Dear Alex, dear friends,

On 04 May 2016, at 02:49, Alex Hankey wrote:

Dear Friends,I was so struck by the group's focus on Gödel's theorems that I wentback to John R. Lucas who originated the idea that Gödel's insightsimply that the human miind is not a machine - and therefore capableof genuine phenomenal experience. You may find the ideas in thefollowing informative and usefulhttps://en.wikipedia.org/wiki/John_Lucas_(philosopher) https://en.wikipedia.org/wiki/Minds,_Machines_and_GödelI noted particularly that I have used complexity (which Lucasmentions towards the end of MM&G) to establish that organisms arenot machines, and out of that I identify the form of informationwhich may explain various aspects of experience.

`I work on this since a long time. It is my domain of investigation`

`actually. Indeed, it is Gödel's proof which made me decide along time`

`ago to become a mathematician instead of a biologist, when I saw that`

`Gödel's technic gave a conceptually clear explanation of how something`

`can self-duplicate, self-refers, self-transforms, etc.`

`The first to get the idea that incompleteness can be used to "prove"`

`that we are not machine was Emil Post in 1922, wen he anticipated`

`incompleteness. Then Emil Post was also the first to see the main`

`error in that argument, and he saw what can still be derived from it`

`(mainly that iF we are a machine THEN we cannot know which machine we`

`are: a key that I have exploited in the derivation of physics from`

`arithmetic and mechanism).`

`Note that Gödel's first incompleteness theorem can be rigorously`

`derived in very few line (indeed just one double diagonalization) from`

`the Church-Turing thesis. This has been seen by Kleene, and is`

`exploited by Judson Webb in his 1980 book to illustrate that not only`

`Gödel's theorem does not refute mechanism, but Gödel's theorem is an`

`incredible chance for mechanism. I have written myself a lot on this,`

`and my work extends this at the extreme, as it shows that mechanism`

`makes incompleteness the roots of both the appearance of qualia and`

`quanta, and this in a precise and unique way, making mechanism`

`empirically refutable.`

`I intended to give here the proof in a few line of Gödel's`

`incompleteness from Church-thesis, but that might wait.`

`Someone (Lou?) said "Proving” that we are not machines is somewhat`

`quixotic from my point of view, in that it should be obvious that we`

`are not machines!"`

`The statement "we are machine" is ambiguous. Does "we" refers to our`

`souls or to our bodies? Does it refers to our third person describable`

`relative bodies or to our private unnameable and non describable first`

`person view.`

`Here, what the Gödel-Löbian machine can already prove about themselves`

`is that IF they are self-referentially correct machine, and if they`

`survive a digital substitution at some level of description, they`

`their soul is not a machine, once we admit to identify the soul with`

`the knower, and translate the Theaetetus' definition of knowing (true`

`opinion, []p & p) in arithmetical terms. If my body is a machine, then`

`my soul is not (says Peano Arithmetic!).`

`I explain the main things in my two JPMB contributions. The key idea,`

`related to Gödel, is that incompleteness separates clearly what is`

`true about the machine and what is justifiable by the machine on one`

`part, and on the other part, it separates clearly the justifiability`

`([]p), knowability ([]p & p), "observability" ([]p & <>t), sensibility`

`([]p & <>t & p) with []p for Gödel's probability predicate, and <>t =`

`~[]~t = ~[]f = consistency (t = constant true, f = constant false). I`

`predicted that []p & p, []p & <>p and []p & <>t & p should give a`

`quantum logic (when p is "computably accessible, or Sigma_1), which I`

`manage to prove after 30 years of research. The whole things leads to`

`a many dream interpretations of arithmetic, from which a derivation of`

`quantum mechanics should be possible (and is partially done).`

`All this has been made possible by Gödel's theorem and its many`

`generalizations in many directions: Löb's theorem, Kleene's theorems`

`and mainly Solovay's completeness theorem for some modal logics, which`

`capture everything in the undecidability field in two modal logics (G`

`and G*). I will just refer everyone interested to my papers.`

`The main point relevant here is that incompleteness saves machine from`

`all reductionism. It shows that a machine (or any effective theory)`

`like Peano Arithmetic (say) is already quite clever. PA can already`

`refute all reductionist theories about its soul, and indeed can`

`already derive physics from Mechanism + Arithmetic.`

`Unfortunately, this uses mathematical logic, which is not well know by`

`non-logicians. Mathematical logic contains important sub-branches,`

`like computation theory, computability, theory, proof theory, and`

`model theory. Model theory is the study of meanings and semantics of`

`formal theories. Physicists used the term "model" for theory, and that`

`often lead to a "dialog of deaf".`

`I don't want be too long so I will stop here. I will reply to possible`

`comments next week, because this is my second post of the week.`

Best regards, have a good week-end, Bruno

