I also think that GEB is a bit of a dangerous book. After reading it, you don’t know that much: you’ve just been subjected to a lot of general ideas with no specifics. The subjects broached are very interesting. TVAA, if you want to learn the specifics, there are a couple books that I recommend wholeheartedly. They are Michael Sipser’s Introduction to the Theory of Computation and Elliot Mendelson’s Introduction to Mathematical Logic…well, I know I got the last names right, and the titles are very close to correct.
I’d recommend that you start with Sipser, as his book places more emphasis on being clear. There’s no lack of rigor, though. Mendelson’s book is very rigorous and not terribly clear, but it’s got a lot of good information.
Unfortunately, neither is terribly cheap, so you may want to see if you can find used copies.
** As far as I can determine, when his statements are interpreted according to his own words about them at the time and the surrounding context of the discussion, they’re wrong.
There’s a reason he complained so much about my refusal to use specific mathematical languages to talk about this debate – he can’t handle the concepts otherwise.
And since he repeatedly complained about my lack of rigor, his falling into error because of a lack of rigor is especially damning.
It is unfortunately the case that many individuals skilled in mathematics are unable to discuss their knowledge except in mathematical terms.
And it’s not merely from me – it’s from me on this particular topic. Before I noticed the significant lack of comprehension and excess of erroneousness in this debate, I regarded Spiritus Mundi as being one of the more rational and intelligent members of these boards. I now realize that – like some others I could mention – he does a better job of simulating intelligence than of actually manifesting it.
People whose intelligence and knowledge on the subjects discussed here cannot be seriously questioned by myself have reviewed the two threads in question, and they’ve repeatedly told me that Spiritus does not in fact know what he’s talking about. He is not able to use words according to their definitions to transmit and receive meaning.
I know that this will not convince you; it’s not intended to convince you. Whatever doubt I may have about my own judgment, I am unable to ignore the conclusions of those I know understand the issues being discussed.
** [sigh] Your conclusions are your right and your own business.
Fair enough.
** Who said the universe is a UTM? The UTMs are merely a special case, but they demonstrate an important point: there is no fundamental difference between computer and program. They’re both informational “structures”, just on different levels of reality. If Spiritus’ claim had been correct, it wouldn’t be possible for UTMs to simulate any other UTM; since we know mathematically that they can, there is a known exception to his statement and it is therefore incorrect.
As I mentioned, I’m working on it. It is difficult – not only do they have more important things to do than I do (and hence can’t “waste their time reading messageboards”), they generally don’t bother wasting time trying to convince people of things. They reviewed the threads and gave me feedback only as a personal favor to me – and one expressly refused to comment on the boards, even though he said I was correct, because he found my tone throughout to be troublesome. (He has a much higher tolerance for idiots than I do.)
** :dubious: I can only conclude that we have vastly different standards for “support”.
I’ve been using the standard meanings… and ambiguity is in the eye of the beholder. I could try again, I suppose…
[sigh] This is going to take a while.
This is classic, and trivial. Give it a set of input equivalent to the problem “determine the truth value of the statement 'this statement cannot be proven by this TM”.
It can’t generate a counter-proof. It can’t generate a proof. And no matter how many steps of computation it carries out, it will never reach any conclusion that will allow it to stop.
What are you talking about? This Turing machine just looks at its input and moves to an accept state no matter what the input is. What do proofs have to do with anything?
I disagree strongly here. He, frankly, will bend over backwards to adopt to someone’s use of terms. But we’re talking about a specific set of concepts here. That, frankly, requires some specificity. You’ve failed to provide that.
He’s damned if he does and damned if he doesn’t against you. If he demands and uses rigor, he’s an idiot savant who can’t understand things except in narrow contexts. If he doesn’t use rigor, he’s a hypocrite. Nice tautology you’ve got there: Spiritus is an ass is true in all cases.
Unimpressive logically. Upsetting personally.
