Re: COMP, Quantum Logic and Gleason's Theorem
Dear John, JM: 'evolutionary' is 'relational' anyway originated in 'human mind capabilities' - D.Bohm: there are no numbers in nature. (Not arguing against Bruno, who IMO stands for nature is IN numbersG) Well yes, that is the interesting question. But if you say that there are no numbers (apart from human invention), then how do you answer Wigner's question? (of the unreasonable effectiveness of mathematics) JM: (misunderstood) conclusions upon (m..) conclusions ((figments)) based on millennia of '(mis)observations' and their explanations within the simplex and ever enriching epistemic cognitive inventory level snip JM: I take it as 'thought experiments' to fabricate unreasonable circumstances to prove (or at least facilitate) the hypothetical snip problem with evolving /structures /at all. Unless one 'believes' in /energy??? /that has become somehow and is directed somehow into doing something. What?? You remain only in the question. Maybe that is a reaction because you feel that society has presented you with answers that weren't any? I suggest taking the middle way: questions, thinking, answers, new questions, criticising and eliminating old answers etc - that is more interesting (more fun!) than remaining only in the question (which is also a bit of a dogmatic position ;-) Cheers, Günther --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-l...@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: COMP, Quantum Logic and Gleason's Theorem
John, my way to the number reality was convoluted, but in looking back maybe two books could give you the central idea: Lakoff and Nunez: Where does mathematics come from, which argues that numbers arise from evolutionary considerations (materialist in tenor, Platonia etc ruled out). The next step then is to realize that modern physics gives us only relational knowledge of the world Ladyman Et Al. Every thing must go. (for an excellent overview and discussion), and that matter is indeed not needed (this was the crossing point into number-reality for me, not the Maudlin thought experiment, because I am somewhat skeptical of thought experiments (you never know if you've forgotten hidden assumptions etc)). Computatations (that's the transition to pure number) then give a more well defined picture than all of mathematics, which gives no handle whatever on white rabbits etc. But then book one (Lakoff Et Al) fits again nicely into the bigger picture, explaining how certain structures can evolve to see numbers (one simply drops the materialist tenor). Best Wishes, Günther John Mikes wrote: Günther and Bruno, am I sorry for not being ~30-40 years younger! I could start to study all those excellent books in diverse kinds of logic (what I missed) and could even have a chance to learn all those advancing ideas over the next 30 or so years... Makes me think of it: 30-40 years ago I WAS that young and did not start. I was busy making 20+ more practical polymer related patents without even thinking of the futility of physical World illusions. I just lived (in it)/(them). I am happy in my scientific agnosticim and would love to read something to bring me closer to the idea that 'numbers' consitute the world and not are the mental products of us, eventuel travellers in this (one) universe. Bruno used the word 'axiomatic', in my vocabulary an axiom is an unjustifiable belief (illusion?) necessary to maintain the validity of a theory - in this case the 'physical world'. Like: 2 + 2 = 4 - Br: AUDA is based on the self-reference logic of axiomatizable or recursively enumerable theories, of machine Who is self-referencing, or even acknowledging self-reference? Or 'Self' for that matter? 'Recursively' I agree with, it is 'within'. Machine (limited capability) is 'us', so the 'enumerable theories' are OK. With such handicap in my thinking it is hard to fully follow the flow of the (A)UDA dicussions. I try. Best regards John M On Wed, Jan 28, 2009 at 12:01 PM, Günther Greindl guenther.grei...@gmail.com mailto:guenther.grei...@gmail.com wrote: Dear Bruno, thanks for the good references, I will integrate them on the resource page (or on a separate page). Some of these books I have already read (Boolos), others are on my list (Rogers). Smullyan's Forever Undecided is unfortunately out of print, but I am on the lookout for used copies ;-) Best Wishes and thanks for your time in thinking about the best references, Günther P.S.: I agree with you that the best way to convey knowledge is discussion - I will keep bugging you with questions concerning COMP and UDA *grin* Bruno Marchal wrote: Günther, AUDA is based on the self-reference logic of axiomatizable or recursively enumerable theories, of machine. Those machines or theories must be rich enough. In practice this means their theorems or beliefs are close for induction.This is the work of Gödel and followers, notably Löb, who found a nice generalzation of Gödel's theorem and Solovay who proves the arithmetical completeness of the logic he will call G and G'. Here is the key paper: Solovay, R. M. (1976). Provability Interpretation of Modal Logic. /Israel Journal of Mathematics/, 25:287-304. I follow Boolos 1979 and Smullyan Forever Undecided in calling such system G and G*. G has got many names K4W, PrL, GL. There are four excellent books on this subject: Boolos, G. (1979). /The unprovability of consistency/. Cambridge University Press, London. This is the oldest book. Probably the best for AUDA. And (very lucky event) it has been reedited in paperback recently; I ordered it, and I got it today (o frabjous day! Callooh! Callay! :). It contains a chapter on the S4 intensional variant of G, and the theorem (in my notation) that S4Grz = S4Grz*. The first person is the same from the divine (true) view and the terrestrial (provable) view. I have it now in three exemplars but two are wandering. Boolos, G. (1993). /The Logic of Provability/. Cambridge University Press, Cambridge. This is the sequel, with the Russians' solutions to virtually
Re: COMP, Quantum Logic and Gleason's Theorem
Dear Bruno, Some of these books I have already read (Boolos), You mean read with pencil and paper? Well no *grin* - it was the adopted textbook in one of the courses I took, and I did the assigned exercises, but now flipping through the book I realize I must go back to it again - more than once :-) There is much to be learned. Machine's theology has no more secret for you? A bit too many secrets even. That is the funny thing about mechanism: the more you learn, the deeper the problems become... Have you read the Plotinus paper? I have read through it once, but as I said, I don't know enough modal logic to appreciate it - I will come back to the paper though, promised :-) Cheers, Günther --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-l...@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: COMP, Quantum Logic and Gleason's Theorem
