[Fis] Fwd: A new discussion session; Self-Other in Biology
Loet Leydesdorff
loet at leydesdorff.net
Wed Dec 28 07:56:59 CET 2022
Dear Joe:
You can phone on +316 27378730 or "Leydesdorff" at skype.
Best. Loet
PS. Given a health condition, I am no longer able to read and write
emails without mistakes. Please, consider using the telephone or skype.
Best, Loet
---------------------------------------------------------------------------------------------
"The Evolutionary Dynamics of Discusive Knowledge"
<https://urldefense.com/v3/__https://link.springer.com/book/10.1007/978-3-030-59951-5__;!!D9dNQwwGXtA!Wbhj4g67Lun_cr1u1y1fj6hAiGuDB-1VOSP6GzcuBX6GVTljSubyIy-2wIpQSdXz8YBdkIuVLZ2DeRAE4cw$ >(Open Access)
Professor emeritus, University of Amsterdam
------ Original Message ------
>From "Loet Leydesdorff" <loet en leydesdorff.net>
To joe.brenner en bluewin.ch; "fis" <fis en listas.unizar.es>
Date 12/26/2022 5:53:15 AM
Subject Re[2]: [Fis] Fwd: A new discussion session; Self-Other in
Biology
>https://urldefense.com/v3/__https://leydesdorff.net/hertzberger/hetty/25dec22.pdf__;!!D9dNQwwGXtA!Wbhj4g67Lun_cr1u1y1fj6hAiGuDB-1VOSP6GzcuBX6GVTljSubyIy-2wIpQSdXz8YBdkIuVLZ2DIGPAFsM$
>
>Dear colleagues, family, and frineds
>
>Pleiase, find a pdf ile at the webpage above.
>This file is in DUtch.
>
>The reference is::
>Leydesorff, L. (2022). „Herr Obersturmbahnführer, gestatten Sie daß Ich
>rede?. (https://urldefense.com/v3/__https://leydesdorff.net/hertzberger/hetty/25dec22.pdf__;!!D9dNQwwGXtA!Wbhj4g67Lun_cr1u1y1fj6hAiGuDB-1VOSP6GzcuBX6GVTljSubyIy-2wIpQSdXz8YBdkIuVLZ2DIGPAFsM$ ), 39 pp.
>Comments are most welcome in this stage:
>
>Best, Loet
>
>PS. Given a health condition, I am no longer able to read and write
>emails without mistakes. Please, consider using the telephone or skype.
>Best, Loet
>
>
>
>---------------------------------------------------------------------------------------------
>
>"The Evolutionary Dynamics of Discusive Knowledge"
><https://urldefense.com/v3/__https://link.springer.com/book/10.1007/978-3-030-59951-5__;!!D9dNQwwGXtA!Wbhj4g67Lun_cr1u1y1fj6hAiGuDB-1VOSP6GzcuBX6GVTljSubyIy-2wIpQSdXz8YBdkIuVLZ2DeRAE4cw$ >(Open Access)
>
>Professor emeritus, University of Amsterdam
>
>
>
>
>
>
>------ Original Message ------
>From "joe.brenner en bluewin.ch" <joe.brenner en bluewin.ch>
>To pedroc.marijuan en gmail.com
>Cc fis en listas.unizar.es
>Date 12/25/2022 10:05:10 PM
>Subject Re: [Fis] Fwd: A new discussion session; Self-Other in Biology
>
>>Dear Pedro, Dear All,
>>
>>Christmas is not even over and already a fascinating disagreement! In
>>my system, Gödel is closer to reality because incompleteness is a
>>feature of reality. Fixed points look pretty inert to me, and in
>>biology??
>>
>>I believe since I don't have all the skills needed to be sure that
>>fixed points eliminate the need for recursive functions but you have
>>just gotten rid of another, probably real mental process. Recursion
>>and anticipation, using dynamic forms of information, are what mind is
>>about.
>>
>>I don't claim that mine is the "right" handle. I just plead for it to
>>humbly accompany the story from computer science.
>>
>>Thank you and Happy New Year!
