|
Author
|
Topic: Cosmogony, Holography and Causality
|
Rex Kerr
Member
Member # 632
|
posted 15. November 2003 05:15
jason asks:
quote:
why can we equate the number of integers able to be defined by a 120 character string in ASCII with the number of integers able to be defined by a 120 character string in English
English is written using letters and a few punctuation marks. Sure, you can make up weird new symbols, but they aren't really English. If you prefer, I could say anything that could be displayed on 1024x768 24-bit monitor that uniquely identifies a number to a given person; then there is some minimal number undefinable to that person, and the example goes on as before, except with 120^256 replaced by 2^(24*1024*768).
Aliet suggests:
quote:
I think Rex is moving to algorithmic complexity and entropy.
But this isn't what I was getting at. jason's idea that I was interested in the definition of definition is closer, but still not it. I'm still hoping that Russel's got it, but I still can't tell whether he does or not.
Russel said:
quote:
----------------------------------
...[Gödel] showed that any logical system capable of supporting Peano's axioms could not be both consistent and complete; and could not prove itself consistent without proving itself inconsistent
----------------------------------
Unsoundness is an inconsistency, in that a proposition A, and its negation ~A are both true. Goedel's incompleteness theorem DOES apply to Rex's problem.
Er, so you are saying that everything we're saying is inconsistent by Gödel's incompleteness theorem? For someone fond of repeating "A or ~A", that's a bizarre position to take.
There is a good reason why Gödel's Incompleteness theorem is called an Incompleteness theorem rather than an Inconsistency or Unsoundness theorem. It is because we do not believe that arithmetic is unsound or inconsistent. Certainly, the inconsistency of arithmetic would completely derail the CTMU (and render it both true and false).
So, no, Gödel's Incompleteness theorem doesn't really get at the problem here.
Maybe you can explain why you provided a link to the ASM machine? If you can do that, maybe it will help you and others understand why I am so worried about precision in definitions.
IP: Logged
|
|
Aliet Jacob
Member
Member # 908
|
posted 15. November 2003 05:35
quote:
syndiffeonic relation must still hold between sub-syntaxes within reality. Covered directly in the paper.
Oh, so now we have common sub-syntaxes?
I thought it was 'common syntax'?
You will have to make up your mind Jason. And I can assist you understand exactly why there is no common "sub-syntax" (or whatever you will choose to call it in your next post) in me and the mountain. How organelles
You needn't read the CTMU paper again to grasp that.Geological activity doesn't get to determine my height.
You could simply define syntax, define human syntax, define a mountanin's syntax, then illustrate how you share a common sytnax with a mountain. Since you are soooo busy, that will make quick your work and will eliminate the necessity to come up with phrases like sub-syntax.
quote:
Yes, but it is complete within itself.
Oh, so now we have complete within itself and incomplete without itself?
And then you turn around and say:
quote:
Aliet, I concede that you make a good point about incompleteness, and the complexity of logical systems. All logical systems of any complexity are, by definition, incomplete.
You will have to make up your mind Russel.
IP: Logged
|
|
chimp
Member
Member # 333
|
posted 15. November 2003 06:29
quote:
Rex:
Maybe you can explain why you provided a link to the ASM machine? If you can do that, maybe it will help you and others understand why I am so worried about precision in definitions.
http://www.eecs.umich.edu/gasm/guide/section3_1.html
quote:
The evolving algebra thesis asserts that evolving algebras are such versatile machines. The thesis suggests an approach to the notorious correctness problem that arises in mathematical modeling of non-mathematical reality: How can one establish that a model is faithful to reality? The approach is to construct an evolving algebra A that reflects the given computer system so closely that the correctness can be established by observation and experimentation. (There are tools for running evolving algebras.) A can then be refined or coarsened and used for numerous purposes.
CTMU is based on propositional logic, not arithmetic. Propositional logic has mathematical closure.
Rex, are you worried about the precision in definitions? So, it all boils down to semantics, not syntax?
Interesting ![[Wink]](wink.gif) If I can understand Rex, and Gedankin's arguments, they are saying that it is totally incorrect to establish first principles "a priori"?
I might agree with Rex and Gedankin, but I must try to approach the CTMU with a totally unbiased perspective, in an effort to understand whether it is, or is not, possible to derive first principles, that correspond to objective reality, using logic.