They are mathematical concepts! You think that mathematics and physics uses rigid symbology to make paragraphs shorter?! The English word “force” is not physic’s “F” in all cases. The symbol “F” called “force” in physics is a highly specialized and more-or-less well defined concept. It is of course no coincidence that we call “F” “force” when we speak of it. That does not mean F == “force[sub]English[/sub]”
!
You want your talk to have all the strength of GIT with none of the development. And who, exactly, did you suppose it would convince like that?
While I am sure this is possible, and while I don’t doubt the content or character of your impressions, I do disagree.
I’m afraid you’ll have to let that, and that alone, let you rest easier, because on that account you have also failed to present a compelling case. Sorry, man. Hopefully your distaste doesn’t translate over my way, but ::shrug:: we all just say what we mean the best we can.
And being on different levels of reality does not invalidate the claim “there is no fundamental difference between them”? Fascinating.
Kenneth Branagh is not me. He never will be. There is a reason why the word simulation is not an indicator of existential equality/identity.
I’m sorry, let me clarify then: statements along the lines of “this is trivially obvious to any idiot” are hand-waving, are they not? As ultrafilter says, if it’s trivially obvious, it should be trivially explainable. And if it can’t be explained trivially, then it wasn’t trivially obvious in the first place.
I might suggest that if no one seems to have a clear idea what your argument is, that might reflect more upon how you’ve made it then upon the entire rest of the SDMB, no?
See, I don’t know too many fourth graders who grasp the concepts of “gradient,” and “second time derivative.” Perhaps the concepts of “rate of change,” or “rate of change of rate of change.” And just out of curiousity: what is the technical definition of “mass,” then?
In the larger sense, though, you’ve missed my point completely. Which probably shouldn’t surprise me, as every time we actually try to engage each other in conversation, this happens, presumably on both our parts.
It doesn’t matter that F=ma is simple enough for a reasonably bright high school student to grasp in some modicum of technicality. It does, however, matter that the technicality is there, in this as in, say, quantum field theory, and plain language variants of technical concepts must first be sure that relevant details are not glossed over, lest thine arguments be thrown into confusion.
Well, if you really must know, F = dp/dt is the correct form, and is not necessarily equivalent. Note also that in relativity, one must use proper time rather than coordinate time, and the four-momentum, not the three-momentum, so that the best generalization you’re likely to come up with is F[sub][symbol]m[/symbol][/sub]=d/d[symbol]t[/symbol](m d/d[symbol]t[/symbol])x[sub][symbol]m[/symbol][/sub].
I hope that came out right; Greek letters don’t display here.
Not that this is relevant, but of course it does illustrate my point, that subtleties ought be understood.
If your TM does nothing but accept data, its programming shouldn’t be sufficiently powerful for this discussion to apply.
The mechanism of the TM (for example, it could be a cellular automata like the Game of Life) still determines what the behavior of the TM it’s simulating is going to be – even if that behavior is as simple (and pointless) of simply moving to an acceptance state.
There are possible inputs to the underlying TM mechanism that could disrupt it – in the GoL example, virtually any glider would terminally disrupt the structure. Only a very small number of possible automata structures are “acceptable” input. Almost anything else will change the behavior of the system radically, utterly wiping out the structure that was acting as the TM.
Moving from one state to another is informationally equivalent to producing a theorem from a set of axioms. You start with the initial conditions (the axioms) and produce a necessary consequence of those conditions (the theorems).
Moving to a state is the same as generating a proof of the statement descibing that state from the statements describing the initial conditions. “Given these conditions, and these operations applied to them, these conditions result.”
There are some states that the initial conditions can never generate, because that would be equivalent to generating a proof of a Godel statement of the system.
You claimed long ago that no computer can be built so that it never crashes. Apparently you only meant Turing-complete computers. That’s fine, but it would’ve been nice if you had said that a long time ago (if you did and I missed it, my apologies). It’s not at all obvious that that’s how we should take “computer” here.
