Re: K the Master Set (+ partial answer to Tom's Diagonalization)
Le 20-juil.-06, à 18:07, John M a écrit : Dear Bruno, I appreciate your efforts to 'enlighten' me (and maybe others as well). my case there is more ignorance interfering with the explanations and I will re-re-read your post before I come to a conclusion. As I tried to tell, when you matter-of-factly handle concepts of your 'daily bread' I have to search after for some meaning I can assign as a key to 'read on'. Even the cardinal points in your theory are not functional parts of my mi nd-content (UD, YesDr, even 'comp') but I get lost with G and G', even I have to translate for my own vocabulary the 1- and 3- features or expressions from 'logics'. All these are raining down in your sentences and I cannot ask you not to use them: I use MY 'words' just the same and others ask back many times using for themselves in other meanings. The field of logic is not so well known, as compared to algebra and calculus. Not your fault. There are very few math\ematically gifted minds among us and it does not help what a post yesterday stated that everybody can learn math (thinking) if diligent. You as math teacher may know pupils who just CANNOT get it. I guess such pupil exists, but I cannot decide, and I don't think the pupil can decide, except for his taste. The fraction of humanity cursed with mathematical imparement (ha ha) looks down to the rest of us, a natural defence of the minority. Many greeks seemed to have fallen in love with numbers when their discovered that the sum of the first even numbers give always perfect square: 1 = 1 1 + 3 = 4 1+ 3 + 5 = 9 1 + 3 + 5 + 7 = 16 1 + 3 + 5 + 7 + 9 = 25 ... A simple drawing can explain why it is necessarily so, thus that is a law of numbers. Is that not cute? But math is just not in fashion today for contingent reasons. Like music, it helps developing the taste and educating it as early as possible. A special case the 'applied math' you mentioned. Mostly physicists (and other scientists as well) - thinking in limited models - learned math and aooky itg equationally to a quantized system of their model-view. It elevates the model content to 'total' I hear that feeling. Incompleteness provides a sort of vaccine against that total apprehension! and the imperfections from neglectimg the 'rest of the world - beyond the model's boundaries' lead to paradoxes and orher misconceptions over millennia. Yes. And I would say, perhaps naively, that now that we know that numbers are antireductionist, we should not fear to come back to some Pythagorean Greek rational theologies. Ah! Perhaps we should wait more people get that point, but then I should advertise more for math I have some understanding in the math0thinking, my problem is that I did not 'learn' and 'continue' enough math after that rudimentary conventional domain necessary for the Ph,D exam as 'elective'. You don't need to say. I have myself been disgusted of logic during ten years, and it is only a very special set of circumstances that throws me back in partially. I am a platonist: if you don't find time for math in this life, prepare yourself to do math in the next one (and take this with a graint of salt ;) In my practical polymer RD including numerous implementations and consulting I did not need 'math' and so it faded over all those decades. I never lear\ned theo. logics. I think I am not the worst candidate for what I proposed, yet it may be more than the burden you might take on. Sorry if I wasted your time and consideration. You didn't, 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-list@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~--~~~~--~~--~--~---
RE: K the Master Set (+ partial answer to Tom's Diagonalization)
Hi all, It's very interesting to see these ideas. Common people can understand common languages (like English, Chinese etc.). So I think even the most difficult math. or physics theories can be translated into other common languages that common people can understand easily. I don't see why common people can not understand the most difficult math. equations. Those math. equations or theorems should be just like one language that can be translated into another common language that everyone can understand. Thanks. WC. From: everything-list@googlegroups.com [mailto:[EMAIL PROTECTED] On Behalf Of John M Sent: Thursday, July 20, 2006 12:01 AM To: everything-list@googlegroups.com Subject: Re: K the Master Set (+ partial answer to Tom's Diagonalization) Bruno, George wrote an admirably wise note and you picked positively on the roadmap with the fruitful mind of a logician. It looks like you both start out from not agreeing because of non-understanding math sufficiently - which may be true, but not necessarily the real root. I think many of us have the wrong information about 'math' in question. You called numbers the series of '1,2,3...many' and we think 'math' is a