Dear Friends, I was so struck by the group's focus on Gödel's theorems that I went back to John R. Lucas who originated the idea that Gödel's insights imply that the human miind is not a machine - and therefore capable of genuine phenomenal experience. You may find the ideas in the following informative and useful https://en.wikipedia.org/wiki/John_Lucas_(philosopher) https://en.wikipedia.org/wiki/Minds,_Machines_and_Gödel
I noted particularly that I have used complexity (which Lucas mentions towards the end of MM&G) to establish that organisms are not machines, and out of that I identify the form of information which may explain various aspects of experience. I found Minds, Machines and Godel very useful to read since it seemed 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 any claim > whatsoever. But I think I have understood from the triangulation 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: "In closing, we conclude > with a remark 'philosophical' suggested during the conference by fellow F. > Rizzo : a result that we can draw from the theorems of exposed is that the > search for truth, both in mathematics and in other sciences, can not be > caged by mechanical rules, nor reduced to a formal calculation, but it > requires inspiration, 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 from encroachment > or domain of the infinitesimal calculus, to declare that 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" of the > capitalization formula V = Rn. 1 / r can be determined or resorting to > "qualitative quantity" of Hegel ( "The Science of Logic") 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 it clearer and more explicit the > isomorphic influence that space-time exercises the capitalization formula > and equation of special relativity, maybe they can illuminate with new > lights the function of the concept of co-efficient of capitalization "(F. > Rizzo," from the 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 and even > to poor people like me to make it, or reduce it to the appropriate size to > understand (and be understood) by all. > Thank you for the opportunity you have given me, I greet you with > intellectual and human friendship. > Francesco > > 2016-05-03 5:28 GMT+02:00 Louis H Kauffman <lou...@gmail.com>: > >> Dear Folks >> I realize in replying to this I surely reach the end of possible comments >> that I can make for a week. But nevertheless … >> I want to comment on Terrence Deacon’s remarks below and also on >> Professor Johnstone’s remark from another email: >> >> "This may look like a silly peculiarity of spoken language, one best >> ignored in formal logic, but it is ultimately what is wrong with the Gödel >> sentence that plays a key role in Gödel’s Incompleteness Theorem. That >> sentence is a string of symbols deemed well-formed according to the >> formation rules of the system used by Gödel, but which, on the intended >> interpretation of the system, is ambiguous: the sentence has two different >> interpretations, a self-referential truth-evaluation that is neither 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 formalization of Arithmetic can be complete. In a >> formal system that distinguishes between the two possible readings of the >> Gödel sentence (an operation that would considerably complicate the >> system), such would no longer be the case. >> ********” >> I will begin with the paragraph above. >> Many mathematicians felt on first seeing Goedel’s argument that it was a >> trick, a sentence like the Liar Sentence that had no real mathematical >> relevance. >> This however is not true, but would require a lot more work than I would >> take in this email to be convincing. Actually the crux of the Goedel >> Theorem is in the fact that a formal system that >> can handle basic number theory and is based on a finite alphabet, has >> only a countable number of facts about the integers that it can produce. >> One can convince oneself on general grounds that there are indeed an >> uncountable number of true facts about the integers. A given formal system >> can only produce a countable number of such facts and so is incomplete. >> This is the short version of Goedel’s Theorem. Goedel worked hard to >> produce a specific statement that could not be proved by the given formal >> system, but the incompleteness actually follows from the deep richness of >> the integers as opposed to the more superficial reach of any given formal >> 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_theorem >> <https://en.wikipedia.org/wiki/Goodstein's_theorem> >> This is an easily understood recursive sequence of numbers that no matter >> how you start it, always ends at zero after some number of iterations. >> This Theorem about the Goodstein recursion is not provable in Peano >> Arithmetic, the usual formalization of integer arithmetic, using standard >> mathematical induction. >> This is a good example of a theorem that is not just a “Liar Paradox” and >> shows that Peano Arithmetic is incomplete. >> >> And by the way, the Goodstein sequence CAN be proved to