Re: Which mathematical structure -is- the universe in Physics?

[Brian Tenneson]

Colin Hales wrote:
> Hi Brian,
> I was wondering if you could connect (in the paper) the maths with our 
> universe? As an example. What set operations or structures correspond to 
> the standard particle model entities, what constitutes a chemical 
> reaction or energy, what space is made of... that kind of thing. Maybe 
> this is supposed to be obvious... if it is then sorry... but you've lost 
> (as an audience) the entire world above mathematical 
> physics...especially biofolks.
>   
I think this is what usual theoretical physics is trying to do.  As an 
example, Torgny mentioned R^4 as being a relevant structure in General 
Relativity.  In string theory, the relevant structure is, as far as I've 
read in the lay literature, some 11 (give or take) dimensional 
manifold.  As such, connecting the ultimate context structure I am 
working towards to specific structures that represent things like 
particle interaction would constitute a complete theory of physics and, 
therefore, I myself am unable to see how this would be done.  It perhaps 
can be done but I lack the knowledge to do so.

Perhaps one thing to keep in mind is that this is a step towards a 
mathematical representation of the so called -level 4- multiverse, by 
which I'm referencing material here:
http://space.mit.edu/home/tegmark/multiverse.html

In Tegmark's "ultimate ensemble" paper, there is a diagram of physics 
and maths structures, part of which is here:
http://space.mit.edu/home/tegmark/toe.gif
All structures, including those in the top row, which are the ones I 
think you're asking about Colin, would have the property of being 
elementarily embeddable within the ultimate structure I'm 
investigating.  (Keep in mind the deficency I mentioned in my previous 
post.)  Roughly speaking, to quote a wiki article, "In model theory 
, an *elementary embedding* 
is a special case of an embedding 
 that preserves all 
first-order formulas."

In short, the sub-structures, so to speak, of the ultimate structure I 
am working towards that are relevant to Quantum Field Theory or General 
Relativity (such as R^4) are covered in other texts.  This paper I am 
working on is to provide an answer to the question which is the subject 
of this thread, raised my Tegmark.  I'm afraid I don't know enough about 
mathematical physics to be more explicit.




> I am a quintessentially visual/spatial thinker.. math does not speak 
> very well to me unless I can 'see' the operations happening. in my mind. 
> I don;t manipulate symbols. I manipulate 'stuff' and then retrofit symbols.
>
> I would also like to see how an observer with qualia might be 
> constructed of it. In other words...how a universe thus constructed 
> might create its own scientist describing it in the way you doHaving 
> looked at the paper I hold some hope that it might contain a formalism I 
> can use to construct the set theoretic basis of my own model... it might 
> be within yoursmaybe... not sure.
>
>   
This is an excellent line of questioning and one I have high hopes to 
one day seeing answered.  In Tegmark's first of two papers along the 
lines of a Mathematical Universe, he mentions what he calls Self Aware 
Structures (SAS's).  I have spent a lot of time wondering what type of 
mathematical structures would have self-awareness.  Two candidates that 
might be just fumbling in the dark are these:
David Wolpert of NASA has written some interesting articles on what he 
calls devices.  These devices are mathematical models of scientists plus 
investigative tools of scientists.  In this mathematical device (not 
completely unlike a Turing machine), it starts with a question and ends 
with an answer; his papers form a theory of how devices operate.  One of 
his papers is entitled "the physical limits of inference."  Anyway, he 
talks at some point about self aware devices, and my understanding is 
that these devices X are ones who correctly answer the question "is X a 
device?"  That is at least some form of self awareness.  For a 
pseudo-second example of mathematical self awareness, I was thinking of 
self-referential first order logical formulas that, in essence, say "I 
have property P," but let P be the property "X is a 1st order formula" 
so these special self-referential 1st order formulas would essentially 
be equivalent to "I am this 1st order formula."  To simplify, that is 
like the sentence "I am this sentence."  The open question is what is 
the nature of SAS's that corresponds to human self-awareness. I think 
constructing an observer with qualia mathematically would be a most 
excellent step and a necessary one to solve that open problem raised in 
Tegmark's first MUH paper.

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Re: Which mathematical structure -is- the universe in Physics?

[Colin Hales]

Hi Brian,
I was wondering if you could connect (in the paper) the maths with our 
universe? As an example. What set operations or structures correspond to 
the standard particle model entities, what constitutes a chemical 
reaction or energy, what space is made of... that kind of thing. Maybe 
this is supposed to be obvious... if it is then sorry... but you've lost 
(as an audience) the entire world above mathematical 
physics...especially biofolks.

I am a quintessentially visual/spatial thinker.. math does not speak 
very well to me unless I can 'see' the operations happening. in my mind. 
I don;t manipulate symbols. I manipulate 'stuff' and then retrofit symbols.

I would also like to see how an observer with qualia might be 
constructed of it. In other words...how a universe thus constructed 
might create its own scientist describing it in the way you doHaving 
looked at the paper I hold some hope that it might contain a formalism I 
can use to construct the set theoretic basis of my own model... it might 
be within yoursmaybe... not sure.