Can the universe be explained as a "mathematical object"? It is the only way that science CAN approach it.
quote:
Aliet:
You will have to make up your mind Russel.
And you will have to "get a clue", Aliet...
[ 15. November 2003, 06:37: Message edited by: Russell E. Rierson ]
IP: Logged
|
|
Rex Kerr
Member
Member # 632
|
posted 15. November 2003 16:29
Yes, you've basically got it.
The reason I chose that paradox is because the meaning seems so astoundingly clear, and the logical manipulations are so straightforward, and yet we end up with a contradiction anyway.
I also chose that paradox because the semantics have implicitly assumed a solution to the Halting problem. Suppose we set a Turing machine on the task: let N be the smallest number not produced by a Turing machine program P of initial length less than L and starting values less than V in no more than S steps. We'll make L long enough so the Turing machine that does this can be coded in less space. But when the Turing machine gets to itself, it'll have to run all over again, causing an infinte recursion that will take more than time S...until the master Turing machine cuts it off at time S and returns "didn't halt in time S, didn't define a number" as a result. The original machine will eventually halt after more than S but fewer than around V^L * S steps, and will return a number.
This strikes me as very similar to the problem of infinitely precise definitions. Infinitely precise definitions are not enough; they also need to avoid the trap we fell into above. Abstract State Machines probably avoid the trap in the same way that Turing machines do, but we have another problem: how do we relate the world to our machines? We're going to have to use some mechanism for that, and the mechanism is presumably us, and then we have exactly the same problem as before we introduced the machines.
The problem with using propositional logic or first-order predicate logic or whatever--with no model!--is that we are trying to form a Cognitive Theoretic Model of the Universe. Logic alone, with no premises, is consistent with every model.
IP: Logged
|
|
chimp
Member
Member # 333
|
posted 17. November 2003 01:52
quote:
Christopher M. Langan:
Perhaps you mean to assert that no matter how hard we try, we are unable to formulate accurate models and descriptions. But in that case, we need merely restrict consideration to raw perception and realize that making an observation immediately rules out any inconsistent observation.
Christopher M. Langan:
Gedanken says that raw perception entails maximum uncertainty and unreliability. Since physical reality is perceptual, this amounts to saying that physical reality itself is unreliable, a scientifically meaningless assertion. Scientific observation is anything but imprecise; experiments are generally undertaken with very precise instruments and duly replicated by other experimenters, something which could not happen if gedanken's notions about reality were correct.
Christopher M. Langan:
Gedanken: "When we are left with only raw perception, we are also left with the greatest uncertanty." [My italics; note that "greatest" means "maximal".]
Gedanken, a few short minutes later: "This of course is highly variable, thus hardly 'maximal'"!
Whatever you say, gedanken! (I guess this is what happens when logic and precision go bye-bye.)
The universe is observed to be logically consistent. A highly confirmed principle that is regarded as a "fact?" is the principle of Conservation of Energy
The total energy for a closed system is constant. A = A .
Gedankin and Rex are refuted again?
[ 17. November 2003, 02:00: Message edited by: Russell E. Rierson ]
IP: Logged
|
|
gedanken
Member
Member # 594
|
posted 17. November 2003 22:09
I refuse to entertain a repetition of the same arguments that have been explained several times.
Yes, indeed, we have considerable degree of accuracy of the model. In principle conservation of energy may obey observation in the limit, if it were possible to perfectly produce measurement. (But of course it is not). So this is still claiming that either the model is crisp thus a matter of models, or claiming that the model is accurate, which is subject of imperfect measurement. And since in principle one cannot actually test "to the limit" the truth of the proposition that the model is accurate in the limit cannot be tested.
But I'm going to quit answering the same question.
[ 17. November 2003, 22:10: Message edited by: gedanken ]
IP: Logged
|
|
chimp
Member
Member # 333
|
posted 18. November 2003 01:26
quote:
Gedankin:
I refuse to entertain a repetition of the same arguments that have been explained several times.
Yes, indeed, we have considerable degree of accuracy of the model. In principle conservation of energy may obey observation in the limit, if it were possible to perfectly produce measurement. (But of course it is not). So this is still claiming that either the model is crisp thus a matter of models, or claiming that the model is accurate, which is subject of imperfect measurement. And since in principle one cannot actually test "to the limit" the truth of the proposition that the model is accurate in the limit cannot be tested.
