Re: Penrose and algorithms
Le 07-juil.-07, à 16:39, LauLuna a écrit : On Jul 7, 12:59 pm, Bruno Marchal [EMAIL PROTECTED] wrote: Le 06-juil.-07, à 14:53, LauLuna a écrit : But again, for any set of such 'physiological' axioms there is a corresponding equivalent set of 'conceptual' axioms. There is all the same a logical impossibility for us to know the second set is sound. No consistent (and strong enough) system S can prove the soundness of any system S' equivalent to S: otherwise S' would prove its own soundness and would be inconsistent. And this is just what is odd. It is odd indeed. But it is. No, it is not necessary so; the alternative is that such algorithm does not exist. I will endorse the existence of that algorithm only when I find reason enough to do it. I haven't yet, and the oddities its existence implies count, obviously, against its existence. If the algorithm exists, then the knowable algorithm does not exist. We can only bet on comp, not prove it. But it is refutable. I'd say this is rather Lucas's argument. Penrose's is like this: 1. Mathematicians are not using a knowably sound algorithm to do math. 2. If they were using any algorithm whatsoever, they would be using a knowably sound one. 3. Ergo, they are not using any algorithm at all. Do you agree that from what you say above, 2. is already invalidate? Not at all. I still find it far likelier that if there is a sound algorithm ALG and an equivalent formal system S whose soundness we can know, then there is no logical impossibility for our knowing the soundness of ALG. We do agree. You are just postulating not-comp. I have no trouble with that. What I find inconclusive in Penrose's argument is that he refers not just to actual numan intellectual behavior but to some idealized (forever sound and consistent) human intelligence. I think the existence of a such an ability has to be argued. A rather good approximation for machine could be given by the transfinite set of effective and finite sound extensions of a Lobian machine. Like those proposed by Turing. They all obey locally to G and G* (as shown by Beklemishev). The infinite and the transfinite does not help the machine with regard to the incompleteness phenomenon, except if the infinite is made very highly non effective. But in that case you tend to the One or truth a very non effective notion). If someone asked me: 'do you agree that Penrose's argument does not prove there are certain human behaviors which computers can't reproduce?', I'd answered: 'yes, I agree it doesn't'. But if someone asked me: 'do you agree that Penrose's argument does not prove human intelligence cannot be simulated by computers?' I'd reply: 'as far as that abstract intelligence you speak of exists at all as a real faculty, I'd say it is far more probable that computers cannot reproduce it'. Why? All you need to do consists in providing more and more time-space-memory to the machine. Humans are universal by extending their mind by pictures on walls, ... magnetic tape I.e. some versions of computationalism assume, exactly like Penrose, the existence of that abstract human intelligence; I would say those formulations of computationalism are nearly refuted by Penrose. There is a lobian abstract intelligence, but it can differentiate in many kinds, and cannot be defined *effectively* (with a program) by any machine. It corresponds loosely to the first non effective or non-nameable ordinal (the OMEGA_1^Church-Kleene ordinal). I hope I've made my point clear. OK. Personally I am just postulating the comp hyp and study the consequences. If we are machine or sequence of machine then we cannot which machine we are, still less which sequence of machines we belong too ... (introducing eventually verifiable 1-person indeterminacies). I argue that the laws of observability (physics) emerges from that comp-indeterminacy. I think we agree on Penrose. Bruno http://iridia.ulb.ac.be/~marchal/ --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: Penrose and algorithms
Le 06-juil.-07, à 14:53, LauLuna a écrit : But again, for any set of such 'physiological' axioms there is a corresponding equivalent set of 'conceptual' axioms. There is all the same a logical impossibility for us to know the second set is sound. No consistent (and strong enough) system S can prove the soundness of any system S' equivalent to S: otherwise S' would prove its own soundness and would be inconsistent. And this is just what is odd. It is odd indeed. But it is. I'd say this is rather Lucas's argument. Penrose's is like this: 1. Mathematicians are not using a knowably sound algorithm to do math. 2. If they were using any algorithm whatsoever, they would be using a knowably sound one. 3. Ergo, they are not using any algorithm at all. Do you agree that from what you say above, 2. is already invalidate? Bruno http://iridia.ulb.ac.be/~marchal/ --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: Penrose and algorithms
Le 06-juil.-07, à 19:43, Brent Meeker a écrit : Bruno Marchal wrote: ... Now all (sufficiently rich) theories/machine can prove their own Godel's theorem. PA can prove that if PA is consistent then PA cannot prove its consitency. A somehow weak (compared to ZF) theory like PA can even prove the corresponding theorem for the richer ZF: PA can prove that if ZF is consistent then ZF can prove its own consistency. Of course you meant ..then ZF cannot prove its own consistency. Yes. (Sorry). So, in general a machine can find its own godelian sentences, and can even infer their truth in some abductive way from very minimal inference inductive abilities, or from assumptions. No sound (or just consistent) machine can ever prove its own godelian sentences, in particular no machine can prove its own consistency, but then machine can bet on them or know them serendipitously). This is comparable with consciousness. Indeed it is easy to manufacture thought experiements illustrating that no conscious being can prove it is conscious, except that consciousness is more truth related, so that machine cannot even define their own consciousness (by Tarski undefinability of truth theorem). But this is within an axiomatic system - whose reliability already depends on knowing the truth of the axioms. ISTM that concepts of consciousness, knowledge, and truth that are relative to formal axiomatic systems are already to weak to provide fundamental explanations. With UDA (Universal Dovetailer Argument) I ask you to implicate yourself in a thought experiment. Obviously I bet, hope, pray, that you will reason reasonably and soundly. With the AUDA (the Arithmetical version of UDA, or Plotinus now) I ask the Universal Machine to implicate herself in a formal reasoning. As a mathematician, I limit myself to sound (and thus self-referentially correct) machine, for the same reason I pray you are sound. Such a restriction is provably non constructive: there is no algorithm to decide if a machine is sound or not ... But note that the comp assumption and even just the coherence of Church thesis relies on non constructive assumptions at the start. Bruno http://iridia.ulb.ac.be/~marchal/ --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: Penrose and algorithms
