Re: ... cosmology? KNIGHT KNAVE
Hi John, At 17:19 26/07/04 -0400, John M wrote: Bruno, (and ClassG) We have an overwhelming ignorance about Ks and Ks. We don't know their logical built, their knowledege-base, their behavior. Indeed. Is the K vs K rule a physical, or rather human statement, when - in the latter case there may be violations (punishable by jail - ha ha). Neither physical, nor human ... (see below). Do K K abide by 100.00% by the ONE rule we know about them, or ~99.999%, when there still may be an aberration? 100,00% Are they robots or humans? Looks like machines. Are machines omniscient? Interesting question (not addressed by Smullyan!). But easy though. From Godel's incompleteness (which we have not yet proved, except in the diagonalisation post some time ago, but on which we will come back: it is the heart of the matter in FU's term), it will be easy to prove that: - Machine cannot be omniscient. - Both knight and knaves are omniscient, and so they cannot be machine. I expect, but will not argue now, that knights cannot exist at all, even in platonia (and this with or without comp). Does this throws doubts on what we can infer from FU's puzzles? No, because the KK island is just a pedagogical tool for building a fictive but easily imaginable situation where reasoners must believe some self-referential propositions. But with the diagonalization lemma (alias the heart of the matter) we will eliminate the need of the KK island. It is the logical fate of the correct machine to meet inescapably true and believable (provable) self-referential propositions, from which we can derive true but unbelievable propositions, ... and much more. Bruno PS: Thanks to those who have send me hard puzzles! I will try to solve them after 16 August. I will be busy until then. I will just answer Hal Finney KK Posts, and then finish my paper. I hope I will get the authorization to make it public soon for it will be a good base to proceed on. It is a step toward the English paper I promised to Wei Dai, a long time ago. http://iridia.ulb.ac.be/~marchal/
Re: ... cosmology? KNIGHT KNAVE
At 09:54 27/07/04 -0700, Hal Finney wrote: I am confused about how belief works in this logical reasoner of type 1. Suppose I am such a reasoner. I can be thought of as a theorem-proving machine who uses logic to draw conclusions from premises. We can imagine there is a numbered list of everything I believe and have concluded. It starts with my premises and then I add to it with my conclusions. OK. In this case my premises might be: 1. Knights always tell the truth 2. Knaves always lie 3. Every native is either a knight or a knave 4. A native said, you will never believe I am a knight. Now we can start drawing conclusions. Let t be the proposition that the native is a knight (and hence tells the truth). Then 3 implies: 5. t or ~t Point 4 leads to two conclusions: 6. t implies ~Bt 7. ~t implies Bt Here I use ~ for not, and Bx for I believe x. I am ignoring some complexities involving the future tense of the word will but I think that is OK. Perfect. Here Hal believes p means that sooner or later Hal will assert, believe or prove p. It means p belongs to the list you mentionned. However now I am confused. How do I work with this letter B? What kind of rules does it follow? I understand that Bx, I believe x, is merely a shorthand for saying that x is on my list of premises/conclusions. Correct. This means Bx is a equivalent with Hal believes x. The only difference is that Bx is supposed to be in the language of the machine. If I ever write down x on my numbered list, I could also write down Bx and BBx and BBBx as far as I feel like going. Is this correct? Well, not necessarily. Unless you are a normal machine, which I hope you are! So let us accept the following definition: a machine is normal when, if it ever assert x, it will sooner or later asserts Bx. Normality is a form of self-awareness: when the machine believes x, it will believe Bx, that is it will believe that it will believe x. But what about the other direction? From Bx, can I deduce x? That's pretty important for this puzzle. If Bx merely is a shorthand for saying that x is on my list, then it seems fair to say that if I ever write down Bx I can also write down x. But this seems too powerful. You are right. It is powerful, but rather fair also. let us define a machine to be stable if that is the case. When the machine believes Bx the machine believes x. So what are the correct rules that I, as a simple machine, can follow for dealing with the letter B? Actually we will be interested in a lot of sort of machine. But I do hope you are both normal and stable. Actually I'm sure you are. The problem is that the rules I proposed here lead to a contradiction. If x implies Bx, then I can write down: 8. t implies Bt Note, this does not mean that if he is a knight I believe it, but rather that if I ever deduce he is a knight, I believe it, which is simply the definition of believe in this context. Here you are mistaken. It is funny because you clearly see the mistake, given that you say 'attention (t implies Bt) does not mean if he is a knight the I believe it. But of course (t implies Bt) *does* mean