cheers
colin


Brian Tenneson wrote:
> An open problem raised in Tegmark's "ensemble TOE" paper (published in 
> '98, if I recall) is to answer the question that is the subject of this 
> thread.
>
> I believe recent work I have done has the potential to be a step towards 
> that answer.
>
> First, a link to the very rough draft (please forgive formatting 
> errors), and second what I think the deficiency is with this document:
>
> http://www.universalsight.org/math/9-26-08/01-03-structure_of_all_structures.pdf
>
> Abstract:
> In this document, the author presents a structure with the
> property that all structures are elementarily embeddable within it. One 
> essen-
> tial tool is a version of New Foundations set theory, ?first developed 
> by Quine,
> as presented by Holmes in [1]. The motivation is given by The Mathematical
> Universe article by Max Tegmark, [2], in which it is hypothesized that 
> physical
> existence is mathematical existence and, consequently, it is 
> hypothesized that
> the structure with the aforementioned property could be central in the Math-
> ematical Universe Hypothesis as being at least keenly connected to the 
> literal
> universe, if not literally being the universe.
> The author assumes some knowledge of mathematical logic such as, for ex-
> ample, the inductive de?finition of a fi?rst-order well-formed formula.
>
>
> The intended audience is primarily Max Tegmark, honestly, but more 
> generally, any physicist interested in Tegmark's self-proclaimed 
> "bananas articles" like the MUH paper, and who have already been exposed 
> to the basics of mathematical logic.
>
> Prior to drafting this document, I contacted Prof. Tegmark regarding the 
> core ideas in the draft. I described what I was attempting and if I 
> recall I sent him the abstract. As I will describe shortly, this is 
> incomplete, so I didn't send him this pdf yet. I hope he wouldn't mind 
> my inclusion of his response, which I think many here might find highly 
> debatable (and worthy of discussion), sent by email:
>
> 
> It sounds to me from what you're saying that A would be the Level IV 
> multiverse, i.e., all of physical reality.
> 
>
> Now for the deficiency I see with my document. -If- there aren't any 
> other errors, then something wrong with my ultimate structure is that it 
> is the ultimate structure with respect to just one symbol set. I need a 
> structure that is ultimate with respect to -all- symbol sets. The basic 
> idea I had which I have not yet tried to formalize is encoding all 
> symbol sets into an ultimate symbol set which in human mathematics is a 
> countable set; so something like the set of natural numbers will encode 
> all possible symbols. One simple way to do this would be to say all 
> numbers congruent to 0 mod 3 are encodings of constant symbols, all 
> numbers congruent to 1 mod 3 are encodings of n-ary relation symbols, 
> and all numbers congruent to 2 mod 3 are encodings of n-ary function 
> symbols.
> So, note that I did not finish what I set out to do in my abstract: "the 
> author presents a structure with the property that all structures are 
> elementarily embeddable within it." I believe what I have done is this: 
> a structure over a fixed symbol set S with the property that all 
> S-structures are elementarily embeddable within it.
>
> Now on to the subject of time.
>
> If Tegmark is correct and an ultimate structure literally is all of 
> physical reality, what strikes me is that this ultimate structure 
> appears quite static. What then is the source of our perceptions of 
> transition, ie, time? This ultimate structure I presume (safely, I 
> believe) is constant yet we perceive things to change. Why and how? IOW, 
> what is the mechanism that converts the static ultimate structure into a 
> fluid appearance of transition? These questions are still valid even if 
> the ultimate structure I have in mind is wr

Re: Which mathematical structure -is- the universe in Physics?

[Torgny Tholerus]

Brian Tenneson skrev:
> Now on to the subject of time.
>
> If Tegmark is correct and an ultimate structure literally is all of 
> physical reality, what strikes me is that this ultimate structure 
> appears quite static. What then is the source of our perceptions of 
> transition, ie, time? This ultimate structure I presume (safely, I 
> believe) is constant yet we perceive things to change. Why and how? IOW, 
> what is the mechanism that converts the static ultimate structure into a 
> fluid appearance of transition? These questions are still valid even if 
> the ultimate structure I have in mind is wrong; Tegmark still 
> hypothesizes that some math structure is all of physical reality.
>   

If you look at our universe as a 4-dimensional space-time structure, 
then that structure will be completely static.  It is nothing peculiar 
with that.  The time is just a direction in that static structure.

-- 
Torgny Tholerus

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Re: Which mathematical structure -is- the universe in Physics?

[Brian Tenneson]

An open problem raised in Tegmark's "ensemble TOE" paper (published in 
'98, if I recall) is to answer the question that is the subject of this 
thread.

I believe recent work I have done has the potential to be a step towards 
that answer.

First, a link to the very rough draft (please forgive formatting 
errors), and second what I think the deficiency is with this document:

http://www.universalsight.org/math/9-26-08/01-03-structure_of_all_structures.pdf

Abstract:
In this document, the author presents a structure with the
property that all structures are elementarily embeddable within it. One 
essen-
tial tool is a version of New Foundations set theory, ?first developed 
by Quine,
as presented by Holmes in [1]. The motivation is given by The Mathematical
Universe article by Max Tegmark, [2], in which it is hypothesized that 
physical
existence is mathematical existence and, consequently, it is 
hypothesized that
the structure with the aforementioned property could be central in the Math-
ematical Universe Hypothesis as being at least keenly connected to the 
literal
universe, if not literally being the universe.
The author assumes some knowledge of mathematical logic such as, for ex-
ample, the inductive de?finition of a fi?rst-order well-formed formula.


The intended audience is primarily Max Tegmark, honestly, but more 
generally, any physicist interested in Tegmark's self-proclaimed 
"bananas articles" like the MUH paper, and who have already been exposed 
to the basics of mathematical logic.

Prior to drafting this document, I contacted Prof. Tegmark regarding the 
core ideas in the draft. I described what I was attempting and if I 
recall I sent him the abstract. As I will describe shortly, this is 
incomplete, so I didn't send him this pdf yet. I hope he wouldn't mind 
my inclusion of his response, which I think many here might find highly 
debatable (and worthy of discussion), sent by email:


It sounds to me from what you're saying that A would be the Level IV 
multiverse, i.e., all of physical reality.


Now for the deficiency I see with my document. -If- there aren't any 
other errors, then something wrong with my ultimate structure is that it 
is the ultimate structure with respect to just one symbol set. I need a 
structure that is ultimate with respect to -all- symbol sets. The basic 
idea I had which I have not yet tried to formalize is encoding all 
symbol sets into an ultimate symbol set which in human mathematics is a 
countable set; so something like the set of natural numbers will encode 
all possible symbols. One simple way to do this would be to say all 
numbers congruent to 0 mod 3 are encodings of constant symbols, all 
numbers congruent to 1 mod 3 are encodings of n-ary relation symbols, 
and all numbers congruent to 2 mod 3 are encodings of n-ary function 
symbols.
So, note that I did not finish what I set out to do in my abstract: "the 
author presents a structure with the property that all structures are 
elementarily embeddable within it." I believe what I have done is this: 
a structure over a fixed symbol set S with the property that all 
S-structures are elementarily embeddable within it.

Now on to the subject of time.

If Tegmark is correct and an ultimate structure literally is all of 
physical reality, what strikes me is that this ultimate structure 
appears quite static. What then is the source of our perceptions of 
transition, ie, time? This ultimate structure I presume (safely, I 
believe) is constant yet we perceive things to change. Why and how? IOW, 
what is the mechanism that converts the static ultimate structure into a 
fluid appearance of transition? These questions are still valid even if 
the ultimate structure I have in mind is wrong; Tegmark still 
hypothesizes that some math structure is all of physical reality.

