You’re welcome, erislover.
What is it with the people in this thread? Are they all so conditioned by domain-specific terminologies that they can’t use their reason?
I find it difficult to accept that the people here don’t understand the meaning of the word “complex”.
I find it difficult to believe that you suppose that in certain contexts a word like “complex” could be ambiguous, undefined, or mean something unexpected from a dictionary definition. No?
In certain contexts? No. In this context? Yes.
That’s not exactly what Godel says, though, is it?
Are we still talking about Turing Machines? You want a TM that enumerates all true statements of number theory? Here’s one: the machine that enumerates all strings. It generates new strings according to a finite set of rules. It will produce any string you want it to.
Perhaps your observation is less earth shaking than you think it is. You observe that a particular TM (the one that tries to enumerate true statements of number theory starting from a set of axioms) will not generate a particular string. And somehow this is a restriction on computation.
Here’s another: there are strings that the TM that enumerates all strings containing only zeros will never produce! Astounding, but true.
I guess I’m not much of a philosopher, so I’ll stay away from that. But I’m pretty sure that this is the sticking point. Nobody but you seems willing to accept this claim.
So you are claiming that, because you interpret the universe as a model of number theory, there is some “state of affairs” that represents a true statement of number theory but can never be reached.
If I find a different encoding, say I swap all your protons and neutrons with each other, then does that suddenly make that “unreachable” state of affairs reachable, and render another unreachable? Do you see the problem here?
And I’m not really much of a physicist, either, I guess. But I wouldn’t be surprised if there were unreachable states of the universe (say the one where I’m traveling down I-494 at rush hour in a Geo Metro at a velocity of c).
I just don’t think that has anything at all to do with math or computation.
But you are talking about the science of computation, no?
TVAA
You have a theorem in mathematics and a construct of computer science.
If you want to draw meaningful conclusions from those things it is a good idea to use the terminology appropriate to those fields. For instance, the “common everyday meaning” of “complexity” is of absolutely no use in showing that set of elements that model a system is necessarily bound by the limitations of the system being modeled.
There is no hard connection between those two concepts, and saying “complexity” does not draw a line of either mathematical or logical implication.
You only have dots. You have drawn no connections. Because you refure to speak in a language specifically designed to make such connections, it is difficult to ever know whether you might have a point.
I have shown that your statements are untrue using the terminology appropriate to the field. You say that you have no interest in using appropriate terminology.
Okay, then the onus is very much upon you to draw the lines in specific detail. The reason that fields of stufy develop specialized usage is so that many things can be communicated subtextually. Since you reject the conventions appropriate to your subject, you don’t have access to any such contextual information.
Personally, it seems a silly choice on your part, but if that’s the way you want to proceed then it is certainly your right to do so.
Stupid?
And what word would you use to describe starting a discussion about mathematics and computer science and then calling mathematicains and computer scientists stupid for using teh terminology appropriate to those fields?
If you like to think “having more parts” is the only way to measure “complexity”, then enjoy your little world of irrelevance. You have hit upon yet another thing that is entirely unconneted to GIT.
Wow, from respectfully disagree to get off the boards. Here is some English language for you.
[ul][li]The OP is wrong.[/li][li]You misunderstand and misapply GIT with rampant abandon.[/li][li]You attempt to disguise this (perhaps even to yourself) by hiding behind ambiguities in language.[/li][li]When pressed, you not only fail to provide a rigorous argument but turn insulting to those who apply the appropriate language to questions of Godel and Turing Machines.[/li][li]You use words that sound like they belong in mathematics or computer science but specifically deny the meaning appropriate to those contexts. Then you pretend that these “pseudo-terms” actually carry some implication for a theorem of mathematics and a construct of computer science.[/li][/ul]
The problem is not the language, though, it is your appalling lack of rigor. Any case that can be made in a formal language can be made in English, but you need to be very rigorous and very specific in your use of language and your demonstration of implication.
You have failed miserably on both fronts.
And how is “similarity” meassured, absent an interpretive consciousness?