if he is a knight the I believe it. You add it means only (and here I add a slight correction) if I ever deduce he is a knight I will deduce I believe he is a knight which really is, in the machine language: (Bt implies BBt) instead of (t implies Bt). To be sure: a machine is normal if for any proposition p, if the machine believes p, it will believe Bp. But this is equivalent with saying that for any proposition p, the proposition (Bp implies BBp) is true *about* the machine. Same remark for stability: you can say a machine is stable if all the propositions (BBp implies Bp) are true about the machine. This does not mean the stable or normal machine will ever believe being stable or normal. You have (momentarily) confuse a proposition being true on a machine, and being believe by a machine. But 6 and 8 together mean that t implies a contradiction, hence I can conclude: 9. ~t He is a knave. 7 then implies 10. Bt I believe he is a knight. And if Bx implies x, then: 11. t and I have reached a contradiction with 9. So I don't think I am doing this right. By taking into account the confusion above, you should be able to prove, with t still the same proposition (that is (t - -Bt)), that (in case you are a normal reasoner of type 1): if you are consistent, then t is not provable (believable, assertable by you) if you are consistent and stable, then -t is not provable either. That's Godel's first incompleteness theorem. (once we eliminate the KK island from the reasoning to be sure). Bravo. To sum up; any normal stable reasoner of type 1 meeting a knight saying you will never believe I'm a knight will be forever incomplete. (incomplete = there is a proposition like t, which is neither provable nor refutable). And so a machine cannot be omniscient because, although the KK island does not exist, the diagonalization lemma will
Re: Quantum Rebel
Oops, I too was a victim of viral paranoia this AM and committed wholesale deletion of all attachment laden emails in my box including, apparently, Russel's. letter. Can someone send or forward me a copy? (of the letter not a virus) ;) Thanks! Please, Russell, for the peace of our minds who believe in 'smart' viruses and have none of the software you indicated: Could you at least put a word in the e-mail that thei comes from you? (eg Safe from RS or from Russ etc.) I wanted to open this attachment in blind face and paranoid shiver, however went first to FILE - Properties - 2nd page and checked the from line at the bottom. Boring. Even there it may be a virus usiing your mailbox - before you detect it. I am not the only one suffering from virus-paranioa. Cheerz John Mikes
Re: Quantum Rebel
I just read the New Scientist article Quantum Rebel last night about Shariar Afshar's work on the double slit experiment. Ingenious as the experiment is, I really don't think it says anything about different interpretations of QM. Indeed, the outcome of the experiment is just what I'd expect from quantum theory, regardless of which interpretation is used. OK - so the claim is that Bohr's complementarity principle (CP) is tested by this experiment and found wanting. I decided to go back to the two text books I learnt quantum mechanics from - Leonard Schiff's book which is the older and more traditional of the two, and Rammamurti Shankar's book which has the more modern approach, but which I found explained things better. Shankar doesn't mention the CP at all, and for Schiff, the CP is basically a restatement of the Heisenberg uncertainty principle, a principle not tested by Afshar's experiment. In the double slit experiment, how I understand the CP to work is that one cannot measure which slit a photon passes through, and retain an interference pattern. Assuming it is possible to do this, one could divide the measures data into those photons that passed through slit A, and those that passed through slit B. The resulting distribution of photons arriving at the screen of the two slit experiment is then the sum of the distributions of the two subsets of data. However, the two sub distributions do not have inteference patterns so how can the sum have an interference pattern. Hence any such measurement of which slit the photon passes through must affect the photons so as to destroy the intereference pattern. Now in the article, Afshar claims to have measured which slit the photon passed through and verified the existence of an interference pattern. However, this is not the case - without the wires in place to detect the presence of the interference pattern, photons arriving at detector A have passed through slit A, and vice-versa with detector B and slit B. However, with the wires in place, some photons are scattered, indeed some photons which passed through slit A will arrive at detector B. With both slits open, and the wire placed exactly at a null point of the interference pattern, the photons passing through slit A and arriving at detector B exactly counteracts the photons passing thoguh slit B that have been lost through scattering. The mathematics of quantum mechanics assures this, coincidental this may seem. It may be a question of interpretations of interpretations of QM, however on the basis of