Talk:Roger Penrose/Archive 1

Introduction
I am a Roger Penrose enthusiast, but certainly not an expert on any of the topics he discusses. However, I will share a few thoughts in the hopes that those better qualified can improve the article:

I believe that Sir Penrose was not saying that humans can solve the halting problem; on the contrary, he is saying that humans can understand why the general halting problem is undecidable, while algorithms (or formal logic)cannot establish the same impossibility. This implies that there is something in human intelligence that is beyond the capability of computers. These arguments are fleshed out in Shadows of the Mind, the sequel to The Emperor's New Mind. In Shadows he phrases his arguments in terms of Godel's revolutionary theorems that describe the limits of formal logic as a foundation for mathematical truth. SRW - That would be an extremely odd claim to make. The proof of the undecidability of the halting problem is a very simple logic proof. Computer logic systems can easily check that proof. Given the axioms and the theorem, they can search and find the proof. If Penrose is saying "formal logic cannot establish the same impossibility", then he might want to read the halting problem article, where formal logic does establish the impossibility. No mathematician would have believed the result if it hadn't been proved by formal logic.


 * I think you are confused between a formal logic proof and a typical human proof. Most substantial proofs are not symbolically formulated and theorem-checked by a computer, that would be hopelessly complicated. Most proofs are written in strict but not formal notation, and consensusly agreed upon by the mathematical community using their own human reasoning. Mathematicians have certainly been believing in proofs long before formal axiomatic proofs were first introduced around a century ago. --rem120


 * Possibly not feasible, but not impossible. "Feasibly impossible" is a contradiction in terms.  Paul Beardsell 19:41, 26 Jul 2004 (UTC)


 * Yes that was badly worded, have now edited it. --rem120

My reading of Penrose differed somewhat. I saw him saying no formal logic system can solve the halting problem, but humans can, therefore humans can go beyond the power of formal logic systems. I've talked with a number of mathematicians who have read the book, and they all had the same understanding of what he was saying. If he really did make the blatant error you suggest, then we could add that to the article, but I think he made the more subtle error that I attributed to him. --LC


 * My reading is that Penrose is not suggesting humans can solve the halting problem for all cases. He is simply stating that there is at least one case where humans can solve it, where no algorithm can give that same answer. This demonstrates that there must be some non-computability in human reasoning. You only need to consider this one halting problem case, not every single case.  --rem120


 * Maybe. But, if so, are you saying that your point has escaped the critics of Penrose's hypothesis?  Paul Beardsell 19:41, 26 Jul 2004 (UTC)


 * I dont think most critics of Penrose make the confusion that it is claimed that humans can solve every single halting problem case. There are many rebuttals, but this is not one of the ones I have seen Penrose bother debating. --rem120


 * I think you'll find while it is something possible to find the halting problem via Termination_analysis, due to Godel's theory it is believed that a complete solution to find all halting problems is not possible. (just my2cents) Pls delete my comment if it is way off base.

He also made an error in his discussion of Goedel's theorem. He said that given a mathematical system of sufficient power, we could use Goedel's method to construct a statement G, which might be thought of as saying "I cannot be proved". Then, G would be unprovable within the formal system, yet we as humans would know G was true, thereby transcending the formal system.

That's incorrect. We don't know G is true. We don't know the statement "G is unprovable" is true. All we know to be true is the statement "either G is unprovable, or this system is inconsistent". That's all we know. And we can prove that within the formal system. Goedel himself proved that using ordinary logic. Maybe this should be added to the Penrose page and the page on Goedel's incompleteness theorem. It seems to be a common misconception. --LC


 * I think we can prove that to prove that "either G is unprovable, or this system is inconsistent" you need meta-matematical arguments that can't be expressed within the formal system. You need to be out of the system. Joao


 * The statement "either G is unprovable, or the system is inconsistent" is exactly what Goedel's incompleteness theorem proves. His proof of the theorem was no more "meta-mathematical" than any other proof.  For any given system of axioms, you can construct G and prove that statement while staying entirely within the system.  --LC

The ideas here are very subtle, but at the risk of seeming foolish I want to point out what seems to me to be the essential argument for conscious knowledge transcending algorithmic behavior (I think this argument is in the spirit of Penrose):

It is possible to construct a sequence of formal statements G1, G2, ? , such that within the formal system each of the statements Gn is provable for every n, but such that the higher order statement ?for every n, Gn? is not provable within the system. As humans looking from outside the formal system we can see that ?for every n, Gn? is, indeed, true (has no counterexamples), even though the formal system is not strong enough to contain a proof.

