[Fis] A new discussion session. Point 6 of LK Note. Goedel without numbers
Francesco Rizzo
13francesco.rizzo at gmail.com
Wed Nov 9 06:36:32 CET 2022
Cari ed egregi colleghi Pedro, Joseph, Sheri e cari tutti,
*credo che siamo in tanti ad aver capito poco o niente*, caro Joseph,
specialmente quando si usano esiziali e speciosi specialismi in un consesso
costituito da cultori di discipline diverse sottese dall’INFORMAZIONE e
capita di apprendere che uno studioso che si dichiara economista dichiara
di non sapere niente di economia quantistica che è proprio quella che
dovrebbe svolgere un compito di mediazione tra le scienze biologiche e
tutte le altre, più o meno esatte. E dire che non avevo nascosto la mia
soddisfazione quando ho appreso che un economista avrebbe introdotto e
coordinato questa nuova sessione Fis.
Comunque mi auguro di (non) avere sbagliato: in proposito ci o mi manca
molto Terry Deacon e tento di penetrare nel silenzio di Pedro, il Maestro.
Ora la smetto e senza voler male a nessuno, penso che sia più opportuno
dedicarmi alla composizione e alla conclusione del mio secondo volume su Lo
sviluppo del mio pensiero economico-valutativo (Aracne, Roma).
Senza offesa per nessuno, un grato e riconoscente saluto a Tutti.
Francesco
Dear and dear colleagues Pedro, Joseph, Sheri and dear all,
Dear and dear colleagues Pedro, Joseph, Sheri and dear all,
I believe that many of us have understood little or nothing, dear Joseph,
especially when deadly and specious specialisms are used in a forum made up
of students of different disciplines underlying INFORMATION and it happens
to learn that a scholar who declares himself an economist says he does not
know anything about quantum economics which is precisely the one that
should play a mediating task between the biological sciences and all the
others, more or less exact. And to say that I did not hide my satisfaction
when I learned that an economist would introduce and coordinate this new
FIS session.
In any case, I hope I have (not) been wrong: in this regard I miss Terry
Deacon very much and I try to penetrate the silence of Pedro, the Master.
Now I stop and without wanting to hurt anyone, I think it is more
appropriate to dedicate myself to the composition and conclusion of my
second volume on “The development of my economic-evaluative thought”
(Aracne, Rome).
No offense to anyone, a grateful and grateful greeting to all.
Francis
Il giorno mar 8 nov 2022 alle ore 18:52 Louis Kauffman <loukau en gmail.com>
ha scritto:
> Please note that the formalities of self reference such as #g —> ~B(#g)
> are done via formal syntax and are seen to embody self reference via the
> interpretation by a human cognition.
> We have not formalized human cognition. It is true that there is an
> additional level of reference in our mind/body relationship with
> formalisms. This additional level cannot be formalized in the way
> we have done so far. This leaves open the further investigation of the
> nature of mind and discrimination.
>
> Please note: I am now closed to communications in this list until next
> week. I will reply to any remarks at that time.
> Best,
> Lou Kauffman
>
> > On Nov 8, 2022, at 9:05 AM, joe.brenner en bluewin.ch wrote:
> >
> > Dear Lou and All,
> > I am not competent to comment on this note in detail, which I
> nevertheless find both coherent and expert.
> >
> > However, in point 6., Lou says that "the Machine is like an organism
> with a limitation. The limitation goes through the semantics of
> self-reference." A human being, then, lacks this limitation. I suggest that
> consciousness adds (at least) one additional level of self-reference by
> recursion which creates the non-computability that makes us human. There is
> Goedelian Principle of incompleteness in reality that operates without
> numbers.
> >
> > Thank you and best wishes,
> > Joseph
> >
> > -----Original Message-----
> > From: Fis <fis-bounces en listas.unizar.es> On Behalf Of Louis Kauffman
> > Sent: Tuesday, November 8, 2022 1:13 AM
> > To: "Pedro C. Marijuán" <pedroc.marijuan en gmail.com>
> > Cc: fis <fis en listas.unizar.es>
> > Subject: Re: [Fis] A new discussion session
> >
> > Dear Pedro,
> > Here are some comments about Goedel numbering and coding.
> >
> > It is interesting to think about Goedel numbering in a biological
> context.