the New Scientist article, I don't believe Afshar have shown a problem with the complementarity principle. Cheers -- *PS: A number of people ask me about the attachment to my email, which is of type application/pgp-signature. Don't worry, it is not a virus. It is an electronic signature, that may be used to verify this email came from me if you have PGP or GPG installed. Otherwise, you may safely ignore this attachment. A/Prof Russell Standish Director High Performance Computing Support Unit, Phone 9385 6967, 8308 3119 (mobile) UNSW SYDNEY 2052 Fax 9385 6965, 0425 253119 () Australia[EMAIL PROTECTED] Room 2075, Red Centrehttp://parallel.hpc.unsw.edu.au/rks International prefix +612, Interstate prefix 02 - Oorspronkelijk bericht - Van: CMR [EMAIL PROTECTED] Aan: [EMAIL PROTECTED] Verzonden: Wednesday, July 28, 2004 03:40 PM Onderwerp: Re: Quantum Rebel Oops, I too was a victim of viral paranoia this AM and committed wholesale deletion of all attachment laden emails in my box including, apparently, Russel's. letter. Can someone send or forward me a copy? (of the letter not a virus) ;) Thanks! Please, Russell, for the peace of our minds who believe in 'smart' viruses and have none of the software you indicated: Could you at least put a word in the e-mail that thei comes from you? (eg Safe from RS or from Russ etc.) I wanted to open this attachment in blind face and paranoid shiver, however went first to FILE - Properties - 2nd page and checked the from line at the bottom. Boring. Even there it may be a virus usiing your mailbox - before you detect it. I am not the only one suffering from virus-paranioa. Cheerz John Mikes
Re: Quantum Rebel
I also deleted everything immediately, fearing the viral possibilities of the attachments. Jeanne - Original Message - From: CMR [EMAIL PROTECTED] To: [EMAIL PROTECTED] Sent: Wednesday, July 28, 2004 9:40 AM Subject: Re: Quantum Rebel Oops, I too was a victim of viral paranoia this AM and committed wholesale deletion of all attachment laden emails in my box including, apparently, Russel's. letter. Can someone send or forward me a copy? (of the letter not a virus) ;) Thanks! Please, Russell, for the peace of our minds who believe in 'smart' viruses and have none of the software you indicated: Could you at least put a word in the e-mail that thei comes from you? (eg Safe from RS or from Russ etc.) I wanted to open this attachment in blind face and paranoid shiver, however went first to FILE - Properties - 2nd page and checked the from line at the bottom. Boring. Even there it may be a virus usiing your mailbox - before you detect it. I am not the only one suffering from virus-paranioa. Cheerz John Mikes
Re: Quantum Rebel
The probability that Russell's message contained a virus was low (he uses linux) but nonzero. So, I guess that's bad news for some of my copies in the multiverse. - Oorspronkelijk bericht - Van: Jeanne Houston [EMAIL PROTECTED] Aan: CMR [EMAIL PROTECTED]; [EMAIL PROTECTED] Verzonden: Wednesday, July 28, 2004 04:20 PM Onderwerp: Re: Quantum Rebel I also deleted everything immediately, fearing the viral possibilities of the attachments. Jeanne - Original Message - From: CMR [EMAIL PROTECTED] To: [EMAIL PROTECTED] Sent: Wednesday, July 28, 2004 9:40 AM Subject: Re: Quantum Rebel Oops, I too was a victim of viral paranoia this AM and committed wholesale deletion of all attachment laden emails in my box including, apparently, Russel's. letter. Can someone send or forward me a copy? (of the letter not a virus) ;) Thanks! Please, Russell, for the peace of our minds who believe in 'smart' viruses and have none of the software you indicated: Could you at least put a word in the e-mail that thei comes from you? (eg Safe from RS or from Russ etc.) I wanted to open this attachment in blind face and paranoid shiver, however went first to FILE - Properties - 2nd page and checked the from line at the bottom. Boring. Even there it may be a virus usiing your mailbox - before you detect it. I am not the only one suffering from virus-paranioa. Cheerz John Mikes
Re: Quantum Rebel
Saibal Mitra fwded It may be a question of interpretations of interpretations of QM, however on the basis of the New Scientist article, I don't believe Afshar have shown a problem with the complementarity principle. I agree. But imagine the usual two-slit set-up. And this unusual screen, to reveal the interference pattern ... _ _ _ _ _ _ They are a set of photographic plates, they can register both the photons and their directions, i.e. if they come from the upper or from the lower slit (photons coming from the upper slit hit the upper face of the photographic plate, photons coming from the lower slit hit the lower face, there is a principle of conservation of momentum after all). Calculations made by Zurek and Wooters show that this kind of screen works, that is it can reveal - in principle - both the welcher weg and the interference pattern. But the complementarity principle forbids that. *PS: A number of people ask me about the attachment to my email, which is of type application/pgp-signature. The problem with that signature is that we (or just me?) do not receive any text, in the body of the msg. Just blank.