>>>----Original Message----
>>>From : pedroc.marijuan en gmail.com
>>>Date : 25/12/2022 - 21:42 (E)
>>>To : fis en listas.unizar.es
>>>Subject : [Fis] Fwd: A new discussion session; Self-Other in Biology
>>>
>>>-------- Mensaje reenviado --------
>>>Asunto: Re: [Fis] A new discussion session; Self-Other in Biology
>>>Fecha: Fri, 23 Dec 2022 09:31:53 +0100
>>>De: guillaume.bonfante <guillaume.bonfante en loria.fr>
>>>Para: Louis Kauffman <loukau en gmail.com>, Markose, Sheri
>>><scher en essex.ac.uk>
>>>CC: Pedro C. Marijuán <pedroc.marijuan en gmail.com>, fis
>>><fis en listas.unizar.es>,
>>>guillaume.bonfante en mines-nancy.univ-lorraine.fr<guillaume.bonfante en mines-nancy.univ-lorraine.fr>,
>>>mikhail.prokopenko en sydney.edu.au<mikhail.prokopenko en sydney.edu.au>,
>>>Neil Gershenfeld <neil.gershenfeld en cba.mit.edu>, Koonin, Eugene
>>>(NIH/NLM/NCBI) [E] <koonin en ncbi.nlm.nih.gov>, Oron Shagrir
>>><oron.shagrir en gmail.com>,
>>>Noson en sci.brooklyn.cuny.edu<Noson en sci.brooklyn.cuny.edu>, Friston,
>>>Karl <k.friston en ucl.ac.uk>, John Mattick <j.mattick en unsw.edu.au>
>>>
>>>
>>>Dear all,
>>>
>>>End of the year, and now some spare time. As a computer scientist,
>>>in my opinion, Goedel's Incompleteness Theorem is touchy but not the
>>>right handle. As some of you mentioned below, I think that fix-points
>>>are much more promising.
>>>
>>>They have various forms. They exist at every layer of the
>>>arithmetical hierarchy (you don't need to implement recursive
>>>functions). So, a very flexible tool.
>>>
>>>All the best,
>>>
>>>Guillaume
>>>
>>>
>>>Le 05/12/2022 à 00:25, Louis Kauffman a écrit :
>>>>Dear Sheri,
>>>>It would indeed be very helpful to have a zoom conversation about
>>>>these themes.
>>>>Please let me know when you would be available to have it.
>>>>We could start with an hour meeting and discussion and then perhaps
>>>>extend to a second meeting with some presentations.
>>>>Included below is a slide show of mine that is a bit cryptic but
>>>>does summarize some points of view.
>>>>I have downloaded the papers you indicated in your email and will
>>>>read them now.
>>>>Very best,
>>>>Lou
>>>>
>>>>
>>>>
>>>>>On Dec 5, 2022, at 2:53 PM, Markose, Sheri < scher en essex.ac.uk>
>>>>>wrote:
>>>>>
>>>>>Dear Louis, Pedro and All –
>>>>>
>>>>>I apologize again for not attending to the comments as soon as they
>>>>>appear. Autumn term is my very busy teaching term. Also, one of
>>>>>the reasons why I got waylaid is that I had to urgently send in my
>>>>>external examiner review of a Thesis of Adam Svahn (University of
>>>>>Sydney) on 25 Nov. I was free to invite folk who I have had some
>>>>>convos with on this topic and may be interested in the Foundations
>>>>>of Information Systems online forum as participants and potential
>>>>>leads on topics… I do this somewhat belatedly, my apologies again.
>>>>>
>>>>>(i) The bulk of Adam Svahn’s work co-authored with
>>>>>Mikhail Prokopenko (S & P) is already published and can be found
>>>>>here
>>>>>https://urldefense.com/v3/__https://doi.org/10.1162/artl_a_00370__;!!D9dNQwwGXtA!Wbhj4g67Lun_cr1u1y1fj6hAiGuDB-1VOSP6GzcuBX6GVTljSubyIy-2wIpQSdXz8YBdkIuVLZ2DafDqFmY$
>>>>><https://urldefense.com/v3/__https://doi.org/10.1162/artl_a_00370__;!!D9dNQwwGXtA!R8lchYgfzUtRK_UvjczZjG4odrY4nCT90ap2vplSNdF7nmHYAjY30NMIkzAIZPRsIn5AxYjgQKn_X6p4BTG0vMTREdnb$>
>>>>> and is relevant to our discussion. While we are still in the dark
>>>>>about how the near universal genomic (ACGT/U) alphabets emerged,
>>>>>the genome clearly manifests an unbroken chain of life with
>>>>>programs encoded in these alphabets and their execution via gene
>>>>>expression produce the somatic and phenotype identity of organisms.
>>>>>S & P share objectives as those that I have given specifically re
>>>>>how self-reference and negator operations known from Gödel (1931)
>>>>>incompleteness theorems and undecidability thereof arise so the
>>>>>software based genomic system is capable of endogenous novelty
>>>>>production and evolvability.