manipulation of such, even if many substitute and functional symbols are used. My question (and I asked it several times here and on diverse other lists and got no satisfactory answer) - still prevails: What are (in the new meaning) NUMBERS - how can we handle the non-number concepts by numbers - (whatever they are)? Rephrased: What is the 'new' meaning of math and how can non-math concepts be handled by math? Norman touched it, 1Z goes around it, David Bohm even went that far as to state: numbers (and so math) are human inventions, probably based on Plato, who made the biggest (philosophical) argument - as the product of HIS mind. Words are loaded with different meanings and people tend to use their favorite - mostly from the mother tongue. I admire George's open mind accepting the diverse positions and I am also no missionary who wants to convert people, but even if I think differently, I like to follow the mental ways of others. It may add usefully to my own thinking. So I propose a 'starting' point to the 'roadmap': How may one consider the new version(s) of number and math instead of the arithmetic-based and binary computer founded conventional ignorance? (It is not a 101 course what this list should be above, it may draw in 'more-sided' opinions into the discussion - which is now pretty much on the math - physics base only. Extending to other planes of 'everything'.) Then we may proceed in understanding the 'stuffy' matter (as e.g.. a photon - ha ha) and the physicists' concepts mostly based on some mathematical application, including the most esoteric 'everything' topics. After all that I may try to speak about my ways how I am not in controversy with all that - only regarding it as a partial view of the totality (which is hard to talk about). Not for converting you or others, just for proving to myself some (Levy-type) sanity. So how should I include the validity of a legal opinion into the numbers? How should I 'comp'(?) the feeling of love? How should I 'materialize' (physically?) the beauty of a sunset? (all without flattening those qualia into a quantitative plane)? Eager to learn John Mikes - Original Message - From: Bruno Marchal To: everything-list@googlegroups.com Sent: Wednesday, July 19, 2006 10:39 AM Subject: Re: K the Master Set (+ partial answer to Tom's Diagonalization) Hi George, A roadmap could be a very good idea. I will think about it. I will keep on your level notions: -kids -grandmother -colleagues (But not in any normative sense: I know kids who are better in math than colleagues, and I know a family where the computer and the net has been installed by the grand-grandmother! So here each one should judge by him/herself on which level they to feel to be. But a roadmap, some summaries ... are in need, sure. Not so easy of course. Just let me think about it. Note also that if I explain in plain english, what I say could appear as a little weird, that is why I tend to be technical. And also, I don't know much people who can swallow both Godel/Church... and Everett/Deutsch ... Quantum information science can help, but this is a bit tricky by itself when you want to be enough precise, and still a long way from Godel-lobian notions. In any case thanks for letting me know when I get too much technical. Thanks to Norman who tries sometimes to convey a similar message, and thanks to Tom for enjoying apparently the more technical posts , and thanks to 1Z for playing the role of the skeptical one, and thanks to all of you, especially Wei Dai, for the kind patience. I will think about some roadmap, but also about some books which could provide helps. Feel
Re: K the Master Set (+ partial answer to Tom's Diagonalization)
one is incorrect. So I propose a 'starting' point to the 'roadmap': How may one consider the new version(s) of number and math instead of the arithmetic-based and binary computer founded conventional ignorance? (It is not a 101 course what this list should be above, it may draw in 'more-sided' opinions into the discussion - which is now pretty much on the math - physics base only. Extending to other planes of 'everything'.) Then we may proceed in understanding the 'stuffy' matter (as e.g.. a photon - ha ha) and the physicists' concepts mostly based on some mathematical application, including the most esoteric 'everything' topics. After all that I may try to speak about my ways how I am not in controversy with all that - only regarding it as a partial view of the totality (which is hard to talk about). Not for converting you or others, just for proving to myself some (Levy-type) sanity. So how should I include the validity of a legal opinion into the numbers? Here I am not sure I follow you. When I talk about (natural) numbers I am really talking about those (non definable) entities that every schoolchild learn about through table of addition, multiplication, etc. They are the same as Euclid's one in the