Günther, *please see inserted in JM: lines* John On Sun, Feb 8, 2009 at 10:02 AM, Günther Greindl guenther.grei...@gmail.com wrote: John, my way to the number reality was convoluted, but in looking back maybe two books could give you the central idea: Lakoff and Nunez: Where does mathematics come from, which argues that numbers arise from evolutionary considerations (materialist in tenor, Platonia etc ruled out). JM: 'evolutionary' is 'relational' anyway originated in 'human mind capabilities' - D.Bohm: there are no numbers in nature. (Not arguing against Bruno, who IMO stands for nature is IN numbersG) The next step then is to realize that modern physics gives us only relational knowledge of the world JM: (misunderstood) conclusions upon (m..) conclusions ((figments)) based on millennia of '(mis)observations' and their explanations within the simplex and ever enriching epistemic cognitive inventory level (still growing) - always keeping the prior art and amend after amendment and so on. The 'physical world' is - as a 'whole' - an [axiomatic?] misconception needed to maintain the theoretical tenets of (conventional) sciences. Ladyman Et Al. Every thing must go. (for an excellent overview and discussion), and that matter is indeed not needed (this was the crossing point into number-reality for me, not the Maudlin thought experiment, because I am somewhat skeptical of thought experiments (you never know if you've forgotten hidden assumptions etc)). JM: I take it as 'thought experiments' to fabricate unreasonable circumstances to prove (or at least facilitate) the hypothetical occurrence of otherwise not realizable theoretical ideas. I would exclude them from the scientific thinking. The EPR kicked physics - now 8 decades - into highly mathematized sci-fi. Nobel prizes notwithstanding. Computatations (that's the transition to pure number) then give a more well defined picture than all of mathematics, which gives no handle whatever on white rabbits etc. But then book one (Lakoff Et Al) fits again nicely into the bigger picture, explaining how certain structures can evolve to see numbers (one simply drops the materialist *tenor*). JM: (pun!) I would drop the mathematicist *terror* as well. I have a problem with evolving *structures *at all. Unless one 'believes' in *energy??? *that has become somehow and is directed somehow into doing something. What?? Best Wishes, Günther JM: Respectfully John John Mikes wrote: Günther and Bruno, am I sorry for not being ~30-40 years younger! I could start to study all those excellent books in diverse kinds of logic (what I missed) and could even have a chance to learn all those advancing ideas over the next 30 or so years... Makes me think of it: 30-40 years ago I WAS that young and did not start. I was busy making 20+ more practical polymer related patents without even thinking of the futility of physical World illusions. I just lived (in it)/(them). I am happy in my scientific agnosticim and would love to read something to bring me closer to the idea that 'numbers' consitute the world and not are the mental products of us, eventuel travellers in this (one) universe. Bruno used the word 'axiomatic', in my vocabulary an axiom is an unjustifiable belief (illusion?) necessary to maintain the validity of a theory - in this case the 'physical world'. Like: 2 + 2 = 4 - Br: AUDA is based on the self-reference logic of axiomatizable or recursively enumerable theories, of machine Who is self-referencing, or even acknowledging self-reference? Or 'Self' for that matter? 'Recursively' I agree with, it is 'within'. Machine (limited capability) is 'us', so the 'enumerable theories' are OK. With such handicap in my thinking it is hard to fully follow the flow of the (A)UDA dicussions. I try. Best regards John M --- truncated Günther Greindl Department of Philosophy of Science University of Vienna guenther.grei...@univie.ac.at Blog: http://www.complexitystudies.org/ Thesis: http://www.complexitystudies.org/proposal/ --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-l...@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: COMP, Quantum Logic and Gleason's Theorem
Kim, Russell I appreciate your concern and propositions. I have a friend who thinks about making a book with a subsubsection only (in french), and I think that you could make hundreds of books from Conscience et Mécanisme. And I believe this could give money to the publishers, and the translators and even the author, and even leads to movie and t-shirts! Everything I say follows from the idea, which can or cannot be in fashion, of self-duplication. I make this clearer in the The secret of the Amoeba (the book ordered by the Grasset french publisher). Self and self-duplication is a perennial thought object with a strong appeal to the art. Have you read Borgess? Also, mechanism is today the most believed idea, but few are aware of the startling consequences, and of the non triviality and generality of the notion of universal machine/number/system/ And my work can also be seen just as enthusiasm in front of the universal machine mathematical world. Well I am afraid I have to die for this being true. For sad and boring circumstantial circumstances. And then Mirek is right too and I should write a book (and a paper), instead. But , and this is part of a problem, I have progressed, and I have progressed on the obviously most delicate point at the heart of computationalism (alas), the fundamental difference it introduced into public provability (about numbers, machines and machine's discourses) and truth about those things. And the fact that universal machine can observe that difference, and can actually *live* that difference. It is the theological part. It is already in Conscience and Mécanisme, with the chapter theology and modality. But there I almost define theology by modal logic following a sort of tradition. The many modal logics have been conceived to help reasoning on fundamental metaphysical and theological issues, and nowadays computer science enlarge that sets of theories. The progress is in the arithmetical interpretation of Plotinus. The Gödel provability predicates illustrates the appearance of a purely mathematical modality, but the yes doctor hypothesis/act of faith justifies, for each machines an abstract mathematical theology, which has 8 natural hypostases, with 3 of them justifying or describing the comp physics (making that theology testable), quanta and qualia being distinguished by the Gödel inherited splitting between the modalities. A toy theology with a complete cosmogony and theogony It is weird. And to be sure, the white rabbit problem has not been solved, only translated into a purely mathematical problem. Anyway I am a bit stuck. Both by boring contingent difficulties and interesting necessary difficulties. Explaining the consequences of comp in this list, like currently, could augment (or diminish) the probability that I write the book, or perhaps I could write the book on-line, so that when more than five or ten people acknowledge a chapter is clear enough I go to the next chapter, I dunno. Anyway many thanks for the interest, and please, you have my permission to translate. I can make links to those translations if I don't find them not correct, but then we can discuss (thanks for crediting). You can be part of the second possible volume of the secret of the amoeba, the story of the thesis, asked by Grasset and the journal LE MONDE in 1998. But I expect before some understanding and acknowledgment of understanding or of not understanding. I like the idea to explain to Kim, because it means starting from zero for the math and computer science part, this could provides possible technical annexes for the book making wider the audience. Kim, to be frank I am not sure you can translate something without understanding it, but I am sure you can understand the main part of it, at least up to the point of trusting results in some books without going in the details. Interdisciplinary research asks for being professionally unprofessional, to smell the level of pertinence and develop a sensibility to the 1004 fallacies, which can grow at the frontiers of the fields. When I say, you can, I admittedly make abstraction of time, things are not easy, Of course when I see the argument for making illegal salvia divinorum I feel a bit depressed about humans and believe it is about time people learn elementary logic. There is a sort of pseudo human science which want to defend irrationalism, in the name of liberty, and which is very useful for arbitrary manipulation of facts and then lifes. We should certainly not prosecute someone for making an arithmetical, or a statistical, or a logical error, given that learning needs errors. But sometimes I think we should be able to prosecute those who makes the *same* error again, and again, and again, and again, ... (generally to rise fear about something or someone or somepeople). Best regards, Bruno
Re: COMP, Quantum Logic and Gleason's Theorem