>>>>>
>>>>>S& R give an interesting and plausible account of how RNA-push down
>>>>>automata with their push and pop rules can produce limit cycle
>>>>>dynamics or the extensively found repetitive motifs sometimes
>>>>>called biological palindromes in the genome. They argue that a
>>>>>2-stack RNA-push down automata is necessary to produce reflexive
>>>>>structures where the automata in addition to simply executing a
>>>>>program can use the 2ndstack to reflect on codes and make changes
>>>>>to them.
>>>>>
>>>>>
>>>>>(ii) Section 5 of S&P relates to undecidability as fixed
>>>>>points of negation functions and has much in common Louis’s 15 Nov
>>>>>email point 5 below ( I have scissored and pasted this in the email
>>>>>trail below) on the ease with which self-negating Gödel sentences
>>>>>can be created by logicians. However, biology unlike Gödel (and
>>>>>other logicians) is not directly concerned about undecidability,
>>>>>incompleteness, or whether a program halts. I have stuck my neck
>>>>>out and said that Gödel machinery used by biology and the
>>>>>formidable genomic self-referential general intelligence is to
>>>>>establish a hack free agenda for the genome geared toward
>>>>>autonomous life.
>>>>>
>>>>>Thus, some further thought needs to be expended as to how the
>>>>>negator operation naturally occurs in biology. I have stated it is
>>>>>the bio malware or the viral software that Eugene Koonin et al have
>>>>>said has been coextensive with life having provided the
>>>>>copy/replicate program possibly in what the computer literature
>>>>>calls Quines. The latter are distinct from online self-assembly of
>>>>>somatic self which requires gene expression or machine execution as
>>>>>in the ribosomal machines. To make out gene-codes that
>>>>>self-assemble the organism have been changed/tampered with by
>>>>>software of non-self other appears to be a pressing matter for
>>>>>homeostasis which is clearly a bio-cybersecurity problem.
>>>>>
>>>>>(iii) The over 85% offline recording in the Thymic MHC
>>>>>receptors of expressed genes in humans for example brings us to the
>>>>>points made by Pedro in his 1 December email on Self and non-self
>>>>>antigen recognition in Adaptive Immune System. Thank you Pedro for
>>>>>the nugget of information of how the MHC receptors has two strips
>>>>>of 8-10 amino acids residues for Class 1, mostly "self", and
>>>>>13-18 amino acids residues for Class 2 for non-self. I did not
>>>>>know this. I will read "Sensing the world and its dangers: An
>>>>>evolutionary perspective in
>>>>>neuroimmunology." By Aurora Krauset al. In, eLife 2021;10:e66706.
>>>>>DOI: https://urldefense.com/v3/__https://doi.org/10.7554/eLife.66706__;!!D9dNQwwGXtA!Wbhj4g67Lun_cr1u1y1fj6hAiGuDB-1VOSP6GzcuBX6GVTljSubyIy-2wIpQSdXz8YBdkIuVLZ2DfVVOpfc$
>>>>><https://urldefense.com/v3/__https://linkprotect.cudasvc.com/url?a=https*3a*2f*2fdoi.org*2f10.7554*2feLife.66706&c=E,1,mnuJARbiz5DP5j0H1X0ciBwcFLUNxlmdaZCNXX6tuWJ7oLj-36Vg9-Wauvxar1tDnTFYRaRF0eqlIxd2zzkL3LoskpUW1kBa2CHZaMqYUapU2iEi&typo=1__;JSUlJSU!!D9dNQwwGXtA!R8lchYgfzUtRK_UvjczZjG4odrY4nCT90ap2vplSNdF7nmHYAjY30NMIkzAIZPRsIn5AxYjgQKn_X6p4BTG0vN18TkB3$>.
>>>>>
>>>>> If you recall how the Recursive Fixed Point
>>>>>Theorem which starts with a mirror mapping between online
>>>>>self-assembly program execution, fg(g)) that have halted and
>>>>>create somatic identity and the offline record of the
>>>>> same in a 2- place function s (g, g) in the
>>>>>MHC receptors creating the Thymic self. I say the first g from the
>>>>>left in s (g, g) and changes thereof relate to what happens to
>>>>>self and the second is self’s record of what
>>>>> the other has done to self. So if the 2nd entry is
>>>>>different from the first entry in s (g, g) it is off diagonal etc.
>>>>>The non-self hostile other is a projection of self g ‘gene codes’
>>>>>and those f ¬ ! which are reactive to the self g-
>>>>> codes denoted as g¬. As we know an astronomic
>>>>>number of potential indexes g¬ are generated by the RAG genes.