Actually, we can only know that ?for every n, Gn? is either true or the axiom system is inconsistent. But, if the axiom system is inconsistent, then every statement is true (and, every statement is false). So, it seems to me that the question of consistency is not relevant to this argument that conscious knowledge transcends algorithmic behavior.

Perhaps someone could ask a logician for clarification. SRW

- There are two problems:

unprovable" and "inconsistent". Rather the choice is between "unprovable"  and "inconsistent."  There are numerous examples, where you can add a statement P to the axiom system and come up with something that is consistent and you can also add the statement not-P to the axiom system and come up with something equally consistent. In these situations, it seems pretty clear to me that being a conscious being doesn't really give you and advantage in deciding what to do.
 * the problem is that the choice is *not* between "true but

axiom system. I might be totally wrong. In any case, there doesn't seem anything magical about "knowing"
 * the other problem is that human beings can be wrong. I can know that an axiom is true.  I can know that an axiom is false.  True might be consistent with the axiom system.  False might also be consistent with the

Can we agree that the essence of the dispute is the supposed existence of a statement P with the following properties?:

·	Anyone who cares to think about it will agree that P is true.

·	Neither P nor ~P is provable within an arbitrarily strong formal logic as consistent as the natural numbers. Equivalently, P is independent of the other axioms.

Clearly, Penrose has not entirely succeeded in providing an example of P, since not everyone accepts that his candidate P is true. In reverse, this reminds me of the Banach-Tarski paradox: Tarski is reputed to have been very disappointed that his theorem did not result in everyone rejecting the axiom of choice as an obviously useful (and true?) tool for the working mathematician.

This is fun; I haven't thought about these things for years. SRW

-

Roger Penrose is *not* saying humans can solve the halting problem for all cases. the halting problem requires the determination of halting for any given algorithm, not just one. instead he is saying there is at least one case where a human can determine whether a specific algorithm will halt, and that no algorithm could determine this. this one case shows that humans must posess some intelligence from beyond the realm of algorithmic behaviour.

Roger Penrose is *not* saying humans can know the truth of unprovable statements. these statements are provable. they are just not decidable by a formal system. proof is not equivalent to an axiomatic theorem. the statements that are claimed to be unprovable can be found in Penrose's The Emperors New Mind and Shadows Of The Mind, with their proofs.

Roger Penrose is *not* saying humans can prove the halting problem while algorithms can not. this is totally missing the point of the proof.

The fundamental semantics of Roger's work is being skewed and misinterpreted time and time again, and is costing his work a lot of credibility. It is amazing how many false statements one can read about his work in this area.

"The fundamental semantics of Roger's work is being skewed and misinterpreted time and time again, and is costing his work a lot of credibility. "

It looks like his critics say more about themselves than they say about Penrose, dissembling is just another form of dishonesty. What I see at the core of Penroses writing is that 'we are not God', for those who would want to play God and create intelligence, he is a thorn of truth in their side. I recently read Hawking said, due to Godel's incompleteness theorum, we can never have a theory of everything, so Penrose is in good company. I quit school at 15 yrs old, can't fathom equations but I know truth when I see it. Regards, Gerard.Jonesg 00:17, 27 December 2005 (UTC)
 * I disagree that, based on what you have said, Penrose and Hawking are in the same company. Hawking is saying that the laws of the Universe are all computable, and hence the Universe can be considered a formal system, which is trapped into only accessing computable truths by Godels Incompleteness Theorem. Penrose however is saying that he can demonstrate our ability to access non-computable truths, and therefore there must be something in the laws of the Universe that is non-computable, which means the laws of the Universe can not representable as a formal system. Penrose does not argue against our ability to ever create intelligence, rather he argues against our ability to ever create computable intelligence (eg. computer programs) that reaches the same level of truth-accessibility as humans. Penrose's arguments allow humans to create human-level intelligent 'machines' (for lack of a better word), as long as they utilise the non-computable laws of physics which he proposes must exist, but of which we have not yet precisely discovered. Remy B 13:41, 28 December 2005 (UTC)