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Re: Which mathematical structure -is- the universe in Physics?

[Brian Tenneson]

I believe I have a working candidate for a plausibility case for a
structure being literally the universe, assuming the MUH.


It is the structure U(U), where the first U is script and the second
is blackboard bold, on page 3 of the following document, listed under
"conjecture 4."




http://www.universalsight.org/conference_abstract/00-02-00.pdf
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Re: Which mathematical structure -is- the universe in Physics?

[Brian Tenneson]

In an attempt to recruit the help of a friend from school, he writes 
this in an email in response:



So, about your question, I've actually never heard
of a lattice-ordered abelian group, so I don't think I
can help you there. I can tell you about the
connection of category theory to physics, though
(although you may already know this): when you talk
about open string theory (i.e. adding D-branes to the
theory), depending on whether you consider the A or B
twist, the D branes are supposed to form a derived
Fukaya category for the A twist, or a category of
derived coherent sheaves on the B twist. In
categorical language, the objects are the D branes,
and the morphisms are (open) strings stretching
between D branes. If you wanted to then make some
(tenuous at best) connection to the real universe,
assuming that string theory is actually true, since
all particles are supposed to be strings (strings are
a subset of D branes), this means that theoretically
the entire universe could be described by a category
of D branes. The problem with this, though, is that D
branes are not fully described by even the derived
Fukaya/coherent sheaf setup, so before that kind of
connection can be made, (1) string theory has to be
proven true, (2) a complete mathematical description
of D branes has to be worked out.




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Re: Which mathematical structure -is- the universe in Physics?

[Brian Tenneson]

Great reference, thanks!

I'm investigating a problem I can phrase two ways, given that the 
category of MV-algebras is equivalent to the category of lattice-ordered 
Abelian groups with a distinguished strong unit.
So one way to phrase my question, and I'm guess it has been answered 
before

Take LOAGDSU to be the set of wffs that define lattice-ordered Abelian 
groups with a distinguished strong unit.  Now consider the collection of 
all models of LOAGDSU.  The question I have to anyone who knows is this: 
How many nonisomophic models can (the completion of) LOAGDSU have?

I might be redundant there if the theory generated by the wffs in 
LOAGDSU is already complete (I haven't looked into that aspect today 
yet).  Either way, for the ease of expression, let's say that my 
question is this: How many nonisomophic models can LOAGDSU have?

Are any of them in an interesting sense non-standard, such as so called 
nonstandard models of arithmetic can give rise to a 'realm' that all 
natural numbers are in and other 'things' are in this realm that are in 
every other way elementarily equivalent to the usual model of (N, +, ., 
<  ), yet these unlimited numbers are larger than every standard natural 
number.  Hyper-natural numbers these are called.  My point is, do 
non-standard models of LOAGDSU exist and what is even standard about 
LOAGDSU that could be pushed into a "non-standard line of thought."

But the main question is how many non-isomorphic models can LOAGDSU have.


In other news, I will try to apply to give a presentation on the 
promising connections between logic, algebra, and the muh in physics at 
this conference:
http://www.mat.unisi.it/~latd2008/

I just need to concoct the best 2 page abstract I can and submit it.  I 
am crossing my fingers.


Back to the point at hand.  Asking how many different models LOAGDSU has 
is in a natural way equivalent to asking how many models MV-algebra 
has.  THat is because of the theorems in chapter 7 of the book 
referenced in my preceding post about their realization that there is a 
deep connection between MV-algebras and those certain l-groups.

When I think of the 'things', denoted with variables, in an MV-algebra I 
think those are elements in the truth set.  Ex/ in Classical Logic, the 
cardinality of the truth set is two.  When I think of the 'things', also 
denoted by letters, in these l-groups, I think of groups (which are 
containers).

However, due to the deep and categorical connection between those two 
systems, and combine that with my suspicion that the universe mentioned 
in Tegmark's paper about the MUH, I then see 'things' in MV-algebras and 
l-groups (with equipment) as -worldlines- of other 'things'.  These 
structures, like MV-algebras, provide some of the laws of Physics as 
they would be under the MUH.


So I guess my next peek will be into what a standard model of the theory 
of MV-algebras is (like) and see if it would be fruitful to investigate 
nonstandard models of the theory of MV-algebras.





Günther Greindl wrote:
> Dear Brian,
>
> have you had a look at Universal logic?
>
> http://en.wikipedia.org/wiki/Universal_logic
>
> Maybe there are points of interest in there for you (the wikipedia 
> article is only a stub, but contains some names to google).
>
> Cheers,
> Günther
>
>   

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Re: Which mathematical structure -is- the universe in Physics?

[Günther Greindl]

Dear Brian,

have you had a look at Universal logic?

http://en.wikipedia.org/wiki/Universal_logic

Maybe there are points of interest in there for you (the wikipedia 
article is only a stub, but contains some names to google).

Cheers,
Günther

Brian Tenneson wrote:
> I was skimming though a book by Roberto Cignoli, Itala D'Ottaviano, and 
> Daniele Mundici called Algebraic Foundations of Many-Valued Reasoning.
> 
> Recall that I conjectured that the Physicist's universe has an 
> MV-algebra structure.  I probably should have said that the Physicist's 
> universe is the category of all MV-algebras, or some such.
> 
> In this book I'm studying, I have lifted some facts which might prove 
> interesting when settling my conjecture (which obviously might be as 
> insignificant as the conjecture 0+1=1).
> 
> 
> 
>  From book:
> Let A be the category of l-groups (lattice-ordered Abelean groups) with 
> a strong distinguished unit.
> 
> Let M be the category of MV-algebras. (I think a briefer way to say that 
> would be "let M be MV-algebra".)
> 
> 
> 
> 
> 
> OK, now... Chapter 7 of the aforementioned book has as its goal proving 
> the following statement:
> There is a natural equivalence between A and M, meaning that there is a 
> functor, call it F, between A and M.  In other words, between A and M, 
> there is a full, faithful, and dense functor F.
> 
> 
> 
> 
> 
> Thus another way to state my conjecture is this:
> The universe is an (or at least has the structure of an) l-group with a 
> strong distinguished unit.  Does this ring any bells with physicists?
> What, "physically" or observably, is this strong distinguished unit, if so?
> 
> > 
> 

-- 
Günther Greindl
Department of Philosophy of Science
University of Vienna
[EMAIL PROTECTED]
http://www.univie.ac.at/Wissenschaftstheorie/

Blog: http://dao.complexitystudies.org/
Site: http://www.complexitystudies.org

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Re: Which mathematical structure -is- the universe in Physics?

