The Mathematics of Self-Awareness

[phoenixthoth]

Has anyone developed a mathematical theory of awareness? For example, if we restrict ourselves to sets, is there a theory/model of awareness? Is the empty set "aware" of the set {a,b,c}? Seems like there should be degrees to which one set is aware of another set; that the awareness should be measurable (not in the sense of measure theory but perhaps).

In regards to Max Tegmark's big TOE, where ME=PE, the next step is to apply the mathematics of awareness of one structure for another to one structure's self-awareness.

It just seems natural to develop awareness in general first and then apply that to one structure and itself.

 

[phoenixthoth]

Perhaps this belongs in TD or Math but I thought some philosopher would like to step in and provide some insights on awareness.

I may have said this earlier, but let's restrict to sets for now.

A is some function of two variables which encapsulates the awareness the first input (variable) has of the second input.

How about this?

Given two sets X and Y, A(X,Y) can be the set of functions from X to Y having at least one fixed point.

(What I want to do is come up with something reasonable, find some properties on it, and then maybe drop what that something is in favor of the properties as axioms.)

Then the complexity and/or cardinality of A(X,Y) is some rough measure of the awareness X has of Y.

Example 1. X={a,b}, Y={1,b}. The set of all functions from X to Y can be listed out:
{(a,1),(b,b)}, {(a,b),(b,1)}.

Then the set of functions having at least one fixed point is this: {{(a,1),(b,b)}}. This is A(X,Y).

Note how this contains some information about the similarities of X and Y, namely the fixed point (and point of intersection) b, and some information about their differences, namely a and 1. The b increases the awareness X has for Y while the a & 1 detract from the awareness X has for Y.

Example 2. X={a,b}, Y={b}. The set of all functions from X to Y can be listed out:
{{(a,b),(b,b)}}. The function in this set has a fixed point, so A(X,Y)={{(a,b),(b,b)}}. Note that the set of all functions from X to Y equals A(X,Y).

Since A(X,Y) is a subset of the set of functions from X to Y, it is "biggest" when it is equal to the set of functions from X to Y. Perhaps when this happens, we can say that X is maximally aware of Y and when A(X,Y) is nonempty, we can say X is aware of Y.

A(R,R) is neither empty nor its maximum which would imply that R does not have maximal self-awareness.

Hmm... I wonder how you could characterize the sets Z for which A(Z,Z) equals the set of functions from Z to Z.

Well this is just a shot in the dark. If you have another characterization of awareness, please do tell.

 

[Hurkyl]

Well, the cardinality of your A(X, Y) can be computed knowing just |X|, |Y|, and |X & Y|. (& means intersection)

There are |Y|^|X| functions from X to Y.

To count the functions without a fixed point, note that anything in X-Y can be mapped to anything, and everything in X&Y can be mapped to anything in Y, except for one element.

I.E. there are |Y|^(|X| - |X&Y|) * (|Y| - 1)^(|X & Y|) functions without fixed point.

The number of functions with fixed point is, of course, simply the difference. Clearly, they can only be the same if |Y| = 1.

(Hrm, there seem to be some assumptions in this argument -- check the boundary cases)



Now that that's out of the way, I really don't get what you're trying to model.

 

[phoenixthoth]

Thanks.

there are |Y|^(|X| - |X&Y|) * (|Y| - 1)^(|X & Y|) functions without fixed point.

There are |Y|^|X| functions from X to Y, right? Then the cardinality of the set of functions with at least one fixed point is (|Y|^|X|)-(|Y|^(|X| - |X&Y|) * (|Y| - 1)^(|X & Y|))?

Now that that's out of the way, I really don't get what you're trying to model.

I really don't get it (awareness), either.

How would you define awarness mathematically? To me, it seems clear that it should be a binary entity; that much models that X is aware of Y.

I don't even have the intuition of awareness down. Should we say that {1,2,3} is not aware at all of {a,b,c} whereas is it aware of R? Should it be a two-valued function that is either "yes" there is awareness and "no" there isn't or should it be something that somehow measures the extent to which X is aware of Y?

I vote for {1,2,3} not being aware of {a,b,c} (at all) but that it is aware of {1,b,c} and even more aware of {1,2,c} and maximally aware of {1,2,3}. Using the formula above:
|A({1,2,3},{a,b,c})|=(27)-(3^(3) * (3 - 1)^(0))=(27)-(27)=0.

|A({1,2,3},{1,b,c})|=(3^3)-(3^(3 - 1) * (3 - 1)^(1)) = (3^3)-(3^(3 - 1) * (3 - 1)^(1))=27-18=9.

|A({1,2,3},{1,2,c})|=(27)-(3^(3 - 2) * (3 - 1)^(2)) = 27-12=15.

|A({1,2,3},{1,2,3})|=27-8=19.

So the awareness goes up as they share more in common. Hmm...