>>>>>
>>>>> How the immune system identifies a yet to happen
>>>>>attack by a novel non-self antigen requires the first part of the
>>>>>fix point of the latter generated in the Thymic T-cell receptors to
>>>>>sync with those generated in the
>>>>> the peripheral MHC receptors when the said
>>>>>expressed genes are attacked (like the lung tissue etc) in real
>>>>>time. The latter is experientially generated and while the former
>>>>>is a spectacular case of predictive coding. So
>>>>> unless the T-cell receptor has cloned the index of
>>>>>the novel bio-malware in advance via V(D) J, the AIS will not be
>>>>>recognize the biomalware should it attack. I have said fixed
>>>>>point for the software/algorithm f ¬ ! requires
>>>>> the full use of say Rogers Second Recursion
>>>>>Theorem and the Gödel Sentence thereof, viz. far more machinery
>>>>>self-referential structures than in the original Gödel (1931)
>>>>>formats.
>>>>>
>>>>>(iv) Finally, there is the problem that the information
>>>>>processing for advanced code based systems is one akin to Formal
>>>>>Systems of Theorems and non-Theorems. For this Raymond Smullyan’s
>>>>>book of the same name is what gave me the idea that a tight grip
>>>>>will be exerted with all inference based recursive reductions and
>>>>>Gödel Sentences when some potential negations to theorems viz. the
>>>>>halting self-assembly gene codes that generate the organism, are in
>>>>>the offing. This self-referential genomic blockchain distributed
>>>>>ledger of the unbroken chain of life has similarities with manmade
>>>>>BCDL, but latter are not self-referential with individual nodes
>>>>>being able to self-report attacks.
>>>>>
>>>>>Louis, I would love to have a zoom chat with you as it will be
>>>>>great to sound you out more. You are right about the mindboggling
>>>>>variants of self-reference ….
>>>>>Ditto for many others who I hope to be in touch with soon.
>>>>>
>>>>>Many thanks again for the great comments.
>>>>>
>>>>>All best
>>>>>
>>>>>Sheri
>>>>>
>>>>>
>>>>>
>>>>>
>>>>>
>>>>>
>>>>>
>>>>>
>>>>>
>>>>>From: Pedro C. Marijuán <pedroc.marijuan en gmail.com>
>>>>>Sent: 01 December 2022 13:03
>>>>>To: Markose, Sheri <scher en essex.ac.uk>; Louis Kauffman
>>>>><loukau en gmail.com>
>>>>>Cc: fis <fis en listas.unizar.es>;
>>>>>guillaume.bonfante en mines-nancy.univ-lorraine.fr
>>>>>Subject: Re: [Fis] A new discussion session
>>>>>
>>>>>Dear Sheri, Lou, and all discussants,
>>>>>
>>>>>It is a pity that this excellent discussion has taken place in
>>>>>complicated academic weeks, as it has been caught in a sort of
>>>>>"punctuated equilibrium" of longer stasis than activities in our
>>>>>evolutionary list. Well, I have a couple of very brief comments:
>>>>>
>>>>>First, emphasizing that one of the references in Youri's last
>>>>>messages should be obligated reading for biologically interested
>>>>>parties: "Sensing the world and its dangers: An evolutionary
>>>>>perspective in
>>>>>neuroimmunology." By Aurora Krauset al. In, eLife 2021;10:e66706.
>>>>>DOI: https://urldefense.com/v3/__https://doi.org/10.7554/eLife.66706__;!!D9dNQwwGXtA!Wbhj4g67Lun_cr1u1y1fj6hAiGuDB-1VOSP6GzcuBX6GVTljSubyIy-2wIpQSdXz8YBdkIuVLZ2DfVVOpfc$
>>>>><https://urldefense.com/v3/__https://linkprotect.cudasvc.com/url?a=https*3a*2f*2fdoi.org*2f10.7554*2feLife.66706&c=E,1,mnuJARbiz5DP5j0H1X0ciBwcFLUNxlmdaZCNXX6tuWJ7oLj-36Vg9-Wauvxar1tDnTFYRaRF0eqlIxd2zzkL3LoskpUW1kBa2CHZaMqYUapU2iEi&typo=1__;JSUlJSU!!D9dNQwwGXtA!R8lchYgfzUtRK_UvjczZjG4odrY4nCT90ap2vplSNdF7nmHYAjY30NMIkzAIZPRsIn5AxYjgQKn_X6p4BTG0vN18TkB3$>.