Re: ... cosmology? KNIGHT KNAVE
This is confusing because I believe p has two different meanings. One is that I have written down p with a number in front of it, as one of my theorems. The other meaning is the string Bp. But that string only has meaning from the perspective of an outside observer. To me, as the machine, it is just a pair of letters. B doesn't have to mean believe. It could mean Belachen, which is German for believe. All I need to know, as a formal system, is what rules the letter B follows. Bruno wrote: At 09:54 27/07/04 -0700, Hal Finney wrote: If I ever write down x on my numbered list, I could also write down Bx and BBx and BBBx as far as I feel like going. Is this correct? Well, not necessarily. Unless you are a normal machine, which I hope you are! So let us accept the following definition: a machine is normal when, if it ever assert x, it will sooner or later asserts Bx. Normality is a form of self-awareness: when the machine believes x, it will believe Bx, that is it will believe that it will believe x. But what about the other direction? From Bx, can I deduce x? That's pretty important for this puzzle. If Bx merely is a shorthand for saying that x is on my list, then it seems fair to say that if I ever write down Bx I can also write down x. But this seems too powerful. You are right. It is powerful, but rather fair also. let us define a machine to be stable if that is the case. When the machine believes Bx the machine believes x. So in my terms, I can add two axioms: 0a. x implies Bx 0b. Bx implies x The first is the axiom of normality, and the second is the axiom of stability. I don't find these words to be particularly appropriate, by the way, but I suppose they are traditional. It also seems to me that these axioms, which define the behavior of the letter B, don't particularly well represent the concept of belief. The problem is that beliefs can be uncertain and don't follow the law of the excluded middle. If p is that there is life on Mars, then (p or ~p) is true. Either there's life there or there isn't. But it's not true that (Bp or B~p). It's not the case that either I believe there is life on Mars or I believe there is no life on Mars. The truth is, I don't believe either way. But axioms 0a and 0b let me conclude (Bp or B~p). Obviously they collectively imply p if and only if Bp. Therefore from (p or ~p) we can immediately get (Bp or B~p). Hence for normal people, the law of the excluded middle applies to beliefs. This proof is pure logic and has no dependence on the meaning of B. If B is Belachen, I have showed that if p implies Belachen(p), then it follows that (Belachen(p) or Belachen(~p)) is true. That's all. It's a step outside the system to say that B follows rules which make it appropriate for us to treat it as meaning believes. But do 0a. and 0b. really capture the meaning of belief? I question that. B looks more like an identity operator under those axioms. The problem is that the rules I proposed here lead to a contradiction. If x implies Bx, then I can write down: 8. t implies Bt Note, this does not mean that if he is a knight I believe it, but rather that if I ever deduce he is a knight, I believe it, which is simply the definition of believe in this context. Here you are mistaken. It is funny because you clearly see the mistake, given that you say 'attention (t implies Bt) does not mean if he is a knight the I believe it. But of course (t implies Bt) *does* mean if he is a knight the I believe it. I don't see this. To me as the machine, there is no meaning. I am just playing with letters. t implies Bt is only a shorthand for if he is a knight then B(if he is a knight). There is no more meaning than that. The letter B is just a letter that follows certain rules. We only get meaning from outside, when we look at what the machine is doing and try to relate the way the rules work to concepts in the real world. It is at this point