I'm certainly open to alternatives.

 

[matt grime]

You're the person who's decided to askl about awareness, so you need to decide what it means. It isn't a mathematical term so we can't even begin to guess what you may mean. AS it is all you seem to be asking is the cardinality of the intersection

 

[PFanalog57]

Since the brain is a nonlinear machine, and self awareness is a form of self reference and even awareness of other things, is made with reference to the "self", it appears that the awareness equations are nonlinear?

[EXTERNAL INPUT]--->[MIND]<--->[INTERNAL INPUT]

 

[matt grime]

Since the brain is a nonlinear machine,

would you care to justify or explain this in a mathematical sense?

and self awareness is a form of self reference and even awareness of other things, is made with reference to the "self", it appears that the awareness equations are nonlinear?

[EXTERNAL INPUT]--->[MIND]<--->[INTERNAL INPUT]

you have equations of awareness? what on Earth is mathematical awareness?

 

[PFanalog57]

would you care to justify or explain this in a mathematical sense?

[...]

you have equations of awareness? what on Earth is mathematical awareness?

By understanding the self reference and self-reflection of formal systems, as exemplified by Goedel's theorem, mathematics can be used to model the reflexivity, associated with self, consciousness, subjectivity, and thus self-awareness.

 

[arildno]

I have a faint hope that eventually, people will develop the attitude that there exist structures so complicated that one can't say anything intelligent about them (as yet..)..

 

[matt grime]

By understanding the self reference and self-reflection of formal systems, as exemplified by Goedel's theorem, mathematics can be used to model the reflexivity, associated with self, consciousness, subjectivity, and thus self-awareness.

I'll take that as a 'no', then.

 

[selfAdjoint]

I'll take that as a 'no', then.

It is known from experiment that neurons respond nonlinearly; you should be able to look that up for yourself if you doubt it. Since they are linked together, they could be idealized as something like the Fermi-Pasta-Ulam (FPU) device, which is mappable into the nonlinear KdV equation on a lattice, and which has a rich variety of non-ergodic, nonlinear solutions including solitons.

 

[matt grime]

I didn't say that some no one has figure out equations that model neural path ways that were non-linear, nor did I imply they ought to be linear if they exist (while applying for a job in a computer science lab I did some research on these things, though to me their fuzzy logic models seem like reinventing the wheel, or probabilitty theory anyway, for people who can't do maths) nor did I say no one had done neural experiments that have non-linear behaviour.

I asked if Russell would mind justifying his post since it contains a number of mathematically dubious statements not to mention undefined terms that scream "crank", and I'm more than a little sceptical following his posts on Fermat's Last Theorem.

 

[phoenixthoth]

arildno, I think mathematicians, or at least this mathematician, already think there are things that defy intelligent discussion.

Just as an example, consider the Juila set of a polynomial over C[x_1,x_2,...,x_n] where n is a googleplex, so there are googleplex variables, whose total degree is the highest number you can think of. I don't know, maybe this can be discussed intelligently but you get the idea. There are already things that defy intelligent discussion.

I hope that awareness is not such a thing that defies intelleligent discussion.

Still restricting ourselves to sets...

Do we want to say {1,2,3} has any awareness of {a,b,c}?
Do we want to say that {1,2,3} has awareness of {1,2,3}?

I figure that working with finite sets will be easier but I could be wrong.

Yes, matt, it has something to do with interesection but it is more than that. IMO, awareness should contain information on the intersection and the symmetric difference.

For example, the awareness {1,2,3} has of {1,b,c} should be drastically different from the awareness (-oo,1] has of [1,oo). The intersection of both pairs of sets is {1} yet the extent to which they differ is "larger" for the second pair.

As an alternate to the fixed point definition which is just a shot in the dark, how about for two sets X and Y, A(X,Y) is the ordered pair (X&Y,X$Y) where $ denotes symmetric difference. I don't know.

Whatever A(X,Y) is, I want there to be some kind of ordering so that A(X_1,Y_1) can be compared to A(X_2,Y_2) while also containing information about how much is in common and how much is different. The original definition does this. A(X,Y) was the subset of Y^X of functions with at least one fixed point. In order for there to be a fixed point, X&Y can't be empty. On the other hand, everything a function in A(X,Y) doesn't fix is potentially a point in the symmetric difference.

Perhaps I jumped the gun by actually writing down a definition. Perhaps the questions should be answered first:
1. Is {1,2,3} aware of {1,b,c}?
2. Is {1,2,3} more aware of {1,2,c}?

3. Is {1,2,3} aware of N? of R?
4. If yes, "more" aware of N than of {1,2,c}?

5. Is {1,2,3} more/less/as aware of {1,2,3} than R is of R?

Once we answer the questions, perhaps we can develop axioms for A(X,Y) and then hunt for a definition of A(X,Y).

