I don’t know about everybody else here, but I’m willing to pass on another dance 'round the mulberry bush, and just go back to reading Deepak Chopra, or listening to Monty Python’s “argument clinic” sketch.
That’s down the hall, FranticMad.
Or we could just rip up the mulberry bush and continue merrily down the Yellow Brick Road…
Has anyone seen my Silver Slippers?
To which statements does this refer. You have made contradictory statements regarding “model” and life/meta-life. You have also made statements that were contradictory in tone regarding “measurement” and its applicability to the current debate. (Have to love those parenthetical attacks on reading comprehension. They’re so much easier than actually demonstrating an error in textual analysis, aren’t they? :rolleyes: )
Well, sometimes you say systems are similar to each other. Other times you say they are not. I will grant, however, that this does nto necessarily imply that you have considered the issue.
Of course not. Rocket science is a rigorous field that rarely contradicts itself.
[ul][li]When placed in a grid with copies of itself, this pattern reproduces the Game of Life.[/li]
[li]I’m saying that, although the two systems exist on different level of implementation, their high-level behavior is the same. They do the same things.[/li]
[li]Two systems are similar if they share properties; the more the properties are shared, the more similar they are. [/li]
[li]There isn’t necessarily a relationship at all. And [Meta-life(A) and life(B)] can’t be described by the same statements.[/ul][/li]So: are meta-life(A) and life(B) similar to each other? If so, what would you call this since you don’t think it is a relationship?
Of course, this might all be irrelevant to your actual argument. It’s difficult to tell since you hav yet to respond to questions about step (1) in your “draw the lines” proof. Any chance you will be doing that in the near future, TVAA?
It’s not a question of whether they’re similar. Everything is somewhat similar to everything else. The question is: how similar are they?
Both Life and Meta-life are manifestations of a specific set of rules. Meta-life necessarily is an emulation run by Life. Any specific configuration of Meta-life may manifest high-level behaviors similar to that of specific configurations of Life.
Are you quite finished bludgeoning this dead equine?
It’s not pining, it’s passed on.
Hey, I’d much prefer it if the parrot would start an eloquent discussion of the issues that have been raised regarding step1 of the “draw the lines” proof. Unfortunately, I can only address the words that Polly squawks.
Still, I think we have now established that:
[ul][li]Everything is similar to everything[/li][li]But some things are more similar to other things.[/li][li]Only TVAA doesn’t want to talk about how one might decide which things are more similar to each other, because that might take “measurement” which is italically irrelevant to this thread![/li][li]Except maybe now it isn’t. “The question is: how similar are they?”[/li][li]“a model” is something that is similar to another thing.[/li][li]But meta-life can’t model life because there’s no relationship between the two, even though they manifest identical high-level behaviors.[/li][li]And when TVAA says “model” he doesn’t mean the noun form that he defined but rather the verb form that he says is the same as “emulate”.[/li][li]But when TVAA sqays “emulate” he doesn’t mean the common English definition (To strive to equal or excel, especially through imitation) or the specific Computer Science definition (To imitate the function of another system), he means something like the English words “compose” or “make up” (as in the atoms compose the molecules, and the molecules compose the compounds.)[/ul][/li]Well, that certainly clears things up. :rolleyes:
Spiritus:
Point 1: GIT shows that there are necessarily certain limitations on logical systems of sufficient complexity; namely, if they can generate arithmetic (which means that they have operations that allow an unlimited number of theorems to be defined), GIT applies.
Point 2: All computers (which is to say, Universal Turing machines; TMs that can’t accept input aren’t computers) are equivalent to a logical system.
Point 3: GIT shows that a sufficiently complex system will contain statements that are true given the axioms of the system but that cannot be shown to be true (proven) using those axioms. In short, operations applied to the system’s axioms cannot generate those statements.
Point 4: The statements that cannot be demonstrated by the system are equivalent to configurations the computer can never generate.
Point 5: Therefore, any computer can be given a program whose resolution requires a configuration the computer cannot reach, causing it to fail or enter an infinite loop.