On Jul 7, 12:59 pm, Bruno Marchal [EMAIL PROTECTED] wrote: Le 06-juil.-07, à 14:53, LauLuna a écrit : But again, for any set of such 'physiological' axioms there is a corresponding equivalent set of 'conceptual' axioms. There is all the same a logical impossibility for us to know the second set is sound. No consistent (and strong enough) system S can prove the soundness of any system S' equivalent to S: otherwise S' would prove its own soundness and would be inconsistent. And this is just what is odd. It is odd indeed. But it is. No, it is not necessary so; the alternative is that such algorithm does not exist. I will endorse the existence of that algorithm only when I find reason enough to do it. I haven't yet, and the oddities its existence implies count, obviously, against its existence. I'd say this is rather Lucas's argument. Penrose's is like this: 1. Mathematicians are not using a knowably sound algorithm to do math. 2. If they were using any algorithm whatsoever, they would be using a knowably sound one. 3. Ergo, they are not using any algorithm at all. Do you agree that from what you say above, 2. is already invalidate? Not at all. I still find it far likelier that if there is a sound algorithm ALG and an equivalent formal system S whose soundness we can know, then there is no logical impossibility for our knowing the soundness of ALG. What I find inconclusive in Penrose's argument is that he refers not just to actual numan intellectual behavior but to some idealized (forever sound and consistent) human intelligence. I think the existence of a such an ability has to be argued. If someone asked me: 'do you agree that Penrose's argument does not prove there are certain human behaviors which computers can't reproduce?', I'd answered: 'yes, I agree it doesn't'. But if someone asked me: 'do you agree that Penrose's argument does not prove human intelligence cannot be simulated by computers?' I'd reply: 'as far as that abstract intelligence you speak of exists at all as a real faculty, I'd say it is far more probable that computers cannot reproduce it'. I.e. some versions of computationalism assume, exactly like Penrose, the existence of that abstract human intelligence; I would say those formulations of computationalism are nearly refuted by Penrose. I hope I've made my point clear. Best --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: Penrose and algorithms
Le 05-juil.-07, à 22:14, Jesse Mazer a écrit : His [Penrose] whole Godelian argument is based on the idea that for any computational theorem-proving machine, by examining its construction we can use this understanding to find a mathematical statement which *we* know must be true, but which the machine can never output--that we understand something it doesn't. But I think my argument shows that if you were really to build a simulated mathematician or community of mathematicians in a computer, the Godel statement for this system would only be true *if* they never made a mistake in reasoning or chose to output a false statement to be perverse, and that therefore there is no way for us on the outside to have any more confidence about whether they will ever output this statement than they do (and thus neither of us can know whether the statement is actually a true or false theorem of arithmetic). I think I agree with your line of argumentation, but you way of talking could be misleading. Especially if people interpret arithmetic by If we are in front of a machine that we know to be sound, then we can indeed know that the Godelian proposition associated to the machine is true. For example, nobody (serious) doubt that PA (Peano Arithmetic, the first order formal arithmetic theory/machine) is sound. So we know that all the godelian sentences are true, and PA cannot know that. But this just proves that I am not PA, and that I have actually stronger ability than PA. I could have taken ZF instead (ZF is Zermelo Fraenkel formal theory/machine of sets), although I must say that if I have entire confidence in PA, I have only 99,9998% confidence in ZF (and thus I can already be only 99,9998% sure of the ZF godelian sentences). About NF (Quine's New Foundation formal theory machine) I have only 50% confidence!!! Now all (sufficiently rich) theories/machine can prove their own Godel's theorem. PA can prove that if PA is consistent then PA cannot prove its consitency. A somehow weak (compared to ZF) theory like PA can even prove the corresponding theorem for the richer ZF: PA can prove that if ZF is consistent then ZF can prove its own consistency. So, in general a machine can find its own godelian sentences, and can even infer their truth in some abductive way from very minimal inference inductive abilities, or from assumptions. No sound (or just consistent) machine can ever prove its own godelian sentences, in particular no machine can prove its own consistency, but then machine can bet on them or know them serendipitously). This is comparable with consciousness. Indeed it is easy to manufacture thought experiements illustrating that no conscious being can prove it is conscious, except that consciousness is more truth related, so that machine cannot even define their own consciousness (by Tarski undefinability of truth theorem). Bruno http://iridia.ulb.ac.be/~marchal/ --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: Penrose and algorithms
On Jul 5, 2:14 pm, LauLuna [EMAIL PROTECTED] wrote: I don't see how to reconcile free will with computationalism either. It seems like you are an incompatibilist concerning free will. Freewill can be reconciled with computationalism (or any deterministic system) if one accepts compatabilism ( http://en.wikipedia.org/wiki/Free_will#Compatibilism ). More worrisome than determinism's affect on freewill, however, is many-worlds (or other everything/ultimate ensemble theories). Whereas determinism says the future is written in stone, many-worlds would say all futures are written in stone. Jason --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: Penrose and algorithms
Le 06-juil.-07, à 14:00, Jason a écrit : On Jul 5, 2:14 pm, LauLuna [EMAIL PROTECTED] wrote: I don't see how to reconcile free will with computationalism either. It seems like you are an incompatibilist concerning free will. Freewill can be reconciled with computationalism (or any deterministic system) if one accepts compatabilism ( http://en.wikipedia.org/wiki/Free_will#Compatibilism ). More worrisome than determinism's affect on freewill, however, is many-worlds (or other everything/ultimate ensemble theories). Whereas determinism says the future is written in stone, many-worlds would say all futures are written in stone. Like comp already say. At least with QM we know that the future are weighted and free-will will correspond to choosing among normal worlds. With comp, there is only promising results in that direction, (which could lead to a refutation of comp). John Bell (the physicist, not the quantum logician) has also crticized the MWI with respect to free-will, but this does not follow from the SWE. The SWE does not say all future are equal. It says that all future are realized, but some have negligible probability, and this left room for genuine free-will. For example I can choose the stairs, the lift or the windows to go outside, but only with the stairs and lift can I stay in relatively normal worlds. By going outside by jumping through the windows, I take the risk of surviving in a white rabbit world and then to remain in the relatively normal world with respect to that not normal world. This is why I think quantum immortality is a form of terrifying thinking ... if you think twice and take it seriously. Of course reality (with or without QM or comp) is more complex in any case, so it is much plausibly premature to panic from so theoretical elaborations. Actually computer science predicts possible unexpectable jump ... Is it worth exploring the possible comp-hell, to search the limit of the unbearable? Well, news indicate humans have some incline to point on such direction. That could be the price of free-will. Have you read the delicious texts by Smullyan (in Mind'sI I think) about the guy who asks God to take away his free-will (and its associated guilt feeling) ? Bruno http://iridia.ulb.ac.be/~marchal/ --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: Penrose and algorithms