that we bring in the interpretation of Bx as the machine believes x. Suppose for some proposition q the machine deduces it on step 117: 117. q Does this mean that q is true? No, it means that that the machine believes q. Does it mean that Bq is true? Yes. Bq is true, because Bq is a shorthand for saying that the machine believes q, and by definition the machine believes something when it writes it down in its numbered list. We can see it right there, number 117. So the machine believes q and Bq is true. But q is not (necessarily) true. The machine writing something down does not mean it is true. By definition, it means the machine believes it. Consider a different example: 191. Bp What does this mean? Does it mean that Bp is true? No, it means that the machine believes Bp, because by definition, what the machine writes down in its numbered list is what it believes. Is BBp true? Yes, it is true, because that says that the machine believes Bp, and that means that Bp is in the
Re: Quantum Rebel
Saibal Mitra wrote: Now in the article, Afshar claims to have measured which slit the photon passed through and verified the existence of an interference pattern. However, this is not the case - without the wires in place to detect the presence of the interference pattern, photons arriving at detector A have passed through slit A, and vice-versa with detector B and slit B. However, with the wires in place, some photons are scattered, indeed some photons which passed through slit A will arrive at detector B. With both slits open, and the wire placed exactly at a null point of the interference pattern, the photons passing through slit A and arriving at detector B exactly counteracts the photons passing thoguh slit B that have been lost through scattering. The mathematics of quantum mechanics assures this, coincidental this may seem. A poster on sci.physics.research elaborates on this point a little with a nice thought-experiment involving enlarging the wires until they are almost touching, at which point you just have a new set of slits: http://makeashorterlink.com/?W3F012BE8 Now I haven't done any calculations or read the New Scientist article except looking at the lab setup graphics, but if I would hazard a quick guess, it would be that it will turn out that even if the wires are placed in the interference fields valleys, the finite width of the wires will diffract just enough photons to erase the which-way information that was gained by focusing the detectors at the holes in the wall through the lens. Consider the limiting case with wires placed with their centres in the interference fields valleys as before, but expand their width so much that they almost touch each other. What you have now is yet another wall with a bunch of slits in! Obviously, almost all which-way information is lost after the wavefronts pass these almost infinitesimal slits since they will diffract the photons equally no matter from which hole in the *first* wall they originated, so any detector placed after this obstacle will be like running a new multiple-slit interference setup (although with the lens now severely defocusing the too-closely placed new slits). And since the which-way information from the first wall is erased, interference is free to happen between the first and the second wall. After the secondary wall the detectors can pick up which-way information causing them to behave as if there was little subsequent interference. Conversely, the other limiting case is with no wires (or secondary wall) present. Then all which-way information is present and again the detectors behave as if there was no interference. The experiment shows a case in between these limits and the effect I guessed at above could (and should, according to traditional QM) turn out to always cancel any attempt to find both 100% interference and 100% which-way information. This would be better showed with some calculations of course...