sense that all what Euclid proved about them is still valid today. How should I 'comp'(?) the feeling of love? In principle there will no be problem for that, although I still cannot explain this without explaining more about the G-G* gap. Later perhaps. Note that such a question is more difficult for a physicalist who believes only in atoms or strings (or quantum gravity loops ...) because they don't have (yet) the equivalent of the G-G* gap (akin to the explanation gap of the philosopher of mind). Try to explain why you like potatoes using only terms from string theory, for example. But comp provides an explanation why anything describable in a seemingly third person way, will automatically be extended into a math structure divided in two parts: a 3-communicable part (deriving from G), and a non-3-communicable part (deriving from the corona G* minus G). I recall that G is a mathematical theory describing completely the (skeleton) of what a correct machine can prove about itself, and G* describes the (skeleton) of all the truth---including the non provable one---concerning what a machine can (and cannot) prove about itself. For those who have read a bit on the difference between programmable function Fi and the total computable function fi could perhaps already smell the mathematical justification of that explanatory gap. How should I 'materialize' (physically?) the beauty of a sunset? (all without flattening those qualia into a quantitative plane)? It is exactly here that it is hard for me non going technical because I find it is worth. Indeed it can be proved that when a universal machine M1 introspects herself, she will discover both sharable (provable) quantitative truth and non sharable qualitative (non quantitative, nor even 3-describable) truth. Actually any much stronger (in term of its set of beliefs) universal machine, M2 say, will be able to show that those non quantitative truth are really disguised form of quantitative truth, but M2 can understand why, from the many points of view of M1 itself (including both the 1 and 3 povs), although quantitative, those truth cannot *appear* to be quantitative. M1 can grasp those personal truth only in a qualitative way. This will explain qualia, but also why in some sense a universal machine cannot know she is a machine, nor even any 3-entity. Later I will come back on the arithmetical notion of persons we encounter through the self-reference theories (G and G*) in computer science. I call them hypostases so that people who read Plotinus can see how close we are, with comp, to Plato, and even to Plotinus' critics of Aristotle misunderstanding of Plato. But that is probably on the last point of the roadmap, so I stop, momentarily here. If I have already been too technical just tell me or ask questions. Hope this helps a bit, Bruno Eager to learn John Mikes x-tad-bigger- Original Message -/x-tad-bigger x-tad-biggerFrom:/x-tad-biggerx-tad-bigger /x-tad-biggerx-tad-biggerBruno Marchal/x-tad-biggerx-tad-bigger /x-tad-bigger x-tad-biggerTo:/x-tad-biggerx-tad-bigger /x-tad-biggerx-tad-biggereverything-list@googlegroups.com/x-tad-biggerx-tad-bigger /x-tad-bigger x-tad-biggerSent:/x-tad-biggerx-tad-bigger Wednesday, July 19, 2006 10:39 AM/x-tad-bigger x-tad-biggerSubject:/x-tad-biggerx-tad-bigger Re: K the Master Set (+ partial answer to Tom's Diagonalization)/x-tad-bigger Hi George, A roadmap could be a very good idea. I will think about it. snip> 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-list@googlegroups.com To unsubscr
Re: K the Master Set (+ partial answer to Tom's Diagonalization)
Dear Bruno, I appreciate your efforts to 'enlighten' me (and maybe others as well). my case there is more ignorance interfering with the explanations and I will re-re-read your post before I come to a conclusion. As I tried to tell, when you "matter-of-factly" handle concepts of your 'daily bread' I have to search after for some meaning I can assign as a key to 'read on'. Even the cardinal points in your theory are not functional parts of my mi nd-content (UD, YesDr, even 'comp') but I get lost with G and G', even I have to translate for my own vocabulary the 1- and 3- features or expressions from 'logics'. All these are raining down in your sentences and I cannot ask you not to use them: I use MY 'words' just the same and others ask back many times using for themselves in other meanings. There are very few math\ematically gifted minds among us and it does not help what a post yesterday stated that "everybody can learn math (thinking) if diligent". You as math teacher may know pupils who "just CANNOT get it. The fraction of humanity cursed with mathematical imparement (ha ha) looks down to the rest of us, a natural defence of the minority. A special case the 'applied math' you mentioned. Mostly physicists (and other scientists as well) - thinking in limited models - learned math and aooky itg equationally