Bruno Marchal wrote: ... Now all (sufficiently rich) theories/machine can prove their own Godel's theorem. PA can prove that if PA is consistent then PA cannot prove its consitency. A somehow weak (compared to ZF) theory like PA can even prove the corresponding theorem for the richer ZF: PA can prove that if ZF is consistent then ZF can prove its own consistency. Of course you meant ..then ZF cannot prove its own consistency. Brent Meeker So, in general a machine can find its own godelian sentences, and can even infer their truth in some abductive way from very minimal inference inductive abilities, or from assumptions. No sound (or just consistent) machine can ever prove its own godelian sentences, in particular no machine can prove its own consistency, but then machine can bet on them or know them serendipitously). This is comparable with consciousness. Indeed it is easy to manufacture thought experiements illustrating that no conscious being can prove it is conscious, except that consciousness is more truth related, so that machine cannot even define their own consciousness (by Tarski undefinability of truth theorem). But this is within an axiomatic system - whose reliability already depends on knowing the truth of the axioms. ISTM that concepts of consciousness, knowledge, and truth that are relative to formal axiomatic systems are already to weak to provide fundamental explanations. Brent Meeker --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: Penrose and algorithms
On 29 jun, 19:10, Jesse Mazer [EMAIL PROTECTED] wrote: LauLuna wrote: On 29 jun, 02:13, Jesse Mazer [EMAIL PROTECTED] wrote: LauLuna wrote: For any Turing machine there is an equivalent axiomatic system; whether we could construct it or not, is of no significance here. But for a simulation of a mathematician's brain, the axioms wouldn't be statements about arithmetic which we could inspect and judge whether they were true or false individually, they'd just be statements about the initial state and behavior of the simulated brain. So again, there'd be no way to inspect the system and feel perfectly confident the system would never output a false statement about arithmetic, unlike in the case of the axiomatic systems used by mathematicians to prove theorems. Yes, but this is not the point. For any Turing machine performing mathematical skills there is also an equivalent mathematical axiomatic system; if we are sound Turing machines, then we could never know that mathematical system sound, in spite that its axioms are the same we use. I agree, a simulation of a mathematician's brain (or of a giant simulated community of mathematicians) cannot be a *knowably* sound system, because we can't do the trick of examining each axiom and seeing they are individually correct statements about arithmetic as with the normal axiomatic systems used by mathematicians. But that doesn't mean it's unsound either--it may in fact never produce a false statement about arithmetic, it's just that we can't be sure in advance, the only way to find out is to run it forever and check. Yes, but how can there be a logical impossibility for us to acknowledge as sound the same principles and rules we are using? But Penrose was not just arguing that human mathematical ability can't be based on a knowably sound algorithm, he was arguing that it must be *non-algorithmic*. No, he argues in Shadows of the Mind exactly what I say. He goes on arguing why a sound algorithm representing human intelligence is unlikely to be not knowably sound. And the impossibility has to be a logical impossibility, not merely a technical or physical one since it depends on Gödel's theorem. That's a bit odd, isn't it? No, I don't see anything very odd about the idea that human mathematical abilities can't be a knowably sound algorithm--it is no more odd than the idea that there are some cellular automata where there is no shortcut to knowing whether they'll reach a certain state or not other than actually simulating them, as Wolfram suggests in A New Kind of Science. The point is that the axioms are exactly our axioms! In fact I'd say it fits nicely with our feeling of free will, that there should be no way to be sure in advance that we won't break some rules we have been told to obey, apart from actually running us and seeing what we actually end up doing. I don't see how to reconcile free will with computationalism either. Regards --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: Penrose and algorithms
LauLuna wrote: On 29 jun, 19:10, Jesse Mazer [EMAIL PROTECTED] wrote: LauLuna wrote: On 29 jun, 02:13, Jesse Mazer [EMAIL PROTECTED] wrote: LauLuna wrote: For any Turing machine there is an equivalent axiomatic system; whether we could construct it or not, is of no significance here. But for a simulation of a mathematician's brain, the axioms wouldn't be statements about arithmetic which we could inspect and judge whether they were true or false individually, they'd just be statements about the initial state and behavior of the simulated brain. So again, there'd be no way to inspect the system and feel perfectly confident the system would never output a false statement about arithmetic, unlike in the case of the axiomatic systems used by mathematicians to prove theorems. Yes, but this is not the point. For any Turing machine performing mathematical skills there is also an equivalent mathematical axiomatic system; if we are sound Turing machines, then we could never know that mathematical system sound, in spite that its axioms are the same we use. I agree, a simulation of a mathematician's brain (or of a giant simulated community of mathematicians) cannot be a *knowably* sound system, because we can't do the trick of examining each axiom and seeing they are individually correct statements about arithmetic as with the normal axiomatic systems used by mathematicians. But that doesn't mean it's unsound either--it may in fact never produce a false statement about arithmetic, it's just that we can't be sure in advance, the only way to find out is to run it forever and check. Yes, but how can there be a logical impossibility for us to acknowledge as sound the same principles and rules we are using? The axioms in a simulation of a