Physics and Consciousness
Here's the current text that I'm concerned about: ''According to Marvin Minsky, because people can construe false ideas to be factual, the process of thinking is not limited to formal logic. Furthermore, he says that artificial intelligence (AI) programs can also conclude that false statements are true, so error is not unique to humans.'' This passage needs work: (1) What sources is this based on? (2) Logic is generally understood as a relationship between propositions. Construing false ideas to be true would not imply that thinking is not limited to formal logic. Is this an accurate representation of Minsky's argument? (3) The fact that error is not unique to humans doesn't bear on Penrose's claim that mathematical insight (and a great deal of human thought generally) is non-computational. I don't want to just delete this material, if someone thinks it can be made sense of. But it shouldn't be left in place in its present form. Any suggestions? Hickorybark (talk) 19:11, 30 December 2009 (UTC)

I rewrote the Minsky passage and took out the unreferenced assertion, "Penrose's views on the human thought process are not widely accepted in scientific circles." Given that we don't have a widely accepted theory about human thought processes, I think a dismissive claim like this needs a great deal of context. Hickorybark (talk) 20:57, 5 January 2010 (UTC)


 * It would appear that passages you edited were sourced, though not attributed. And sourced too well...they appear to have been directly plagarized from Tetlow's book, "The Web's Awake". Fladrif (talk) 21:50, 6 January 2010 (UTC)


 * Interesting. What we have now seems pretty balanced to me.Hickorybark (talk) 21:08, 15 January 2010 (UTC)
 * Re today's deletion of the paragraph on Tetlow by the anon IP editor. The broader issue covered in this paragraph is how widely Penrose's ideas are accepted, and not the narrower disagreement between Tetlow and Minsky. So, it's not repetitive. I don't pretend to know all the details of the subject matter, but Hickorybark seems to be up on this stuff, and thinks the prior version was balanced. That's why I restored the deleted paragraph. Fladrif (talk) 21:38, 26 January 2010 (UTC)
 * I don't see any harm in leaving it in. I personally think the quantum mind scientists are on the right track, but it's certainly true that their's is a minority point of view. The broader scientific community is pretty conservative, and it will take some overwhelming evidence for them to get enthusiastic about the quantum mind approach. It's truthful and balanced to acknowledge that.Hickorybark (talk) 19:21, 30 January 2010 (UTC)


 * A little more discussion: It's misleading to say that Penrose's ideas are "not widely accepted in scientific circles." Most scientists, including those active in research relating to human intelligence and consciousness, are working on well-defined, solvable problems, and they try not to speculate on the bigger issues unless they have to. Among those who are energetically engaged with the larger questions, like quantum mind, Penrose's ideas are at least given a respectful hearing by more people than these skeptical quotes indicate. But this is bordering on OR, and it's probably best to leave the text as it is.Hickorybark (talk) 19:43, 30 January 2010 (UTC)

Emperors New Mind
I've rewritten the section covering the arguments in the emporers new mind, I think it was innacurate and misleading as it was, I believe it's much better now. Any thoughts? --Starx 16:05, 26 Mar 2005 (UTC)


 * no, sorry, I haven't read it

On the subject of the book The Large, the Small and the Human Mind: I am curious as to why virtually the entire paragraph is devoted to Marvin Minsky's refutation absent information given about Penrose' book other than its broad purpose. This is not meant to be a Marvin Minsky pet-theory page. Inclusion of his quote is not unjustified; however, as it is, it reads as prejudiced against Roger Penrose' ideas absent any detailed context - detail the Minsky quote offers. —Preceding unsigned comment added by Horsie (talk • contribs) 13:08, 14 February 2010 (UTC)

The rebuttal is not as well represented on this page as it is in the literature. Why is there no citation of http://www.mth.kcl.ac.uk/~llandau/Homepage/Math/penrose.html or even just Putnam's review of TENM? Spot (talk) 05:21, 23 July 2010 (UTC)

THE MAGIC OF GOEDEL
I have to admit that after some reading I had to accept that Goedel's sentence is true only when it refers to consistent formal system F. Because we cannot say at first glance is a system F consistent or not, we cannot say also whether the sentence G that we propose is true or not. So, there is no magic, just confusion. Penrose is wrong to say that understanding of Godel's theorem is proof for our non-algorithmic thinking. Godel's theorem can be formally proved therefore F understands it also. Danko Georgiev, 9 jan 2006
 * Is it really not possible to prove the consistency of F? If that could be done (or has already been done) then wouldnt that imply that we know G to be true? Remy B 04:52, 9 January 2006 (UTC)
 * Immediate reply - by "F" we denote "any formal system", so we must decide "for each concrete F" whether it is true or not. The fact that we denote with "F" - "any formal system", expresses our wish to speak about "all finite size F's". F is variable, so if you prove F true, it means you say it is true for all possible F values. I hope this clarifies why your question must be made more concrete - if you mean by F - "all F's" then such algorithm does not exist, so we cannot reply some PC to do the "dirty work" for free instead of us. Danko Georgiev MD 03:06, 12 January 2006 (UTC)

Dear Remy, actually I have found something very curious, but I am still thinking over. After the "shock" that the first Goedel theorem is theorem in F also, and that actually G is not always true in our language, I decided to move on the other side, and check "what if the strong AI were true?".