Okay. A model is something that behaves like another thing. So, each is equally a model of the othe, correct?
It is not a consequence for which you have provided a valid argument. Thus, it is an assertion.
The assertion was: the movements of electrons are sufficiently complex that they “contain” arithmetic. Is this another of your “plain English meanings”? Does “contain” now mean “can be used to model”? Does a pen and paper contain the Mona Lisa?
Is it therfore valid to say that the Mona Lisa restricts the behavior of pen and paper?
It is not my task to connect the dots of your argument. That task is yours.
Nevertheless, if it makes you feel better to imagine that I am simply too stupid to see the obvious connection that make your case, feel free. I care not at all what whims of today rule your image of me. Please, enlighten this poor stupid Mundi. Draw the lines.
You can use whatever language you wish, but do remember that I am stupid. I shall demand that you make every line very clear and very explicit. Otherwise, I am sure I will lack sufficient intelligence to finish the picture.
Well, since you object to the terminology of mathematics and computer science, you had better define “system”, too.
You are wrong, even under your own definition of complexity. For instance, the psudo code:
[ol][li]Read X from tape[/li][li]write X[/li][li]loop to 1 until end of tape[/ol][/li]can be implemented in any numbe rof cimputer languages. The amount of data required to perfectly define that algorithm does not change based upon the scale of input.
I know it is too much to ask that you use the language of mathematics of computer theory appropriately. Is it too much to ask that you use your own chosen language consistently?
Actually, I am one among several who have been arguing that field-specific concepts are best addressed with the appropriate terminology. Apparently, your own definition of “appropriate” is “what TVAA wants.”
Okay, I will stop trying to get you to use the terminology of GIT and TM correctly. Despite what you might think, this will not make your task easier.
Prove it.
GIT is absolutely no help to you here. GIT says nothing about “possibilities not capable of being manifested within a system”. You appear to be suffering from the misunderstanding that every true statement in a Peano Axiomatization has a “possible proof”, but GIT just won’t let us manifest them. This is incorrect. GIT says that no possible proof exists for some true statements.
In other words, the pen and paper can write anything that it is possible to write. Similarly, every possible statement of arithmetic can be generated. GIT stops neithe rof those. It just stops any elements of the first from proving all true elements of the second.
Sorry, I’m too stupid. Why don’t you connect the dots. After all, your the one who imagines that he sees a picture.
You keep saying things like this.
You keep forgetting to prove them.
Yes.
By the definitions that I offered, my statement is rigorously correct. Apply your own definitions if you wish, but please stop making ignorant statements about the proper application of terms as used in computer theory.
I understand. I fear that you have now lost all hope of ever understanding the proper scope of GIT and the limitations of TM, but I confess that it seemd there was little enough hope to be lost on that front.
Do go ahead and start drawing nice rigorous and well-defined lines using regular English, though. I would be happy to have my pessimism proved wrong.
I wish that I could find it difficult to believe that someone would use the word “complex” in a dicussion about computer theory and not only demand that it not be understood in the context of comuter theory but call someone stupid for using it in that context.
Unfortunately, I have all too much evidence that some people do indeed embrace that particular juxtaposition of behaviors. Lucky me.
** A system can be emulated by another only if the elements of the emulating system can represent the elements of the emulated system. The emulated system must therefore contain fewer elements than the emulating system. If it contained as many, it would be identical to the emulating system – or in other words, the system would represent itself, which is the trivial case. If it could consider more, the emulated system would contain elements not represented in the emulating system, which violates our premise.
So you don’t like the word “complex”? How about “simple”? The emulated system must be simpler than the emulating system.
** You’ve shown that my statements are invalid in a certain language. I say that I’m not speaking that language.
** You’re not being stupid because you’re using that terminology. You’re being stupid because you’re acting as if I’m using it, then claiming that my statements are wrong because they’re inconsistent with the terminology.
** The more “parts” the system has, the more relationships exist within it, and the more complex it is. The fewer parts, the fewer relationships, and the simpler it is. These are not difficult concepts to grasp.