brain would have nothing to do with the high-level conceptual principles and rules we use when thinking about mathematics, they would be axioms concerning the most basic physical laws and microscopic initial conditions of the simulated brain and its simulated environment, like the details of which brain cells are connected by which synapses or how one cell will respond to a particular electrochemical signal from another cell. Just because I think my high-level reasoning is quite reliable in general, that's no reason for me to believe a detailed simulation of my brain would be sound in the sense that I'm 100% certain that this precise arrangement of nerve cells in this particular simulated environment, when allowed to evolve indefinitely according to some well-defined deterministic rules, would *never* make a mistake in reasoning and output an incorrect statement about arithmetic (or even that it would never choose to intentionally output a statement it believed to be false just to be contrary). But Penrose was not just arguing that human mathematical ability can't be based on a knowably sound algorithm, he was arguing that it must be *non-algorithmic*. No, he argues in Shadows of the Mind exactly what I say. He goes on arguing why a sound algorithm representing human intelligence is unlikely to be not knowably sound. He does argue that as a first step, but then he goes on to conclude what I said he did, that human intelligence cannot be algorithmic. For example, on p. 40 he makes quite clear that his arguments throughout the rest of the book are intended to show that there must be something non-computational in human mental processes: I shall primarily be concerned, in Part I of this book, with the issue of what it is possible to achieve by use of the mental quality of 'understanding.' Though I do not attempt to define what this word means, I hope that its meaning will indeed be clear enough that the reader will be persuaded that this quality--whatever it is--must indeed be an essentail part of that mental activity needed for an acceptance of the arguments of 2.5. I propose to show that the appresiation of these arguments must involve something non-computational. Later, on p. 54: Why do I claim that this 'awareness', whatever it is, must be something non-computational, so that no robot, controlled by a computer, based merely on the standard logical ideas of a Turing machine (or equivalent)--whether top-down or bottom-up--can achieve or even simulate it? It is here that the Godelian argument plays its crucial role. His whole Godelian argument is based on the idea that for any computational theorem-proving machine, by examining its construction we can use this understanding to find a mathematical statement which *we* know must be true, but which the machine can never output--that we understand something it doesn't. But I think my argument shows that if you were really to build a simulated mathematician or community of mathematicians in a computer, the Godel statement for this system would only be true *if* they never made a mistake in
Re: Penrose and algorithms
Le 28-juin-07, à 16:32, LauLuna a écrit : This is not fair to Penrose. He has convincingly argued in 'Shadows of the Mind' that human mathematical intelligence cannot be a knowably sound algorithm. Assume X is an algorithm representing the human mathematical intelligence. The point is not that man cannot recognize X as representing his own intellingence, it is rather that human intellingence cannot know X to be sound (independently of whether X is recognized as what it is). And this is strange because humans could exhaustively inspect X and they should find it correct since it contains the same principles of reasoning human intelligence employs! As far as human intelligence is sound and finitely describable as X, human intelligence cannot recognize X as being human intelligence. One way out is claiming that human intelligence is insonsistent. Another, that such a thing as human intelligence could not exist, since it is not well defined. The latter seems more of a serious objection to me. So, I consider Penrose's argument inconclusive. Of course this will not work assuming comp, i.e. the (non constructive) assumption that there is a level of description such that I can be described correctly at that level. The conclusion is only that I cannot prove to myself that such a level is the correct one, so the yes doctor has to be a non-constructive bet. Practically it needs some platonic act of faith. Assuming comp, we don't have to define what is intelligence or consciousness ... to make reasoning. Anyway, the use Lucas and Penrose make of Gödel's theorem make it seem less likely that human reason can be reproduced by machines. This must be granted. The Lucas Penrose (of the Emperor's new clothes) argument is just incorrect, and its maximal correct reconstruction just shows that human reason/body cannot build machine provably or knowably endowed with human reason/body. It can do that in some non provable way, and we could there is a case that animals do something similar since the invention of asexual and sexual reproduction. Penrose is correct in the shadows of the mind, (by adding the knowably you refer to above) but he does not take seriously the correction into account. But the whole of the' arithmetical interpretation of Plotinus hypostases including its matter theory is build on that nuance. Assuming comp, we really cannot know (soundly) which machine we are, and thus which computations support us. This gives the arithmetical interpretation of the first person comp indeterminacies. It predicts also that any sound lobian machine looking at itself below its substitution level will discover a sharable form of indeterminacy, like QM confirms (and illustrates). I do appreciate Penrose (I talked with him in Siena). Unlike many physicist, he is quite aware of the existence and hardness of the mind body problem, and agrees that you cannot have both materialism and computationalism (but for different reason than me, and as I said slightly incorrect one which forces him to speculate on the wrongness of both QM and comp). I get the same conclusion by keeping comp and (most probably) QM, but then by abandoning physicalism/materialism. Regards, Bruno On 9 jun, 18:40, Bruno Marchal [EMAIL PROTECTED] wrote: Hi Chris, Le 09-juin-07, à 13:03, chris peck a écrit : Hello The time has come again when I need to seek advice from the everything-list and its contributors. Penrose I believe has argued that the inability to algorithmically solve the halting problem but the ability of humans, or at least Kurt Godel, to understand that formal systems are incomplete together demonstrate that human reason is not algorithmic in nature - and therefore that the AI project is fundamentally flawed. What is the general consensus here on that score. I know that there are many perspectives here including those who agree with Penrose. Are there any decent threads I could look at that deal with this issue? All the best Chris. This is a fundamental issue, even though things are clear for the logicians since 1921 ... But apparently it is still very cloudy for the physicists (except Hofstadter!). I have no time to explain, but let me quote the first paragraph of my Siena papers (your question is at the heart of the interview of the lobian machine and the arithmetical interpretation of Plotinus). But you can find many more explanation in my web pages (in french and in english). In a nutshell, Penrose, though quite courageous and more lucid on the mind body problem than the average physicist, is deadly mistaken on Godel. Godel's theorem are very lucky event for mechanism: eventually it leads to their theologies ... The book by Franzen on the misuse of Godel is quite good. An deep book is also the one by Judson Webb, ref in my thesis). We will have the opportunity to come back on this deep issue, which illustrate a gap