Amazingly I arrive at paradoxical result that follows from second Goedel theorem, that says a self-consistent formal system cannot neither prove, nor disprove its own consistency. According to strong AI all we know, believe, etc. must be result of computation [i.e. proof, formally written as |=]. So if we were really as strong AI says, then there exists no algorithm that can make us "believe in our consistency", because the believe must be result by computation [proof]. Note: by definition the strong AI advocated require we to be self-consistent algorithms, otherwise they will have hard time explaining why we are not "looping" or "cycling"? Paradoxically if all strong AI advocates change their minds and say that really neither strong AI, nor Penrose's thesis is true, THEY WILL SAY EXACTLY THAT THEY ARE CONSISTENT. There is online textbook by Karlis Podnieks on Goedel theorem, and briefly IF you PROVE that in some system F a given sentence is not provable, this is EQUIVALENT to prove the CONSISTENCY of the system F. Why?? Isn't it absurd? NO! If the system is controversial, from controversy follows EVERYTHING, so there will be NO sentence that you cannot prove. If such sentence exists in a system, then the system is consistent. This is interesting trick! This is exactly what the second Godel's theorem says. The situation is really interesting - if we are algorithms, according to second Godel's theorem we can never be able to have theory of our minds, and every theory that suggests that we are consistent [like strong AI] will BE WRONG. If however the strong AI were true, if you were asked "are you consistent?", then you should enter an eternal "looping". On contrast, we may have some real but incomplete theory if we use not algorithmic processes. Now I answer directly your question - the problem is that we cannot recursively prove that any arbitrary F is consistent or not. The believe whether F is consistent or not does not solve also the truthness of G, because we believe, we do not really know. We may find that G is false if we occasinally find some inconsistency like formula 0=1 in F, so in F everything will be provable, and G is false, because it says it is not provable. We may also occasionally prove that the theory F is consistent hence G is true, but possibly we will need new and stronger axioms. To argue about axioms is useless however, so we do not some special powers to see the truth. We however are not algorithms, and we pay for this "freedom" by making errors. Actually the discovery of a single important axiom takes a whole human life, and that is why Penrose's idea that we see the truthness of G [which is not recursive] is not true. One of the fresh examples is an axiom discovered by Woodin, which if it is enough beautiful and useful to be generally accepted, will disprove the Continuum Hypothesis, that is not solved for a hundred years. Danko Georgiev MD 10:27, 11 January 2006 (UTC)
 * Your point about proving the unprovability of a sentence implying the consistency of the formal system is indeed an interesting trick, at the least it is not something I can think of anything wrong with. You go on to say 'The believe whether F is consistent or not does not solve also the truthness of G, because we believe, we do not really know.'... doesnt Godels second theorem essentially state that 'either G (expressed in F) is true, or F is inconsistent'? If so, and we can prove the consistency of a single F (which is sufficiently complex to follow Godels proof), then doesnt that mean we have proven G for that single F (because if A or B, and not B, then A), and so we have performed something non-algorithmic in our proof? Remy B 08:22, 12 January 2006 (UTC)