Where has my use of language been vague? Where has my use of language been inconsistent?
** “Similarity” is not measured – it is, just as the properties of the system are, regardless of whether a mind observes them. Do you think that the Game of Life acts differently when no one’s looking at the computer screen?
** No. The meta-grid can easily be disrupted without disrupting the grid – adding or removing a single active square in a crucial location is often sufficient. Disrupting the grid (by interfering with its implementation in the computer, for example) immediately disrupts the meta-grid. The higher-level system is dependent on the lower-level in the same sense that the floors of a tower are dependent on their foundation. Take away the foundation… and the tower falls.
** No, of course not. Don’t be ridiculous. The possible relationships between atoms contains the Mona Lisa. The Mona Lisa is just a particular configuration (technically a category of configurations… it’s never the same twice, after all).
** No. The limitations of atomic interactions place inherent limits on the nature of possible Mona Lisas.
** Very well.
**
If X is variable, the larger it becomes, the more cycles the above program will run, and the more memory will be required to represent it. The implementation of the algorithm will change as X changes.
** You’re aware that I’m not using that terminology. Do you feel that it’s appropriate to apply its definitions to statements not made within it?
** If the extent of the Game of Life grid is infinite, the meta-grid can represent any configuration that the grid can. If the grid is finite, the grid will ways be able to take on more configurations than a meta-grid emulated by that grid could. In order for a meta-grid to be able to handle the same configurations as a finite grid, it would need to be emulated on another finite grid much larger than the first. It would therefore necessarily require more resources to function.
Oh, for fuck’s sake. It can be done, therefore it’s necessary? I know better than to ask you to support your assertions, so go on, think you’re right.
** NO.
GIT shows that, for any given system of axioms capable of representing arithmetic, there will be statements are true but that the system cannot prove. It does NOT follow that there are statements that are true but cannot be proven in ANY system.
It may well be the case that some statements can never be shown to be true, but GIT doesn’t imply this.
The statement “This statement cannot be proven in system X” might be provable in system Y, which adds one or more axioms to system X.
Ultrafilter: It’s known that systems for generating theorem models can be constructed within the physical world. Specifically, it’s known that it’s possible to make systems that can demonstrate arithmetic. Therefore, we know that the underlying rules of the physical world are capable of demonstrating arithmetic. This is necessarily the case.
I’d love to see proof of the above. And it seems to me that if you succeed in proving the above, you will have disproved Church’s hypothesis. But I haven’t studied this stuff in years, so please enlighten me.
TVAA
I’m still watiing for those lines to be drawn.
For the rest, it was obviously not apparent to me (or several others) that you were rejecting the teminology of the fields appropriate to your questions in all cases. Your latest post does so again by using “algorithm” in a manner that is inconsistent with the definition specific to computer theory. Your latest post also continues to “chastise” me for past statements made when I thought that you might wish to apply the terminology standard ot the fields you make broad pronouncements about.
So, you have proven nothing except that you like to insult people.
Bravo.
There are holes in your latest post, too, but until you actually provide the detailed proof that you have promised it seems silly for me to expend any effort refuting hand-waving and broad declarations. Present your proof.
Until then , you are simply making unsupported pronouncements.
Well, the use of the word “model” has been inconsistent:
[ul][li]A model is a system – a set of interactions – that manifests properties similar to those of another system. This is inherent in the dictionary definition of the word.[/li][li]No. The meta-grid can easily be disrupted without disrupting the grid (In response to “they each model the other”).[/ul][/li]So, here you are adding another requirement to your definition for “model”. I know you like dictionaries, so please provide a dictionary citation that injects a requirement upon “disruption” into the word “model”.
Then you can answer the question: Do life and “meta-life” model each other if the “meta-life” is not being instantiated on the same grid as the game of life? That removes any potential issues of “disuption” and concentrates on your definition of “model” as it relates to your definition of “complexity”.