Re: Penrose and algorithms
On 29 jun, 02:13, Jesse Mazer [EMAIL PROTECTED] wrote: LauLuna wrote: For any Turing machine there is an equivalent axiomatic system; whether we could construct it or not, is of no significance here. But for a simulation of a mathematician's brain, the axioms wouldn't be statements about arithmetic which we could inspect and judge whether they were true or false individually, they'd just be statements about the initial state and behavior of the simulated brain. So again, there'd be no way to inspect the system and feel perfectly confident the system would never output a false statement about arithmetic, unlike in the case of the axiomatic systems used by mathematicians to prove theorems. Yes, but this is not the point. For any Turing machine performing mathematical skills there is also an equivalent mathematical axiomatic system; if we are sound Turing machines, then we could never know that mathematical system sound, in spite that its axioms are the same we use. And the impossibility has to be a logical impossibility, not merely a technical or physical one since it depends on Gödel's theorem. That's a bit odd, isn't it? Regards --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: Penrose and algorithms
LauLuna wrote: On 29 jun, 02:13, Jesse Mazer [EMAIL PROTECTED] wrote: LauLuna wrote: For any Turing machine there is an equivalent axiomatic system; whether we could construct it or not, is of no significance here. But for a simulation of a mathematician's brain, the axioms wouldn't be statements about arithmetic which we could inspect and judge whether they were true or false individually, they'd just be statements about the initial state and behavior of the simulated brain. So again, there'd be no way to inspect the system and feel perfectly confident the system would never output a false statement about arithmetic, unlike in the case of the axiomatic systems used by mathematicians to prove theorems. Yes, but this is not the point. For any Turing machine performing mathematical skills there is also an equivalent mathematical axiomatic system; if we are sound Turing machines, then we could never know that mathematical system sound, in spite that its axioms are the same we use. I agree, a simulation of a mathematician's brain (or of a giant simulated community of mathematicians) cannot be a *knowably* sound system, because we can't do the trick of examining each axiom and seeing they are individually correct statements about arithmetic as with the normal axiomatic systems used by mathematicians. But that doesn't mean it's unsound either--it may in fact never produce a false statement about arithmetic, it's just that we can't be sure in advance, the only way to find out is to run it forever and check. But Penrose was not just arguing that human mathematical ability can't be based on a knowably sound algorithm, he was arguing that it must be *non-algorithmic*. I think my thought-experiment shows why this doesn't make sense--we can see that Godel's theorem doesn't prove that an uploaded brain living in a closed computer simulation S would think any different from us, just that it wouldn't be able to correctly output a theorem about arithmetic equivalent to the simulation S will never output this statement. But this doesn't show that the uploaded mind somehow is not self-aware or that we know something it doesn't, since *we* can't correctly judge that statement to be true either! It might very well be that the simulated brain will slip up and make a mistake, giving that statement as output even though the act of doing so proves it's a false statement about arithmetic--we have no way to prove this will never happen, the only way to know is to run the program forever and see. And the impossibility has to be a logical impossibility, not merely a technical or physical one since it depends on Gödel's theorem. That's a bit odd, isn't it? No, I don't see anything very odd about the idea that human mathematical abilities can't be a knowably sound algorithm--it is no more odd than the idea that there are some cellular automata where there is no shortcut to knowing whether they'll reach a certain state or not other than actually simulating them, as Wolfram suggests in A New Kind of Science. In fact I'd say it fits nicely with our feeling of free will, that there should be no way to be sure in advance that we won't break some rules we have been told to obey, apart from actually running us and seeing what we actually end up doing. Jesse _ Need a break? Find your escape route with Live Search Maps. http://maps.live.com/default.aspx?ss=Restaurants~Hotels~Amusement%20Parkcp=33.832922~-117.915659style=rlvl=13tilt=-90dir=0alt=-1000scene=1118863encType=1FORM=MGAC01 --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: Penrose and algorithms
This is not fair to Penrose. He has convincingly argued in 'Shadows of the Mind' that human mathematical intelligence cannot be a knowably sound algorithm. Assume X is an algorithm representing the human mathematical intelligence. The point is not that man cannot recognize X as representing his own intellingence, it is rather that human intellingence cannot know X to be sound (independently of whether X is recognized as what it is). And this is strange because humans could exhaustively inspect X and they should find it correct since it contains the same principles of reasoning human intelligence employs! One way out is claiming that human intelligence is insonsistent. Another, that such a thing as human intelligence could not exist, since it is not well defined. The latter seems more of a serious objection to me. So, I consider Penrose's argument inconclusive. Anyway, the use Lucas and Penrose make of Gödel's theorem make it seem less likely that human reason can be reproduced by machines. This must be granted. Regards On 9 jun, 18:40, Bruno Marchal [EMAIL PROTECTED] wrote: Hi Chris, Le 09-juin-07, à 13:03, chris peck a écrit : Hello The time has come again when I need to seek advice from the everything-list and its contributors. Penrose I believe has argued that the inability to algorithmically solve the halting problem but the ability of humans, or at least Kurt Godel, to understand that formal systems are incomplete together demonstrate that human reason is not algorithmic in nature - and therefore that the AI project is fundamentally flawed. What is the general consensus here on that score. I know that there are many perspectives here including those who agree with Penrose. Are there any decent threads I could look at that deal with this issue? All