My answers would be:
1. yes
2. yes
3. yes and yes
4. more
5. less

I would also want to speculate that under some assumptions the Whitney embedding theorm implies that if the universe is a 11 D manifold then it can be embedded in R^n where n=2(11)+1=23. Thus I would argue that if Max is correct, we will find human-style self awareness structure in a subset of R^23. I hope that's not "overly" speculative, whatever is meant by "overly"...

 

[Problem+Solve=Reason]

arildno, I think mathematicians, or at least this mathematician, already think there are things that defy intelligent discussion.

Just as an example, consider the Juila set of a polynomial over C[x_1,x_2,...,x_n] where n is a googleplex, so there are googleplex variables, whose total degree is the highest number you can think of. I don't know, maybe this can be discussed intelligently but you get the idea. There are already things that defy intelligent discussion.

I hope that awareness is not such a thing that defies intelleligent discussion.

Still restricting ourselves to sets...

Do we want to say {1,2,3} has any awareness of {a,b,c}?
Do we want to say that {1,2,3} has awareness of {1,2,3}?

I figure that working with finite sets will be easier but I could be wrong.

Yes, matt, it has something to do with interesection but it is more than that. IMO, awareness should contain information on the intersection and the symmetric difference.

For example, the awareness {1,2,3} has of {1,b,c} should be drastically different from the awareness (-oo,1] has of [1,oo). The intersection of both pairs of sets is {1} yet the extent to which they differ is "larger" for the second pair.

As an alternate to the fixed point definition which is just a shot in the dark, how about for two sets X and Y, A(X,Y) is the ordered pair (X&Y,X$Y) where $ denotes symmetric difference. I don't know.

Whatever A(X,Y) is, I want there to be some kind of ordering so that A(X_1,Y_1) can be compared to A(X_2,Y_2) while also containing information about how much is in common and how much is different. The original definition does this. A(X,Y) was the subset of Y^X of functions with at least one fixed point. In order for there to be a fixed point, X&Y can't be empty. On the other hand, everything a function in A(X,Y) doesn't fix is potentially a point in the symmetric difference.

Perhaps I jumped the gun by actually writing down a definition. Perhaps the questions should be answered first:
1. Is {1,2,3} aware of {1,b,c}?
2. Is {1,2,3} more aware of {1,2,c}?

3. Is {1,2,3} aware of N? of R?
4. If yes, "more" aware of N than of {1,2,c}?

5. Is {1,2,3} more/less/as aware of {1,2,3} than R is of R?

Once we answer the questions, perhaps we can develop axioms for A(X,Y) and then hunt for a definition of A(X,Y).

My answers would be:
1. yes
2. yes
3. yes and yes
4. more
5. less

I would also want to speculate that under some assumptions the Whitney embedding theorm implies that if the universe is a 11 D manifold then it can be embedded in R^n where n=2(11)+1=23. Thus I would argue that if Max is correct, we will find human-style self awareness structure in a subset of R^23. I hope that's not "overly" speculative, whatever is meant by "overly"...

Can you give me a detailed explanation of truth?

 

[Philocrat]

LET'S PUT ALL YOUR MATHEMATICAL MINDS TO GOOD USE!

You have all dsiplayed an impressive set of mathematical minds. Now, let's put them to good use. Since this project is about mathematicisation of 'AWARENESS'. Let's design a set of 'PARAPLEXES'. A Paraplex is simply a perfect part of any design, or system or entity. It has the following fundamental characteristics:

1) It can only perform one function, and one function only. It cannot perform more than one function.

2) It is 'FUNCTION-CRITICAL. This means that a paraplex is a part that when removed from a system renders the entire system to a hault. The system just stops working.

2) Because it is a perfect part of a whole, it is structurally and functionally indestructible. You cannot reverse-engineer it.


The standard theory is that:

A system made only of paraplexes is structurally and functionally perferct.

The question now is, can you guys formulate and formalise procedures for designing a paraplex, let alone a set of them? My argument is that, you cannot axiomatise, mathematicise a mind in a perfect way without formulating and formalising paraplexorial procedures. Note that this is only possible if you are one of those who accept the notion of 'DIVISIBILITY OF THE MIND' or the notion of a 'MULTI-PARTITE MIND'. This is so true because some versions of DUALISM hold that not only is the MIND 'NON-PHYSICAL' or 'IMMATERIAL' but also that the mind is 'INDIVISIBLE'. So, axiomatixing or mathematicising Awareness, you will have take all these into account.

So, here are the key points.

* you cannot design a mind or awarenes proper, let alone a 'PERFECT' one, without paraplexes,
* you cannot axiomatise or mathematicise steps for designing a paraplex.
without accepting the concept of a divisible mind.

If you fall into the trap of joining the bandwagon of accepting the notion of a NON-DIVISIBLE MIND, then kiss goodbye to the dream of mathematicising awareness, let alone designing a 'PARAPLEXED MIND'.

