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