Cari Terrence, Louis, Maxine e Tutti, premetto che sono un "poverino esponenziale" che non ha la pretesa di menare alcun vanto. Ma mi pare di aver capito dalla triangolazione dei tre colleghi,che non credo si sia conclusa, quello che: - Rosario Strano, un valente matematico dell'Università di Catania, tenendo una conferenza su "Goodel, Tarski e il mentitore" alla fine ha affermato: "In chiusura concludiamo con un'osservazione 'filosofica' suggerita durante la conferenza dal collega F. Rizzo: una conseguenza che possiamo trarre dai teoremi su esposti è che la ricerca della verità, sia nella matematica che nelle altre scienze, non può essere ingabbiata da regole meccaniche, nè ridursi a un calcolo formale, ma richiede estro, intuizione e genialità, tutte caratteristiche proprie dell'intelletto umano ("Bollettino Mathesis" della sezione di Catania, Anno V, n. 2, 28 aprile 2000); - anche a me è capitato, per difendere la scienza economica dall'invadenza o dominio del calcolo infinitesimale, di dichiarare che i modelli matematici assomigliano a dei simulacri, in parte veri (secondo la logica) e in in parte falsi (secondo la realtà): cfr. ultimamente, Rizzo F., ..."Economi(c)a", Aracne editrice, Roma, aprile 2016; -nella teoria e nella pratica economica il saggio di capitalizzazione "r" della formula di capitalizzazione V = Rn. 1/r si può determinare o ricorrendo alla "quantità qualitativa" di hegel ("La scienza della logica") oppure ai numeri complessi o immaginari che, fra l'altro consentirono al matematico polacco Minkowski, maestro di A. Einstein, di aggiustare la teoria della relatività generale, tanto che ho scritto: "I numeri immaginari e/o complessi usati per concepire l''universo di Minkowiski' che trasforma il tempo in spazio, rendendo più chiara ed esplicita l'influenza isomorfica che lo spazio-tempo esercita sulla formula di capitalizzazione e sull'equazione della relatività ristretta, forse possono illuminare di luce nuova la funzione del concetto di co-efficiente di capitalizzazione" (Rizzo F., "Dalla rivoluzione keynesiana alla nuova economia", FrancoAngeli, Milano, 2002, p. 35). Come vedete il mondo sembra grande ma in fondo spetta a Voi e, anche ai poverini come me di renderlo o ridurlo alla dimensione adeguata per comprendere (ed essere compreso) da tutti. Nel ringraziarVi per l'opportunità che mi avete dato, Vi saluto con amicizia intellettuale ed umana. 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 list >> Fis@listas.unizar.es >> http://listas.unizar.es/cgi-bin/mailman/listinfo/fis >> > > > > -- > Professor Terrence W. Deacon > University of California, Berkeley > _______________________________________________ > Fis mailing list > Fis@listas.unizar.es > http://listas.unizar.es/cgi-bin/mailman/listinfo/fis > > > > _______________________________________________ > Fis mailing list > Fis@listas.unizar.es > http://listas.unizar.es/cgi-bin/mailman/listinfo/fis > >

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