So to make this a little more specific, I dug up the state table for a UTM. Which state can it never reach?
I’m just trying to reconcile these two statements by TVAA–
Sounds like either you don’t understand GIT after all, or you think it’s a relatively basic concept.
And this:
Is that like writing up a list of things you did this weekend, write “To Do List” at the top, check everything, and say you’ve completed your To Do List?
Wait a minute – you’re giving me a table that lists the elementary internal states of a Turing machine and then asking me to show you the configuration it can’t reach?
I can’t show you the sequence of steps the machine can’t reach by manipulating the elements of that table. That’s the whole point!
No, I don’t know the methods for finding a Godel statement for any particular system.
Ethilrist: GIT is relatively simple to understand; just not in its formal form. It is extremely elegant (or so I’m assured), but if it were simple it wouldn’t be regarded as one of the greatest triumphs in modern mathematics and philosophy. Every mathematician would have thought of it.
The basic principles and significance of GIT… those I can understand.
TVAA would you do me a favor and state every conclusion you’re trying to get us to accept? You may be right about something, but I need to see it in context.
This is my feeling too. I tried, in GD, to envision where you are going. Somewhat clumsily I suggested that your context is anti-non-deterministic (for lack of a better word). Like ultrafilter, I would appreciate a listing of your conclusions.
I know this is frustrating for you, but it is pretty common for abstract concepts to be misunderstood by others. You are bridging between a mathematical proof and English, and adding your own comments on top of that. Therefore, it strikes me a being entirely reasonable that most people would have trouble following you.
On the other hand it’s about 4:15 pm where I am. Almost time for to put that keyboard down and – Karaoke until tomorrow!
----- flowbark said, “I would assert that when disagreements over the meaning of the original text arise, that it is necessary to evaluate the original source, in its native (mathematical) language.”
------ TVAA replies, “Here is where I’m forced to disagree. If someone claims that elliptical orbits are not the result of an attractive force that varies with the inverse square of distance, we don’t have to go back to Newton’s Principia to resolve the dispute.”
Your reply muddies the waters, I fear. If somebody says Newton says thus and so, recourse to Newton’s Principia is necessary to confirm or deny that claim.
In contrast, if somebody says that, F[sub]g[/sub]=G*(m1*m2)/r[sup]2[/sup], one would not need to pull out any ancient text. One would, however have to 1) define his terms, 2) muster empirical evidence and 3) understand elementary algebra.
Furthermore, phrases such as “inverse square” presume a certain level of mathematical knowledge. If you are not familiar with exponents, you really can’t evaluate such a claim.
----- I’ve read lots and lots of standard-English explanations of what GIT is and what it implies.
Uh, great. But you can’t judge the accuracy of your standard-English explanations of GIT without recourse to the original mathematical argument. (Secondary sources are ok, provided they contain sufficient mathematical rigor, or so I claim).
You see, there is the problem of trendiness. If “Potential-energy” was a trendy concept, I could imagine all sorts of plausible applications of it. One could say that somebody with a good night’s sleep has more potential energy than a hypercaffeinated insomniac. Those who are more awake can get more work done, right? Furthermore, I could argue potential energy is directly related to the square of the amount of sleep displaced. Provided I defined my terms and was willing to test them directly, that would even be acceptable, if a little weird.
But I wouldn’t want to make to much of the connections between “Potential sleep energy” and U=.5kx[sup]2[/sup], as defined in mechanics (kinetics?).
Still, you have my sympathies. There are lots of people who have read about Heisenberg’s Uncertainty Principle and rather fewer who have grasped its relationship to frequency distributions.
So. How many examples of deceitful posting does it take to make a pattern?
3?
4?
5?
EXHIBIT 1: Deliberate repetition of false claims
Please note the dates and times in the following sequence of posts, some of them in the GD Godel thread and some of them in this thread.