to a quantized system of their model-view. It elevates the model content to 'total' and the imperfections from neglectimg the 'rest of the world - beyond the model's boundaries' lead to paradoxes and orher misconceptions over millennia. I have some understanding in the math0thinking, my problem is that I did not 'learn' and 'continue' enough math after that rudimentary conventional domain necessary for the Ph,D exam as 'elective'. In my practical polymer RD including numerous implementations and consulting I did not need 'math' and so it faded over all those decades. I never lear\ned theo. logics. I think I am not the worst candidate for what I proposed, yet it may be more than the burden you might take on. Sorry if I wasted your time and consideration. John - Original Message - From: Bruno Marchal To: everything-list@googlegroups.com Sent: Thursday, July 20, 2006 8:22 AM Subject: Re: K the Master Set (+ partial answer to Tom's Diagonalization) John,Le 19-juil.-06, à 18:01, John M a écrit : Bruno,George wrote an admirably wise note and you picked positively on the roadmap with the fruitful mind of a logician.It looks like you both start out from "not agreeing because of non-understanding math sufficiently" - which may be true, but not necessarily the "real" root.I think many of us have the wrong information about 'math' in question. You called "numbers" the series of '1,2,3...many' and "we" think 'math' is a manipulation of such, even if many substitute and functional symbols are used.All right. My question (and I asked it several times here and on diverse other lists and got no satisfactory answer) -still prevails:What are (in the new meaning) NUMBERS - how can we handle the non-number concepts by numbers - (whatever they are)? Rephrased: What is the 'new' meaning of "math" and how can non-math concepts be handled by math?OK, OK, but this is a difficult question, John. Let me give you a standard answer, which should be simple, and then add a comp nuance, which is probably a little bit more subtle.First I don't think there is new meaning of math. Just new branch of math like mathematical logics, philosophical logics, metamathematics, computer science, etc. Since Euler I think mathematician are more and more aware that the numbers are mysterious, and since Godel we have results which somehow explain why numbers are necessarily mysterious. Such limitation results are made *general* (machine or formalism independent) with Church thesis. And then with comp above, those results will bear on the limitation of *humans*: in that sense we can say that we begin to understand why the numbers are mysterious, why we cannot find unifying theory for the numbers, etc.Now for the question "How can non-math concept be handled by math?" The standard answer goes trough the label "applied mathematics". You just need to make a correspondence between some term of the theory and some element of the "reality" you want to modelize with the math theory. This is what physicists do all the time, and this what theologians have done during one millenia (before "religion" has been used as a political power (say)(*))It just applied mathematics.Unfortunately with comp there is a big nuance here.Indeed, when you are using some theory (model in the physicist sense) to predict the whether (say), it
Re: K the Master Set (+ partial answer to Tom's Diagonalization)
Bruno, George wrote an admirably wise note and you picked positively on the roadmap with the fruitful mind of a logician. It looks like you both start out from "not agreeing because of non-understanding math sufficiently" - which may be true, but not necessarily the "real" root. I think many of us have the wrong information about 'math' in question. You called "numbers" the series of '1,2,3...many' and "we" think 'math' is a manipulation of such, even if many substitute and functional symbols are used. My question (and I asked it several times here and on diverse other lists and got no satisfactory answer) -still prevails: What are (in the new meaning) NUMBERS - how can we handle the non-number concepts by numbers - (whatever they are)? Rephrased: What is the 'new' meaning of "math" and how can non-math concepts be handled by math? Norman touched it, 1Z goes around it, David Bohm even went that far as to state: numbers (and so math) are human inventions, probably based on Plato, who made the biggest (philosophical) argument - as the product of HIS mind. Words are loaded with different meanings and people tend to use their favorite - mostly from the mother tongue. I admire George's open mind accepting the diverse positions and I am also no missionary who wants to convert people, but even if I think differently, I like to follow the mental ways of others. It may add usefully to my own thinking. So I propose a 'starting' point to the 'roadmap': How may one consider the new version(s) of number and math instead of the