> > Actually we are talking about how a given entity has semantics that can
> vary from context to context.
> > It is not simply a matter of assigning a code number. If g —> F is the
> relation of a Goedel number g to a statement F, then we have two contexts
> for F.
> > 1. F as a well formed formula in a formal system S.
> > 2. g as a number in either a number system for an observer of S or g as
> a number in S, but g, as a representative for F can be regarded in the
> system S with the meanings so assigned.
> > Thus we have produced by the assignment of Goedel numbers a way for a
> statement F to exist in the semantics of more than one context.
> > This is the key to the references and self-references of the Goedelian
> situations.
> >
> > Lets look at this more carefully. Recall that there is a formal system S
> and that to every well formed formula in S, there is a code number g =
> g(S). The code number can be produced in many ways.
> > For example, one can assign different index numbers n(X) to each
> distinct generating symbol in S. Then with an expression F regarded as an
> ordered string of symbols, one can assign to F the product of the prime
> numbers, in their standard order, with exponents the indices of the
> sequence of characters that compose F. For example, g(~ x^2 = 2) = 2^{n(~)}
> 3^{n(x)}5^{n(^)}7^{n(2)}11^{n(=)}13^{n(2)}. From such a code, one can
> retrieve the original formula in a unique way.
> > The system S is a logical system that is assumed to be able to handle
> logic and basic number theory. Thus it is assumed that S can encode the
> function g: WFFS(S) —> N where N denotes the natural numbers.
> > And S can decode a number to find the corresponding expression as well.
> It is assumed that S as a logical system, is consistent.
> >
> > With this backgound, let g —> F denote the condition that g = g(F). Thus
> I write a reference g —> F for a mathematical discussion of S, to indicate
> that g is the Goedel number of F.
> >
> > Now suppose that F(x) is a formula in S with a free variable x. Free
> variables refer to numbers. Thus if I write x^2 = 4 then this statement can
> be specialized to 2^2 = 4 with x =2 and the specialization is true.
> > Or I can write 3^2 = 4 and this is a false statement. Given F(x) and
> some number n, I can make a new sentence F(n).
> > Now suppose that
> > g —> F(x).
> > Then we can form F(g) and this new statement has a Goedel number. Let #g
> denote the Goedel number of F(g).
> > #g —> F(g).
> > This # is a new function on Goedel numbers and also can be encoded in
> the system S. I will abbreviate the encoding into S by writing #n for
> appropriate numbers n handled by S.
> > Then we can consider
> > F(#x) and it has a Goedel number
> > h —> F(#x)
> > And we can shift that to
> > #h —> F(#h).
> > This is the key point.
> > Now we have constructed a number #h so that F(#h) discusses its own
> Goedel number.
> >
> > This construction allows the proof of the Goedel Incompleteness Theorem
> via the sentence B(x) that states
> > B(x) = “The statement with Goedel number x is provable in S.” (This can
> also be encoded in S.)
> >
> > We then construct
> > h—> ~B(#x)
> > and
> > #h —> ~B(#h)
> > and obtain the statement
> > G= ~B(#h).
> > G states the unprovability of the Goedel decoding of #h.
> > But the Goedel decoding of #h is the statement G itself.
> > Thus G states its own unprovability.
> > Therefore, S being consistent, cannot prove G.
> >
> > By making these arguments we have have proved that G cannot be proved by
> S.
> > Thus we have shown that G is in fact true.
> > We have shown that there are true statements in number theory unprovable
> by system S..
> > ##########################
> >
> > The above is a very concise summary of the proof of Goedel’s
> Incompleteness Theorem, using Goedel number encoding.
> >
> > It is a very interesting question whether such encoding or such multiple
> relationships to context occur in biology. Here are some remarks.
> >
> > 1. In biology is is NORMALLY the case that certain key structures have
> multiple interpretations and uses in various contexts.
> > The understanding of such multiple uses and the naming of them requires
> an observer of the biology. Thus we see the action of a cell membrane and
> we see the action of mitosis, and so on.
> >
> > 2. There are implicit encodings in biology such as the sequence codes in
> DNA and RNA and their unfoldment. To what extent do they partake of the
> properties of Goedel coding?
> >
> > 3. The use of the Goedel coding in the Incompleteness theorem depends
> crucially on the relationship of syntax and semantic in the formal system
> and in the mathematician’s interpretation of the workings of that system.