Dear Remy, thanks for your question. I still use the double notation that says sentence in our human understandable arithmetic A, and sentence in the formal sytem F. I will make this clear - in our human arithmetic A we say G(g), but in F you say G([g]), where [g] is the numeral of g (example the numeral [2] is s(s(0))). Now do you see the difference between G(g) and G([g]) - they say the same thing "G([g]) is not provable in F", but they are said in two different languages. For example the german word 'zeit' is not understandable in english, but if you translate it as 'time' it will be understandable. So the same is true for F, if you write the number 2, not as s(s(0)) then F will NOT understand it [indeed F might not use at all symbols like 1,2,3...]. Well keep in mind this basic difference, and you will see that in A G(g) says "G([g]) is not provable in F" and since the G([g]) after the translation says the same thing, then ONLY G([g]) refers to itself, not G(g). So initially I was also misleaded by this fact, and thought there is some magic in it. BY DEFINITION in order the sentence S to be true in F, it must be proved by F, formally F|= S. So sentence G([g]) is UNDECIDABLE in F, when F is consistent, not true. True is the sentence G(g) in A, when F is consistent. Now if F is not consistent then the sentence G([g]) is both true and false in F, while G(g) is false in A. Everything is fine up to this point, but A is still without definition - we say it is human understandable, and that is all. We did not say that A is finite size formal system too. Now you will see several things - first, nobody makes difference between G(g) and G([g]) - this is huge mistake, because G(g) might be true since human understandable truthness is still not defined, but G([g]) must be proved in order to be considered formally true - i.e. in F all that matters is "formal truthness = provability". Second, surprize is that we do not see directly the truthness of every G(g) in A. Actually see above that the truth value of G(g) in A depends on the consistency of F. So very bad for Penrose's argument, which is false. Third, one might hope that we know that "G(g) is true when F is consistent", yet in F the first Goedel theorem is PROVABLE "Consis => G([g])". Actually exactly because the first Goedel theorem is provable in F, follows the second Goedel theorem that the formula Consis is UNDECIDABLE in F, if F is consistent, otherwise from first Goedel theorem, plus the formula Consis you will be able to prve G([g]) in the consistent F, which is not possible. Well actually in F, the implication "follows" has different meaning from "prove=true". There still maybe some hope that the Goedel's first theorem in F is not exactly "translated" but this is very tricky issue, and needs very cool estimation of all these "loopings" between A and F, and semantical translations between A and F. I personally don't know what to reply to possible Formalist objection, "So what, implication (denoted with =>)is different from prove (formal truthness, denoted with |=), so in A you must somewhere confused these two notions acceptin both ideas as "informal truthness". Now you see that first Godel theorem is also provable in F, so knowing it, gives us no advange against computers, they also know it. So this second Penrose-like idea is also false. Now answering your question - even if we see for some F, to be consistent, we prove a single case when G(g) is true in A, and we have not proved that G(g) is always true [which is impossible, because of the Goedel's first theorem that says G(g) is false when F is not consistent]. Suppose now you want to derive some conclusion from a single case when you have true G(g) and consistent F. From this does not follow that A is not algorithmic, it follows only that A sees the truthness of F, therefore A is either bigger formal system than F, or A is not algorithmic. So your hope vanishes - the individual truthness of G(g) in some particular case, can be interpreted by strong AI advocate - "well, you have proved you are bigger formal system than F." Yet the possible self-contradiction is the strong AI might be due to second Goedel's theorem as suggested in my previous post. Danko Georgiev MD 05:18, 13 January 2006 (UTC)
 * Thanks for the large response. I am fairly sure I follow what you are saying, and I do not immediately disagree with you. I have followed this topic informally for a few years now, and have only very recently begun to appreciate the role proof of consistency may have in unravelling Penrose' argument. I have to admit that I do not grasp the concepts any more concretely than is found in popular literature (although I understand concepts you have used such as the successor function, A, F, G(g), G([g]), etc.). I would be extremely interested in a direct response by Penrose to the arguments you have presented, as it is a rebuttal to his reasoning that I have not yet (or do not remember having) seen him argue against. Remy B 08:58, 13 January 2006 (UTC)

Dear Remy, thanks for your reply. I work neuroscience now, but the question what is mind bothers me since 2002. I systematically studied math logic in order to understand Penrose's argument [that I hoped is revolutionary], and although I still do not have an absolutely certain opinion on the topic, there are facts that are true, and I understand the deep roots of why they are true. I hope I will finish my draft, and I will send you a link when ready. Hurrying does not help, because the topic is too "slippery". Nevertheless I hope in one month I will have finalized the first version of my draft. Danko Georgiev MD 14:16, 13 January 2006 (UTC) p.s. I am lucky because I have participated in a very interesting discussion on the quantum transactions and free will, and why the idea of OR where consciousness is result from the collapse is clearly wrong. All this happened in Tucson 2003 and in the discussion were 4 persons - me, prof. Penrose, Scott Hagan and prof. Chris C. King from New Zealand. Then I have seen that prof. Penrose is prone to improve his views, if better alternative is found. Yet, I do not think he remembers me, nor I think he will be interested to collaborate on these 2 loopholes. I currently read his book Emperor's New Mind [revisited edition, 1999] and I hope I will find direct quotations of his main thesis, so I do not to accuse him unfairly without direct quotation. All Penrose-like ideas described above are from second-hand web resources.Danko Georgiev MD 14:32, 13 January 2006 (UTC)