>>>>>In this vein, I will follow with the argument that the
>>>>>multicellular self is a composite, an association with a microbial
>>>>>consortium that probably was the big evolutionary cause to create a
>>>>>defense system of such a great complexity. The innate immune
>>>>>system would represent the evolutionary learning about those
>>>>>dangers, with scores of different components and pattern
>>>>>recognition strategies...
>>>>>
>>>>>And second, about the adaptive immune system, it is where the
>>>>>ongoing mostly formal discussion would apply (can we agree with
>>>>>that?). Then, it seems that the core of this adaptive immune branch
>>>>>is the Major Histocompatibility Complex molecule (MHC). This MHC
>>>>>molecules of two major classes are highly complex (polygenic and
>>>>>polymorphic) and they are in charge of presenting to lymphocyte T
>>>>>cells the protein fragments churned out from the proteosomes inside
>>>>>cells (fragments of variable lenght: 8-10 amino acids residues for
>>>>>Class 1, mostly "self", and 13-18 amino acids residues for Class
>>>>>2, mostly "non self"). Then, the thymus is in charge of
>>>>>deactivating the T cells loaded with self stuff. My point is that
>>>>>the defense in front of the non-self is based on indirect products
>>>>>of protein translation. This causes me some uneasiness, as protein
>>>>>translation (see Youri's presentation months ago) introduces a
>>>>>layer of extra complexity, not to speak the processing via
>>>>>proteosomes. Further, with just 10 or 12 amino acids can we
>>>>>faithfully ascertain algorithmic non-self provenance??
>>>>>
>>>>>Well, Sheri is far more acknowledged with all this stuff. And
>>>>>perhaps Lou can say something about the formal distinguishability
>>>>>of 10-12 aa.
>>>>>Best--Pedro
>>>>>
>>>>>
>>>>>
>>>>>
>>>>>From: Louis Kauffman loukau en gmail.com
>>>>>Sent: 15 November 2022 23:02
>>>>>To: Markose, Sheri scher en essex.ac.uk
>>>>>Cc: "Pedro C. Marijuán" pedroc.marijuan en gmail.com; fis
>>>>>fis en listas.unizar.es;
>>>>>guillaume.bonfante en mines-nancy.univ-lorraine.fr
>>>>>Subject: Re: [Fis] A new discussion session
>>>>>
>>>>>Sheri,
>>>>>I will try to respond to your letter about the post Goedel
>>>>>structures by first quoting the last part of my previous letter
>>>>>that discusses Goedelian ideas from the point of view of fixed
>>>>>points.
>>>>>My letter was quite long, and it is possible to not get to the
>>>>>second half.
>>>>>Note also that the first half is based on a referential situation g
>>>>>—> F where #g ——> Fg is what I call the Indicative Shift of g —> F.
>>>>>This is formal and does not assume anythng other than arrow
>>>>>structure.
>>>>>With g —> F# we have #g —> F#g making F#g refer to its own name.
>>>>>There is more to say herd and references that I cannot send to the
>>>>>list, so I will get a dropbox for it and further discussion later
>>>>>today.
>>>>>Best,
>>>>>Lou K.
>>>>>
>>>>>##########
>>>>>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.
>>>>> ###############
>>>>>Best,
>>>>>Lou Kauffman
>>>>>
>>>>>
>>>>>
>>>>>El 15/11/2022 a las 21:19, Markose, Sheri escribió:
>>>>>>Dear Louis, dear Colleagues -
>>>>>>
>>>>>>Louis has given an excellent exposition of Gödel Numbering (g.n)
>>>>>>(your point number 2 on coding and semantics is giving me food for
>>>>>>thought) , giving example of prime factorization and also of Gödel
>>>>>>Sentence as one that states its own unprovability. Unlike
>>>>>>statements like "this is false", GS is not paradoxical and in a
>>>>>>consistent system it is a theorem with a constructive g.n. The
>>>>>>latter in terms of the prime factorization format, it is indeed a
>>>>>>Hilbert 10 Diophantine equation with no integer solutions. A
>>>>>>remarkable achievement in maths, considering Gödel was only 23
>>>>>>years of age .... But what has this got to do with Biology and
>>>>>>novelty production, the objectives of the my FIS discussion ?