the best Chris. This is a fundamental issue, even though things are clear for the logicians since 1921 ... But apparently it is still very cloudy for the physicists (except Hofstadter!). I have no time to explain, but let me quote the first paragraph of my Siena papers (your question is at the heart of the interview of the lobian machine and the arithmetical interpretation of Plotinus). But you can find many more explanation in my web pages (in french and in english). In a nutshell, Penrose, though quite courageous and more lucid on the mind body problem than the average physicist, is deadly mistaken on Godel. Godel's theorem are very lucky event for mechanism: eventually it leads to their theologies ... The book by Franzen on the misuse of Godel is quite good. An deep book is also the one by Judson Webb, ref in my thesis). We will have the opportunity to come back on this deep issue, which illustrate a gap between logicians and physicists. Best, Bruno -- (excerp of A Purely Arithmetical, yet Empirically Falsifiable, Interpretation of Plotinus¹ Theory of Matter Cie 2007 ) 1) Incompleteness and Mechanism There is a vast literature where G odel¹s first and second incompleteness theorems are used to argue that human beings are different of, if not superior to, any machine. The most famous attempts have been given by J. Lucas in the early sixties and by R. Penrose in two famous books [53, 54]. Such type of argument are not well supported. See for example the recent book by T. Franzen [21]. There is also a less well known tradition where G odel¹s theorems is used in favor of the mechanist thesis. Emil Post, in a remarkable anticipation written about ten years before G odel published his incompleteness theorems, already discovered both the main ³G odelian motivation² against mechanism, and the main pitfall of such argumentations [17, 55]. Post is the first discoverer 1 of Church Thesis, or Church Turing Thesis, and Post is the first one to prove the first incompleteness theorem from a statement equivalent to Church thesis, i.e. the existence of a universalPost said ³complete²normal (production) system 2. In his anticipation, Post concluded at first that the mathematician¹s mind or that the logical process is essentially creative. He adds : ³It makes of the mathematician much more than a clever being who can do quickly what a machine could do ultimately. We see that a machine would never give a complete logic ; for once the machine is made we could prove a theorem it does not prove²(Post emphasis). But Post quickly realized that a machine could do the same deduction for its own mental acts, and admits that : ³The conclusion that man is not a machine is invalid. All we can say is that man cannot construct a machine which can do all the thinking he can. To illustrate this point we may note that a kind of machine-man could be constructed who would prove a similar theorem for his mental acts.² This has probably constituted his motivation for lifting the term creative to his set theoretical formulation of mechanical universality [56]. To be sure, an
Re: Penrose and algorithms
LauLuna wrote: This is not fair to Penrose. He has convincingly argued in 'Shadows of the Mind' that human mathematical intelligence cannot be a knowably sound algorithm. Assume X is an algorithm representing the human mathematical intelligence. The point is not that man cannot recognize X as representing his own intellingence, it is rather that human intellingence cannot know X to be sound (independently of whether X is recognized as what it is). And this is strange because humans could exhaustively inspect X and they should find it correct since it contains the same principles of reasoning human intelligence employs! But why do you think human mathematical intelligence should be based on nothing more than logical deductions from certain principles of reasoning, like an axiomatic system? It seems to me this is the basic flaw in the argument--for an axiomatic system we can look at each axiom individually, and if we think they're all true statements about mathematics, we can feel confident that any theorems derived logically from these axioms should be true as well. But if someone gives you a detailed simulation of the brain of a human mathematician, there's nothing analogous you can do to feel 100% certain that the simulated brain will never give you a false statement. It helps if you actually imagine such a simulation being performed, and then think about what Godel's theorem would tell you about this simulation, as I did in this post: http://groups.google.com/group/everything-list/browse_thread/thread/f97ba8b290f7/5627eb66017304f2?lnk=gstrnum=1#5627eb66017304f2 Jesse _ Make every IM count. Download Messenger and join the i'm Initiative now. It's free. http://im.live.com/messenger/im/home/?source=TAGHM_June07 --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: Penrose and algorithms
For any Turing machine there is an equivalent axiomatic system; whether we could construct it or not, is of no significance here. Reading your link I was impressed by Russell Standish's sentence: 'I cannot prove this statement' and how he said he could not prove it true and then proved it true. Isn't it more likely that the sentence is paradoxical and therefore non propositional. This is what could make a difference between humans and computers: the correspinding sentence for a computer (when 'I' is replaced with the description of a computer) could not be non propositional: it would be a gödelian sentence. Regards On Jun 28, 10:05 pm, Jesse Mazer [EMAIL PROTECTED] wrote: LauLuna wrote: This is not fair to Penrose. He has convincingly argued in 'Shadows of the Mind' that human mathematical intelligence cannot be a knowably sound algorithm. Assume X is an algorithm representing the human mathematical intelligence. The point is not that man cannot recognize X as representing his own intellingence, it is rather that human intellingence cannot know X to be sound (independently of whether X is recognized as what it is). And this is strange because humans could exhaustively inspect X and they should find it correct since it contains the same principles of reasoning human intelligence employs! But why do you think human mathematical intelligence should be based on nothing more than logical deductions from certain principles of reasoning, like an axiomatic system? It seems to me this is the basic flaw in the argument--for an axiomatic system we can look at each axiom individually, and if we think they're all true statements about mathematics, we can feel confident that any theorems derived logically from these axioms should be true as well. But if someone gives you a detailed simulation of the brain of a human mathematician, there's nothing analogous you can do to feel 100% certain that the simulated brain will never give you a false statement. It helps if you actually imagine such a simulation being performed, and then think about what Godel's theorem would tell you about this simulation, as I did in this post: http://groups.google.com/group/everything-list/browse_thread/thread/f... Jesse _ Make every IM count. Download Messenger and join the i'm Initiative now. It's free.http://im.live.com/messenger/im/home/?source=TAGHM_June07 --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: Penrose and algorithms