arithmetic-based and binary computer founded conventional ignorance? (It is not a 101 course what this list should be above, it may draw in 'more-sided' opinions into the discussion - which is now pretty much on the math - physics base only. Extending to other planes of 'everything'.) Then we may proceed in understanding the 'stuffy' matter (as e.g.. a photon - ha ha) and the physicists' concepts mostly based on some mathematical application, including the most esoteric 'everything' topics. After all that I may try to speak about my ways how I am not in controversy with all that - only regarding it as a partial view of the totality (which is hard to talk about). Not for converting you or others, just for proving to myself some (Levy-type) sanity. So how should I include the validity of a legal opinion into the numbers? How should I 'comp'(?) the feeling of love? How should I 'materialize' (physically?) the beauty of a sunset? (all without flattening those qualia into a quantitative plane)? Eager to learn John Mikes - Original Message - From: Bruno Marchal To: everything-list@googlegroups.com Sent: Wednesday, July 19, 2006 10:39 AM Subject: Re: K the Master Set (+ partial answer to Tom's Diagonalization) Hi George,A roadmap could be a very good idea. I will think about it. I will keep on your level notions:-kids-grandmother-colleagues(But not in any normative sense: I know kids who are better in math than colleagues, and I know a family where the computer and the net has been installed by the grand-grandmother! So here each one should judge by him/herself on which level they to feel to be.But a roadmap, some summaries ... are in need, sure. Not so easy of course. Just let me think about it.Note also that if I explain in plain english, what I say could appear as a little weird, that is why I tend to be technical. And also, I don't know much people who can swallow both Godel/Church... and Everett/Deutsch ... Quantum information science can help, but this is a bit tricky by itself when you want to be enough precise, and still a long way from Godel-lobian notions.In any case thanks for letting me know when I get too much technical. Thanks to Norman who tries sometimes to convey a similar message, and thanks to Tom for enjoying apparently the more technical posts , and thanks to 1Z for playing the role of the skeptical one, and thanks to all of you, especially Wei Dai, for the kind patience.I will think about some roadmap, but also about some books which could provide helps.Feel free to say more on your "relativity"-information theory. Everyone can talk I certainly don't want to monopolize the threads (but then I got a result and I like to share with motivated people ...)...Now I will leave my office before I liquefy completely BrunoLe 19-juil.-06, à 00:32, George Levy a écrit : Hi BrunoEach one of us like to do what we do best and we apply our preferred techniques to the problem at hand. Thus a mechanic may solve the pollution problem by building electric cars, and the cook may solve the same problem by preparing vegetarian meals.As a mathematician you are trying to compose a theory of everything using mathematics, this is understandable, and you ca
K the Master Set (+ partial answer to Tom's Diagonalization)
Hi Tom, Hi George, George, and others, you can skip the partial answer to Tom, and go directly to K, the master set below. Tom seems to propose an alternate proof, which does not convince me, although I cannot right now provide a full counter-example. Note that the section K, the Master Set could already put some light on that matter. 1) Partial answer to Tom: Le 17-juil.-06, 22:42, Tom Caylor a crit : Now *your* G is just defined by G(n) = GEN2(n). But doesn't G output the range of one of the set of *all* partial recursive functions, whereas GEN2 outputs the code of a *fortran* program? So shouldn't it be the following, where execute() actually executes the fortran program generated by GEN2(n)? G(n) = execute(GEN2(n)) I should have written G(n) = Gen2(n) (n) (= execute Pn on n) Tell me if you are convince that your and my G are programmable. They are both programmable, but I think they are both non-*executable* on k (if G=Fk), for the same reason, self-reference. Let me give you a counterexample with a sequence of total functions. Let Hi be a RE sequence of (codes) of total functions. (so the seq. Hi is from the seq. Fi) Let GBruno be defined by GBruno(n) = Hn(n) +1 Let GTom be defined by GTom(n) = Hn(n) Could GBruno belongs to the sequence Hi? If GBruno belongs to the Hi, it means there is a number kbr such that GBruno = Hkbr, thus GBruno(kbr) = Hkbr(kbr) = Hkbr(kbr)+1. So I can be sure that GBruno does not belong to the sequence Hi. OK? (the usual simple subtraction would lead to 0 = 1) Does GTom belongs to Hi? If GTom belongs to the Hi, it means there is a number kto such Gtom = Hkto, thus Gtom(kto) = Hkto(kto), which is the case by