>>>>>>
>>>>>>In view of brevity and also urged by Pedro, I dropped a couple of
>>>>>>paragraphs in my FIS kick off submission as to why we need to
>>>>>>exceed Gödel (1931) and couch the Gödel Incompleteness Results and
>>>>>>the Gödel Sentence with a fuller understanding of algorithms as
>>>>>>encoded instructions and as machine executable codes, of the
>>>>>>notion of recursive enumeration (re) and re sets that was
>>>>>>developed in the Emil Post (1944). I hope Louis Kauffman can
>>>>>>comment on the the application of the fuller Gödel-Turing
>>>>>>-Post-Rogers framework mentioned in my FIS note and in my papers
>>>>>>cited there.
>>>>>>
>>>>>>1. I have found the following statement by Joel Hamkins ( :
>>>>>>https://urldefense.com/v3/__http://jdh.hamkins.org/wp-content/uploads/A-review-of-several-fixed-point-theorems-1.pdf__;!!D9dNQwwGXtA!Wbhj4g67Lun_cr1u1y1fj6hAiGuDB-1VOSP6GzcuBX6GVTljSubyIy-2wIpQSdXz8YBdkIuVLZ2DWDbPCgc$
>>>>>><https://urldefense.com/v3/__https://linkprotect.cudasvc.com/url?a=http*3a*2f*2fjdh.hamkins.org*2fwp-content*2fuploads*2fA-review-of-several-fixed-point-theorems-1.pdf&c=E,1,bKIlk9p4sIB5v1zLhbA_VCdX_aoMSPljj6KZdLjCesxOjPwYqUF5PkC4wqvoWq0qqGndGHjZ6ELzpZ8IhqbUDEGNINdm7Da4GNcSgCn3k0us&typo=1__;JSUlJSUl!!D9dNQwwGXtA!R8lchYgfzUtRK_UvjczZjG4odrY4nCT90ap2vplSNdF7nmHYAjY30NMIkzAIZPRsIn5AxYjgQKn_X6p4BTG0vHnETe5i$>
>>>>>>) useful as it makes an important observation that the original
>>>>>>Gödel (1931) framework permits an encodable proposition to make
>>>>>>statements about itself while Second Recursion Theorems (SRT) also
>>>>>>called Fixed Point Theorems are needed “to construct
>>>>>>programs/algorithms that refer to themselves”. The terms programs
>>>>>>and algorithms will be used interchangeably.
>>>>>>
>>>>>>I choose Rogers Fixed Point Theorem of (total) computable
>>>>>>functions starting with the staple I have already indicated Diag
>>>>>>(g) (RHS of (8) below) is what Neil Gerschenfeld calls ribosomal
>>>>>>self-assembly machines in gene expression where the program g
>>>>>>builds the machine that runs g.
>>>>>>
>>>>>>
>>>>>>II. The first requirement of a system to identify Fixed Points
>>>>>>viz. self-referential constructions of algorithms/programs is (8)
>>>>>>viz to identify what function/algorithm has altered the Diag (g).
>>>>>>
>>>>>>
>>>>>><image001.png>
>>>>>>
>>>>>>When online gene expression takes place on RHS of (8), viz. these
>>>>>>programs have halt commands and builds the somatic and phenotype
>>>>>>identity of vertebrates online, the offline record of this is made
>>>>>>in the Thymus that can not only represent the Thymic/immune self
>>>>>>but also concatenate changes thereof.
>>>>>>
>>>>>>I have suggested that the Adaptive Immune System and the Mirror
>>>>>>Neuron System have these structures in (8). And the domain of
>>>>>>self-halting machines as in (8) are the Theorems of the system and
>>>>>>a subset of Post (1944) Creative Set. The non-Theorems have
>>>>>>codes say g¬ which cannot halt in a formal system that is
>>>>>>consistent. To my mind, the embodiment via the physical self
>>>>>>being self-assembled and an offline record of this on LHS of (8)
>>>>>>is what fuses syntax and semantics.
>>>>>>
>>>>>>II. Once, (8) is in place, the Adaptive immune system has to
>>>>>>identify novel negation software function f¬! of non-self
>>>>>>antigens which is an uncountable infinite possibilities. Hence the
>>>>>>close to astronomic search with V(D) J of 10 20 – 10 30 ) of
>>>>>>non-self antigens that can hijack the self- assembly machines as
>>>>>>recorded on RHS of (8). Only from knowledge of self can the
>>>>>>hostile other, in the case of the AIS, be identified.