LauLuna wrote: For any Turing machine there is an equivalent axiomatic system; whether we could construct it or not, is of no significance here. But for a simulation of a mathematician's brain, the axioms wouldn't be statements about arithmetic which we could inspect and judge whether they were true or false individually, they'd just be statements about the initial state and behavior of the simulated brain. So again, there'd be no way to inspect the system and feel perfectly confident the system would never output a false statement about arithmetic, unlike in the case of the axiomatic systems used by mathematicians to prove theorems. Reading your link I was impressed by Russell Standish's sentence: 'I cannot prove this statement' and how he said he could not prove it true and then proved it true. But prove does not have any precisely-defined meaning here. If you wanted to make it closer to Godel's theorem, then again, you'd have to take a detailed simulation of a human mind which can output various statements, and then look at the statement The simulation will never output this statement--certainly the simulated mind can see that if he doesn't make a mistake he *will* never output that statement, but he can't be 100% sure he'll never make a mistake, and the statement itself is only about the well-defined notion of what output the simulation gives, not in more ill-defined notions of what the simulation knows or can prove in its own mind. Jesse _ Get a preview of Live Earth, the hottest event this summer - only on MSN http://liveearth.msn.com?source=msntaglineliveearthhm --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: Penrose and algorithms
You could look up Murmurs in the Cathedral, Daniel Dennett's review of Penrose's The Emperor's New Mind, in the Times literary supplement (and maybe online somewhere?) Here's an excerpt from a review of the review: -- However, Penrose's main thesis, for which all this scientific exposition is mere supporting argument, is that algorithmic computers cannot ever be intelligent, because our mathematical insights are fundamentally non-algorithmic. Dennett is having none of it, and succinctly points out the underlying fallacy, that, even if there could not be an algorithm for a particular behaviour, there could still be an algorithm that was very very good (if not perfect) at that behaviour: Dennett The following argument, then, in simply fallacious: X is superbly capable of achieving checkmate. There is no (practical) algorithm guaranteed to achieve checkmate, therefore X does not owe its power to achieve checkmate to an algorithm. So even if mathematicians are superb recognizers of mathematical truth, and even if there is no algorithm, practical or otherwise, for recognizing mathematical truth, it does not follow that the power of mathematicians to recognize mathematical truth is not entirely explicable in terms of their brains executing an algorithm. Not an algorithm for intuiting mathematical truth - we can suppose that Penrose has proved that there could be no such thing. What would the algorithm be for, then? Most plausibly it would be an algorithm - one of very many - for trying to stay alive, an algorithm that, by an extraordinarily convoluted and indirect generation of byproducts, happened to be a superb (but not foolproof) recognizer of friends, enemies, food, shelter, harbingers of spring, good arguments - and mathematical truths. /Dennett it is disconcerting that he does not even address the issue, and often writes as if an algorithm could have only the powers it could be proven mathematically to have in the worst case. On Jun 9, 2007, at 4:03 AM, chris peck wrote: Hello The time has come again when I need to seek advice from the everything-list and its contributors. Penrose I believe has argued that the inability to algorithmically solve the halting problem but the ability of humans, or at least Kurt Godel, to understand that formal systems are incomplete together demonstrate that human reason is not algorithmic in nature - and therefore that the AI project is fundamentally flawed. What is the general consensus here on that score. I know that there are many perspectives here including those who agree with Penrose. Are there any decent threads I could look at that deal with this issue? All the best Chris. _ PC Magazine's 2007 editors' choice for best Web mail--award-winning Windows Live Hotmail. http://imagine-windowslive.com/hotmail/?locale=en- usocid=TXT_TAGHM_migration_HM_mini_pcmag_0507 --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to [EMAIL PROTECTED] To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: Penrose and algorithms
cheers Bruno. :) From: Bruno Marchal [EMAIL PROTECTED] Reply-To: [EMAIL PROTECTED] To: [EMAIL PROTECTED] Subject: Re: Penrose and algorithms Date: Sat, 9 Jun 2007 18:40:50 +0200 Hi Chris, Le 09-juin-07, à 13:03, chris peck a écrit : Hello The time has come again when I need to seek advice from the everything-list and its contributors. Penrose I believe has argued that the inability to algorithmically solve the halting problem but the ability of humans, or at least Kurt Godel, to understand that formal systems are incomplete together demonstrate that human reason is not algorithmic in nature - and therefore that the AI project is fundamentally flawed. What is the general consensus here on that score. I know that there are many perspectives here including those who agree with Penrose. Are there any decent threads I could look at that deal with this issue? All the best Chris. This is a fundamental issue, even though things are clear for the logicians since 1921 ... But apparently it is still very cloudy for the physicists (except Hofstadter!). I have no time to explain, but let me quote the first paragraph of my Siena papers (your question is at the heart of the interview of the lobian machine and the arithmetical interpretation of Plotinus). But you can find many more explanation in my web pages (in french and in english). In a nutshell, Penrose, though quite courageous and more lucid on the mind body problem than the average physicist, is deadly mistaken on Godel. Godel's theorem are very lucky event for mechanism: eventually it leads to their theologies ... The book by Franzen on the misuse of Godel is quite good. An deep book is also the one by Judson Webb, ref in my thesis). We will have the opportunity to come back on this deep issue, which illustrate a gap between logicians and physicists. Best, Bruno -- (excerp of A Purely Arithmetical, yet Empirically Falsifiable, Interpretation of Plotinus¹ Theory of Matter Cie 2007 ) 1) Incompleteness and Mechanism There is a vast literature where G odel¹s first and second incompleteness theorems are used to argue that human beings are different of, if not superior to, any machine. The most famous attempts have been given by J. Lucas in the early sixties and by R. Penrose in two famous books [53, 54]. Such type of argument are not well supported. See for example the recent book by T. Franzen [21]. There is also a less well known tradition where G odel¹s theorems is used in favor of the mechanist thesis. Emil Post, in a remarkable anticipation written about