I found Minds, Machines and Godel very useful to read since itseemed to confirm aspects of my offering to you all.On 3 May 2016 at 07:49, Francesco Rizzo<13francesco.ri...@gmail.com> wrote:Cari Terrence, Louis, Maxine e Tutti,I state that I am a "exponential poor" who do not claim to lead anyclaim whatsoever. But I think I have understood from thetriangulation of three colleagues, I do not think it is concluded,that:- Rosario Strano, a mathematician at the University of Catania,lecturing on "Goodel, Tarski and liar" in the end he said: "Inclosing, we conclude with a remark 'philosophical' suggested duringthe conference by fellow F. Rizzo : a result that we can draw fromthe theorems of exposed is that the search for truth, both inmathematics and in other sciences, can not be caged by mechanicalrules, nor reduced to a formal calculation, but it requiresinspiration, intuition and genius, all features own human intellect( "Bulletin Mathesis" section of Catania, Year V, n. 2, April 28,2000);- Also it happened to me, to defend the economic science fromencroachment or domain of the infinitesimal calculus, to declarethat the mathematical models resemble simulacra, partly true(according to the logic) and partly false (according to reality) :see. lately, F. Rizzo, ... "Bursars (c) to" Arachne publishing,Rome, April 2016;-in theory and in economic practice the capitalization rate "r" ofthe capitalization formula V = Rn. 1 / r can be determined orresorting to "qualitative quantity" of Hegel ( "The Science ofLogic") or to complex or imaginary numbers that, among other things,allowed the Polish mathematician Minkowski, master of A. Einstein,to adjust the general theory of relativity, so much so that I wrote:"the imaginary and / or complex numbers used to conceive the'Minkowiski of' universe that transforms time into space, making itclearer and more explicit the isomorphic influence that space-timeexercises the capitalization formula and equation of specialrelativity, maybe they can illuminate with new lights the functionof the concept of co-efficient of capitalization "(F. Rizzo," fromthe Keynesian revolution to the new economy ", Franco Angeli, Milan,2002, p . 35).How you see the world looks great but basically it is up to you andeven to poor people like me to make it, or reduce it to theappropriate size to understand (and be understood) by all.Thank you for the opportunity you have given me, I greet you withintellectual and human friendship.Francesco 2016-05-03 5:28 GMT+02:00 Louis H Kauffman <lou...@gmail.com>: Dear FolksI realize in replying to this I surely reach the end of possiblecomments that I can make for a week. But nevertheless …I want to comment on Terrence Deacon’s remarks below and also onProfessor Johnstone’s remark from another email:"This may look like a silly peculiarity of spoken language, one bestignored in formal logic, but it is ultimately what is wrong with theGödel sentence that plays a key role in Gödel’s IncompletenessTheorem. That sentence is a string of symbols deemed well-formedaccording to the formation rules of the system used by Gödel, butwhich, on the intended interpretation of the system, is ambiguous:the sentence has two different interpretations, a self-referentialtruth-evaluation that is neither true nor false or a true statementabout that self-referential statement. In such a system, Gödel’sconclusion holds. However, it is a mistake to conclude that nopossible formalization of Arithmetic can be complete. In a formalsystem that distinguishes between the two possible readings of theGödel sentence (an operation that would considerably complicate thesystem), such would no longer be the case.********” I will begin with the paragraph above.Many mathematicians felt on first seeing Goedel’s argument that itwas a trick, a sentence like the Liar Sentence that had no realmathematical relevance.This however is not true, but would require a lot more work than Iwould take in this email to be convincing. Actually the crux of theGoedel Theorem is in the fact that a formal system thatcan handle basic number theory and is based on a finite alphabet,has only a countable number of facts about the integers that it canproduce. One can convince oneself on general grounds that there areindeed an uncountable number of true facts about the integers. Agiven formal system can only produce a countable number of suchfacts and so is incomplete. This is the short version of Goedel’sTheorem. Goedel worked hard to produce a specific statement thatcould not be proved by the given formal system, but theincompleteness actually follows from the deep richness of theintegers as opposed to the more superficial reach of any givenformal system.Mathematicians should not be perturbed by this incompleteness.Mathematics is paved with many formal systems.In my previous email I point to the Goldstein sequence. https://en.wikipedia.org/wiki/Goodstein%27s_theoremThis is an easily understood recursive sequence of numbers that nomatter how you start it, always ends at zero after some number ofiterations.This Theorem about the Goodstein recursion is not provable in PeanoArithmetic, the usual formalization of integer arithmetic, usingstandard mathematical induction.This is a good example of a theorem that is not just a “LiarParadox” and shows that Peano Arithmetic is incomplete.And by the way, the Goodstein sequence CAN be