Contributions to cosmology
Penrose has made signifigant contributions to the theory of general relativity and to cosmology. In cosmology it was Roger Penrose and Stephen Hawking who proved in 1970 that our universe must have had a spacetime singularity at the beggining of its big bang expansion and that if the universe recollapses it will have a spacetime singularity at it's end. I think that's a pretty signifigant result, so it seems to me to quite accurate to comment in the article that Penrose made signifigant contributions to cosmology. --Starx 20:14, 14 September 2005 (UTC)


 * Hi, Starx, you are preaching to the converted! Indeed, I am a member of the fledgling WikiProject GTR.  See Contributors to general relativity for why I feel that Penrose's contribs to cosmology (huge, I agree!) are via his contributions to general relativity.  In particular, the singularity theorem is just one of his major contributions in this area.  Actually, AFAIK I am the only person other than Penrose with even modest claim to being expert both on Penrose tilings (I wrote a Ph.D. diss. on them) and general relativity.  (In recent years, Penrose has focused on twistor theory and explaining his speculations on quantum theory, but he was a major contributor to the Golden Age of General Relativity (c. 1960-1975).


 * Penrose tilings have mostly been a sidelight in his career; many in the tiling theory field credit N. G. de Bruijn and John Horton Conway with being the first mathematicians to recognize the extraordinary mathematical interest of this construction.) I once had the pleasure of showing him a highly amusing interpretation of a certain construction involving Penrose tilings in terms of closed timelike curves which unfortunately doesn't seem to be anything more than a kind of mathematical visual joke.  However, David Hillman (no relation) is a mathematical physicist (working outside academia, last I knew) interested in applications of cellular automata to gravitation theory, and Penrose tilings can be interpreted in terms of symbolic dynamics and therefore have a close connection with cellular automata.  Right now I am concentrating on Wikiproject GTR, but in future (if wikiservice improves) I would like to greatly improve the Wikipedias coverage of symbolic dynamics, tiling theory, and related topics.


 * Anyway, I will let the cosmology reference stand until the gtr pages are in sufficiently good shape for me to try to rewrite parts of his biography. Cheers---CH  (talk) 21:08, 14 September 2005 (UTC)

Reply to Critic of Penrose

The claims that Penrose has interpreted the Halting rpoblem incorrectly fails to understand the big picture. First, the problem is not simple problem of logic, but an amazing insight by Alan Turing who used formal logic to represent his insight. The fact, the very nature of the problem states that a computer will not halt given an incomplete program, thus a computer cannot solve the problem, nor can it even in principle generate enough insight to derive a corrollary to the halting problem proof. Again, the work of Turing is not a simple act of computation, otherwise it would have been easily discovered and never challenged, but rather is an example of mathematical insight that is codified in formal symbolic logic, just like Godel's theorems. Next there is this interpretation of Godel's theorem and Penrose's explanation of it which again is incorrect: '''That's incorrect. We don't know G is true. We don't know the statement "G is unprovable" is true. All we know to be true is the statement "either G is unprovable, or this system is inconsistent". That's all we know. And we can prove that within the formal system. Goedel himself proved that using ordinary logic. Maybe this should be added to the Penrose page and the page on Goedel's incompleteness theorem. It seems to be a common misconception.''' The entire representation of Penrose's interpretation is false. Penrose essentially says that Godel's first theorem shows that given a sound formal system powerful enough to include arithemtic, it will produce statements which are true but cannot be proved so according to the axioms of the system. This is unquestionably a mathematical truth and represents the limitations of Formal Logic. Indeed the strongest response to this claim by penrose regarding AI, is that human intellectual processes are not necessarily sound so therefore, Godel's proofs do not apply. This response explicitly accepts the limits of "sound" computational systems and argues that to replicate human insight and consciousness we simply need to construct unsound computer programs. Not very appealing from the perspective of a hopefully sound mind! --Gshenks 05:33, 7 November 2005 (UTC)

Elitzur-Vaidman bomb-testing problem
I have written an article about the Elitzur-Vaidman bomb-testing problem, which is widely discussed in Penrose's books. Would it be appropriate to add a link to that article? RCSB 05:37, 7 December 2005 (UTC)