 

[PFanalog57]

I'll take that as a 'no', then.

Nonlinear Science:


http://www.faqs.org/faqs/sci/nonlinear-faq/

 

[matt grime]

You could then define the awareness A(X,Y) to be the triple (card(XnY),card(X),card(Y)) which seems to contain all the information you wish, and can be lexicograhpically ordered.

and russell, why are you pointing me to a basic introduction to something I have a degree in?

 

[PFanalog57]

You could then define the awareness A(X,Y) to be the triple (card(XnY),card(X),card(Y)) which seems to contain all the information you wish, and can be lexicograhpically ordered.

and russell, why are you pointing me to a basic introduction to something I have a degree in?

http://sulcus.berkeley.edu/FreemanWWW/manuscripts/IF8/99.html

Abstract

To explain how stimuli cause consciousness, we have to explain causality. We can't trace linear causal chains from receptors after the first cortical synapse, so we use circular causality to explain neural pattern formation by self-organizing dynamics. But an aspect of intentional action is causality, which we extrapolate to material objects in the world. Thus causality is a property of mind, not matter.

http://sulcus.berkeley.edu/FreemanWWW/manuscripts/wjfmanuscripts.html

 

[matt grime]

And, again, what's your point? Question about "awareness" between the two sets of real numbers and you point out something like this without explanation of what one is supposed to think you're talking about, and all aimed at me apparently. I've read plenty of papers like this, what's your point?

 

[PFanalog57]

Epistemological Problems of Perception:

http://plato.stanford.edu/entries/perception-episprob/

 

[arildno]

Excuse me, Russel:
But are you not able to provide arguments on your own??

 

[PFanalog57]

And, again, what's your point? Question about "awareness" between the two sets of real numbers and you point out something like this without explanation of what one is supposed to think you're talking about, and all aimed at me apparently. I've read plenty of papers like this, what's your point?

I am thinking that "awareness" between sets of numbers is hard to define. Awareness might be non-computable?

 

[matt grime]

They aren't even arguments, really. That last link points to a philsophical discussion about the nature of things in the physical world, and nothing to do with mathematical objects.

 

[phoenixthoth]

A(X,Y)=

(card(XnY),card(X),card(Y))

.

I want the class {A(X,Y): X and Y are sets} to be orderable. I don't wish it to be a well ordering or anything; just an ordering with trichotomy (is that what's called a linear order?).

You mentioned lex ordering; so would you say
A(X_1,Y_1) < A(X_2,Y_2) iff
card(X_1 n Y_1) < card(X_2 n Y_2) or
if card(X_1 n Y_1) = card(X_2 n Y_2) then
card(X_1) < card(X_2) or
if card(X_1 n Y_1) = card(X_2 n Y_2) and card(X_1) = card(X_2),
card(Y_1) < card(Y_2) ELSE
A(X_1,Y_1) >= A(X_2,Y_2) ?

Alternately, I want awareness to go up the more the two have in common and go down the more they differ (as in XOR). I know you can't subtract card(X'xor'Y) from card(XnY) but maybe some contrived 'thing' would go up with card(XnY) goes up and go down when card(X'xor'Y) goes down. Maybe the set of functions from X to Y having at least one fixed point does this. I was also thinking of the set of functions from X to Y for which XnY is a confining set (ie f(XnY) is a subset of XnY). Will play around and see...

 

[matt grime]

up, down, implies you want a total ordering: that is to say that given two objects either one is less than the other or they are the same. Well, that ain't going to work, is it? at least it seems more than unreasonablwe to ask about since you want to totally order something with 3 variables, and seeing as we can't even totally order the complex numbers in a way that agrees with the real numbers and their natural ordering this seems more than a little unrealistic.

all the sensible things you are asking for are encoded in that triple in some way, if you wish to attempt to totally order that in a sensible consistent way then good luck to you, but it will be almost totally arbitrary. The point about lexicographic ordering is that if you fix X and XnY then the various Y's will be totally ordered, and vice versa. It also records the cardinality of the symmetric difference, approximately, and it is not symmetric, so that the "awareness" X of Y is not the same as the "awareness" of Y of X, necessarily.

 

[PFanalog57]

Excuse me, Russel:
But are you not able to provide arguments on your own??

Yes, at the beginning of this interesting thread, phoenix wrote about Max Tegmark's ensemble hypothesis where mathematical existence equals physical existence. Tegmark's proposal is very appealing due to the symmetry considerations, since all mathematical structures would exist physically as well as mathematically. There would be no preferred existence over another, or no preferred laws!

But it appears that it will be impossible to mathematically model physical existence due to its non-computability.

For one, there is no way to precisely model the position and momentum of physical particles in OUR universe due to Heisenberg Uncertainty.