I request more development on points 2 and 4, please.
Point 2: It’s the accurate description of the computer. Having a computer that accepts initial input and derives output is equivalent to taking a series of statements and generating theorems from them.
Point 4: See Point 2. Generating a theorem from axioms is the same as generating an output configuration from input. Since the logical system accurately represents the operations performed by the computer, showing that there are statements that can’t be generated by the system is equivalent to showing that the computer can’t produce the output that corresponds with those statements.
[ul][li]**Point 1: GIT shows that there are necessarily certain limitations on logical systems of sufficient complexity; namely, if they can generate arithmetic (which means that they have operations that allow an unlimited number of theorems to be defined), GIT applies. **Maybe. GIT applies to any logical system that can derive a Peano axiomatization of arithmetic. Given your non-standard usage I have no idea whether it is true that GIT applies to things that can “generate arithmetic”.[/li]
[li]**Point 2: All computers (which is to say, Universal Turing machines; TMs that can’t accept input aren’t computers) are equivalent to a logical system. **Your parenthetical note is incorrect. For one thing, a TM that “can’t accept input” is not a TM. For another, a TM is closer in analogy to a program than to a computer. A UTM, then, would be a program that is simulating the behavior of a computer. The statement “All computers are equivalent to a logical system” cannot really be evaluated until you tell me what you mean by “equivalent”. In simple English the statement is false. [/li]
Also, at some point you should be explicit whether you are defining “All computers” to mean “computers that can be simluated by a UTM (modern digital computers, for instance)” or “computers that are equivalent in computational power to a UTM (things that are Turing Complete. If the Church-Turing Thesis is correct, things that can solve algorithms.)”
[li]**Point 3: GIT shows that a sufficiently complex system will contain statements that are true given the axioms of the system but that cannot be shown to be true (proven) using those axioms. In short, operations applied to the system’s axioms cannot generate those statements. **Not quite. GIT holds for a sufficiently complex and consistent system. The distinction is not inconsequential when you take your system to be “the Universe”.[/li]
[li]**Point 4: The statements that cannot be demonstrated by the system are equivalent to configurations the computer can never generate. **False. Statements that cannot be proven (but are true) would correspond to outputs that cannot be generated by a given computer program (UTM). [/li]
The only limit to the configurations possible for the computer are hardware related. For a modern binary computer, any physical state that can map a set of binary numbers that are representable in its memory space are “possible configurations”. “Impossible configurations”, then , would be “component states that are either not binary or excede the memory space of the hardware.”
[li]Point 5: Therefore, any computer can be given a program whose resolution requires a configuration the computer cannot reach, causing it to fail or enter an infinite loop. No. Therefore any computer program sufficently complex to derive a Peano Axiomatization will be unable to generate certain outputs that are consistent with all outputs that it can gnerate.[/ul][/li]
For the record, the first point of your “draw the lines” proof and the questions that I asked were:
I have to agree with the other posters’ complaints about your vague use of terminology. Honestly, I doubt that you have a clear idea of what you mean by “equivalent.”
**
I’m not sure what you mean by “fail,” but it’s obviously true that any computer can be programmed to go into an infinite loop. So what?
In any event, your “Point 5” doesn’t seem to follow from your earlier points. Not that it really matters.
[QUOTE]
*Originally posted by Spiritus Mundi *
**Maybe. GIT applies to any logical system that can derive a Peano axiomatization of arithmetic. Given your non-standard usage I have no idea whether it is true that GIT applies to things that can “generate arithmetic”.
[quote]
** No, it applies to any logical system that can generate arithmetic. Peano’s axioms may be axioms of the system.
As other posters have pointed out, it is possible to have a trivial Turing Machine that can’t accept input, but it can’t be a computer.
** What’s the difference?
No, the statement is true in simple English.
And how can the Universe be inconsistent?
It’s the same thing.