terminate by >> using ‘imaginary values’ as Professor Deacon describes (with a tip of the >> hat to Spencer-Brown). >> In this case the imaginary values are a segment of Cantor’s transfinite >> ordinals. Once these transfinite numbers are admitted into the discussion >> there is an elegant proof available for the termination of the Goodstein >> sequence. Spencer-Brown liked to talk about 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 is very >> very fruitful from my point of view and I could bend your ear on that for a >> long time. Here is a recent paper of mine on that subject. Start in Section >> 2 if you want to start with the mathematics 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.pdf >> >> More generally, the idea is that one significant way to move out of >> paradox is to move into a state of time. >> I feel that this is philosophically a deep remark on the nature of time >> and that i as an oscillation is the right underlying mathematical metaphor >> for time. >> It is, in this regard, not an accident that the Minkowski metric is X^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 previous >> stories! >> 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-logician G. >> Spencer Brown and the anthropologist-systems theorist Gregory Bateson, >> reframed variants of the Liar’s paradox as it might apply to real world >> phenomena. Instead of being stymied by the undecidability of the logic or >> the semantic ambiguity, they focused on the very process of analyzing these >> relationships. The reason these forms lead to undecidable results is that >> each time they are interpreted it changes the context in which they must be >> interpreted, and so one must inevitably alternate between true and false, >> included and excluded, consistent and inconsistent, etc. So, although there >> is no fixed logical, thus synchronic, status of the matter, the process of >> following these implicit injunctions results in a predictable pattern >> across time. In logic, the statement “if true, then false” is a >> contradiction. In space and time, “if on, then off” is an oscillation. >> Gregory Bateson likened this to a simple electric buzzer, such as the bell >> in old ringer telephones. The basic design involves a circuit that includes >> an electromagnet which when supplied with current attracts a metal bar >> which pulls it away from an electric contact that thereby breaks the >> circuit cutting off the electricity to the electromagnet which allows the >> metal bar to spring back into position where the electric contact re-closes >> the circuit re-energizing the electromagnet, and so on. The resulting >> on-off-on-off activity is what produces a buzzing sound, or if attached to >> a small mallet can repeatedly ring a bell. >> Consider another variant of incompletability: the concept of >> imaginary number. The classic formulation involves trying to determine the >> square root of a negative number. The relationship of this to the liar’s >> paradox and the buzzer can be illustrated by stepping through stages of >> solving the equation *i *x* i* = *-1*. Dividing both sides by *i* >> produces *i* = *-1/i, *and then substituting the value of *i *one gets* >> i = -1/-1/i *and then again* i = -1/-1/-1/i *and so forth, indefinitely. >> With each substitution the value alternates from negative to positive and >> cannot be resolved (like the true/false of the liar’s paradox and the >> on/off of the buzzer). But if we ignore this irresolvability and just >> explore the properties of this representation of an irresolvable value, as >> have mathematicians for centuries, it can be shown that *i* can be >> treated as a form of unity and subject to all the same mathematical >> principles as can 1 and all the real numbers derived from it. So *i *+ >> *i* = 2*i* and *i* - 2*i* = *-**i* and so on. Interestingly, 0 x *i* = 0 >> X 1 = 0, so we can conceive of the real number line and the imaginary >> number line as two dimensions intersecting at 0, the origin. Ignoring the >> many uses of such a relationship (such as the use of complex numbers with a >> real and imaginary component) we can see that this also has an open-ended >> consequence. This is because the very same logic can be used with respect >> to the imaginary number line. We can thus assign *j *x *j = -i *to >> generate a third dimension that is orthogonal to the first two and also >> intersecting at the origin. Indeed, this can be done again and again, >> without completion; increasing dimensionality without end (though by >> convention we can at any point restrict this operation in order to use >> 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 restricts such >> operations altogether, somewhat parallel to Bertrand Russell’s proposed >> restriction on logical type violation. But mathematicians have discovered >> that the concept of imaginary number is remarkably useful, without which >> some of the most powerful mathematical tools would never have been >> discovered. And, similarly, we could discount Gödel’s discovery because we >> can’t see how it makes sense