[Brian Tenneson]

I was skimming though a book by Roberto Cignoli, Itala D'Ottaviano, and 
Daniele Mundici called Algebraic Foundations of Many-Valued Reasoning.

Recall that I conjectured that the Physicist's universe has an 
MV-algebra structure.  I probably should have said that the Physicist's 
universe is the category of all MV-algebras, or some such.

In this book I'm studying, I have lifted some facts which might prove 
interesting when settling my conjecture (which obviously might be as 
insignificant as the conjecture 0+1=1).



 From book:
Let A be the category of l-groups (lattice-ordered Abelean groups) with 
a strong distinguished unit.

Let M be the category of MV-algebras. (I think a briefer way to say that 
would be "let M be MV-algebra".)





OK, now... Chapter 7 of the aforementioned book has as its goal proving 
the following statement:
There is a natural equivalence between A and M, meaning that there is a 
functor, call it F, between A and M.  In other words, between A and M, 
there is a full, faithful, and dense functor F.





Thus another way to state my conjecture is this:
The universe is an (or at least has the structure of an) l-group with a 
strong distinguished unit.  Does this ring any bells with physicists?
What, "physically" or observably, is this strong distinguished unit, if so?

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Re: Which mathematical structure -is- the universe in Physics?

[Bruno Marchal]

Le 26-avr.-08, à 06:55, nichomachus a écrit :

>
>
>
> On Apr 25, 5:27 am, Bruno Marchal <[EMAIL PROTECTED]> wrote:
>> Le 24-avr.-08, à 18:26, nichomachus a écrit :
>>
>>
>>> On Apr 22, 11:28 pm, "Brian Tenneson" <[EMAIL PROTECTED]> wrote:
 Perhaps Hilbert was right and Physics ought to have been axiomatized
 when he
 suggested it.  ;)  Then again, there might not have been a 
 motivation
 to
 until recently with Tegmark's MUH paper and related material (like 
 by
 David
 Wolpert of NASA).
>>
>>> The logical positivists were motivated to axiomatize in the predicate
>>> calculus the laws of scientific theories in the early 20th century,
>>> first because they believed that it would guarantee the cognitive
>>> significance of theoretical terms in the theory (such as the
>>> unphysical ether of maxwell's electromagnetism), and then later
>>> because it had evolved into an attempt to specify the proper form of 
>>> a
>>> scientific theory. In practice this had too many problems and was
>>> eventually abandoned. One of the consequences of this program was 
>>> that
>>> axiomatizing the laws of a theory in first order predicate calculus
>>> with equality was that such a formulation of a theory always implied
>>> various unintended interpretations. The amount of effort needed to
>>> block these unintended interpretations was out of proportion with the
>>> benefit received by axiomatization.
>>
>> It is a bit weird because it is just logically impossible to block
>> those unintended interpretations. And This should not be a problem.
>> The reason why physical theories are not axiomatize is more related to
>> the fact that axiomatization does not per se solve or even address the
>> kind of conceptual problem raised by physics.
>
> Also to this point, that it is impossible to identify a theory with
> any particular linguistic formulation of it. Theories are not
> linguistic entities.
>
> And since we’re on the subject: according to Max Tegmark, given the
> apparent direction of inter-theoretic reduction, one may assume that
> the foundational physics of our universe should be able to be
> expressed in a completely “baggage-free” description that is without
> reference to any human-specific concepts.



This is vague. Do you think that natural numbers are human-specific 
concepts?
You cannot axiomatize the natural numbers in a way such that it avoids 
other objects obeying your axioms.
Even arithmetical truth (the set of first order true arithmetical 
propositions seen as a theory) has no standard models.
Computability theory/ recursion theory is the best, imo, way to get a 
human independent, even a machine or formalism independent, mathematics 
(despite non standardness). ... doubly so with the explicit use of the 
(classical) Church's thesis.





>  This presumed most basic
> law of the universe would be capable of being axiomatized without
> unintended implications since the mathematical structure expressing
> the most basic law would be isomorphic with the law itself to the
> degree that it may appropriately be identified with it.

If you say "yes" to the doctor, accepting a digital brain/body, you 
identify yourself (your 3-self) locally with a finite linguistic (et 
least finitely 3-person presentable) structure.



>  The
> mathematical laws which describe the phenomena of all of the emergent
> levels or organization diverge from this ideal more and more the
> further one proceeds from this unknown foundational theory.


This is hard to interpret because I don't know your theoretical 
background. I say a few more words below.


>
>>> Also, I
>>> personally remain unconvinced that there is anything problematic 
>>> about
>>> the exitence of the universe of universes, or the ensemble of all
>>> possible mathematical structures, thought it may not be well defined
>>> at present. I don't believe that this is simply the union of all
>>> axiomatic systems. If trying to define the Everything as a set 
>>> implies
>>> a contradiction, then fine -- it isn't a set, it's an ensemble, which
>>> doesn't carry any of the connotations that are implied by the use of
>>> "set" in the mathematical sense. Therefore each entity in the 
>>> ensemble
>>> is a unique collection of n axioms that has no necessary relationship
>>> to any other axiom collection. What happens in an axiom system stays
>>> in that axiom system, and can't bleed over to the next one on the
>>> list. Some of these may be equivalent to each other.
>>
>>> A = The collection of all finite axiom systems
>>> B = The collection of all consistent finite axiom systems
>>
>> I guess you mean "recursively enumerable" instead of finite. You would
>> loose first order Peano Arithmetic (my favorite lobian machine :).
>
> Really? It would seem that all recursively enumerable (RE) axiom 
> systems
> would exist in A.

"A" is ambiguous. Strictly speaking Peano Arithmetic is an 
axiomatization, in first order predicate logic, of element

Re: Which mathematical structure -is- the universe in Physics?