** No, the “hardware” can be considered as another system. There are configurations that this system can never reach.
How can you consider the ‘program’ without considering the configuration of the hardware that represents it? Given that configuration, there are other configurations that the system cannot reach.
** There’s no difference between a ‘program’ and the ‘hardware’. Your inability to grasp this simple concept is astounding.
** Equivalent:
But that’s precisely the point. No computer can be made so that it can’t be programmed not to fail. Either it will become trapped in an infinite logical loop, or it will pass out of its standard configuration and give junk output.
But it does follow. I’ve demonstrated that there’s always statements that the system of the computer can’t handle.
You’ve offered a bunch of different definitions of “equivalent.” Which one were you using when you said that “all computers are equivalent . . . to a logical system”?
Actually, strike that question. If you seriously believe that the definition of “equivalent” you posted is satisfactory vis-a-vis your claim that “All computers . . . are equivalent to a logical system,” there’s really no point in continuing the discussion.
If you are trolling, then I congratulate you on doing a magnificent job.
**
Whatever.
:rolleyes:
All UTMs can emulated all other UTMs. A UTM CANNOT BE DISTINGUISHED FROM A COMPUTER PROGRAM!
There’s not only no way to determine whether any given example of a UTM is being emulated by another UTM or not, there’s no conceptual difference between an emulated and a “real” UTM.
Sweet, merciful Buddha on a pogo-stick! Don’t any of you people actually understand the way computers work? Do you just accept their functioning on faith? Are they just magic boxes that conveniently do what you want some of the time?
All of those definitions do apply… although I think you’re right about one thing: continuing this discussion is pointless.
I honestly had no idea that so many people who appeared relatively knowledgeable and intelligent could be so profoundly ignorant and downright stupid.
Do you come to the Straight Dope to find out information and participate in intelligent discussions, or are you just here to inflate your egos and make yourself feel smart? I’ve never seen a group of people be so underservedly arrogant.
Erislover, if you’d like to continue this discussion, you know how to contact me.
buh-bye
Again, until you specify exactly what you mean by “generate arithmetic” it is impossible to judge whether this system is correct. I suspect, though, that it is not.
No, this is sloppy language on your part. Others have given examples or Turing Machines whose internal states are independent of the particulars of an input string. These two statements are not idential.
What’s the difference between a blueprint and a collection of building materials?
We disagree. Perhaps you would be so kind as to demonstrate exactly the equivalence that you find between computers and logical systems.
[ul][li]States that a computer can achieve.[/li][li]States that can be generated by a computer program generating true statements in a Peano Arithmetic.[/ul][/li]I say these two sets are not identical. You say that they are. My argument is as follows:
[ol][li]Assume a binary computer program that generates statements in a P.A.[/li][li]Each true statement will correspond to a binary number.[/li][li]Some true statements in a P.A. will require a negation symbol “!”.[/li][li]this negation symbol, too, must be representable as a binary number.[/li][li]Thus, some true statement !(T) will be generated, and the result will be represented by the number !(T)[sub]bin[/sub].[/li][li]Since !(T) is true, T is false (else our P.A. is inconsistent).[/li][li]Since !(T)[sub]bin[/sub] is representable by the binary computer, the substring T[sub]bin[/sub] is representable on the binary computer.[/li][li]But T will never be generated by our program, which derives only true statements from our Peano Axiomatization.[/li][li]:. the two sets are not identical.[/ol][/li]There are easier ways to demonstrate the difference, of course. But I know that at least one person in this thread has already mentioned that it is trivial to have an algorithm generate all strings in a language. Since that didn’t make the point clear to you, I thought I would try a different tack.
Of course, if you still think that a computer can never generate a state that some particular computer program cannot achieve as a result, then you will have to come up with a specific argument of your own.
You keep saying that.
You keep neglecting to provide any specific argument that supports your assertion. This goes back to my questions about point 1 of your “draw the lines” proof. Do you have any intention of ever answering those questions?