Kim, beware of your heroic offer! I read some books in both the original and translated formats and KNOW that they are different. Not only has the translator his 1st person understanding of WHAT to translate, the words convey the new language's ambiguity for the reader's OWN 1st person interpretation. (Lately about Self-organization in(orig) German and an excellent English version. German is my 2nd mother-tongue, English I live with - although only in the US - since 1965) Communication is Glücksache (matter of luck). (The orig. German proverb sais it for foreign words - Fremdwörter sind Glucksache). In scientific discours it is even worse. Bruno's difficulty stems from the urge to explain *his own* ideas to people with less knowledge than his own (not the regular 'teacher's problem' who conveys mostly only *general*knowledge) and 'you' (we all) do our best to follow. It is never enough. I think the trick of the 'Old Man' at the Tower of Babel was a dirty one: to mix up 'peoples' languages. This is why 'national' sciences are dfferent, even in physical sciences. Not the equations: the meaning/conclusions. The figments. JohnM On Fri, Jan 30, 2009 at 9:20 PM, Kim Jones kimjo...@ozemail.com.au wrote: On 31/01/2009, at 3:37 AM, Bruno Marchal wrote: I've also tried to dig through both Bruno's thesis with the help of google translator. It works for a while but soon one hits a wall with a difficult sentence/paragraph which is hard to understand even if it stands as the author inteded - and extra hard to understand if its meaning is corrupted by the translation. Bruno, I'd love to read your thesis in english, but I fully understand how hard it must be to get a good translation that you would be happy with. At the end, it might be easier to start from scratch, take the essential from both thesis, update a little bit and write a book in english on your own directly. Is that an option for you? Bruno reads beautifully in French. I have offered to translate some of his stuff - the Brussels thesis is a wonderful read in French, I can't really understand the stuff about the construction of the computer because I have no background in computer science, but I can translate all the text into good, idiomatic English if I could generate some little income in the process. He has said The road to hell is paved with the best of intentions to me in the past, and I agree with him on that, also that publishing deals will benefit the publisher, not the author, but there are many people (me included) who love his stuff now and wish it could be presented to a wider audience. Failing that, a few of you might have to learn French, which would benefit your brain cells anyway. French is just English pronounced wrongly anyway ;-) K --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-l...@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: COMP, Quantum Logic and Gleason's Theorem
Hi Mirek, I would certainly like to read the book - I managed a bit the Lille thesis (with my French), but it was hard going and I think I only understood the stuff because we have had many discussions here on the list - so it was easy to translate. I am not so sure I can manage the huge Bruxelles work, but I will try someday when I have more time :-)) Maybe you can find a publisher who is prepared to translate the book into english? Excellent idea. For reason I don't want to bore you with, I am a bit stuck on those sort of issue. But I am sure a good publishing could make rich the publisher :) I've also tried to dig through both Bruno's thesis with the help of google translator. It works for a while but soon one hits a wall with a difficult sentence/paragraph which is hard to understand even if it stands as the author inteded - and extra hard to understand if its meaning is corrupted by the translation. Bruno, I'd love to read your thesis in english, but I fully understand how hard it must be to get a good translation that you would be happy with. At the end, it might be easier to start from scratch, take the essential from both thesis, update a little bit and write a book in english on your own directly. Is that an option for you? I should do that. I will do that. I appreciate very much your encouragement. Best wishes, 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 everything-l...@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: COMP, Quantum Logic and Gleason's Theorem
On 31/01/2009, at 3:37 AM, Bruno Marchal wrote: I've also tried to dig through both Bruno's thesis with the help of google translator. It works for a while but soon one hits a wall with a difficult sentence/paragraph which is hard to understand even if it stands as the author inteded - and extra hard to understand if its meaning is corrupted by the translation. Bruno, I'd love to read your thesis in english, but I fully understand how hard it must be to get a good translation that you would be happy with. At the end, it might be easier to start from scratch, take the essential from both thesis, update a little bit and write a book in english on your own directly. Is that an option for you? Bruno reads beautifully in French. I have offered to translate some of his stuff - the Brussels thesis is a wonderful read in French, I can't really understand the stuff about the construction of the computer because I have no background in computer science, but I can translate all the text into good, idiomatic English if I could generate some little income in the process. He has said The road to hell is paved with the best of intentions to me in the past, and I agree with him on that, also that publishing deals will benefit the publisher, not the author, but there are many people (me included) who love his stuff now and wish it could be presented to a wider audience. Failing that, a few of you might have to learn French, which would benefit your brain cells anyway. French is just English pronounced wrongly anyway ;-) K --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-l...@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: COMP, Quantum Logic and Gleason's Theorem
I would certainly like to read the book - I managed a bit the Lille thesis (with my French), but it was hard going and I think I only understood the stuff because we have had many discussions here on the list - so it was easy to translate. I am not so sure I can manage the huge Bruxelles work, but I will try someday when I have more time :-)) Maybe you can find a publisher who is prepared to translate the book into english? Excellent idea. For reason I don't want to bore you with, I am a bit stuck on those sort of issue. But I am sure a good publishing could make rich the publisher :) I've also tried to dig through both Bruno's thesis with the help of google translator. It works for a while but soon one hits a wall with a difficult sentence/paragraph which is hard to understand even if it stands as the author inteded - and extra hard to understand if its meaning is corrupted by the translation. Bruno, I'd love to read your thesis in english, but I fully understand how hard it must be to get a good translation that you would be happy with. At the end, it might be easier to start from scratch, take the essential from both thesis, update a little bit and write a book in english on your own directly. Is that an option for you? Cheers, mirek --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-l...@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: COMP, Quantum Logic and Gleason's Theorem
John, Who is self-referencing, or even acknowledging self-reference? Gödel and All. It is a major discovery of the 20th century: a completely clear notion of third person self-reference. A first person self-reference theory follows naturally, accepting Theaetetus' definition of knowledge. The first person self, like the big one, has really no name. No identity card, nor even a body. Paradoxes and even shit happens when we name them. Each of the 8 hypostases (or of the 16, ...) can be seen as a notion of self-reference, but acting on different facet of the arithmetical reality. There are nameable selves and unnameable one, and they obey to different logics. More later. I wish you the best, 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 everything-l...@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: COMP, Quantum Logic and Gleason's Theorem