[nichomachus]

On Apr 25, 5:27 am, Bruno Marchal <[EMAIL PROTECTED]> wrote:
> Le 24-avr.-08, à 18:26, nichomachus a écrit :
>
>
>
>
>
>
>
> > On Apr 22, 11:28 pm, "Brian Tenneson" <[EMAIL PROTECTED]> wrote:
> >> Perhaps Hilbert was right and Physics ought to have been axiomatized
> >> when he
> >> suggested it.  ;)  Then again, there might not have been a motivation
> >> to
> >> until recently with Tegmark's MUH paper and related material (like by
> >> David
> >> Wolpert of NASA).
>
> > The logical positivists were motivated to axiomatize in the predicate
> > calculus the laws of scientific theories in the early 20th century,
> > first because they believed that it would guarantee the cognitive
> > significance of theoretical terms in the theory (such as the
> > unphysical ether of maxwell's electromagnetism), and then later
> > because it had evolved into an attempt to specify the proper form of a
> > scientific theory. In practice this had too many problems and was
> > eventually abandoned. One of the consequences of this program was that
> > axiomatizing the laws of a theory in first order predicate calculus
> > with equality was that such a formulation of a theory always implied
> > various unintended interpretations. The amount of effort needed to
> > block these unintended interpretations was out of proportion with the
> > benefit received by axiomatization.
>
> It is a bit weird because it is just logically impossible to block
> those unintended interpretations. And This should not be a problem.
> The reason why physical theories are not axiomatize is more related to
> the fact that axiomatization does not per se solve or even address the
> kind of conceptual problem raised by physics.

Also to this point, that it is impossible to identify a theory with
any particular linguistic formulation of it. Theories are not
linguistic entities.

And since we’re on the subject: according to Max Tegmark, given the
apparent direction of inter-theoretic reduction, one may assume that
the foundational physics of our universe should be able to be
expressed in a completely “baggage-free” description that is without
reference to any human-specific concepts.  This presumed most basic
law of the universe would be capable of being axiomatized without
unintended implications since the mathematical structure expressing
the most basic law would be isomorphic with the law itself to the
degree that it may appropriately be identified with it. The
mathematical laws which describe the phenomena of all of the emergent
levels or organization diverge from this ideal more and more the
further one proceeds from this unknown foundational theory.

> > Also, I
> > personally remain unconvinced that there is anything problematic about
> > the exitence of the universe of universes, or the ensemble of all
> > possible mathematical structures, thought it may not be well defined
> > at present. I don't believe that this is simply the union of all
> > axiomatic systems. If trying to define the Everything as a set implies
> > a contradiction, then fine -- it isn't a set, it's an ensemble, which
> > doesn't carry any of the connotations that are implied by the use of
> > "set" in the mathematical sense. Therefore each entity in the ensemble
> > is a unique collection of n axioms that has no necessary relationship
> > to any other axiom collection. What happens in an axiom system stays
> > in that axiom system, and can't bleed over to the next one on the
> > list. Some of these may be equivalent to each other.
>
> > A = The collection of all finite axiom systems
> > B = The collection of all consistent finite axiom systems
>
> I guess you mean "recursively enumerable" instead of finite. You would
> loose first order Peano Arithmetic (my favorite lobian machine :).

Really? It would seem that all recursively enumerable axiom systems
would exist in A.

> Note also that SAS occurs very quickly. SAS occur in theories which are
> much weaker than the SAS themselves (ex: SAS occur in Robinson
> Arithmetic, i.e. when you can define successor, addition and
> multiplication. SAS themselves need induction.

I don’t understand. Are you saying that Self Aware Substructures exist
in the Robinson Arithmetic?
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Re: Which mathematical structure -is- the universe in Physics?

[nichomachus]

On Apr 24, 12:08 pm, "Brian Tenneson" <[EMAIL PROTECTED]> wrote:
> I was attempting to -invalidate- that argument against the existence of the
> universe, actually, by saying that in three truth values, which the
> Physicists can't rule out as being the more accurate logic of their
> universe, the argument "reductio ad absurdum" is not a tautology and,
> therefore, can't necessarily be applied.
>
> However, in binary logic, the Physicist's universe (or whatever Everything
> means) can't exist.

I take your point about the reductio not working in three valued
logic.

I am not convinced that the problem you are attempting to solve is
necessarily a problem since I haven’t been able to construe the
proposed reductio ad absurdum argument in a way that seems coherent to
my way of thinking.

However, you may be on to something with the general program that you
have embarked upon. Maybe there is a need for a mathematics to
describe the everything ensemble. Something along those lines is
likely the only way to define the everything with any sort of rigor. I
think it is a good idea.

Set theory does seem to be too rich for the job. Determining what type
of formalism is apropriate is a task. I think that such a mathematical
formalism may be precisely what is called for in order to define the
everything, as well as analyze it any useful sort of way.

I am still confused by what you mean by certain terms. What is meant
by the Physicist’s universe? Even more to the point, what is meant by
saying that it cannot exist in binary logic? The propositional
calculus, for example, does not even satisfy the conditions the Godel
theorems, i.e. there are no undecidable propositions possible in it.
To think that the axioms of any two valued logic could be sufficient
to produce a physical existence for self-aware substructures is
distinctly overstepping what Max Tegmark suggests in his metaphysical
theory.


>
> I doubt self-reference is inherently the problem in light of things like
> Tarski's fixed point theorems which provide concrete examples of wffs that
> are self-referencing, in terms of Godel numbers, if I recall.  That proof I
> was exposed to was not an existence proof of self-referencing wffs merely by
> "logical flamboyancy" but by the providing an example of an actual -class-
> of self-referencing wffs.  Obviously, the above argument does not explicitly
> involve wffs (it does, however, implicitly), and I am -only- making a case
> for plausibility at this particular moment.
>
> I see no problems with the argument given that in binary logic, their
> universe can't exist; this, to me, convinces me that the Physicist's
> universe can't operate on binary logic by Occam's Razor as -none- of the
> data in any experiment would fit the result that confirms their speculation
> that their universe exists.
>
> Ergo, the Physicist's universe must operate on at least three truth values.
> (Consequently, it exists.)  This to me is a more elegant solution to the
> argument than citing self-referencing issues as automatically damning.  If
> natural language can be used to prove the Heine-Borel theorem, without the
> need for wffs, then why must a statement about Everything be formalized in
> machine-level code with wffs?
>
> If there is further objection to my line of thinking, -please- point it out
> to Everyone (which I hope is well-defined or else no one would know what I
> mean, right?)  ;)
>
> Thank you for your remarks; I find all input extremely productive!!