Because a “program” corresponds to a “method” or an “algorithm” or a “turing machine” and can thus be implemented on many different hardware platforms. The specifics of the hardware implementation are (at a theoretical level) entirely irrelevant to the bahavior of the program.
Really–you have just asked how one can do work in Computer Theory. If you don’t know, then you should really stop talking about Turing Machines.
Sure, if you mean that there a states that it is physically impossible for circuitry to achieve. (For example: A single bit representation of 3 mutually exclusive values.)
Wrong, if you are talking about a logical restriction on the physically possible states that can be reached. (For example: A specific sequence of binary digits in memory.
Really?
What I find astounding is your obnoxious prediliction for using words in a non-standard manner and then castigating others for not grasping “simple concepts” that are trivially false when parsed in standard English.
[ul][li]A program is an abstraction. [/li][li]Hardware is a physical construct.[/li][/ul]
There is a difference between the two.
Here is a simple concept for you, though, I have lost patience with both your insults and your perverse fixation on expressing things in a language that only you understand. As it stands now, this thread has demonstrated several things:
[ul][li]The OP is unsubstantiated and highly suspect as an implication of GIT[/li][li]TVAA lacks the ability to discuss either GIT or Cpmuter theory in teh terms appropriate to those fields.[/li]TVAA either fails to understand the work of Godel and Turing or is incapable of expressing his understanding in a rigorous and intelligible manner. Or both.
[li]My money is on both.[/li][li]TVAA likes to insult posters who disagree with him.[/li][li]But TVAA does not like to answer direct questions about his arguments.[/li][li]Spiritus Mundi has concluded that it is extremely unlikely that any of the above points will change as this thread continues.[/ul][/li]:wally
An on preview, I see one of the more ironic posts I have ever come across on this board.
You are some piece of work, TVAA.
** Is the system capable of generating the (infinitely many) theorems of arithmetic? If so, it generates arithmetic.
** Then they’re not UTMs and they’re not computers. A “computer” is the term once used to describe people who performed arithmetic for a living; it’s now used to refer to things that perform computation.
If the blueprint is really, really complex? Nothing. The program is not distinguishable from the thing.
Jesus Christ, didn’t you see The Matrix? I thought that movie had managed to make even the dimmest individuals understand that a reality is whatever interacts with you. A computer simulation that accurately represented the physics of the building materials is the building materials, at least from inside the simulation.
This reality can be considered to be a simulation, a program being run by the fundamental physics.
** The states of the computer evolve according to certain rules – the rules of physics. This is equivalent to a system that applies operations to specific statements – when the statements accurately represent the configuration of the computer.
You really don’t know how computers work, do you?
But that algorithm is inconsistent. Your example assumes that this system can actually take a set of axioms and derive statement T. If the axioms are consistent, this isn’t possible. If the axioms are inconsistent, GIT doesn’t apply.
** There are many ways that a computation can be performed, but it’s not the case that any configuration of the hardware will do. Regardless, there will be statements that the system represented by the hardware will not be able to derive – and there will be configurations that the hardware will therefore not be able to reach. These configurations will differ between representational methods, but they will all represent those forbidden statements.
Circuitry in a particular configuration will never be able to enter certain other states. Adding outside influences can expand the set of configurations it can reach, but the new system will still have configurations it can’t reach. Without positing an infinite universe that makes all possible influences possible, there will still be unavailable configurations!
The computer will never reach it on its own – given only the input and the rules that govern the computer, there will be memory sequences it can never have.
** But they’re not false – that’s the important part.
** No, you idiot, there is no difference. Every UTM can be emulated by another, and that UTM can be emulated by another, and so on ad infinitum. Which is the hardware and which is the program?
Not at all. I do not suffer fools gladly – I do not suffer fools at all.
And I do not suffer people who can neither obey the rules of this forum not engage in a reasoned debate. If it were within your capability to demonstrate that I am a fool, then you should have done so. Then again, your entire argumentative approach in this thread has been to make strong assertions and then provide no specific support for them. At least you are consistent in your varied modes of ignorance.