>>>>>>
>>>>>>
>>>>>>III. Roger Fixed Point assures us that the indexes of the fixed
>>>>>>point for f¬! be generated. I have cced Guillame Bonfante who I
>>>>>>think was among the first (with coauthors, 2006) to suggest how
>>>>>>SRT can be used to identify computer viruses. But they do not use
>>>>>>the full force of Self-Ref and Self -Rep and only implicitly use
>>>>>>Post Creative and Productive Sets. The index of the Godel Sentence
>>>>>>for the fixed point will endogenously lie outside of Post listable
>>>>>>or recusively enumerable set for Theorems and known non-Theorems.
>>>>>>
>>>>>>IV. From these Gödel Sentences produced in the immune-cognitive
>>>>>>systems, the explicit use of Post (1944) Theorems indicates how
>>>>>>novel antibodies cannot be produced in the absence of the Gödel
>>>>>>Sentence which allows a biotic element to self-report it is under
>>>>>>attack.
>>>>>>
>>>>>>V. In conclusion, while it has become fashionable for some like
>>>>>>Jurgen Schmidhuber to claim that there can be endogenous self
>>>>>>improving recursive novelty (he calls them Gödel machines) , the
>>>>>>Gödel Logic says that the original theorems and self-codes are
>>>>>>kept unchanged/hack free and novelty is produced only in response
>>>>>>to adversarial attacks of self codes. So the AIS story is somatic
>>>>>> hypermutation so that nothing in the genome changes. As to how
>>>>>>the germline itself changes, needs more investigation, in
>>>>>>Biosystems paper, I suggest something very briefly.
>>>>>>
>>>>>>So thankyou all again for your in depth comments and interest.
>>>>>>
>>>>>>Best Regards
>>>>>>
>>>>>>Sheri
>>>>>>
>>>>>>-----Original Message-----
>>>>>>From: Fis < fis-bounces en listas.unizar.es> On Behalf Of Louis
>>>>>>Kauffman
>>>>>>Sent: 08 November 2022 00:13
>>>>>>To: "Pedro C. Marijuán" < pedroc.marijuan en gmail.com>
>>>>>>Cc: fis < fis en listas.unizar.es>
>>>>>>Subject: Re: [Fis] A new discussion session
>>>>>>
>>>>>>CAUTION: This email was sent from outside the University of Essex.
>>>>>>Please do not click any links or open any attachments unless you
>>>>>>recognise and trust the sender. If you are unsure whether the
>>>>>>content of the email is safe or have any other queries, please
>>>>>>contact the IT Helpdesk.
>>>>>>
>>>>>>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
>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>>
>>>>>>_______________________________________________
>>>>>>Fis mailing list
>>>>>>Fis en listas.unizar.es
>>>>>>https://urldefense.com/v3/__https://linkprotect.cudasvc.com/url?a=http*3a*2f*2flistas.unizar.es*2fcgi-bin*2fmailman*2flistinfo*2ffis&c=E,1,1A1lgz03IPLQ2hs74kfiKoMaeMUB45427CkA4or9aZPmd25ZxHPE88KK0k9wHh7-Un8A9g25n5WMXHIu8yhAyMhiouezvcso3GGs3inouA,,&typo=1__;JSUlJSUlJQ!!D9dNQwwGXtA!Wbhj4g67Lun_cr1u1y1fj6hAiGuDB-1VOSP6GzcuBX6GVTljSubyIy-2wIpQSdXz8YBdkIuVLZ2D6cM25Lw$
>>>>>><https://urldefense.com/v3/__https://linkprotect.cudasvc.com/url?a=http*3a*2f*2flistas.unizar.es*2fcgi-bin*2fmailman*2flistinfo*2ffis&c=E,1,1A1lgz03IPLQ2hs74kfiKoMaeMUB45427CkA4or9aZPmd25ZxHPE88KK0k9wHh7-Un8A9g25n5WMXHIu8yhAyMhiouezvcso3GGs3inouA,,&typo=1__;JSUlJSUlJQ!!D9dNQwwGXtA!R8lchYgfzUtRK_UvjczZjG4odrY4nCT90ap2vplSNdF7nmHYAjY30NMIkzAIZPRsIn5AxYjgQKn_X6p4BTG0vHlkTv5Q$>
>>>>>>----------
>>>>>>INFORMACIN SOBRE PROTECCIN DE DATOS DE CARCTER PERSONAL
>>>>>>
>>>>>>Ud. recibe este correo por pertenecer a una lista de correo
>>>>>>gestionada por la Universidad de Zaragoza.