But if there are no preferred laws, since all mathematical structures exist physically as well as mathematically, then there exist universes where the position and momentum of particles CAN be known precisely. But that would mean that there are universes preferred over others, in that position and momentum can be precisely known.

The ensemble hypothesis becomes self contradictory.

So only a finite[or denumerably infinite] number of universes can physically exist. Mathematical existence does not equal physical existence.

 

[arildno]

What do you mean by "prefer"??
Why is a "universe" where momentum and positions can be know exactly "preferable" above other universes?


And, on another note, why should one even deign to consider the possible truth of, what, to me, is a patently silly idea? (That to each possible mathematical construct there necessarily must be a correlated physical object?)

 

[loseyourname]

And, on another note, why should one even deign to consider the possible truth of, what, to me, is a patently silly idea? (That to each possible mathematical construct there necessarily must be a correlated physical object?)

Maybe this idea becomes obviously ridiculous when one learns more about math, but to me, given the incredible diversity and complexity of physical processes in the universe, it seems plausible to suggest that any given mathematical construction can be used to at least approximately model some kind of physical phenomenon.

 

[PFanalog57]

What do you mean by "prefer"??
Why is a "universe" where momentum and positions can be know exactly "preferable" above other universes?


And, on another note, why should one even deign to consider the possible truth of, what, to me, is a patently silly idea? (That to each possible mathematical construct there necessarily must be a correlated physical object?)

http://home.si.rr.com/phyzika/Scientific%20American%20Parallel%20Universes.htm

The Platonic paradigm raises the question of why the universe is the way it is. To an Aristotelian, this is a meaningless question: the universe just is. But a Platonist cannot help but wonder why it could not have been different. If the universe is inherently mathematical, then why was only one of the many mathematical structures singled out to describe a universe? A fundamental asymmetry appears to be built into the very heart of reality.

As a way out of this conundrum, I have suggested that complete mathematical symmetry holds: that all mathematical structures exist physically as well. Every mathematical structure corresponds to a parallel universe. The elements of this multiverse do not reside in the same space but exist outside of space and time. Most of them are probably devoid of observers. This hypothesis can be viewed as a form of radical Platonism, asserting that the mathematical structures in Plato's realm of ideas or the "mindscape" of mathematician Rudy Rucker of San Jose State University exist in a physical sense. It is akin to what cosmologist John D. Barrow of the University of Cambridge refers to as "π in the sky," what the late Harvard University philosopher Robert Nozick called the principle of fecundity and what the late Princeton philosopher David K. Lewis called modal realism. Level IV brings closure to the hierarchy of multiverses, because any self-consistent fundamental physical theory can be phrased as some kind of mathematical structure.

 

[matt grime]

Maybe this idea becomes obviously ridiculous when one learns more about math, but to me, given the incredible diversity and complexity of physical processes in the universe, it seems plausible to suggest that any given mathematical construction can be used to at least approximately model some kind of physical phenomenon.

It does, in some sense, become ridiculous. It seems to me highly unlikely that the continuum hypothesis will have any use in a model, the notion of uncountable regular cradinals doesn't seem very useful, perhaps you have a use for the stable homotopy category of sphere spectra? This isn't to say there isn't one, or that one won't be fuond, but the objects of mathematical interest are very strange things indeed. The ones which people populalry know about, such as chaos theory, really aren't as big as people tend to think they are.

 

[arildno]

Now, on the evidence, it still seems somewhat likely that any real phenomenon can (in principle, at least) be modeled by a mathematical construct.

But if this is true , it does not at all follow that the converse holds!
On the face of it (and matt grime has a much richer reservoir to draw upon than myself), the converse seems very unlikely indeed.

 

[selfAdjoint]

Now, on the evidence, it still seems somewhat likely that any real phenomenon can (in principle, at least) be modeled by a mathematical construct.

But if this is true , it does not at all follow that the converse holds!
On the face of it (and matt grime has a much richer reservoir to draw upon than myself), the converse seems very unlikely indeed.

Well, Tegmark's level 4 multiverse permits anything to be regarded as physical, so maybe there's a universe for every math theory. I admit that's a whopping big stretch!

 

[matt grime]

Universe has a strict meaning in mathematics. Though I'll concede that, say a homotopy colimit exists in this universe, in the non-mathematical sense, roughly at the point when I stub my toe on one. Actually, that isn't unlikely, I suppose, so I'll make it one where the filtered index has cardinality Aleph-t where t is some very big ordinal.

 

[phoenixthoth]

this may amount to just a bunch of handwaving and numerology

1
If I continue to define random structures...

A(x,y) is the set of functions from x to y having at least one fixed point. For f in A(x,y), define Fix(f) to be the set of points in x that f fixes.

Define a relation (which is an equivalence), ~, so that f~g iff Fix(f)=Fix(g).

Then A(x,y)/~ is closer to what I call awareness than A(x,y). Each class in A(x,y)/~ is called (just for fun) an awareness tendril.