ten years before G odel published his incompleteness theorems, already discovered both the main ³G odelian motivation² against mechanism, and the main pitfall of such argumentations [17, 55]. Post is the first discoverer 1 of Church Thesis, or Church Turing Thesis, and Post is the first one to prove the first incompleteness theorem from a statement equivalent to Church thesis, i.e. the existence of a universalPost said ³complete²normal (production) system 2. In his anticipation, Post concluded at first that the mathematician¹s mind or that the logical process is essentially creative. He adds : ³It makes of the mathematician much more than a clever being who can do quickly what a machine could do ultimately. We see that a machine would never give a complete logic ; for once the machine is made we could prove a theorem it does not prove²(Post emphasis). But Post quickly realized that a machine could do the same deduction for its own mental acts, and admits that : ³The conclusion that man is not a machine is invalid. All we can say is that man cannot construct a machine which can do all the thinking he can. To illustrate this point we may note that a kind of machine-man could be constructed who would prove a similar theorem for his mental acts.² This has probably constituted his motivation for lifting the term creative to his set theoretical formulation of mechanical universality [56]. To be sure, an application of Kleene¹s second recursion theorem, see [30], can make any machine self-replicating, and Post should have said only that man cannot both construct a machine doing his thinking and proving that such machine do so. This is what remains from a reconstruction of Lucas-Penrose argument : if we are machine we cannot constructively specify which machine we are, nor, a fortiori, which computation support us. Such analysis begins perhaps with Benacerraf [4], (see [41] for more details). In his book on the subject, Judson Webb argues that Church Thesis is a main ingredient of the Mechanist Thesis. Then, he argues that, given that incompleteness is an easyone double diagonalization step, see aboveconsequence of Church Thesis, G odel¹s 1931 theorem, which proves incompleteness without appeal to Church Thesis, can be taken as a confirmation of it. Judson Webb concludes that G odel¹s incompleteness theorem is a very lucky event for the mechanist
Re: Penrose and algorithms
Hi Chris, Le 09-juin-07, à 13:03, chris peck a écrit : Hello The time has come again when I need to seek advice from the everything-list and its contributors. Penrose I believe has argued that the inability to algorithmically solve the halting problem but the ability of humans, or at least Kurt Godel, to understand that formal systems are incomplete together demonstrate that human reason is not algorithmic in nature - and therefore that the AI project is fundamentally flawed. What is the general consensus here on that score. I know that there are many perspectives here including those who agree with Penrose. Are there any decent threads I could look at that deal with this issue? All the best Chris. This is a fundamental issue, even though things are clear for the logicians since 1921 ... But apparently it is still very cloudy for the physicists (except Hofstadter!). I have no time to explain, but let me quote the first paragraph of my Siena papers (your question is at the heart of the interview of the lobian machine and the arithmetical interpretation of Plotinus). But you can find many more explanation in my web pages (in french and in english). In a nutshell, Penrose, though quite courageous and more lucid on the mind body problem than the average physicist, is deadly mistaken on Godel. Godel's theorem are very lucky event for mechanism: eventually it leads to their theologies ... The book by Franzen on the misuse of Godel is quite good. An deep book is also the one by Judson Webb, ref in my thesis). We will have the opportunity to come back on this deep issue, which illustrate a gap between logicians and physicists. Best, Bruno -- (excerp of A Purely Arithmetical, yet Empirically Falsifiable, Interpretation of Plotinus¹ Theory of Matter Cie 2007 ) 1) Incompleteness and Mechanism There is a vast literature where G odel¹s first and second incompleteness theorems are used to argue that human beings are different of, if not superior to, any machine. The most famous attempts have been given by J. Lucas in the early sixties and by R. Penrose in two famous books [53, 54]. Such type of argument are not well supported. See for example the recent book by T. Franzen [21]. There is also a less well known tradition where G odel¹s theorems is used in favor of the mechanist thesis. Emil Post, in a remarkable anticipation written about ten years before G odel published his incompleteness theorems, already discovered both the main ³G odelian motivation² against mechanism, and the main pitfall of such argumentations [17, 55]. Post is the first discoverer 1 of Church Thesis, or Church Turing Thesis, and Post is the first one to prove the first incompleteness theorem from a statement equivalent to Church thesis, i.e. the existence of a universalPost said ³complete²normal (production) system 2. In his anticipation, Post concluded at first that the mathematician¹s mind or that the logical process is essentially creative. He adds : ³It makes of the mathematician much more than a clever being who can do quickly what a machine could do ultimately. We see that a machine would never give a complete logic ; for once the machine is made we could prove a theorem it does not prove²(Post emphasis). But Post quickly realized that a machine could do the same deduction for its own mental acts, and admits that : ³The conclusion that man is not a machine is invalid. All we can say is that man cannot construct a machine which can do all the thinking he can. To illustrate this point we may note that a kind of machine-man could be constructed who would prove a similar theorem for his mental acts.² This has probably constituted his motivation for lifting the term creative to his set theoretical formulation of mechanical universality [56]. To be sure, an application of Kleene¹s second recursion theorem, see [30], can make any machine self-replicating, and Post should have said only that man cannot both construct a machine doing his thinking and proving that such machine do so. This is what remains from a reconstruction of Lucas-Penrose argument : if we are machine we cannot constructively specify which machine we are, nor, a fortiori, which computation support us. Such analysis begins perhaps with Benacerraf [4], (see [41] for more details). In his book on the subject, Judson Webb argues that Church Thesis is a main ingredient of the Mechanist Thesis. Then, he argues that, given that incompleteness is an easyone double diagonalization step, see aboveconsequence of Church Thesis, G odel¹s 1931 theorem, which proves incompleteness without appeal to Church Thesis, can be taken as a confirmation of it. Judson Webb concludes that G odel¹s incompleteness theorem is a very lucky event for the mechanist philosopher [70, 71]. Torkel Franzen, who concentrates mainly on the negative (antimechanist in general) abuses of G odel¹s theorems, notes, after