Dear Bruno, thanks for the good references, I will integrate them on the resource page (or on a separate page). Some of these books I have already read (Boolos), others are on my list (Rogers). Smullyan's Forever Undecided is unfortunately out of print, but I am on the lookout for used copies ;-) Best Wishes and thanks for your time in thinking about the best references, Günther P.S.: I agree with you that the best way to convey knowledge is discussion - I will keep bugging you with questions concerning COMP and UDA *grin* Bruno Marchal wrote: Günther, AUDA is based on the self-reference logic of axiomatizable or recursively enumerable theories, of machine. Those machines or theories must be rich enough. In practice this means their theorems or beliefs are close for induction.This is the work of Gödel and followers, notably Löb, who found a nice generalzation of Gödel's theorem and Solovay who proves the arithmetical completeness of the logic he will call G and G'. Here is the key paper: Solovay, R. M. (1976). Provability Interpretation of Modal Logic. /Israel Journal of Mathematics/, 25:287-304. I follow Boolos 1979 and Smullyan Forever Undecided in calling such system G and G*. G has got many names K4W, PrL, GL. There are four excellent books on this subject: Boolos, G. (1979). /The unprovability of consistency/. Cambridge University Press, London. This is the oldest book. Probably the best for AUDA. And (very lucky event) it has been reedited in paperback recently; I ordered it, and I got it today (o frabjous day! Callooh! Callay! :). It contains a chapter on the S4 intensional variant of G, and the theorem (in my notation) that S4Grz = S4Grz*. The first person is the same from the divine (true) view and the terrestrial (provable) view. I have it now in three exemplars but two are wandering. Boolos, G. (1993). /The Logic of Provability/. Cambridge University Press, Cambridge. This is the sequel, with the Russians' solutions to virtually all open problems in Boolos 1979. The main problem was the question of the axiomatizability of the first-order extension of G and G* (which I note sometimes qG and qG*). And the answers, completely detailed in Boolos' book, are as negative as they can possibly be. qG is PI_2 complete, and qG* is PI_1 complete *in* the Arithmetical Truth. The divine intelligible of Peano Arithmetic is far more complex than Peano Arithmetic's ONE, or God, in the arithmetical interpretation of Plotinus. Smoryński, P. (1985). /Self-Reference and Modal Logic/. Springer Verlag, New York. I have abandon this one sometimes ago, because of my eyes sight defect, but with spectacles I have been able to distinguish tobacco product from indices in formula, and by many tokens, it could be very well suited for AUDA. The reason is that it develops the theory in term of (computable) function instead of assertions, showing directly the relation between computability and SIGMA_1 provability. Nice intro from Hilbert's program to Gödel and Löb's theorem, and the Hilbert Bernays versus Löb derivability conditions. It contains a chapter, a bit too much blazed in the tone, on the algebraic approach to self-reference, which indeed initiates originally the field in Italy (Roberto Magari). It contains also chapter on the Rosser intensional variants. Smullyan, R. (1987). /Forever Undecided/. Knopf, New York. This is a recreative introduction to the modal logic G. I was used some times ago in this list to refer to that book by FU, and I don't hesitate to use some of Smullyan's trick to ease the way toward self-reference. It helps some, but can irritate others. Note that Smullyan wrote *many* technical books around mathematical self-reference, Gödel's theorems in many systems. Modal logic is not so well known that such book can presuppose it, and all those books introduce modal logic in a rather gentle way. But all those books presuppose some familiarity with logic. Boolos Et Al. is OK. It is difficult to choose among many good introduction to Logic. By some aspect Epstein and Carnielly is very good too for our purpose. Note that the original papers are readable (in this field). All this for people who does not suffer from math anxiety which reminds me I have to cure Kim soon or later. The seventh step requires some math. AUDA requires to understand that those math are accessible to all universal machine 'grasping the induction principle, this is the work of Gödel and Al. I think the book by Rogers is also fundamental. Cutland's book is nice, but it omits the study of the Arithmetical Hierarchy (SIGMA_0, PI_0, SIGMA_1, PI_1, SIGMA_2, PI_2, ...). AUDA without math = Plotinus (or Ibn Arabi or any serious and rational mystic). Roughly speaking. I will think about a layman explanation of AUDA without math, and different from UDA. Best regards,
Re: COMP, Quantum Logic and Gleason's Theorem
Bruno, theoretical computer science and mathematical logic. Rereading Conscience et Mécanisme I realize Russell Standish was right, and that book should be translated in english because it contains an almost complete (self-contained) explanation of logic (for the physicists), including the historical foundations which are genuine, and a detailed explanation of the measurement problem in quantum physics, for the logicians. (beyong the most detailed account of the UD). It renders also justice to all the contributors in the debate on Gödel (like Benacerraf, Reinhardt, Webb, Wang, and many others). There are many misunderstandings, which reminds me the book by Torkel Franzen ... I would certainly like to read the book - I managed a bit the Lille thesis (with my French), but it was hard going and I think I only understood the stuff because we have had many discussions here on the list - so it was easy to translate. I am not so sure I can manage the huge Bruxelles work, but I will try someday when I have more time :-)) Maybe you can find a publisher who is prepared to translate the book into english? Also, it is hard to *believe* in the plausibility of the conclusion of UDA without having a good understanding of Everett's Quantum Mechanics. What could be a good introduction to Everett? ... Deutch' FOR book, but also Albert's one, D'Espagnat, . Of course, yes - I will include Albert and Deutsch on the references page concerning the UDA. Albert especially is good in his account of weird quantum experiments. Cheers, Günther --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-l...@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: COMP, Quantum Logic and Gleason's Theorem
On 28 Jan 2009, at 18:07, Günther Greindl wrote: Bruno, theoretical computer science and mathematical logic. Rereading Conscience et Mécanisme I realize Russell Standish was right, and that book should be translated in english because it contains an almost complete (self-contained) explanation of logic (for the physicists), including the historical foundations which are genuine, and a detailed explanation of the measurement problem in quantum physics, for the logicians. (beyong the most detailed account of the UD). It renders also justice to all the contributors in the debate on Gödel (like Benacerraf, Reinhardt, Webb, Wang, and many others). There are many misunderstandings, which reminds me the book by Torkel Franzen ... I would certainly like to read the book - I managed a bit the Lille thesis (with my French), but it was hard going and I think I only understood the stuff because we have had many discussions here on the list - so it was easy to translate. I am not so sure I can manage the huge Bruxelles work, but I will try someday when I have more time :-)) Maybe you can find a publisher who is prepared to translate the book into english? Excellent idea. For reason I don't want to bore you with, I am a bit stuck on those sort of issue. But I am sure a good publishing could make rich the publisher :) Also, it is hard to *believe* in the plausibility of the conclusion of UDA without having a good understanding of Everett's Quantum Mechanics. What could be a good introduction to Everett? ... Deutch' FOR book, but also Albert's one, D'Espagnat, . Of course, yes - I will include Albert and Deutsch on the references page concerning the UDA. Albert especially is good in his account of weird quantum experiments. Perhaps not. It could give the wrong impression that QM is needed for the UDA. Perhaps yes: Everett illustrates that physics has already the most startling comp feature. I don't know. Best, 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 everything-l...@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: COMP, Quantum Logic and Gleason's Theorem