It wasn't clear to me before but is now that A(x,x)/~ is isomorphic to P(x)\{Ø}; for infinite sets, |A(x,x)/~|=|P(x)|.

R^23 is the set I conjecture to have human style self awareness (or at least a subset of R^23); this set is the flat space an 11D universe is diffeomorphic to.

|A(R^23,R^23)|=|P(R^23)|.

Question: is |P(R^23)|=|P(R)|? Henceforth, I will assume it is yes.

2
Consider this bunch of handwaving: on the DVD for the movie the matrix (or your favorite movie), the people and dialogue are representable by a string of 1's and 0's. Let's say this string is denoted s. Neo's dialogue is contained on some substring s'. Neo appears self-aware to me, how about you? And since he appears self-aware, then he might as well be self aware.

Another example is a teleconference between two beings. Their voices can be encoded in a sequence of 0's and 1's. This sequence, this string, is, I would say, a self-aware-structure (SAS).

Back to 1, I conjecture that within a set isomorphic to P(R), there is enough complexity for self-awareness. My plausibility argument is that myself, presumably a SAS, can be encoded in EVERY way in some element of P(R), or some set isomorphic to P(R).

Well, these are just ideas and I obviously am shooting blanks in the dark with a broken machine gun. Feel free to jump in and define awareness your own way if you don't like A(x,x)/~.

EDIT: From http://www.hep.upenn.edu/~max/multiverse.html

A digital universe?
From Ninad Jog, ninad@wam.umd.edu, Jul 21, 2003, at 2:09,
I believe that self-aware-substructures can arise in spacetimes with fewer than 3 space dimensions (n < 3) despite the absence of gravity. These SAS will evolve from what are currently known as Artificial Life forms or Digital Organisms that reside in habitable universes such as the Avida and Tierra artificial life software platforms. DOs can evolve only on specialized platforms with minimum-length instruction sets, so that any arbitrary mutation in an organism's genome (instruction) results in a different legitimate instruction from the set. [...] The cyber universe is qualitatively different from our own, but does that mean it's a separate type of universe (another level), or is it part of the level-II multiverse? I'll be most interested in your comments. Yes, the n<3 argument applies only for universes otherwise identical to ours, not to the sort you are simulating, which need indeed not have any meaningful dimensionality. I would term the DO "Cyber Universe" you simulate as part of our own, since we can interact with it even though the DO's, if they were complex enough to be self-aware, would as you say be unaware of our existence. They would derive that their universe obeyed "laws of physics" that were simply the rules that you had programmed. My guess is that the Level IV multiverse also contains such a cyber universe existing all on its own, without it being simulated on a "physical" computer. It's DO/SAS inhabitants couldn't tell the difference, of course. However, such a cyber universe could have an infinite implementation space and an infinite number of evolution steps; I suspect that any DO we can simulate on our current computers is way too simple to be self-aware in any interesting sense, and this would require a much larger implementation space to allow greater DO complexity.

 

[loseyourname]

It does, in some sense, become ridiculous. It seems to me highly unlikely that the continuum hypothesis will have any use in a model, the notion of uncountable regular cradinals doesn't seem very useful, perhaps you have a use for the stable homotopy category of sphere spectra? This isn't to say there isn't one, or that one won't be fuond, but the objects of mathematical interest are very strange things indeed. The ones which people populalry know about, such as chaos theory, really aren't as big as people tend to think they are.

I don't have any clue what any of those things are, so I'll just have to take your word that it's unlikely they'll ever find any application. Modern math may be far weirder than I can imagine.

 

[matt grime]

You weren't suppposed to understand them, hell, I don't really understand them. And these are on small part of one small part of... of mathematics.

 

[selfAdjoint]

I recall that Paul Cohen proved that the continuum hypothesis and the axiom of choice are independent of the other axoms of set theory and of each other. Taking the maximal multiverse view, there would exist four types of rational universes: Zornian Cantorian, Zornian non-Cantorian, non-Zornian Cantorian, and non-Zornian non-Cantorian. The last one is the one constrictivists prefer.

Stable homotopy categories and other categorical structures may exist in our universe! In the sense of being important defining properties of a TOE. See the new research by John Baez and Urs Schreiber.

 

[matt grime]

hmm, the categories are models of these phenomenon, and they've been known for a while, for instance cobordims are the morphisms in a triangulated category. It isnt' even clear that arbitrary homotopy colimits exist in triangulated categories. In fact we know that they don't as there are obstructions called Toda classes.

Would you say compact special unitary groups exist since that's all elementary particles seem to be... (joke)

 

[phoenixthoth]

Fact:
|A(x,y)/~|=|P(x&y)\{Ø}|

 

[phoenixthoth]

"continuum hypothesis will have any use in a model" I think I might have a use for it. Give me two months and I think I can connect it to Awareness.