So you keep repeating the same dishonest claim more than a week after acknowledging that our disagreement had to do with the proper application of the word “proof” to demonstrations in extended logical systems. This has nothing to do with understanding GIT.
Q.E.D.
EXHIBIT 2: Hypocritical aplication of a double standard
TVAA has taken great glee in screaming that I must be ignorant about GIT because after 7 pages of discussion about GIT I neglected to specify that a system must be meet specific requirements in order for GIT to reply. What I said was:
[ul][li] Here’s another point that you don’t understand: a system cannot be proven to be consistent from within itself. That is GIT[sup]2[/sup]. Thus, you can never demonstrate your claim that the Universe itself is a consistent axiomatic system.[/ul][/li]TVAA has taken great relish in slamming me for not supplying the habitual specification upon when GIT[sup]1 or 2[/sup] applies. He calls me stupid for omitting it, and goes on to argue that this omission means that I obviously do not understand the theory. But let’s take a look at how TVAA himself introduced GIT in his GD thread
Strangely enough, he doesn’t immediately declare himself stupid and conclude that he obviouslt knows nothign about GIT. How interesting.
Q.E.D.
EXHIBIT 3: Deliberate distortion of another poster’s position (with an added dash of hypocricy.)
TVAA has berated me for demanding that all discussions of GIT be held in a formal symbolic language. His most recent repetition of that claim was:
Of course, I have never demanded any such thing. The disagreement that we have had over language begins on page 3 of the GD thread. There, in a part of our discussion relating to computers and computer theory, I question his use of the words “model” and “complexity”. Because the “model” discussion dragged on a very long time, I’ll just repost the “complexity” exchange here. Anyone interested can follow TVAA’s twisting of the word “model” in the other thread.
Okay. so what do we see here:
[ol][li]At no time did I demand that GIT be discussed only in formal mathematical language.[/li][li]I did assume, upon seeing terms like “complexity” in a discussion about computer theory that the terms were being used to descrive the recognized concepts of computer theory. [/li][li]TVAAinitially claimed that his usage was in accord with the definition of “complexity” in computer theory.[/li][li]TVAA then changed tactics and declared that he was not using “complexity” in a manner consistent with the context of computer theory. And, of course, I was stupid for thinking that he was.[/li][li]Even then, I did not demand that he use the terminology appropriate to the field, I simply asked him to explicitely and clearly define his terms.[/li][li]Which got me called stupid again.[/ol][/li]And now for the dash of irony. Consider this exchange:
Notice:
[ul][li]TVAA doesn’t say it is stupid to expect the terms to be used in their technically specific sense (else it would destroy his little example).[/li][li]TVAA doesn’t rant and rage that only someone who didn’t understand physics would expect the terms to be used in their technically specific sense.[/li][li]TVAA, in fact, appears to very much support the idea that it is not too much to ask that someone use terms in their technically specific sense when discussing ideas in a specific context.[/ul][/li]
Q.E.D.
EXHIBIT 4: Equivocation in argument: changing the charges as evidence shifts
TVAA and I have been round a bit on whether GIT deals with the truth or falsehood of statements. Witness the exchange:
So we see that:
[ul][li]I initially pointed out that “truth” was not really required by GIT but is often used as a gloss by people sriting about GIT in natural language. GIT is, in fact, entirely mute on the cubject of truth, caring only about consistency and completeness. [/li][li]TVAA disagreed, saying that GIT specifically required that some unproveable statements be true.[/li][li]TVAA posted a cite to back up his claim.[/li][li]I showed that the cite actually specifically refuted his claim in a note.[/ul][/li]Now, some might think that this would inspire TVAA to acknowledge that my initial statement was correct. But such folks fail to recon with the power of cognitive dossonance personified. It cannot be possible that my statement was correct, because then it would weaken his vehement belief that I am an idiot who doesn’t know what I am talking about. Thus . . . TVAA shifts tactics and says that he knew all along that GIT did not require some unproveable statements to be true, but that I was misusing, “soundness”. (I will address the factual claim later. It is unrelated to this exhibit of TVAA’s dishonesty in argument.)