Dear Günther, thanks for the good references, I will integrate them on the resource page (or on a separate page). Some of these books I have already read (Boolos), You mean read with pencil and paper? Machine's theology has no more secret for you? Have you read the Plotinus paper? others are on my list (Rogers). This one is transcendental. Even from the pedagogical and philosophical point of view I think. Only one critic: I prefer the Kleene's version of the second recursion theorem, far better suited for abstract biology (like in my planaria paper). But it is a three line reasoning to go from one form to the another. Note that Kleene version is more general: it works on the subcreative sets, meaning that it can have something to say on tractability issues too. Smullyan's Forever Undecided is unfortunately out of print, but I am on the lookout for used copies ;-) Best Wishes and thanks for your time in thinking about the best references, Günther P.S.: I agree with you that the best way to convey knowledge is discussion - I will keep bugging you with questions concerning COMP and UDA *grin* Please do. With pleasure, Bruno Bruno Marchal wrote: Günther, AUDA is based on the self-reference logic of axiomatizable or recursively enumerable theories, of machine. Those machines or theories must be rich enough. In practice this means their theorems or beliefs are close for induction.This is the work of Gödel and followers, notably Löb, who found a nice generalzation of Gödel's theorem and Solovay who proves the arithmetical completeness of the logic he will call G and G'. Here is the key paper: Solovay, R. M. (1976). Provability Interpretation of Modal Logic. /Israel Journal of Mathematics/, 25:287-304. I follow Boolos 1979 and Smullyan Forever Undecided in calling such system G and G*. G has got many names K4W, PrL, GL. There are four excellent books on this subject: Boolos, G. (1979). /The unprovability of consistency/. Cambridge University Press, London. This is the oldest book. Probably the best for AUDA. And (very lucky event) it has been reedited in paperback recently; I ordered it, and I got it today (o frabjous day! Callooh! Callay! :). It contains a chapter on the S4 intensional variant of G, and the theorem (in my notation) that S4Grz = S4Grz*. The first person is the same from the divine (true) view and the terrestrial (provable) view. I have it now in three exemplars but two are wandering. Boolos, G. (1993). /The Logic of Provability/. Cambridge University Press, Cambridge. This is the sequel, with the Russians' solutions to virtually all open problems in Boolos 1979. The main problem was the question of the axiomatizability of the first-order extension of G and G* (which I note sometimes qG and qG*). And the answers, completely detailed in Boolos' book, are as negative as they can possibly be. qG is PI_2 complete, and qG* is PI_1 complete *in* the Arithmetical Truth. The divine intelligible of Peano Arithmetic is far more complex than Peano Arithmetic's ONE, or God, in the arithmetical interpretation of Plotinus. Smoryński, P. (1985). /Self-Reference and Modal Logic/. Springer Verlag, New York. I have abandon this one sometimes ago, because of my eyes sight defect, but with spectacles I have been able to distinguish tobacco product from indices in formula, and by many tokens, it could be very well suited for AUDA. The reason is that it develops the theory in term of (computable) function instead of assertions, showing directly the relation between computability and SIGMA_1 provability. Nice intro from Hilbert's program to Gödel and Löb's theorem, and the Hilbert Bernays versus Löb derivability conditions. It contains a chapter, a bit too much blazed in the tone, on the algebraic approach to self-reference, which indeed initiates originally the field in Italy (Roberto Magari). It contains also chapter on the Rosser intensional variants. Smullyan, R. (1987). /Forever Undecided/. Knopf, New York. This is a recreative introduction to the modal logic G. I was used some times ago in this list to refer to that book by FU, and I don't hesitate to use some of Smullyan's trick to ease the way toward self- reference. It helps some, but can irritate others. Note that Smullyan wrote *many* technical books around mathematical self-reference, Gödel's theorems in many systems. Modal logic is not so well known that such book can presuppose it, and all those books introduce modal logic in a rather gentle way. But all those books presuppose some familiarity with logic. Boolos Et Al. is OK. It is difficult to choose among many good introduction to Logic. By some aspect Epstein and Carnielly is very good too for our purpose. Note that the original papers are readable (in this field). All this for people who does not
Re: COMP, Quantum Logic and Gleason's Theorem
Günther and Bruno, am I sorry for not being ~30-40 years younger! I could start to study all those excellent books in diverse kinds of logic (what I missed) and could even have a chance to learn all those advancing ideas over the next 30 or so years... Makes me think of it: 30-40 years ago I WAS that young and did not start. I was busy making 20+ more practical polymer related patents without even thinking of the futility of physical World illusions. I just lived (in it)/(them). I am happy in my scientific agnosticim and would love to read something to bring me closer to the idea that 'numbers' consitute the world and not are the mental products of us, eventuel travellers in this (one) universe. Bruno used the word 'axiomatic', in my vocabulary an axiom is an unjustifiable belief (illusion?) necessary to maintain the validity of a theory - in this case the 'physical world'. Like: 2 + 2 = 4 - Br: AUDA is based on the self-reference logic of axiomatizable or recursively enumerable theories, of machine Who is self-referencing, or even acknowledging self-reference? Or 'Self' for that matter? 'Recursively' I agree with, it is 'within'. Machine (limited capability) is 'us', so the 'enumerable theories' are OK. With such handicap in my thinking it is hard to fully follow the flow of the (A)UDA dicussions. I try. Best regards John M On Wed, Jan 28, 2009 at 12:01 PM, Günther Greindl guenther.grei...@gmail.com wrote: Dear Bruno, thanks for the good references, I will integrate them on the resource page (or on a separate page). Some of these books I have already read (Boolos), others are on my list (Rogers). Smullyan's Forever Undecided is unfortunately out of print, but I am on the lookout for used copies ;-) Best Wishes and thanks for your time in thinking about the best references, Günther P.S.: I agree with you that the best way to convey knowledge is discussion - I will keep bugging you with questions concerning COMP and UDA *grin* Bruno Marchal wrote: Günther, AUDA is based on the self-reference logic of axiomatizable or recursively enumerable theories, of machine. Those machines or theories must be rich enough. In practice this means their theorems or beliefs are close for induction.This is the work of Gödel and followers, notably Löb, who found a nice generalzation of Gödel's theorem and Solovay who proves the arithmetical completeness of the logic he will call G and G'. Here is the key paper: Solovay, R. M. (1976). Provability Interpretation of Modal Logic. /Israel Journal of Mathematics/, 25:287-304. I follow Boolos 1979 and Smullyan Forever Undecided in calling such system G and G*. G has got many names K4W, PrL, GL. There are four excellent books on this subject: Boolos, G. (1979). /The unprovability of consistency/. Cambridge University Press, London. This is the oldest book. Probably the best for AUDA. And (very lucky event) it has been reedited in paperback recently; I ordered it, and I got it today (o frabjous