Someone I came across told me that the awareness X has of Y is at least superficially covered in information theory in the context of "mutual information".

If I can just get a definition for the entropy of a set, H(X) and some kind of joint entropy, H(X,Y), then
A(X,Y), the awareness of X of Y could be modeled by
H(X)+H(Y)-H(X,Y).

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

I will now google for definitions of entropy of a set, if possible.

 

[phoenixthoth]

Seems as though entropy is only defined for random variables. Any clue how to define entropy for a set?

 

[phoenixthoth]

a new idea in SASs

From this site: http://star.tau.ac.il/~inon/wisdom1/node8.html
We get this quote of interest to this subject:

To refer to the genome as being self-aware is a very strong statement with far-reaching implications. The issue will be presented in a forthcoming publication [60]. I briefly describe here the main points needed for this presentation. Our logic and mathematics are based on the notion of a set composed of elements. Implicitly, the set is closed and static, the elements have a fixed identity (it does not change due to the fact that they are part of the set) and they either do not have internal structure or, if they do, it is not relevant to the definition of the set. The set is defined by an external observer, i.e., it is not a result of self-assembly of the elements under a common goal. The elements, being passive and of no structure, do not have any information about the set. The definition of sets leads to logical paradoxes (Russel-type, like the famous barber paradox) when we try to include a notion of self-reference. Russel and others have devoted much effort to construct formal axiomatic systems free of inherent logical paradoxes. Gödel's theorem [62,63] proved that they all have to be "incomplete", including the Principa Mathematica of Russell and Whitehead. It is important to emphasize that Gödel's theorem applies to closed systems which are also fixed in time. I propose that one has to take an entirely different approach and not start with the notion of sets of elements. I believe that here is exactly where the reductionist approach fails. We cannot reach self-awareness starting from passive elements, no matter how intricate their assembly. I propose to replace elements by agents, that possesses internal structure, purpose and some level of self-interest, and whose identity is not fixed. The notion of a set is replaced by a cell, which refers to a collection of agents with a common goal and mutual dependence. It also implies that the system of agents is open, i.e., it exchanges energy and information with the environment. I argue that, in order for a cell of agents to be self-aware, it must have an advanced language, i.e., a language which permits self-reference to sentences and to its grammar. The language also enables the individual agents to have information about the entire system.

So if I understand it right, a SAS is a structure equipped to make self-referential statements.

Do self-referential statements even exist in mathematics, I wondered. Then I cracked open "Mathematical Logic" (which is really a math theory of language in my opinion). In chapter 10, section 7, we have, "self-referential statements and Godel's Incompleteness theorem." I tried reading this and it will take some time to remember what all the notation means. There's a "fixed point theorem" which says something I don't yet understand. But just below the theorem, it says,

Intuitively, phi says "I have the property psi."

That means that phi is self aware!

phi and psi are wffs.

To go further, so we have wffs that are "self aware." These are precisely the self-referential wffs. We could then define a set to be self-aware if it is of the form {x|phi(x)} where phi is self aware.

Thoughts? Comments?

 

[selfAdjoint]

You missed the word "intuitively" by which the author is signalling you that he's going to lie for pedagogical purposes. Just like ascribing will-power to genes when talking about adaptation. It ain't so, McGee.

The statements are self-referent; their inner nature is determined by a refenerence to itself. They don't have awareness, but a great interrelated congeries of them might generate a kind of awareness as an "emergent" property.

 

[phoenixthoth]

You missed the word "intuitively" by which the author is signalling you that he's going to lie for pedagogical purposes. Just like ascribing will-power to genes when talking about adaptation. It ain't so, McGee.

The statements are self-referent; their inner nature is determined by a refenerence to itself. They don't have awareness, but a great interrelated congeries of them might generate a kind of awareness as an "emergent" property.

We can define self-awareness to mean, in relation to wffs, self-referentiality. But then, of course, we could define self-awareness to mean, in relation to functions, differentiability. What the who, right? The difference is that defining self-awareness to mean self-referentiality has roots in intuition, as indicated in the quote.

What makes you say that statements don't possesses awareness yet a great interrelated congeries of them might generate a kind of awareness as an emergent property? You have to then deal with the issue about when it becomes self-aware. How many interrelated self-referential statements are required to produce a self-aware structure; and what makes it self-aware and not the statements themselves?

In the plan offered where we define self-awareness to mean self-referentiality, you don't need to worry about emergent properties or how to interrelate the statements, etc. To top it off, it has a basis in intuition in the sense that the fixed point theorem says that there are wffs (statements) that when translated to English say, "I have property psi."

 

[selfAdjoint]

If a law is untenable then it won't last long, no matter who wants it too. If it works well as a part of physical reality, it could stick around for a while to come.