Re: Quantum Rebel
Actually, looking at the diagram and explanation of the experiment posted at http://www.kathryncramer.com/wblog/archives/000674.html I think Saibal Mitra and the sci.physics.research poster I quoted may have misunderstood what happened in this experiment. I may have misunderstood, but it sounded as if both were arguing that the finite width of the wires could erase some of the which-path information and explain why you'd see interference at the final detectors. But the diagram seems to say that *no* interference was found at the detectorsthe interference Afshar is talking about was just in the fact that no photons were scattering against the wires because they were all placed in the interference valleys. So the idea seems to be that interference is the explanation for why no photons scatter against the wires, but the focusing lens behind the wires makes sure that photons from the left slit always go to the left detector and the photons from the right slit always go to the right detector--this is the violation of complementarity, that the photons behave like a wave in avoiding the wires but behave like particles when arriving at the detectors. I'm not sure that the notion of complementarity has ever been sufficiently well-defined to say that this experiment violates it though, and in any case, as long as the results of the experiment match the predictions made by the standard theory of quantum mechanics, it cannot be taken as a disproof of the Everett interpretation, since the basic idea of the Everett interpretation is to keep the standard rules for wavefunction evolution but just to drop the collapse idea (the projection postulate). Jesse
Re: Quantum Rebel
Not me but Russell wrote that. I should have made that clear better when I posted Russell's attachment (Sorry Russell!). - Oorspronkelijk bericht - Van: Jesse Mazer [EMAIL PROTECTED] Aan: [EMAIL PROTECTED] Verzonden: Wednesday, July 28, 2004 08:59 PM Onderwerp: Re: Quantum Rebel Saibal Mitra wrote: Now in the article, Afshar claims to have measured which slit the photon passed through and verified the existence of an interference pattern. However, this is not the case - without the wires in place to detect the presence of the interference pattern, photons arriving at detector A have passed through slit A, and vice-versa with detector B and slit B. However, with the wires in place, some photons are scattered, indeed some photons which passed through slit A will arrive at detector B. With both slits open, and the wire placed exactly at a null point of the interference pattern, the photons passing through slit A and arriving at detector B exactly counteracts the photons passing thoguh slit B that have been lost through scattering. The mathematics of quantum mechanics assures this, coincidental this may seem. A poster on sci.physics.research elaborates on this point a little with a nice thought-experiment involving enlarging the wires until they are almost touching, at which point you just have a new set of slits: http://makeashorterlink.com/?W3F012BE8 Now I haven't done any calculations or read the New Scientist article except looking at the lab setup graphics, but if I would hazard a quick guess, it would be that it will turn out that even if the wires are placed in the interference fields valleys, the finite width of the wires will diffract just enough photons to erase the which-way information that was gained by focusing the detectors at the holes in the wall through the lens. Consider the limiting case with wires placed with their centres in the interference fields valleys as before, but expand their width so much that they almost touch each other. What you have now is yet another wall with a bunch of slits in! Obviously, almost all which-way information is lost after the wavefronts pass these almost infinitesimal slits since they will diffract the photons equally no matter from which hole in the *first* wall they originated, so any detector placed after this obstacle will be like running a new multiple-slit interference setup (although with the lens now severely defocusing the too-closely placed new slits). And since the which-way information from the first wall is erased, interference is free to happen between the first and the second wall. After the secondary wall the detectors can pick up which-way information causing them to behave as if there was little subsequent interference. Conversely, the other limiting case is with no wires (or secondary wall) present. Then all which-way information is present and again the detectors behave as if there was no interference. The experiment shows a case in between these limits and the effect I guessed at above could (and should, according to traditional QM) turn out to always cancel any attempt to find both 100% interference and 100% which-way information. This would be better showed with some calculations of course...
Re: Quantum Rebel
Can anyone tell me why the body of the email is blank to some people? Is it some overzealous defang program the removes the body as well as the attachment? I don't care if the attachment is removed - it doesn't contain information - its purpose is to authenticate the letter only, and can be safely ignored by anyone who doesn't care. I'll send this email unsigned, to make sure the body gets through... Cheers On Wed, Jul 28, 2004 at 06:08:40PM +0200, scerir wrote: *PS: A number of people ask me about the attachment to my email, which is of type application/pgp-signature. The problem with that signature is that we (or just me?) do not receive any text, in the body of the msg. Just blank. -- *PS: A number of people ask me about the attachment to my email, which is of type application/pgp-signature. Don't worry, it is not a virus. It is an electronic signature, that may be used to verify this email came from me if you have PGP or GPG installed. Otherwise, you may safely ignore this attachment. A/Prof Russell Standish Director High Performance Computing Support Unit, Phone 9385 6967, 8308 3119 (mobile) UNSW SYDNEY 2052 Fax 9385 6965, 0425 253119 () Australia[EMAIL PROTECTED] Room 2075, Red Centrehttp://parallel.hpc.unsw.edu.au/rks International prefix +612, Interstate prefix 02