The use of the word “model” has also been vague:
[ul][li]A model is a system – a set of interactions – that manifests properties similar to those of another system. This is inherent in the dictionary definition of the word.[/li][li]“Similarity” is not measured – it is, just as the properties of the system are, regardless of whether a mind observes them. Do you think that the Game of Life acts differently when no one’s looking at the computer screen?[/ul][/li]So a model is a system “similar” to another system, but “similarity” is not measured. It “just is”. Thus, I could say that a system defining the orbital motions of planets models a the digestion of a snickers bar. Hey–similarity “just is”, right. We have no way to measure it, no way to create a cutoff, and no human decision is required to judge one system as “similar” to another.
TVAA: It’s known that inconsistent formal systems can be constructed within the physical world. Specifically, it’s known that it’s possible to make a system that can claim both A and ~A. Therefore, we know that the underlying rules of the physical world are inconsistent. This is necessarily the case.
See, you claim that there must be some unprovable “true” state of the universe (whatever that means). I say “no way”, everything’s provable. So who’s right?
I am, because I’m not actually so silly as to think that the physical universe is a formal axiomatic system governed by a finite set of axioms, and ruled by the classical laws of non-contradiction and the excluded middle.
Here are some more for you:
The universe is capable of “modeling” (via my brain) the axiom of infinity from ZF set theory: there exists a set that contains zero and the successor of each of its element. Does that mean that the physical universe necessarily contains an infinite set?
The universe is capable of “modeling” (via Zermelo’s brain) the axiom of choice: that given a set of mutually exclusive non-empty sets there exists a set that contains exactly one element in common with each of the sets. Does that mean that the axiom of choice is necessarily true of the physical universe?
But wait! The universe is capable of “modeling” (via Cohen’s brain) that if ZF set theory with the axiom of choice is consistent, then ZF set theory with choice replaced by the negation of the axiom of choice is also consistent. Does that mean that the physical universe is necessarily inconsistent? Or does it mean that it isnot necessary at all that either the axiom of choice or its negation is “true” of the physical universe?
And we can go on and on and on. The universe is capable of modeling, via a set of Star Trek episodes that people travel faster than light, can be “beamed” through space, and that portals exist that allow travel into the past. The universe is capable of modeling leprechauns, pots of gold at the end of rainbows, and magic genies in lamps.
For some reason, you want the limits of a particular sort of formal axiomatic system to have something to do with the physical universe. It is unclear to me that they do, and your only argument has been “it’s obvious”.
Well, it’s obvious to me that axiomatic number theory doesn’t limit “the universe” in the way that you think it does. And moreover, it’s obvious that your claims are silly. Don’t you think?
**
It’s not “another requirement”. According to dictionary.com, the primary meaning of the word disrupt is “to throw into confusion or disorder”. If a model of a system is changed so that its properties no longer correspond with some of the properties of the system, it’s no longer a model. The order or pattern that it previously manifested is gone. It has been disrupted.
** Life can be used to model meta-Life, but meta-Life can never be used to model Life. To have meta-Life, it’s necessary to have Life in the first place. If the meta-Life is modeled without the underlying system of Life, it’s not meta-Life: it’s just Life.
I never said there’s no way to measure it. It simply does not require human measurement. And it’s entirely possible that a system modeling the orbital motions of the planets might also model the digestioni of a Snickers. We know for a fact that the system underlying orbital motion is the same system the underlies digestion.
** It’s entirely possible for a consistent system to emulate an inconsistent system. Godel, Escher, Bach points out several examples of this, in fact. However, an inconsistent system can’t emulate a consistent one.
** Everything’s provable? Fine. There is therefore a proof that everything isn’t provable.
** Then you’ve just claimed that the concept of truth doesn’t apply to the universe. Nothing can be either true or false, because anything can be proven true and proven false.
Brilliant, Einstein.
I refuse to repeat the reasoning that falsifies the examples you offer – I’ll only state it once.
It’s certainly true that the system underlying the universe is capable of emulating episodes of Star Trek, and systems that state the axiom of choice, stories of leprechauns, etc. We know this because these things exist in the universe. Just because episodes of Star Trek exist does not imply that the people and events described in those episodes exist in the universe.