But I have an an awareness that isn't just self reference in the wff sense. Where did that come from? Not from the wffs! Apparently it's emergent; and yes, with all emergent properties, since they cannot be tracked back to the elements, there arise difficulties of bounding. How many grains of sand does it take to make a pile, if a pile can slump, but a grain can't?

 

[phoenixthoth]

But I have an an awareness that isn't just self reference in the wff sense. Where did that come from? Not from the wffs! Apparently it's emergent; and yes, with all emergent properties, since they cannot be tracked back to the elements, there arise difficulties of bounding. How many grains of sand does it take to make a pile, if a pile can slump, but a grain can't?

I never said that all SASs have to be self-referential wffs; at least, I would withdraw that if I did. I said that self-referential wffs are self-aware. I (no longer) want to imply that human self-awareness stems from wffs directly.

However, you say that your awareness is not from wffs. How do you know this? I would say that our psyche may be reducible to many, many wffs such as:
if I am pricked then I will say "ouch"
I am made of mostly water (this would come from a self-referential wff)
if the blood is low in oxygen, then increase breathing rate
etc. Obviously, proving this should not entail actually reducing the psyche to many wffs by explicitly spelling out what the wffs are. If it's true that our psyche is a collection of wffs (or just one wff that is the conjunction of component wffs), then this will be proved some other way.

Well, actually the heap of sand dilemma can be solved, not necessarily to your satisfaction, by fuzzy logic. Same with self-awareness. I didn't expect to find anything remotely obviously self-aware and I once expected that self-awareness should be a measureable and would vary between 0 and 1 (for example). But now considering self referential statements I would say that if a statement can refer to itself it has self-awareness. Likewise, if a wff can refer to another formula, then that wff has awareness of the other formula.

Obviously, or at least apparently, human awareness is not like this. But we wouldn't really expect it to because our awareness changes from moment to moment while a wff's selfawareness is not subject to the passage of time.

 

[selfAdjoint]

Do you mean real fuzzy logic or just the buzz word that gets tossed around? If the sandpile problem, which is well known, has been solved by any method, I have not heard of it. Have you any citations?

 

[phoenixthoth]

If you feel inclined to pay:
click here.
I'd like to know what those "natural assumptions" are.

I think some of this article is summarized here.

The key seems to be this so called "almost true" unary relation.

But I'd rather not talk about Sorites Paradox.

What's your opinion: does the following statement demonstrate on some level self awareness:
This statement consists of letters and spaces and punctuation.
?

When I tell some people that statement, their opinion is that it is self-aware. What is awareness anyway? I read somewhere that awareness is "poised for appropriate interaction with the immediate environment." Here, appropriate interaction with the immediate environment means appropriate interaction with itself. Then the appropriate interaction would be to say something, which it does: about itself. And if you prefer, substitute the word "mean" for "say."

 

[selfAdjoint]

What's your opinion: does the following statement demonstrate on some level self awareness:
This statement consists of letters and spaces and punctuation.
?

I do not find the statement to be self-aware. In this case I think it is because the self-reference is STATIC. The statement does not transform in any way because of the reference. Whereas the kind of self-reference generated by Goedel numbering, where an arithmetic statement, in expressing its arithmetic value, turns out to also be making a statement about its own provability, that is much closer to awareness, but I don't think Goedel would have thought so. Certainly Wittgenstein who disdained all work with wffs as cheap tricks, wouldn't have.

I guess my minimum and not claimed to be sufficient for awareness would be something you could call background-free dynamic recursion. Whatever that means!

 

[phoenixthoth]

I do not find the statement to be self-aware. In this case I think it is because the self-reference is STATIC. The statement does not transform in any way because of the reference. Whereas the kind of self-reference generated by Goedel numbering, where an arithmetic statement, in expressing its arithmetic value, turns out to also be making a statement about its own provability, that is much closer to awareness, but I don't think Goedel would have thought so. Certainly Wittgenstein who disdained all work with wffs as cheap tricks, wouldn't have.

I guess my minimum and not claimed to be sufficient for awareness would be something you could call background-free dynamic recursion. Whatever that means!

When I say: "I am a man," I suppose that does change me in that while up to ten minutes ago, my memory did not contain myself writing that I am a man, now it does. My conclusion isn't that a self-referential wff (or perhaps even a statement in natural language) is not self-aware; it is that the two awarenesses are different.

Your example of a self-referential wff of the kind you listed being "closer to awareness" is precisely the type of wff mentioned in the very first post when I resurrected this thread. The one in my logic book quoted as intuitively meaning, "I have the property psi." For example, we can show that a phi exists that says, "This sentence has at most 1000 symbols". And the theorem I'm referring to is Tarski's self-reference lemma which was not named as such in my book where it was called a fixed point theorem.

Funny you should mention background free dynamic recursion. I wonder if that's like the ideas of Chaotic Logic:
http://www.goertzel.org/books/logic/chapter_seven.htm