day! Callooh! Callay! :). It contains a chapter on the S4 intensional variant of G, and the theorem (in my notation) that S4Grz = S4Grz*. The first person is the same from the divine (true) view and the terrestrial (provable) view. I have it now in three exemplars but two are wandering. Boolos, G. (1993). /The Logic of Provability/. Cambridge University Press, Cambridge. This is the sequel, with the Russians' solutions to virtually all open problems in Boolos 1979. The main problem was the question of the axiomatizability of the first-order extension of G and G* (which I note sometimes qG and qG*). And the answers, completely detailed in Boolos' book, are as negative as they can possibly be. qG is PI_2 complete, and qG* is PI_1 complete *in* the Arithmetical Truth. The divine intelligible of Peano Arithmetic is far more complex than Peano Arithmetic's ONE, or God, in the arithmetical interpretation of Plotinus. Smoryński, P. (1985). /Self-Reference and Modal Logic/. Springer Verlag, New York. I have abandon this one sometimes ago, because of my eyes sight defect, but with spectacles I have been able to distinguish tobacco product from indices in formula, and by many tokens, it could be very well suited for AUDA. The reason is that it develops the theory in term of (computable) function instead of assertions, showing directly the relation between computability and SIGMA_1 provability. Nice intro from Hilbert's program to Gödel and Löb's theorem, and the Hilbert Bernays versus Löb derivability conditions. It contains a chapter, a bit too much blazed in the tone, on the algebraic approach to self-reference, which indeed initiates originally the field in Italy (Roberto Magari). It contains also chapter on the Rosser intensional variants. Smullyan, R. (1987). /Forever Undecided/. Knopf, New York. This is a recreative introduction to the
Re: COMP, Quantum Logic and Gleason's Theorem
Günther, AUDA is based on the self-reference logic of axiomatizable or recursively enumerable theories, of machine. Those machines or theories must be rich enough. In practice this means their theorems or beliefs are close for induction.This is the work of Gödel and followers, notably Löb, who found a nice generalzation of Gödel's theorem and Solovay who proves the arithmetical completeness of the logic he will call G and G'. Here is the key paper: Solovay, R. M. (1976). Provability Interpretation of Modal Logic. Israel Journal of Mathematics, 25:287-304. I follow Boolos 1979 and Smullyan Forever Undecided in calling such system G and G*. G has got many names K4W, PrL, GL. There are four excellent books on this subject: Boolos, G. (1979). The unprovability of consistency. Cambridge University Press, London. This is the oldest book. Probably the best for AUDA. And (very lucky event) it has been reedited in paperback recently; I ordered it, and I got it today (o frabjous day! Callooh! Callay! :). It contains a chapter on the S4 intensional variant of G, and the theorem (in my notation) that S4Grz = S4Grz*. The first person is the same from the divine (true) view and the terrestrial (provable) view. I have it now in three exemplars but two are wandering. Boolos, G. (1993). The Logic of Provability. Cambridge University Press, Cambridge. This is the sequel, with the Russians' solutions to virtually all open problems in Boolos 1979. The main problem was the question of the axiomatizability of the first-order extension of G and G* (which I note sometimes qG and qG*). And the answers, completely detailed in Boolos' book, are as negative as they can possibly be. qG is PI_2 complete, and qG* is PI_1 complete *in* the Arithmetical Truth. The divine intelligible of Peano Arithmetic is far more complex than Peano Arithmetic's ONE, or God, in the arithmetical interpretation of Plotinus. Smoryński, P. (1985). Self-Reference and Modal Logic. Springer Verlag, New York. I have abandon this one sometimes ago, because of my eyes sight defect, but with spectacles I have been able to distinguish tobacco product from indices in formula, and by many tokens, it could be very well suited for AUDA. The reason is that it develops the theory in term of (computable) function instead of assertions, showing directly the relation between computability and SIGMA_1 provability. Nice intro from Hilbert's program to Gödel and Löb's theorem, and the Hilbert Bernays versus Löb derivability conditions. It contains a chapter, a bit too much blazed in the tone, on the algebraic approach to self- reference, which indeed initiates originally the field in Italy (Roberto Magari). It contains also chapter on the Rosser intensional variants. Smullyan, R. (1987). Forever Undecided. Knopf, New York. This is a recreative introduction to the modal logic G. I was used some times ago in this list to refer to that book by FU, and I don't hesitate to use some of Smullyan's trick to ease the way toward self- reference. It helps some, but can irritate others. Note that Smullyan wrote *many* technical books around mathematical self-reference, Gödel's theorems in many systems. Modal logic is not so well known that such book can presuppose it, and all those books introduce modal logic in a rather gentle way. But all those books presuppose some familiarity with logic. Boolos Et Al. is OK. It is difficult to choose among many good introduction to Logic. By some aspect Epstein and Carnielly is very good too for our purpose. Note that the original papers are readable (in this field). All this for people who does not suffer from math anxiety which reminds me I have to cure Kim soon or later. The seventh step requires some math. AUDA requires to understand that those math are accessible to all universal machine 'grasping the induction principle, this is the work of Gödel and Al. I think the book by Rogers is also fundamental. Cutland's book is nice, but it omits the study of the Arithmetical Hierarchy (SIGMA_0, PI_0, SIGMA_1, PI_1, SIGMA_2, PI_2, ...). AUDA without math = Plotinus (or Ibn Arabi or any serious and rational mystic). Roughly speaking. I will think about a layman explanation of AUDA without math, and different from UDA. Best regards, Bruno On 25 Jan 2009, at 18:45, Günther Greindl wrote: Hi Bruno, Goldblatt, Mathematics of Modality Note that it is advanced stuff for people familiarized with mathematical logic (it presupposes Mendelson's book, or Boolos Jeffrey). Two papers in that book are part of AUDA: the UDA explain to the universal machine, and her opinion on the matter. I would like to add a guide to AUDA section on the resources page. Maybe you could specify the core references necessary for understanding the AUDA (if you like and have the time)? Here a first suggestion of what I am thinking of: Boolos Et
Re: COMP, Quantum Logic and Gleason's Theorem
I will think about it. Somehow, the best layman intro to UDA and AUDA are in this list. The first 15-step version of UDA was a reply to Russell Standish a long time ago. UDA is the logical guide to AUDA, which is just a deeper second pass on UDA. AUDA *is* UDA explained to the dummy, with the dummy played by the machine. Can we still access the everything list posts individually through web address? Give me time to think on the best book for the technical understanding of AUDA, I have already made some advertising on some books like the one by Boolos, or (better for the layman) Smullyan, especially Forever Undecided, for a recreative introduction to the modal logic G. Any good textbook in mathematical logic is a necessary companion. AUDA uses the most standard notion and results there. Probably a key book (even for just the seventh step of UDA) is the book by Webb. See the reference of the paper linked below). For UDA, good popular training are SIMULACRON 3, MATRIX, but also Plato, and many other up to the book Minds'I edited by Dennett and Hofstadter. The original paper on the UD and UDA (and MGA) is my 1991 paper. It contains the seeds of AUDA. It contains a shorter bibliography, which could help... Marchal B., 1991, Mechanism and Personal Identity, proceedings of WOCFAI 91, M. De Glas D. Gabbay (Eds), Angkor, Paris. Except that for understanding the UD