I too appreciate the chance to talk about such interesting ideas.
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Re: Which mathematical structure -is- the universe in Physics?

[Brian Tenneson]

I think we have no choice in the matter (once we assume the
"unbelievable comp hyp."). The physical is not just a mathematical
structure among others. The physical emerged from a sort of sum
pertaining on the whole of the mathematical possible histories. If this
does not give the empirical physics, then comp will be refuted. But
preliminary results give already a sort of quantum topology. The one I
have more or less extracted from the comp hyp, at the modest
propositional level, has not yet been prove to be be equivalent to
universal quantum topology, but they are clues indicating that comp
could be the promising path. It is quasi obvious that comp entails many
consistent histories, and the math gives reasons why such histories
interferes statistically in a "quantum way", i.e. with a
perpendicularity relation on the possible incompatible states/stories.
Ah yes the "truly" parallel realities are perpendicular, but this is
already the case with quantum mechanics and its "scalar product".
What is hard, and on which I am stuck since years is to find the
(arithmetical) needed tensor product, or how does a first person plural
reality occur. Mathematically it is enough to assume at some place a
linearity condition. But this is cheating; we have to justify that
linearity from comp only, as comp justifies we have to do. Sorry if I
am a bit short.
bruno


In the sense of David Wolerpt's (of NASA) omniscient devices and 
oracles, I think a theorem is this:

Inconsistency in some sense (like answering a question as neither yes 
nor no, but something like MU in Eastern thought), is a -necessary- 
condition for omniscience.

Or, phrased differently, omniscience implies inconsistency.

In a -binary- logical universe, that is.

What is "This"?

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Re: Which mathematical structure -is- the universe in Physics?

[Bruno Marchal]

inite axiom systems
> B = The collection of all consistent finite axiom systems


I guess you mean "recursively enumerable" instead of finite. You would 
loose first order Peano Arithmetic (my favorite lobian machine :).
Note also that SAS occurs very quickly. SAS occur in theories which are 
much weaker than the SAS themselves (ex: SAS occur in Robinson 
Arithmetic, i.e. when you can define successor, addition and 
multiplication. SAS themselves need induction.



>
> The "cardinality" of B is not greater than the "cardinality" of A.
> (Scare qutoes since cardinality is a property of sets and these may
> not be sets if that would imply contradiction.) It is B that is
> interesting from the point of this discussion since it is believed (I
> don't know of any proof of this) that only systems in B could produce
> the type of rational and orderly physical existence capable of
> containing observers who can think about their existence as we do
> (SASs, or Self-Aware Substructures). The collection of all those
> systems capable of containing SASs is the most interesting from the
> point of view of the present discussion, and must have a "cardinality"
> not greater than that of B, since many axiom systems are too simple to
> contain SAS, and the ones with them are expected to predominate.
>
> The idea of this ensemble so propounded does not seem to entail an ad
> absurdum paradox such as you gave above. Further, didn't I see you say
> somewhere that you don't even believe in sets? I apologize if I am
> mistaken, but if that is true, I can't see how that statement would
> reconcile with sincere belief in the validity of the agument you gave
> above.
>
> If there is some genuine logical inconsistency in the above, please
> point it out to me as to me this (which is Tegmark) seems like a good
> direction to go in trying to formulate a proper definition of the
> Everything.



I suggest you read my papers (published in French a long time ago, in 
english a bit more recently) or my posts to this list. Those should 
help you for making clear what is missing in Tegmark or Schmidhuber 
(and many others): the fact that they do not take into account the 
first person (plural or not) indeterminacy.
With comp (and its transfinitely numerous weakenings) we cannot know 
which computational consistent histories (in case we bet there is one) 
do support our experiences, and this put total constraints on the 
nature of any observable and sharable realities.
This *changes* everything ... Like in Plato, the physical world can 
only be the border of "our" ignorance or first person plenitude (as 
George Levy called it). "our" is "us the (hopefully) sound machines". 
(or alpha-machine, a constructive ordinal based weakening of the notion 
of machine, but I use "omega-machine" (that is: machine) for reason of 
simplicity).





>
>>
>> By a simple logical tautology (a variant of ad absurdum), this proves 
>> that
>> "Everything is well defined" is a false statement.  It also raises 
>> doubts as
>> to the existence of this so called Everything.  Maybe this google 
>> group
>> should end?
>>
>> I don't think so.
>>
>> My quip was something along the lines of, "however, in any ternary 
>> logic, ad
>> absurdum is not a tautology and therefore, can't be used here."
>>
>> That discussion got me going and while mostly off task, I've been 
>> thinking
>> about this on and off since then.  Basically, my motivation to 
>> "rescue" a
>> universal set is so that Cantor's dream of formalizing in a 
>> mathematical way
>> some type of deity could be realized.  The analogy would be Abraham 
>> Robinson
>> is to Issac Newton (on infinitesimals) as Quinne (et al) are to 
>> Cantor (on a
>> universal set).  Right idea, but never considered using fuzzy logic 
>> not to
>> be delved into much until Lukaseiwicz, Zadeh, and others revitalized 
>> FL.  As
>> it took an army of giants to "rescue" Newton's intuition which was
>> criticized by another philosopher (Berkeley, akin to Russell) to 
>> develop
>> enough tools (compactness theorem), it is taking an army of logicians 
>> to
>> "rescue" Cantor's intuition about God which, and this may be 
>> apocrypha, he
>> believed to be his maximally infinite set.  He thought infinity must 
>> be an
>> attribute of God and therefore delved into infinite sets, hoping, I 
>> assume,
>> to reach some type of Omega set that contains all sets and would then 
>> be
>> necessarily the "biggest" infin

Re: Which mathematical structure -is- the universe in Physics?

[Russell Standish]

On Thu, Apr 24, 2008 at 10:08:16AM -0700, Brian Tenneson wrote:
> I was attempting to -invalidate- that argument against the existence of the
> universe, actually, by saying that in three truth values, which the
> Physicists can't rule out as being the more accurate logic of their
> universe, the argument "reductio ad absurdum" is not a tautology and,
> therefore, can't necessarily be applied.
> 
> However, in binary logic, the Physicist's universe (or whatever Everything
> means) can't exist.
> 

...

> 
> If there is further objection to my line of thinking, -please- point it out
> to Everyone (which I hope is well-defined or else no one would know what I
> mean, right?)  ;)
> 
> Thank you for your remarks; I find all input extremely productive!!