definition of your Gtom. No contradiction occurs, so in principle the total function Gtom could belongs to the list, and indeed is equal to the sequence Hi, despite self-reference. The same could be true for the partial recursive Fi. I don't see any reason why, if G(n) is defined by Fn(n), G should be necessarily undefined on its own index. Your argument could rely on the way you implement G. Actually I could perhaps build an ad hoc counterexample working for some particular enumeration of the Fi, but I need some time to do it, if it is possible So I propose we come back on this after a while. Probably you will figure out what is happening by yourself. Actually your intuition is right: something happens with self-application (see below). If I try to explain all of it here, this could be a little confusing. What you need to be sure of is the fact that when G(n) is defined by Fn(n)+1, then G(k) will be necessarily undefined on all k such that G = Fk. (Independently of the fact that you could be right that G'(k') is also undefined when G' (n) is defined by Fn(n), and k' is a code or index of G'; but your argument is not a proof because it depends on the precise way G is implemented). I must think ... 2) K, the Master set Emil Post, the founder of Recursion Theory, introduced the following set (of numbers) which will appears to be fundamental. It will correspond, in term of set, to the universal machine. K will be an universal RE set, capable of generating all RE sets. I recall the code of the RE sets are generable, and the RE sets are the domain Wi of the Fi. Definition: K is the set of numbers x such that Fx(x) is defined. So K is the set of natural number x such that the xth programs in the enumeration of the codes of all programs does stop when apply on itself. I prefer to talk about self-application instead of self-reference (to follow standard terminology). I give exercises (if only because my office is an oven and my brain is boiling hot): 1) Is K an RE set? Answer: yes (why?) 2) Is N \ K an RE set? Answer: no (why? Hint: diagonalization) 3) From this conclude that the halting problem is insoluble. 4) try to justify that someone having an algorithm for generating K will be able to generate any Wi. Put in another way, from a mechanical solution to the problem does Fx(x) stop we can construct an algorithm solving the apparent more general problem does Fx(y) stop. 5) From 2) show that N \ K is productive (like the set of codes of the computable growing functions). That is N \ K is not only not-RE, but is constructively not-RE. You need to find an algorithm A such that for any Wi included in N \ K, A(i) will give an element in N \ K which is not in Wi. If you look at that Wi as an attempt to enumerate all N \ K, you can see the algorithm A as providing a counter-example. Conclude that N \ K can be better and better approximated by iterations in the constructive transfinite (like we done with the fairy). MAIN DEFINITION (Emil Post): A set E (of numbers) is called CREATIVE if 1) E is RE 2) N \ E is productive So the exercise can be sum up into: show that K is a creative set. There are deep relations between creative sets, universal machines, and lobian
Re: K the Master Set (+ partial answer to Tom's Diagonalization)
Hi Bruno Each one of us like to do what we do best and we apply our preferred techniques to the problem at hand. Thus a mechanic may solve the pollution problem by building electric cars, and the cook may solve the same problem by preparing vegetarian meals. As a mathematician you are trying to compose a theory of everything using mathematics, this is understandable, and you came up with COMP which is strongly rooted in mathematics and logic. I came up independently with my own concept involving a generalization of relativity to information theory ( my background is engineering/physics) and somehow we seem to agree on many points. Unfortunately I do not have the background and the time to give my ideas a formal background. It is just an engineering product and it feels right. I believe that what you are saying is right, however I am having some trouble following you, just like Norman Samish said. It would help if you outlined a roadmap. Then we would be able to follow the roadmap without having to stop and admire the mathematical scenery at every turn even though it is very beautiful to the initiated, I am sure. For example you could use several levels of explanation: a first level would be as if your were talking to your grandmother; a second level, talking to your kids (if they listen); a last level, talking to your colleagues. George --~--~-~--~~~---~--~~ You received this message because you are subscribed to the Google Groups Everything List group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to [EMAIL PROTECTED] For more options, visit this group at http://groups.google.com/group/everything-list -~--~~~~--~~--~--~---