itself, and thus the seventh step, and to comprehend its generality, you have to know a bit of theoretical computer science and mathematical logic. Rereading Conscience et Mécanisme I realize Russell Standish was right, and that book should be translated in english because it contains an almost complete (self-contained) explanation of logic (for the physicists), including the historical foundations which are genuine, and a detailed explanation of the measurement problem in quantum physics, for the logicians. (beyong the most detailed account of the UD). It renders also justice to all the contributors in the debate on Gödel (like Benacerraf, Reinhardt, Webb, Wang, and many others). There are many misunderstandings, which reminds me the book by Torkel Franzen ... Also, it is hard to *believe* in the plausibility of the conclusion of UDA without having a good understanding of Everett's Quantum Mechanics. What could be a good introduction to Everett? ... Deutch' FOR book, but also Albert's one, D'Espagnat, . There are many good books, working at different levels. Let me think a bit, Best, Bruno On 25 Jan 2009, at 18:45, Günther Greindl wrote: Hi Bruno, Goldblatt, Mathematics of Modality Note that it is advanced stuff for people familiarized with mathematical logic (it presupposes Mendelson's book, or Boolos Jeffrey). Two papers in that book are part of AUDA: the UDA explain to the universal machine, and her opinion on the matter. I would like to add a guide to AUDA section on the resources page. Maybe you could specify the core references necessary for understanding the AUDA (if you like and have the time)? Here a first suggestion of what I am thinking of: Boolos Et Al. Computability and Logic. 2002. 4th Edition Chellas. Modal Logic. 1980. Goldblatt, Semantic Analysis of Orthologic and Arithmetical Necessity, Provability and Intuitionistic Logic to be found in Goldblatt, Mathematics of Modality. 1993. What do you think? Best Wishes, Günther 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 everything-l...@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: COMP, Quantum Logic and Gleason's Theorem
Goldblatt 1993, Mathematics of Modality this book is available online: http://standish.stanford.edu/bin/detail?fileID=458253745 mirek Goldblatt, Mathematics of Modality http://www.amazon.com/Mathematics-Modality-Center-Language-Information/dp/1881526240/ref=sr_1_1?ie=UTF8s=booksqid=1232402154sr=8-1 (the book contains the full paper) Not only that! It contains also his paper on the arithmetical intuitionist, alias the arithmetical knower, alias the universal first person, alias the arithmetical interpretation of Plotinus' third hypostase (the universal soul), alias the epistemical temporal arithmetical modal logic S4Grz (pronounce: S four Grzegorczyk). --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-l...@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: COMP, Quantum Logic and Gleason's Theorem
Hi Günther, The paper is not online, but I found it in this book which is at our University Library, maybe interesting also for other people: Goldblatt, Mathematics of Modality http://www.amazon.com/Mathematics-Modality-Center-Language-Information/dp/1881526240/ref=sr_1_1?ie=UTF8s=booksqid=1232402154sr=8-1 (the book contains the full paper) Not only that! It contains also his paper on the arithmetical intuitionist, alias the arithmetical knower, alias the universal first person, alias the arithmetical interpretation of Plotinus' third hypostase (the universal soul), alias the epistemical temporal arithmetical modal logic S4Grz (pronounce: S four Grzegorczyk). A key paper for the AUDA, except that Boolos found those results, on SAGrz about the same time, see the reference to Boolos in any of my theses. Or see the S4 chapters in the Boolos 1993, book or in the recent paperback reedition of Boolos 1979. It is the logic of provable and true. It leads to a notion of person which the machine cannot named or define. The arithmetical knower is not arithmetical! The book contains also a very interesting study of the Diodorean modality in the Minkowski Space-time, and a logical approach to Groethendieck topology. Note that it is advanced stuff for people familiarized with mathematical logic (it presupposes Mendelson's book, or Boolos Jeffrey). Two papers in that book are part of AUDA: the UDA explain to the universal machine, and her opinion on the matter. 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 everything-l...@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: COMP, Quantum Logic and Gleason's Theorem
On 16 Jan 2009, at 22:04, Günther Greindl wrote: Hi all, the question goes primarily to Bruno but all other input is welcome :-)) Bruno, you said you have already arrived at a quantum logic in your technical work? Yes. The hypostases, with p restrict to the Sigma-1 sentences (the UD) given by Bp p (the knower certainty) Bp Dp (the observer certainty) Bp Dp p (the feeler certainty), with B the Godel Beweisbar predicate, and Da = ~B~a. gives rise to Brouwersche like modal logics with natural quantization (BDp) which act like quantum projector, except that I loose the Brouwersche necessitation rule, which formally makes things more complex, more rich also. May I refer to the following two paragraphs?: We can read here: http://plato.stanford.edu/entries/qt-quantlog/ The Reconstruction of QM From the single premise that the “experimental propositions” associated with a physical system are encoded by projections in the way indicated above, one can reconstruct the rest of the formal apparatus of quantum mechanics. The first step, of course, is Gleason's theorem, which tells us that probability measures on L(H) correspond to density operators. There remains to recover, e.g., the representation of “observables” by self-adjoint operators, and the dynamics (unitary evolution). The former can be recovered with the help of the Spectral theorem and the latter with the aid of a deep theorem of E. Wigner on the projective representation of groups. See also R. Wright [1980]. A detailed outline of this reconstruction (which involves some distinctly non-trivial mathematics) can be found in the book of Varadarajan [1985]. The point to bear in mind is that, once the quantum-logical skeleton L(H) is in place, the remaining statistical and dynamical apparatus of quantum mechanics is essentially fixed. In this sense, then, quantum mechanics — or, at any rate, its mathematical framework — reduces to quantum logic and its attendant probability theory. Very nice text. I agree, but it is a difficult matter. You can extract the quantum of 1 bit, but the quibit needs a good tensor product, which is not easy to derive (unless in ad hoc way) from quantum logic. With comp, I think we will need the first order extension of the hypostases, and it could be that special feature of computability theory will need to be discovered to complete the derivation. In my 1991 paper I sum by saying that comp is in search of its Gleason theorem. A lot of work remains, of course. And here we read: http://en.wikipedia.org/wiki/Gleason%27s_theorem Quantum logic treats quantum events (or measurement outcomes) as logical propositions, and studies the relationships and structures formed by these events, with specific emphasis on quantum measurement. More formally, a quantum logic is a set of events that is closed under a countable disjunction of countably many mutually exclusive events. The representation theorem in quantum logic shows that these logics form a lattice which is isomorphic to the lattice of subspaces of a vector space with a scalar product. It remains an open problem in quantum logic to prove that the field K over which the vector space is defined, is either the real numbers, complex numbers, or the quaternions. This is a necessary result for Gleason's theorem to be applicable, since in all these cases we know that the definition of the inner product of a non-zero vector with itself will satisfy the requirements to make the vector space in question a Hilbert space. Application The representation theorem allows us to treat quantum events as a lattice L = L(H) of subspaces of a real or complex Hilbert space. Gleason's theorem allows us to assign probabilities to these events. END QUOTE So I wonder - how much are you still missing to construct QM out of the logical results you have arrived at? I have the formal systems. In a sense, nothing is missing. Except enough competent and interested people in those weird self-referential logics. It is a sequence of open math problems. It is normal. When the research is driven by high level question, you don't choose the mathematical objects you have to handle. You discover them. I could later give more explanation, but here we are at the end of the AUDA (!). It would be too much technical right now. you can take a look at Goldblatt 1974, one or the clearest paper on the Brouwersche Modal quantum logic. Best, 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 everything-l...@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en
