Hi Bruno Marchal and Brian, 

"Bigness" can only limit physical entities (those extended in space), 
but is irrelevant with regard to nonphysical or mental entities,  
as these are not extended in space. 

[Roger Clough], [rclo...@verizon.net] 
"Forever is a long time, especially near the end." - Woody Allen 
----- Receiving the following content -----  
From: Bruno Marchal  
Receiver: everything-list  
Time: 2012-12-30, 08:57:29 
Subject: Re: Ten top-of-my-head arguments against multiverses 

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 

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 

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 

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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