According to your side of the argument Gedankin, the world[universe] is not really logical, it only appears to be? Yet, the universe does not contradict itself, because if the universe DID contradict itself, perception, or existence, could not be trusted. Therefore, I can put one foot in front of the other, without worrying about suddenly flying off at superluminal speeds...
Certain logical rules hold for certain conditions.
So I am trying to imagine how the universe contradicts itself, since Gedankin explains that reality is not actually logical? and perception is unreliable?
Analytic propositions follow from mathematical definitions, while synthetic propositions make connections between things in the world not related strictly by definition. Synthetic propositions make statements about things that can be empirically tested. So the real challenge is to discover the intersection between the definitively axiomatic, and the testably contingent, phenomena.
Hopefully, the "useful fictions" will be replaced by necessary truths.
[1.]Necessary Truth: If the universe did not hold itself together, or was not held together by some constraint, we would not exist.
YES or NO ?
[ 18. November 2003, 01:30: Message edited by: Russell E. Rierson ]
IP: Logged
|
|
DN
Member
Member # 996
|
posted 18. November 2003 09:50
quote:
[1.]Necessary Truth: If the universe did not hold itself together, or was not held together by some constraint, we would not exist.
YES or NO ?
The quick reductio:
[1.]Necessary Truth: If the universe did not repel itself apart, or was not being pulled apart by some antigravitational force, we would not exist because the universe would have imploded.
YES or NO?
The principle of conservation of energy is not logical any more than the principle of superposition is.
[ 18. November 2003, 09:53: Message edited by: DN ]
IP: Logged
|
|
Rex Kerr
Member
Member # 632
|
posted 18. November 2003 19:25
There is a difference between reality and a model of reality, at least in my model of reality.
It is not so much that the universe contradicts itself, because the universe (unless we believe the CTMU) isn't a bunch of truth-statements floating around. We are the ones who introduce the idea of infinitely precise categories and then operate on these categories.
The universe contradicts itself all the time, according to our models of it. A man is inside a room or outside of a room. A hand is a part of a man. If X is in Y, and Y is in Z, then X is in Z. Therefore, the man and his hand are inside the room, or the man and his hand are outisde. Uh-oh, the man is in the room and he stuck his hand outside. Contradiction in the universe!
Whenever there is a contradiction, it is up to the modeler to choose how to resolve it. We resolve it by changing our categories. For example, with the man, we can change the binary "inside or outside" to "inside or outside or (part inside, part outside, and possibly a measure zero volume at the inside-outside border)". But the universe isn't doing this inherently. We are. The universe does not contradict itself because we don't let it!
But we mustn't get the nature of the universe confused with our strategy when faced with contradictory observations. As a matter of speculation, one could suppose that the universe has inherent categories and it refines those much like we do. But even if it didn't, we still could refuse to let the universe contradict itself.
(Indeed, when you have a single electron taking two paths simultaneously and then interfering with itself, that's quite a contradiction. Yet we stubbornly refuse to allow that to be a contradiction and instead redefine terms like "electron", "path", and "self" until the contradiction goes away.)
IP: Logged
|
|
chimp
Member
Member # 333
|
posted 19. November 2003 02:20
quote:
DN:
The principle of conservation of energy is not logical any more than the principle of superposition is.
The universal constraint that governs the "anti-gravity" also governs the incorrect description of "flying apart"? The universe holds itself together.
Are you saying that the principle of conservation of energy is not logical?
Total energy for a closed system is constant: E_s = E_s
A = A
Conservation of energy, is logical.
quote:
Rex:
The universe contradicts itself all the time, according to our models of it. A man is inside a room or outside of a room. A hand is a part of a man. If X is in Y, and Y is in Z, then X is in Z. Therefore, the man and his hand are inside the room, or the man and his hand are outisde. Uh-oh, the man is in the room and he stuck his hand outside. Contradiction in the universe!
Rex, is this a deliberate oversimplification? A real contradiction would be where the living being is completely-physically in room A and also completely-physically in room B, and rooms A and B are separated by arbitrary distance d, that is to say, A and B are not next to each other. Let them be 1000 miles apart, to really illustrate the paradox. It does not happen. Model or no model, there is no contradictory universe, only people that contradict themselves with irrational thinking...
quote:
Rex:
(Indeed, when you have a single electron taking two paths simultaneously and then interfering with itself, that's quite a contradiction. Yet we stubbornly refuse to allow that to be a contradiction and instead redefine terms like "electron", "path", and "self" until the contradiction goes away.)