Re: COMP, Quantum Logic and Gleason's Theorem
On 17 Jan 2009, at 07:52, Brent Meeker wrote: Günther Greindl wrote: Hi all, the question goes primarily to Bruno but all other input is welcome :-)) Bruno, you said you have already arrived at a quantum logic in your technical work? May I refer to the following two paragraphs?: We can read here: http://plato.stanford.edu/entries/qt-quantlog/ The Reconstruction of QM From the single premise that the “experimental propositions” associated with a physical system are encoded by projections in the way indicated above, one can reconstruct the rest of the formal apparatus of quantum mechanics. The first step, of course, is Gleason's theorem, which tells us that probability measures on L(H) correspond to density operators. There remains to recover, e.g., the representation of “observables” by self-adjoint operators, and the dynamics (unitary evolution). The former can be recovered with the help of the Spectral theorem and the latter with the aid of a deep theorem of E. Wigner on the projective representation of groups. See also R. Wright [1980]. A detailed outline of this reconstruction (which involves some distinctly non-trivial mathematics) can be found in the book of Varadarajan [1985]. The point to bear in mind is that, once the quantum-logical skeleton L(H) is in place, the remaining statistical and dynamical apparatus of quantum mechanics is essentially fixed. In this sense, then, quantum mechanics — or, at any rate, its mathematical framework — reduces to quantum logic and its attendant probability theory. And here we read: http://en.wikipedia.org/wiki/Gleason%27s_theorem Quantum logic treats quantum events (or measurement outcomes) as logical propositions, and studies the relationships and structures formed by these events, with specific emphasis on quantum measurement. More formally, a quantum logic is a set of events that is closed under a countable disjunction of countably many mutually exclusive events. The representation theorem in quantum logic shows that these logics form a lattice which is isomorphic to the lattice of subspaces of a vector space with a scalar product. It remains an open problem in quantum logic to prove that the field K over which the vector space is defined, is either the real numbers, complex numbers, or the quaternions. This is a necessary result for Gleason's theorem to be applicable, since in all these cases we know that the definition of the inner product of a non-zero vector with itself will satisfy the requirements to make the vector space in question a Hilbert space. Application The representation theorem allows us to treat quantum events as a lattice L = L(H) of subspaces of a real or complex Hilbert space. Gleason's theorem allows us to assign probabilities to these events. END QUOTE So I wonder - how much are you still missing to construct QM out of the logical results you have arrived at? Best Wishes, Günther I don't think this form of QM is consistent with Bruno's ideas. Quantum logic takes the projection operation as be fundamental which is inconsistent with unitary evolution and the MWI. But in QM the unitary evolution gives a third person point of view. UDA shows (or is supposed to show) that Physics is first person (plural). A logic of projection is interesting for just that reason. Quantum logic and many world/dream are related by a relation akin to the difference between a ket Ia, and a projection on that ket IaaI. The relation of proximity on the worlds is the anti-relation of perpendicularity among the states (this transform Kripke semantics of quantum logic into Kripke semantics of the Brouwersche modal logic). I know some have used QL to solve (or hide) the conceptual problems of QM, like if QL could evacuate the many worlds, but this is not the case. The modal (à-la-Goldblatt) view of QL invites the many alternate realties. 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 everything-l...@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
Re: COMP, Quantum Logic and Gleason's Theorem
Günther Greindl wrote: Hi all, the question goes primarily to Bruno but all other input is welcome :-)) Bruno, you said you have already arrived at a quantum logic in your technical work? May I refer to the following two paragraphs?: We can read here: http://plato.stanford.edu/entries/qt-quantlog/ The Reconstruction of QM From the single premise that the “experimental propositions” associated with a physical system are encoded by projections in the way indicated above, one can reconstruct the rest of the formal apparatus of quantum mechanics. The first step, of course, is Gleason's theorem, which tells us that probability measures on L(H) correspond to density operators. There remains to recover, e.g., the representation of “observables” by self-adjoint operators, and the dynamics (unitary evolution). The former can be recovered with the help of the Spectral theorem and the latter with the aid of a deep theorem of E. Wigner on the projective representation of groups. See also R. Wright [1980]. A detailed outline of this reconstruction (which involves some distinctly non-trivial mathematics) can be found in the book of Varadarajan [1985]. The point to bear in mind is that, once the quantum-logical skeleton L(H) is in place, the remaining statistical and dynamical apparatus of quantum mechanics is essentially fixed. In this sense, then, quantum mechanics — or, at any rate, its mathematical framework — reduces to quantum logic and its attendant probability theory. And here we read: http://en.wikipedia.org/wiki/Gleason%27s_theorem Quantum logic treats quantum events (or measurement outcomes) as logical propositions, and studies the relationships and structures formed by these events, with specific emphasis on quantum measurement. More formally, a quantum logic is a set of events that is closed under a countable disjunction of countably many mutually exclusive events. The representation theorem in quantum logic shows that these logics form a lattice which is isomorphic to the lattice of subspaces of a vector space with a scalar product. It remains an open problem in quantum logic to prove that the field K over which the vector space is defined, is either the real numbers, complex numbers, or the quaternions. This is a necessary result for Gleason's theorem to be applicable, since in all these cases we know that the definition of the inner product of a non-zero vector with itself will satisfy the requirements to make the vector space in question a Hilbert space. Application The representation theorem allows us to treat quantum events as a lattice L = L(H) of subspaces of a real or complex Hilbert space. Gleason's theorem allows us to assign probabilities to these events. END QUOTE So I wonder - how much are you still missing to construct QM out of the logical results you have arrived at? Best Wishes, Günther I don't think this form of QM is consistent with Bruno's ideas. Quantum logic takes the projection operation as be fundamental which is inconsistent with unitary evolution and the MWI. Brent --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-l...@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~--~~~~--~~--~--~---
