[Fis] _ Re: _ Gödel discussion
Bruno Marchal
marchal at ulb.ac.be
Thu May 12 16:24:22 CEST 2016
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 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 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 it
> seemed to confirm aspects of my offering to you all.
>
> On 3 May 2016 at 07:49, Francesco Rizzo
> <13francesco.rizzo at 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 <loukau at 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
> 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
>> <deacon at 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 = 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 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 <msj at 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 at listas.unizar.eshttp
>> ://listas.unizar.es/cgi-bin/mailman/listinfo/fis
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>>
>> --
>> Professor Terrence W. Deacon
>> University of California, Berkeley
>> _______________________________________________
>> Fis mailing listFis at listas.unizar.eshttp://listas.unizar.es/cgi-bin/mailman/listinfo/fis
>>
>>
>
>
> _______________________________________________
> Fis mailing listFis at listas.unizar.eshttp://listas.unizar.es/cgi-bin/mailman/listinfo/fis
>
>
>
>
>
> _______________________________________________
> Fis mailing listFis at 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 at listas.unizar.es
> http://listas.unizar.es/cgi-bin/mailman/listinfo/fis
http://iridia.ulb.ac.be/~marchal/
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://listas.unizar.es/pipermail/fis/attachments/20160512/7edf7f52/attachment.html>
More information about the Fis
mailing list