Isn't the sort of everything you have in mind a bit like omnipotence
(which has problems such as creating the immovable object, then moving
it).

Perhaps such an everything really is logically impossible. The sorts
of everything we've discussed here on the list are much more modest
beasts - even Tegmark's all mathmatics tends to be viewed in terms of
recursive enumerable structures (or finite axiomatic systems).

Cheers

-- 


A/Prof Russell Standish  Phone 0425 253119 (mobile)
Mathematics  
UNSW SYDNEY 2052 [EMAIL PROTECTED]
Australiahttp://www.hpcoders.com.au


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Re: Which mathematical structure -is- the universe in Physics?

[Brian Tenneson]

> Then Everything does not exist.  Proof:
> > Consider the thing, call it "this something," that is the qualia of all
> > things that do not contain themselves.
> > Then this something contains itself if and only if this something does
not
> > contain itself.
>
> I am suspect of the claim that a logical argument such as the above,
> which relies on a paradox of self-reference, could be used to
> demonstrate the non-existence of the so-called Everything. Also, I
> personally remain unconvinced that there is anything problematic about
> the exitence of the universe of universes, or the ensemble of all
> possible mathematical structures, thought it may not be well defined
> at present. I don't believe that this is simply the union of all
> axiomatic systems. If trying to define the Everything as a set implies
> a contradiction, then fine -- it isn't a set, it's an ensemble, which
> doesn't carry any of the connotations that are implied by the use of
> "set" in the mathematical sense. Therefore each entity in the ensemble
> is a unique collection of n axioms that has no necessary relationship
> to any other axiom collection. What happens in an axiom system stays
> in that axiom system, and can't bleed over to the next one on the
> list. Some of these may be equivalent to each other.
>
> A = The collection of all finite axiom systems
> B = The collection of all consistent finite axiom systems
>
> The "cardinality" of B is not greater than the "cardinality" of A.
> (Scare qutoes since cardinality is a property of sets and these may
> not be sets if that would imply contradiction.) It is B that is
> interesting from the point of this discussion since it is believed (I
> don't know of any proof of this) that only systems in B could produce
> the type of rational and orderly physical existence capable of
> containing observers who can think about their existence as we do
> (SASs, or Self-Aware Substructures). The collection of all those
> systems capable of containing SASs is the most interesting from the
> point of view of the present discussion, and must have a "cardinality"
> not greater than that of B, since many axiom systems are too simple to
> contain SAS, and the ones with them are expected to predominate.
>
> The idea of this ensemble so propounded does not seem to entail an ad
> absurdum paradox such as you gave above. Further, didn't I see you say
> somewhere that you don't even believe in sets? I apologize if I am
> mistaken, but if that is true, I can't see how that statement would
> reconcile with sincere belief in the validity of the agument you gave
> above.
>
> If there is some genuine logical inconsistency in the above, please
> point it out to me as to me this (which is Tegmark) seems like a good
> direction to go in trying to formulate a proper definition of the
> Everything.
>
>
>
>
>
> > By a simple logical tautology (a variant of ad absurdum), this proves
that
> > "Everything is well defined" is a false statement.  It also raises
doubts as
> > to the existence of this so called Everything.  Maybe this google group
> > should end?
>
> > I don't think so.
>
> > My quip was something along the lines of, "however, in any ternary
logic, ad
> > absurdum is not a tautology and therefore, can't be used here."
>
> > That discussion got me going and while mostly off task, I've been
thinking
> > about this on and off since then.  Basically, my motivation to "rescue"
a
> > universal set is so that Cantor's dream of formalizing in a mathematical
way
> > some type of deity could be realized.  The analogy would be Abraham
Robinson
> > is to Issac Newton (on infinitesimals) as Quinne (et al) are to Cantor
(on a
> > universal set).  Right idea, but never considered using fuzzy logic not
to
> > be delved into much until Lukaseiwicz, Zadeh, and others revitalized
FL.  As
> > it took an army of giants to "rescue" Newton's intuition which was
> > criticized by another philosopher (Berkeley, akin to Russell) to develop
> > enough tools (compactness theorem), it is taking an army of logicians to
> > "rescue" Cantor's intuition about God which, and this may be apocrypha,
he
> > believed to be his maximally infinite set.  He thought infinity must be
an
> > attribute of God and therefore delved into infinite sets, hoping, I
assume,
> > to reach some type of Omega set that contains all sets and would then be
> > necessarily the "biggest" infinity.  Cantor proved that the power set of
any
> > set is "larger," however, a

Re: Which mathematical structure -is- the universe in Physics?

[nichomachus]

ded does not seem to entail an ad
absurdum paradox such as you gave above. Further, didn't I see you say
somewhere that you don't even believe in sets? I apologize if I am
mistaken, but if that is true, I can't see how that statement would
reconcile with sincere belief in the validity of the agument you gave
above.

If there is some genuine logical inconsistency in the above, please
point it out to me as to me this (which is Tegmark) seems like a good
direction to go in trying to formulate a proper definition of the
Everything.

>
> By a simple logical tautology (a variant of ad absurdum), this proves that
> "Everything is well defined" is a false statement.  It also raises doubts as
> to the existence of this so called Everything.  Maybe this google group
> should end?
>
> I don't think so.
>
> My quip was something along the lines of, "however, in any ternary logic, ad
> absurdum is not a tautology and therefore, can't be used here."
>
> That discussion got me going and while mostly off task, I've been thinking
> about this on and off since then.  Basically, my motivation to "rescue" a
> universal set is so that Cantor's dream of formalizing in a mathematical way
> some type of deity could be realized.  The analogy would be Abraham Robinson
> is to Issac Newton (on infinitesimals) as Quinne (et al) are to Cantor (on a
> universal set).  Right idea, but never considered using fuzzy logic not to
> be delved into much until Lukaseiwicz, Zadeh, and others revitalized FL.  As
> it took an army of giants to "rescue" Newton's intuition which was
> criticized by another philosopher (Berkeley, akin to Russell) to develop
> enough tools (compactness theorem), it is taking an army of logicians to
> "rescue" Cantor's intuition about God which, and this may be apocrypha, he
> believed to be his maximally infinite set.  He thought infinity must be an
> attribute of God and therefore delved into infinite sets, hoping, I assume,
> to reach some type of Omega set that contains all sets and would then be
> necessarily the "biggest" infinity.  Cantor proved that the power set of any
> set is "larger," however, and settled his own quest in his own way though
> I'm guessing he -desired- the opposite conclusion to have been reached.
>
> Others in the FL army are trying to reach that conclusion which Cantor,
> chronologically, would have to have re-discovered much mathematics to
> realize in the way this army is doing.
>
> So the basic motivation is to find some type of thing with maximality in
> some important sense and study it.  With the MUH, now I suspect that
> Everything would be a likely candidate for a literal God and atheism might
> have to suddenly be the irrational side to be on.
>
> So on this note, the works of David Hawkins (a psychiatrist and
> spiritualist) inspired me to ponder the following question, along with
> Tegmark's articulation of the MUH, of course.
> Which mathematical structure -is- the universe in Physics?
>
> I suspect it might already exist and has been studied.  

