On 12/29/2012 2:51 PM, 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.

Hi Brian,

Check out this proposed structure: a compressed PS file: boole.stanford.edu/pub/*gamut*.ps.gz or pdf: http://boole.stanford.edu/pub/gamut.pdf


    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?



    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? What are the actual flaws of a mathematical universe? 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.

Is it possible that this isomorphism is one example of a more general relation?



    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?


    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"?

ISTM, that to transcend from 'inside' would be to contradict Godel's incompleteness theorems, no?




    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?
--

I think that Bruno is assuming an ensemble or collection of possible encodings within the relations between numbers (or equivalent) to account for every possible description and thus would include any observer, its points of view and its many distributions. All of it exists a priori in Platonia. No?

--
Onward!

Stephen

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