuss this, but the embedding notion does not seem the right think to study, compared to emulation, at least with the comp hypothesis.

But if you assume comp, elementary arithmetic is enough, and it is better to keep the infinities and categories into the universal machine's mind tools.

Enough for what, in what sense?

Enough for a basic ontology (and notion of existence) to explain all the different sort of existence, notably of persons, consciousness, matter appearances, etc. See my papers, as I pretend that with comp we have no choice in those matter, except for pedagogical variants and practice.

To have a single mathematical object that all mathematical structures can be embedded would give us an object that, in a sense, contains all structures. If one follows Tegmark's idea that ME=PE, then a definition for universe just might be a mathematical object (which by ME=PE is a physical object) that contains, in a sense, all mathematical objects (i.e., all physical objects).

I think that this is deeply flawed. We cannot identify the physical and the mathematical. We might try theory on the physical, or on the mental, or on the mathematical, which might suggest relation between those thing, but I doubt any non trivial theory would identify them, unless enlarging the sense of the words like mental, physical.

Isn't it simpler to assume there is only one type of existence?

It seems to me part of the data that this is not the case. My pain in a leg has a type of existence different from a quark. The game of bridge as a different type of existence than the moon material constitution. Then for machine, once we distinguish their different points of view (intuoitively like in UDA) or formally like in AUDA, we get many different sort of existence. The ontic one is the simpler ExP(x), but we have also []ExP(x), []Ex[]P(x), []<>P(x), []<>Ex[]<>P(x), etc. All this in 8 different modal logics extracted from self-reference.

What are the actual flaws of a mathematical universe?

Too big. It is a metaphor.

A physical system can be mathematically encoded by its corresponding set of world lines. This encoding is an isomorphism. A very simple example of what I mean is the nearly parabolic path taken by a projectile. The set of world lines would be some subset of R^4 or R^n if it turns out that n != 4. I am aware that indeterminacy due to Heisenberg's uncertainty principle kicks in here so we may never "know" which subset of R^n a physical system is isomorphic to but by a pigeonhole principle, the physical system must be isomorphic to some subset of R^n, several in fact.

May be. But I am driven by the mind-body problem, and what you show above is mathematical physics. With comp, by UDA, we have to extract the belief in such physical idea by ultimately explaining them in term probabilities on computations (that the result I invite you to study and criticize).

With computationalism, the coupling consciousness/physical is a phenomenon, person perceptible through numbers relations when they (the persons) bet on their relative self-consistency. This explains the appearance of the physical, without going out of the arithmetical. It works thanks to Church thesis and the closure of the comp everything (UD*, sigma_1 completeness).

How are you defining consciousness here?

I can't define it. I just hope you know what I mean. Basically something true but non provable about yourself, and, by comp, invariant for some local digital substitution.

It's not super clear to me that the cocompletion of the category of all structures C exists though since C is not a small category and thus Yoneda's lemma doesn't apply. I would have to fine-tune the argument to work in the case of the category C I have in mind.

The n-categories might be interesting, but we don't need so rich ontology. If we are machine, the cardinality of the basic TOE is absolutely undecidable from inside. Omega is enough.

Do you have an argument that proves that our minds can't transcend "inside"?

The mind can do that. Math, by diagonalization, does that, actually, even in a 3p way. But the fact that "number's mind" can do that invite us to not reify the transcendental. This is what lead to superstition and non necessarily complex ontologies.

If the cocompletion of C is the One, that which all mathematical structures can be embedded, then the parallel universe question would be a matter of logic and category theory; it would depend on how you defined "the visible universe" and "parallel" universe.

You will have to define an observer, its points of view, and to take into account its many distributions in that super-mathematical structure, but you can't do that, as you will need an even bigger structure to define and study the indeterminacy. So you will have to limit your notion of observer and use some "comp" hypothesis (an infinite variant if you want).

With comp, it is easier: you cannot really take more than arithmetic. God created the Natural Numbers, all the rest belong to the (singular and collective) number's imagination. If nature refutes this, it will still remain time to add the infinities needed. I think.

How is the arithmetical structure going to give rise to a description of reality that takes into account observer, its points of view, and its many distributions without the need to study the indeterminacy?

You might take a look on my paper(s)(°), or my posts here, as this is what I keep trying to explain here. I am not sure how you can study the relation between the first person and its possible realities without using the first person indeterminacy, which is the building block of the physical realities. The observer are the (Turing) universal numbers. Physics is given by the measure on the computations going through their state. This extends Everett on arithmetic. It leads to a many-dream view of arithmetic, and can be shown to be developed by almost all universal numbers.


(°) http://iridia.ulb.ac.be/~marchal/publications/SANE2004MARCHALAbstract.html

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