>>>>>>Puede encontrar toda la informacin sobre como tratamos sus datos
>>>>>>en el siguiente enlace:
>>>>>>https://urldefense.com/v3/__https://linkprotect.cudasvc.com/url?a=https*3a*2f*2fsicuz.unizar.es*2finformacion-sobre-proteccion-de-datos-de-caracter-personal-en-listas&c=E,1,fozeJ_L1c5tT22-_XAnl69C5WGhrrENGO-y2mO0uH3X4Bbm3EnwS5CaEussDHCR05GDKiVPAM9G4jQaY0kVhqsc4vdv55TdLJ2956rnsNTuETjVx&typo=1__;JSUlJQ!!D9dNQwwGXtA!Wbhj4g67Lun_cr1u1y1fj6hAiGuDB-1VOSP6GzcuBX6GVTljSubyIy-2wIpQSdXz8YBdkIuVLZ2DDLQWzwU$
>>>>>><https://urldefense.com/v3/__https://linkprotect.cudasvc.com/url?a=https*3a*2f*2fsicuz.unizar.es*2finformacion-sobre-proteccion-de-datos-de-caracter-personal-en-listas&c=E,1,fozeJ_L1c5tT22-_XAnl69C5WGhrrENGO-y2mO0uH3X4Bbm3EnwS5CaEussDHCR05GDKiVPAM9G4jQaY0kVhqsc4vdv55TdLJ2956rnsNTuETjVx&typo=1__;JSUlJQ!!D9dNQwwGXtA!R8lchYgfzUtRK_UvjczZjG4odrY4nCT90ap2vplSNdF7nmHYAjY30NMIkzAIZPRsIn5AxYjgQKn_X6p4BTG0vJLaKpZp$>
>>>>>>Recuerde que si est suscrito a una lista voluntaria Ud. puede
>>>>>>darse de baja desde la propia aplicacin en el momento en que lo
>>>>>>desee.
>>>>>>https://urldefense.com/v3/__https://linkprotect.cudasvc.com/url?a=http*3a*2f*2flistas.unizar.es&c=E,1,3TvXH92hrTfzt-a8xmVthnhgYDIEoQe6-G0P6rC6QRkjfvtNsCmkhdLTIB3yp7fRPc9B_8iQu5fWOkBGz-j3blB0p3sUtmf6XMK2hwJsC8gB1kGLD5vipYwnBGfi&typo=1__;JSUl!!D9dNQwwGXtA!Wbhj4g67Lun_cr1u1y1fj6hAiGuDB-1VOSP6GzcuBX6GVTljSubyIy-2wIpQSdXz8YBdkIuVLZ2DJLukYtI$
>>>>>><https://urldefense.com/v3/__https://linkprotect.cudasvc.com/url?a=http*3a*2f*2flistas.unizar.es&c=E,1,3TvXH92hrTfzt-a8xmVthnhgYDIEoQe6-G0P6rC6QRkjfvtNsCmkhdLTIB3yp7fRPc9B_8iQu5fWOkBGz-j3blB0p3sUtmf6XMK2hwJsC8gB1kGLD5vipYwnBGfi&typo=1__;JSUl!!D9dNQwwGXtA!R8lchYgfzUtRK_UvjczZjG4odrY4nCT90ap2vplSNdF7nmHYAjY30NMIkzAIZPRsIn5AxYjgQKn_X6p4BTG0vDDD0K5c$>
>>>>>>----------
>>>>>
>>>>>
>>>>>
>>>>><~WRD0001.jpg>
>>>>><https://urldefense.com/v3/__https://www.avast.com/sig-email?utm_medium=email&utm_source=link&utm_campaign=sig-email&utm_content=emailclient__;!!D9dNQwwGXtA!R8lchYgfzUtRK_UvjczZjG4odrY4nCT90ap2vplSNdF7nmHYAjY30NMIkzAIZPRsIn5AxYjgQKn_X6p4BTG0vNvQg6vb$>
>>>>>Libre de virus.https://urldefense.com/v3/__http://www.avast.com__;!!D9dNQwwGXtA!Wbhj4g67Lun_cr1u1y1fj6hAiGuDB-1VOSP6GzcuBX6GVTljSubyIy-2wIpQSdXz8YBdkIuVLZ2DrIYpvao$
>>>>><https://urldefense.com/v3/__https://www.avast.com/sig-email?utm_medium=email&utm_source=link&utm_campaign=sig-email&utm_content=emailclient__;!!D9dNQwwGXtA!R8lchYgfzUtRK_UvjczZjG4odrY4nCT90ap2vplSNdF7nmHYAjY30NMIkzAIZPRsIn5AxYjgQKn_X6p4BTG0vNvQg6vb$>
>>>>>
>>>>
>>>
>>>
>>
>>
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