> The Goedel argument depends upon the formal system S being seen as a
> mathematical object that itself can be studied for its properties and
> behavior.
> > When we speak of the truth of G, we are speaking of our assessment of
> the possible behaviour of S, given its consistency. We are reasoning about
> S just as Euclid reasons about the structure of right triangle.
> >
> > 4. In examining biological structures we take a similar position and
> reason about what we know about them. Sufficiently complex biological
> structures can be seen as modeled by certain logical formal systems.
> > And then Goedelian reasoning can be applied to them. This can even be
> extended to ourselves. Suppose that I am modeled correctly in my
> mathematical reasoning by a SINGLE CONSISTENT FORMAL SYSTEM S.
> > Then “I” can apply the above proof of Goedel’s Therem to S and deduce
> that G cannot be proven by S. Thus “I” have exceeded the capabilities of S.
> Therefore it is erroneous to assume that my mathematical reasoning is
> encapsulated by a single formal system S. If I am a formal system, that
> system must be allowed to grow in time. Such reasoning as this is subtle,
> but the semantics of the relationship of mathematicians and the formal
> systems that they study is subtle and when biology is brought in the whole
> matter becomes exceedingly interesting.
> >
> > 5. We man not need numbers to have these kinds of relationships. And
> example is the Smullyan Machine that prints sequences of symbols from the
> alphabet {~,P,R} on a tape. Sequences that begin with P,~P,PR and ~PR are
> regarded as meaningful, with the meanings:
> > PX: X can be printed.
> > ~PX: X cannot be printed.
> > PRX: XX can be printed.
> > ~PRX: XX cannot be printed.
> > Here X is any string of the symbols {~,P,R}.
> > Thus PR~~P means that XX can be printed where X = ~~P. Thus PR~~P means
> that ~~P~~P can be printed.
> > By printed we mean on one press of the button on the Machine, a string
> of characters is printed.
> > IT IS ASSUMED THAT THE SMULLYAN MACHINE ALWAYS TELLS THE TRUTH WHEN IT
> PRINTS A MEANINGFUL STATEMENT.
> > Then we have the
> >
> > Theorem. There are meaningful true strings that the Smullyan Machine
> cannot print.
> >
> > This is a non-numerical analog of the Goedel Theorem. And the string
> that cannot be printed is G = ~PR~PR.
> > For you see that G is meaningful and since G = ~PRX, G says that XX
> cannot be printed. But X = ~PR and XX = ~PR~PR = G. So G says that G cannot
> be printed.
> > If the machine were to print G, it would lie. And the machine does not
> lie.
> > Therefore G is unprintable.
> > But this is what G says.
> > So we have established the truth of G and proved the Theorem.
> >
> > 6. Examine this last paragraph 5. The Machine is like an organism with a
> limitation. This limitation goes through the semantics of reference. ~PRX
> refers to XX and so can refer to itself if we take X = ~PR. ~PX refers to X
> and cannot refer to itself since it is longer than X. In biological coding
> the DNA code is fundamentally smaller or equal to the structure to which it
> refers.
> > Thus the self-reproduction of the DNA is possible since DNA = W+C the
> convention of the Watson and Crick strand and each of W and C can by
> themselves engage in an action to encode, refer to, the other strand. W can
> produce a copy of C in the form W+C and C can produce a copy of W in the
> form W+C each by using the larger environment. Thus W+C refers to itself,
> reproduces itself by a method of encoding quite similar to the self
> reference of the Smullyan Machine.
> >
> > 7. Von Neuman devised a machine that can build itself. B is the von
> Neuman machine and B.x —> X,x where x is the plan or blueprint or code for
> and entity X. B builds X with given the blueprint x.
> > Then we have B,b —> B,b where b is the blueprint for B. B builds itself
> from its own blueprint. I hope you see the analogy with the Goedel code.
> >
> > 8. I will stop here. The relationships with biology are very worth
> discussing.
> > Before stopping it is worth remarking that the Maturana Uribe Varela
> autopoeisis is an example of a system arising into a form of self-reference
> that has a lifetime due to the probabilisitic dynamics of its process.
> >
> > Very best,
> > Lou Kauffman
> >
> >
> >
> >
> >
> > _______________________________________________
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