I recall reading that not only electrons interfere with themselves, but also neutrons and atoms.
I found a page that explains wave-particle duality for carbon 60 molecules :
http://physicsweb.org/article/news/3/10/12/1
quote:
Wave-particle duality seen in carbon-60 molecules
15 October 1999
The formation of an interference pattern when a beam of particles passes through a double slit is the classic signature of the wave-particle duality of quantum particles. Wave-particle duality has been observed with electrons, atoms and small molecules. Now Markus Arndt, Anton Zeilinger and co-workers at the University of Vienna in Austria have observed wave-like behaviour in a beam of carbon-60 molecules - which are an order of magnitude larger than any other particles for which quantum interference effects have been observed (M Arndt et al. 1999 Nature 401 680).
Wavelength = [Planck's constant]/[momentum]
The electron has a specific wavelength determined by its momentum and Planck's constant. Certain rules must hold, for certain conditions. A bowling ball has an extremely short wavelength, shorter than the electron, which prevents it from quantum mechanically interfering with itself.
DxDp >= hbar/2
One cannot physically be at the North Pole and the South Pole at the same "time", relatively speaking of course. Logical parameters
[ 19. November 2003, 02:29: Message edited by: Russell E. Rierson ]
IP: Logged
|
|
Rex Kerr
Member
Member # 632
|
posted 19. November 2003 05:02
Ah, well, never mind. I seem unable to convey the critical distinctions between content and structure, and between model-of-reality and reality-itself.
IP: Logged
|
|
DN
Member
Member # 996
|
posted 19. November 2003 10:17
quote:
Are you saying that the principle of conservation of energy is not logical?
Is the principle of superposition logical?
Is madness logical? Is illogic logical?
If illogic is not logical, then we would be incapable of (re)cognizing it - right?
Therefore illogic is logical and non-existence exists.
IP: Logged
|
|
DN
Member
Member # 996
|
posted 19. November 2003 10:33
Oh, and Russel?
While you are at it, explain why the principle of superposition is logical. And how you arrive at that judgement, because so far, the attribution of phenomena as logical seems to be done in an entirely arbitrary and subjective manner.
CTMU Argument
1. The syntax of cognition is logical.
2. Logical syntax can only process/recognize that which is real or logical.
3. We can recognize reality.
4. Therefore reality is logical
Refutation
The reductio ad absurdum of the CTMU argument therefore is:
We can recognize illogicity, therefore the syntax of cognition is illogical.
Therefore CTMU is illogical.
On the other hand, when an argument yields absurdity when put to test, thats the sure test for whether or not the argument is sound.
IP: Logged
|
|
DN
Member
Member # 996
|
posted 19. November 2003 10:43
quote:
The M=R or Mind Equals Reality Principle asserts that mind and reality are ultimately inseparable to the extent that they share common rules of structure and processing...To put it another way: if the “noumenal” (perceptually independent) part of reality were truly
unrelated to the phenomenal (cognition-isomorphic) part, then these two “halves” of reality would neither be coincident nor share a joint medium relating them. In that case, they would simply fall apart, and any integrated “reality” supposedly containing both of them would fail for lack of an integrated model. Where M (mind) is identified with cognition and R (reality) with physically-embodied information, M=R says that reality everywhere consists of a common substance,
infocognition, having the dual nature of mind and (informational) reality.
Langan, M.,The Cognitive-Theoretic Model of the Universe: A New Kind of Reality Theory, p.22
Based on the above reasoning, since cognition can recognize illogic, then illogic is isomorphic to cognition: because there must be a joint between them to integrate them. If there was no medium linking cognition to illogic, cognition wouldnt recognize illogic.
Also, they must have a common syntax (common rules of structure and processing), otherwise illogic would be unrecognizable.
Would you disagree Russel?
[ 19. November 2003, 10:48: Message edited by: DN ]
IP: Logged
|
|
chimp
Member
Member # 333
|
posted 19. November 2003 14:59
DN, you remind me very much of Aliet's posts.
According to the principle of multiplex unity, reality is a self resolving paradox. Thus cognition recognizes paradox and resolves it.
IP: Logged
|
|
|
Printer-friendly view of this topic