[sigh]
I’ll deal with this when I “connect the dots” for Spiritus… I see no reason to waste time dealing with the same objection twice.
But you have not demonstrated in what way the physical universe is emulating anything.
I had some people over last night to watch hockey. At one point during the evening, I observed that there were three people in the living room, and that two people were in the kitchen. When those two people entered the living room, I counted five people in the living room.
Does this mean that my living room is a formal system capable of modeling arithmetic? That my living room is necessarily inconsistent or incomplete, because it is the sort of formal system GIT talks about? No, it does not.
And a proof that the moon is made of green cheese! And one that it’s not! Big whup.
Ummm, no, I did not claim that. Notice how wishy-washy you are about “proven true” and “being true” in everything you write? That’s a pretty big problem, when you want to talk about GIT.
Here’s the nub. I think you want to claim that the universe is “doing arithmetic”. Well, it’s not. What is the universe doing? It’s just doing whatever universes do. You know, universe stuff.
I don’t think you can construct any kind of Godel sentence about the universe–something that’s “true about the universe” but can’t be “proven by the universe”, because the universe lacks the sort of self reference you need to make the stuff that happens in the universe be about the universe.
Without that, you’ve got dick.
I’m eagerly awaiting that.
Hey, TVAA, I’ve got a question for you. In your understanding, what does GIT not apply to?
Well, this is another example of your inconsistent usage. Or perhaps inconsistent reading. The question I asked was: does the “Meta-life” game model the life game, not will the “meta-life” game be forever and ever a model of the life game.
So, either you misread the question and anwered whether some future, post-disruption, Meta-life game would be a model of the life game, or else you have added another requirement to “model” that considers the possibility of future disruption.
This one I am certain is a discrepancy in reading. I explicitely asked about a “Meta-life” that is not being instantiated on the same grid. I thought that was clear, but obviously not. Consider the following:
[ul][li]Comuter(A) -> life(A) -> Meta-life(A)[/li][li]Computer(B) -> life(B) -> Meta-life(B)[/ul][/li]Can Meta-life(A) model Meta-life(B)?
Can life(A) model life(B)?
Can Meta-life(A) model life(B)?
The bolded question, of course, is the one I asked in my last post.
Ooops. There you go being inconsistent in your use of “model” again. We know for a fact that the same system underlies a “disrupoted” game of Meta-life and a game of life, and just a few lines ago you said that Meta-life could never model life.
As I said some time ago, you do yourself no favors when you reject the well-defined terminology of the field(s).
And, for the record, what you said was:
Now, if you meant that statement to include the possibility that “similarity” can be measured, then I’m afraid I find your word choice to be misleading.
Well, this is another example of your inconsistent usage. Or perhaps inconsistent reading. The question I asked was: does the “Meta-life” game model the life game, not will the “meta-life” game be forever and ever a model of the life game.
So, either you misread the question and anwered whether some future, post-disruption, Meta-life game would be a model of the life game, or else you have added another requirement to “model” that considers the possibility of future disruption.
This one I am certain is a discrepancy in reading. I explicitely asked about a “Meta-life” that is not being instantiated on the same grid. I thought that was clear, but obviously not. Consider the following:
[ul][li]Comuter(A) -> life(A) -> Meta-life(A)[/li][li]Computer(B) -> life(B) -> Meta-life(B)[/ul][/li]Can Meta-life(A) model Meta-life(B)?
Can life(A) model life(B)?
Can Meta-life(A) model life(B)?
The bolded question, of course, is the one I asked in my last post.
Ooops. There you go being inconsistent in your use of “model” again. We know for a fact that the same system underlies a “disrupoted” game of Meta-life and a game of life, and just a few lines ago you said that Meta-life could never model life.
As I said some time ago, you do yourself no favors when you reject the well-defined terminology of the field(s).
And, for the record, what you said was:
Now, if you meant that statement to include the possibility that “similarity” can be measured, then I’m afraid I find your word choice to be misleading.
Also, would you mind taking my statement of GIT on page 2 and translating it into plain English?