proved to terminate byusing ‘imaginary values’ as Professor Deacon describes (with a tipof the hat to Spencer-Brown).In this case the imaginary values are a segment of Cantor’stransfinite ordinals. Once these transfinite numbers are admittedinto the discussion there is an elegant proof available for thetermination of the Goodstein sequence. Spencer-Brown liked to talkabout the possibility of proofs by using “imaginary Boolean values”.Well, the Goodstein proof is an excellent example of this philosophy.A further comment, thinking about i (i^2 = -1) as an oscillation isvery very fruitful from my point of view and I could bend your earon that for a long time. Here is a recent paper of mine on thatsubject. Start in Section 2 if you want to start with themathematics of the matter.http://arxiv.org/pdf/1406.1929.pdf And here is an older venture on the same theme. http://homepages.math.uic.edu/~kauffman/SignAndSpace.pdfMore generally, the idea is that one significant way to move out ofparadox is to move into a state of time.I feel that this is philosophically a deep remark on the nature oftime and that i as an oscillation is the right underlyingmathematical metaphor for time.It is, in this regard, not an accident that the Minkowski metric isX^2 + Y^2 + Z^2 + (iT)^2.TIME = iT This is an equation with double meaning. Time is measured oscillation. Time is rotated ninety degrees from Space. And one can wonder: How does i come to multiply itself and return -1?Try finding your own answers before you try mine or all the previousstories!Best, Lou (See you next week.)On May 2, 2016, at 9:31 PM, Terrence W. DEACON<dea...@berkeley.edu> wrote:A number of commentators, including the philosopher-logicianG. Spencer Brown and the anthropologist-systems theorist GregoryBateson, reframed variants of the Liar’s paradox as it might applyto real world phenomena. Instead of being stymied by theundecidability of the logic or the semantic ambiguity, they focusedon the very process of analyzing these relationships. The reasonthese forms lead to undecidable results is that each time they areinterpreted it changes the context in which they must beinterpreted, and so one must inevitably alternate between true andfalse, included and excluded, consistent and inconsistent, etc. So,although there is no fixed logical, thus synchronic, status of thematter, the process of following these implicit injunctions resultsin a predictable pattern across time. In logic, the statement “iftrue, then false” is a contradiction. In space and time, “if on,then off” is an oscillation. Gregory Bateson likened this to asimple electric buzzer, such as the bell in old ringer telephones.The basic design involves a circuit that includes an electromagnetwhich when supplied with current attracts a metal bar which pullsit away from an electric contact that thereby breaks the circuitcutting off the electricity to the electromagnet which allows themetal bar to spring back into position where the electric contactre-closes the circuit re-energizing the electromagnet, and so on.The resulting on-off-on-off activity is what produces a buzzingsound, or if attached to a small mallet can repeatedly ring a bell.Consider another variant of incompletability: the concept ofimaginary number. The classic formulation involves trying todetermine the square root of a negative number. The relationship ofthis to the liar’s paradox and the buzzer can be illustrated bystepping through stages of solving the equation i x i = -1.Dividing both sides by i produces i = -1/i, and then substitutingthe value of i one gets i = -1/-1/i and then again i = -1/-1/-1/iand so forth, indefinitely. With each substitution the valuealternates from negative to positive and cannot be resolved (likethe true/false of the liar’s paradox and the on/off of the buzzer).But if we ignore this irresolvability and just explore theproperties of this representation of an irresolvable value, as havemathematicians for centuries, it can be shown that i can be treatedas a form of unity and subject to all the same mathematicalprinciples as can 1 and all the real numbers derived from it. So i+ i = 2i and i - 2i = -i and so on. Interestingly, 0 x i = 0 X 1 =0, so we can conceive of the real number line and the imaginarynumber line as two dimensions intersecting at 0, the origin.Ignoring the many uses of such a relationship (such as the use ofcomplex numbers with a real and imaginary component) we can seethat this also has an open-ended consequence. This is because thevery same logic can be used with respect to the imaginary numberline. We can thus assign j x j = -i to generate a third dimensionthat is orthogonal to the first two and also intersecting at theorigin. Indeed, this can be done again and again, withoutcompletion; increasing dimensionality without end (though byconvention we can at any point restrict this operation in order touse multiple levels of imaginaries for a particular application,there is no intrinsic principal forcing such a restriction).One could, of course, introduce a rule that simply restrictssuch operations altogether, somewhat parallel to Bertrand Russell’sproposed restriction on logical type violation. But mathematicianshave discovered that the concept of imaginary number is remarkablyuseful, without which some of the most powerful mathematical toolswould never have been discovered. And, similarly, we coulddiscount Gödel’s discovery because we can’t see how it makes sensein some interpretations of semiosis. On the other hand, like G.Spencer Brown, Doug Hofstadter, and many others, thinking outsideof the box a bit when considering these apparent dilemmas mightlead to other useful insights. So I’m not so willing to brand theLiar, Gödel, and all of their kin as useless nonsense. It’s not abug, it’s a feature.On Mon, May 2, 2016 at 2:19 PM, Maxine Sheets-Johnstone <m...