I agree. We could exist in the Mandlebrot set for all we know.
Determining which mathematical structure is our own universe is likely
practically impossible, though determining which classes of
mathematical structures are more likely candidates may be doable.


It's like finding
> the correct non-Euclidean Geometry applicable to the universe we perceive
> gets us to a GR that coincides with observation (for the most part?).  I am
> guessing that the universe must have an MV-algebra 
> structure.http://en.wikipedia.org/wiki/MV-algebra
>
> I was trying to rejoin Bruno's "too rich" -valid- (imho) objection to
> Tegmark's approach in his MUH paper by concocting a theory that was far less
> rich.  All I need are things and a notion of containment.  I was going to
> call it container theory.  Then there'd be no need to develop something
> strong enough to do numbers, infinite sets, and such, so with those goals
> gone, so much more is available to Physics without having to squeeze any set
> theory or logic into Physics.  It's there, I suspect, in -classical logic-
> and recent -algebra- in the guise of MV-algebra.  This area is exactly what
> I mean by thing and containment.  Now if you look at the wiki article above,
> observe, firstly, how little there is reliance on sets or non-classical
> logic.
>
> Secondly, I could view all the letters that would normally be variables as
> things in the "container theory" I was trying to work on.  In MV-algebras,
> the variables represent truth degrees and the carrier of the MV-algebra is
> the truth set, the set of all truth values which ha

Which mathematical structure -is- the universe in Physics?

[Brian Tenneson]

Perhaps Hilbert was right and Physics ought to have been axiomatized when he
suggested it.  ;)  Then again, there might not have been a motivation to
until recently with Tegmark's MUH paper and related material (like by David
Wolpert of NASA).





I was trying to answer Bruno's objections regarding set theory being too
rich to be the 'ultimate math' the MUH needs to propose what the universe is
and I quipped that that was because math is invented or discovered to
further its own end by logicians, for the most part, and that
metamathematicians such as Cantor had no apparent interest in physical
things or furthering the pursuit of Physics.

Another question of Bruno's was my motivation.  I started this quest hoping
that three truth values were sufficient to develop a set theory with a
universal set that was in a classical logic sense consistent to ZFC set
theory.  Or, if not true, prove that and figure out why.  Perhaps more truth
values would solve that.  My main motivation has definitely not been to
"rescue" a major apparent shortcoming in the MUH as I started this
on-and-off quest in 2003 with no internet connection or resources such as a
deluge of journals (ie, a good library).  How it started was that someone
online in a place such as this used Russell-like arguments to -prove- that
the Physic's universe -does not exist- for essentially the same reasons a
universal set can't seem to be non-antimonious.

Suppose Everything is well defined along with its partner, containment (such
as the earth is contained in the solar system by the definitions of both).
Then Everything does not exist.  Proof:
Consider the thing, call it "this something," that is the qualia of all
things that do not contain themselves.
Then this something contains itself if and only if this something does not
contain itself.

By a simple logical tautology (a variant of ad absurdum), this proves that
"Everything is well defined" is a false statement.  It also raises doubts as
to the existence of this so called Everything.  Maybe this google group
should end?



I don't think so.




My quip was something along the lines of, "however, in any ternary logic, ad
absurdum is not a tautology and therefore, can't be used here."






That discussion got me going and while mostly off task, I've been thinking
about this on and off since then.  Basically, my motivation to "rescue" a
universal set is so that Cantor's dream of formalizing in a mathematical way
some type of deity could be realized.  The analogy would be Abraham Robinson
is to Issac Newton (on infinitesimals) as Quinne (et al) are to Cantor (on a
universal set).  Right idea, but never considered using fuzzy logic not to
be delved into much until Lukaseiwicz, Zadeh, and others revitalized FL.  As
it took an army of giants to "rescue" Newton's intuition which was
criticized by another philosopher (Berkeley, akin to Russell) to develop
enough tools (compactness theorem), it is taking an army of logicians to
"rescue" Cantor's intuition about God which, and this may be apocrypha, he
believed to be his maximally infinite set.  He thought infinity must be an
attribute of God and therefore delved into infinite sets, hoping, I assume,
to reach some type of Omega set that contains all sets and would then be
necessarily the "biggest" infinity.  Cantor proved that the power set of any
set is "larger," however, and settled his own quest in his own way though
I'm guessing he -desired- the opposite conclusion to have been reached.

Others in the FL army are trying to reach that conclusion which Cantor,
chronologically, would have to have re-discovered much mathematics to
realize in the way this army is doing.




So the basic motivation is to find some type of thing with maximality in
some important sense and study it.  With the MUH, now I suspect that
Everything would be a likely candidate for a literal God and atheism might
have to suddenly be the irrational side to be on.






So on this note, the works of David Hawkins (a psychiatrist and
spiritualist) inspired me to ponder the following question, along with
Tegmark's articulation of the MUH, of course.
Which mathematical structure -is- the universe in Physics?



I suspect it might already exist and has been studied.  It's like finding
the correct non-Euclidean Geometry applicable to the universe we perceive
gets us to a GR that coincides with observation (for the most part?).  I am
guessing that the universe must have an MV-algebra structure.
http://en.wikipedia.org/wiki/MV-algebra


I was trying to rejoin Bruno's "too rich" -valid- (imho) objection to
Tegmark's approach in his MUH paper by concocting a theory that was far less
rich.  All I need are things and a notion of containment.  I was going to
call it container theory.  Then there'd be no ne