Q.E.D.
EXHIBIT 5: Repeated offensive claims offered without substantiation
I’ll keep this one short. This is TVAA on a very simple issue of fact:
Now, his claim here is plain: he has already posted something that specifically cites a passage of mine that introduces symbology into our debate. Of course, he doesn’t deign to post it again, even though it would be an easy way to demonstrate that I actually amm in error. No, he simply says: I already did it and caps it with an insult. Why not simply quote the relevant post?
I think we can all figure out why.
Q.E.D.
Questions of fact
Just a couple, since this is too long already.
[ol][li]Because whether or not consistency can be proved does not change the nature of the Universe.[/li][li]Because I am not even saying that the consistency cannot be proved within the Universe. I am saying that it cannot be proved IF the Universe is a deductive system powerful enough to invoke GIT. [/li][li]:wally[/ol][/li]
I could have chosen any number of others, but the basic assertion is that I am misusing soundness because something can be true but not proveable. Well, in the general sense you are absolutely correct that somethign can be true but not proveable. What that means for a statement in a deductive system, though, is that there is an external, non-deductive model to which the system maps and the isomorphic partner for the statement is present in the model. Absent the correspondence with such a model (and I use the term here in the symbolic logic sense, not in the TVAA sense), the only way to demonstrate truth is through deduction from sound axioms.
Think about that for a second, and recall that this entire argument about “truth” and “soundness” took place within the very specific context of how one could sat that something is true “in a Peano Axiomatization.” So, if you can tell me what serves as an external, non-deductive model for a Peano Axiomatization, then I might have to change my mind and agree with you that a statement can be erifiably true “in a Peano axiomatization” without being proveable “in a Peano Axiomatization”.
Until then, I stand by my statements that in the precise context that we are discussing, there is no non-deductive external model from which to infer truth, so the only statements that can be properly called “true within the system” are those which can be proven within the system. “Statements can be true without being proveable” is certainly true of many systems to which GIT applies, but it is not true of all systems to which GIT applies. GIT is a very fundamental theorem. It applies to many, many systems–not just those for which an external model of truth exists.
**Now, having said that, I have no intention of engaging in any further discussion of GIT with you. I will address the examples I gave above of your dishonesty and deceitfulness, since I think it is unfair to level charges without being prepared to support them. But the pattern of those examples is clear: you lack either the intention or the capability to debate with me honestly. You may continue to hurl invectives and level false charges as you wish, but do so only for masturbatory self-gratification. I am through dignifying them with responses. As I said before:
Your esteem was worhtless then. Your contempt is insignificant now. You lack the integrity to make either a concern.**
Spiritus picks up a misconception, forces me to beat it to death over several pages, and then claims I’m changing my argument when I cease referencing his inanities?
Remarkably, for all that Spiritus claims to be well-educated in the fields of mathematical logic and computer science, he missed the obvious weakness in my argument – one place that he might actually be able to mount a successful attack against.
Will he notice it, folks? Only time will tell. Let’s watch…
And for the last time, we don’t even need to use a PA, only an axiomatic system at least as powerful as PA!
**
Oi.
Rememberthis site ? Where it’s shown that a general Godel statement is true in a proof by contradiction? Where it’s clearly stated that the statement is true in the theory but cannot be proven (shown to be true) within the theory?
If you want the statement to be “verifiably” true, then you’re demanding a proof of its truth. But we’ve already established that there can be no proof of G within the system we’re discussing… so to verify the statement, we’ll have to use a different set of axioms (or extend the ones we have) to do so, which is just what that site does.
Thus, another system can show that the statement is true within the first system, but there can be no such demonstration using only the first system. It’s true within the system without being provable within the system.
Do you understand now, or do I have to explain this yet again and in even simpler language?