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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