@uoregon.edu> wrote:Many thanks for your comments, Lou and Bruno. I read and pondered, and finally concluded that the paths taken by each of you exceed my competencies. I subsequently sent your comments to Professor Johnstone—-I trust this is acceptable—asking him if he would care torespond with a brief sketch of the reasoning undergirding hiscritique,which remains anchored in Gödel’s theorem, not in the writings ofothersabout Gödel’s theorem. Herewith his reply: ******** Since no onecommented on the reasoning supporting the conclusions reached inthe two cited articles, let me attempt to sketch the crux of thecase presented. The Liar Paradox contains an important lesson aboutmeaning. A statement that says of itself that it is false, givesrise to a paradox: if true, it must be false, and if false, it mustbe true. Something has to be amiss here. In fact, what is wrong isthe statement in question is not a statement at all; it is a pseudo-statement, something that looks like a statement but is incompleteor vacuous. Like the pseudo-statement that merely says of itselfthat it is true, it says nothing. Since such self-referential truth-evaluations say nothing, they are neither true nor false. Indeed,the predicates ‘true’ and ‘false’ can only be meaningfully appliedto what is already a meaningful whole, one that already sayssomething. The so-called Strengthened Liar Paradox features apseudo-statement that says of itself that it is neither true norfalse. It is paradoxical in that it apparently says something thatis true while saying that what it says it is not true. However, theparadox dissolves when one realizes that it says something that isapparently true only because it is neither true nor false. However,if it is neither true nor false, it is consequently not astatement, and hence it says nothing. Since it says nothing, itcannot say something that is true. The reason why it appears to saysomething true is that one and the same string of words may be usedto make either of two declarations, one a pseudo-statement, theother a true statement, depending on how the words refer. Considerthe following example. Suppose we give the name ‘Joe’ to what I amsaying, and what I am saying is that Joe is neither true nor false.When I say it, it is a pseudo-statement that is neither true norfalse; when you say it, it is a statement that is true. Thesentence leads a double life, as it were, in that it may be used tomake two different statements depending on who says it. A similarsituation can also arise with a Liar sentence: if the liar saysthat what he says is false, then he is saying nothing; if I saythat what he says is false, then I am making a false statementabout his pseudo-statement. This may look like a silly peculiarityof spoken language, one best ignored in formal logic, but it isultimately what is wrong with the Gödel sentence that plays a keyrole in Gödel’s Incompleteness Theorem. That sentence is a stringof symbols deemed well-formed according to the formation rules ofthe system used by Gödel, but which, on the intended interpretationof the system, is ambiguous: the sentence has two differentinterpretations, a self-referential truth-evaluation that isneither true nor false or a true statement about that self-referential statement. In such a system, Gödel’s conclusion holds.However, it is a mistake to conclude that no possible formalizationof Arithmetic can be complete. In a formal system thatdistinguishes between the two possible readings of the Gödelsentence (an operation that would considerably complicate thesystem), such would no longer be the case. ******** Cheers, Maxine_______________________________________________ Fis mailing listFis@listas.unizar.eshttp://listas.unizar.es/cgi-bin/mailman/listinfo/fis-- Professor Terrence W. Deacon University of California, Berkeley _______________________________________________ Fis mailing listFis@listas.unizar.eshttp://listas.unizar.es/cgi-bin/mailman/listinfo/fis_______________________________________________ Fis mailing listFis@listas.unizar.eshttp://listas.unizar.es/cgi-bin/mailman/listinfo/fis _______________________________________________ Fis mailing listFis@listas.unizar.eshttp://listas.unizar.es/cgi-bin/mailman/listinfo/fis -- Alex Hankey M.A. (Cantab.) PhD (M.I.T.) Distinguished Professor of Yoga and Physical Science, SVYASA, Eknath Bhavan, 19 Gavipuram Circle Bangalore 560019, Karnataka, India Mobile (Intn'l): +44 7710 534195 Mobile (India) +91 900 800 8789 ____________________________________________________________2015 JPBMB Special Issue on Integral Biomathics: Life Sciences,Mathematics and Phenomenological Philosophy_______________________________________________ Fis mailing list Fis@listas.unizar.es http://listas.unizar.es/cgi-bin/mailman/listinfo/fis

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