in some interpretations of semiosis. On the >> other hand, like G. Spencer Brown, Doug Hofstadter, and many others, >> thinking outside of the box a bit when considering these apparent dilemmas >> might lead to other useful insights. So I’m not so willing to brand the >> Liar, Gödel, and all of their kin as useless nonsense. It’s not a bug, 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 to >>> respond with a brief sketch of the reasoning undergirding his critique, >>> which remains anchored in Gödel’s theorem, not in the writings of others >>> about Gödel’s theorem. Herewith his reply: ******** Since no one >>> commented on the reasoning supporting the conclusions reached in the >>> two cited articles, let me attempt to sketch the crux of the case presented. >>> The Liar Paradox contains an important lesson about meaning. A statement >>> that says of itself that it is false, gives rise to a paradox: if true, it >>> must be false, and if false, it must be true. Something has to be amiss >>> here. In fact, what is wrong is the statement in question is not a >>> statement at all; it is a pseudo-statement, something that looks like a >>> statement but is incomplete or vacuous. Like the pseudo-statement that >>> merely says of itself that 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 >>> applied to what is already a meaningful whole, one that already says >>> something. The so-called Strengthened Liar Paradox features a >>> pseudo-statement that says of itself that it is neither true nor false. It >>> is paradoxical in that it apparently says something that is true while >>> saying that what it says it is not true. However, the paradox dissolves >>> when one realizes that it says something that is apparently true only >>> because it is neither true nor false. However, if it is neither true nor >>> false, it is consequently not a statement, and hence it says nothing. Since >>> it says nothing, it cannot say something that is true. The reason why it >>> appears to say something true is that one and the same string of words may >>> be used to make either of two declarations, one a pseudo-statement, the >>> other a true statement, depending on how the words refer. Consider the >>> following example. Suppose we give the name ‘Joe’ to what I am saying, 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 nor false; when you say it, it >>> is a statement that is true. The sentence leads a double life, as it were, >>> in that it may be used to make two different statements depending on who >>> says it. A similar situation can also arise with a Liar sentence: if the >>> liar says that what he says is false, then he is saying nothing; if I say >>> that what he says is false, then I am making a false statement about his >>> pseudo-statement. This may look like a silly peculiarity of spoken >>> language, one best ignored in formal logic, but it is ultimately what is >>> wrong with the Gödel sentence that plays a key role in Gödel’s >>> Incompleteness Theorem. That sentence is a string of symbols deemed >>> well-formed according to the formation rules of the system used by Gödel, >>> but which, on the intended interpretation of the system, is ambiguous: the >>> sentence has two different interpretations, a self-referential >>> truth-evaluation that is neither 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 formalization >>> of Arithmetic can be complete. In a formal system that distinguishes >>> between the two possible readings of the Gödel sentence (an operation that >>> would considerably complicate the system), such would no longer be the case. >>> ******** Cheers, Maxine _______________________________________________ >>> Fis mailing listFis@listas.unizar.eshttp://listas.unizar.es/cgi- >>> <http://listas.unizar.es/cgi-bin/mailman/listinfo/fis> >>> bin/mailman/listinfo/fis >>> <http://listas.unizar.es/cgi-bin/mailman/listinfo/fis> >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> <Fis@listas.unizar.es> >>> <http://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- >> <http://listas.unizar.es/cgi-bin/mailman/listinfo/fis> >> bin/mailman/listinfo/fis >> <http://listas.unizar.es/cgi-bin/mailman/listinfo/fis> >> <Fis@listas.unizar.es> >> <http://listas.unizar.es/cgi-bin/mailman/listinfo/fis> >> >> >> >> _______________________________________________ >> Fis mailing listFis@listas.unizar.eshttp://listas.unizar.es/cgi- >> <http://listas.unizar.es/cgi-bin/mailman/listinfo/fis> >> bin/mailman/listinfo/fis >> <http://listas.unizar.es/cgi-bin/mailman/listinfo/fis> >> <Fis@listas.unizar.es> >> <http://listas.unizar.es/cgi-bin/mailman/listinfo/fis> >> >> > > _______________________________________________ > Fis mailing listFis@listas.unizar.eshttp://listas.unizar.es/cgi- > <http://listas.unizar.es/cgi-bin/mailman/listinfo/fis> > bin/mailman/listinfo/fis > <http://listas.unizar.es/cgi-bin/mailman/listinfo/fis> > <Fis@listas.unizar.es> > <http://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 <http://www.sciencedirect.com/science/journal/00796107/119/3>
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