> 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
> 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
> 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.
> 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
> 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?
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