[Fis] Fwd: A new discussion session; Self-Other in Biology
Pedro C. Marijuán
pedroc.marijuan at gmail.com
Sun Dec 25 21:42:26 CET 2022
-------- 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 at loria.fr>
Para: Louis Kauffman <loukau at gmail.com>, Markose, Sheri
<scher at essex.ac.uk>
CC: Pedro C. Marijuán <pedroc.marijuan at gmail.com>, fis
<fis at listas.unizar.es>, guillaume.bonfante at mines-nancy.univ-lorraine.fr
<guillaume.bonfante at mines-nancy.univ-lorraine.fr>,
mikhail.prokopenko at sydney.edu.au <mikhail.prokopenko at sydney.edu.au>,
Neil Gershenfeld <neil.gershenfeld at cba.mit.edu>, Koonin, Eugene
(NIH/NLM/NCBI) [E] <koonin at ncbi.nlm.nih.gov>, Oron Shagrir
<oron.shagrir at gmail.com>, Noson at sci.brooklyn.cuny.edu
<Noson at sci.brooklyn.cuny.edu>, Friston, Karl <k.friston at ucl.ac.uk>, John
Mattick <j.mattick at 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 at 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!R8lchYgfzUtRK_UvjczZjG4odrY4nCT90ap2vplSNdF7nmHYAjY30NMIkzAIZPRsIn5AxYjgQKn_X6p4BTG0vMTREdnb$
>> <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 abouthow 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 givenspecifically 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
>> automatawith 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 2^nd stack to reflect on codes and make changes to them.
>> (ii)Section 5 of S&P relates to undecidability as fixed points of
>> negation functionsand 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 onSelf 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!R8lchYgfzUtRK_UvjczZjG4odrY4nCT90ap2vplSNdF7nmHYAjY30NMIkzAIZPRsIn5AxYjgQKn_X6p4BTG0vKoTwbdV$
>> <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,/f//_g //(g)/) that have halted and create somatic identity
>> and the/offline/record of the
>> same in a2-place function s(g, g) in the MHC
>> receptors creating the Thymic self. I say the first g from the left
>> ins(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 2^nd entry is
>> different from the first entry ins(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 at gmail.com>
>> *Sent:*01 December 2022 13:03
>> *To:*Markose, Sheri <scher at essex.ac.uk>; Louis Kauffman
>> <loukau at gmail.com>
>> *Cc:*fis
>> <fis at listas.unizar.es>;guillaume.bonfante at 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!R8lchYgfzUtRK_UvjczZjG4odrY4nCT90ap2vplSNdF7nmHYAjY30NMIkzAIZPRsIn5AxYjgQKn_X6p4BTG0vKoTwbdV$
>> <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 Kauffmanloukau at gmail.com <mailto:loukau at gmail.com>
>> *Sent:*15 November 2022 23:02
>> *To:*Markose, Sherischer at essex.ac.uk <mailto:scher at essex.ac.uk>
>> *Cc:*"Pedro C. Marijuán"pedroc.marijuan at gmail.com
>> <mailto:pedroc.marijuan at gmail.com>; fisfis at listas.unizar.es
>> <mailto:fis at listas.unizar.es>;guillaume.bonfante at mines-nancy.univ-lorraine.fr
>> <mailto:guillaume.bonfante at 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!R8lchYgfzUtRK_UvjczZjG4odrY4nCT90ap2vplSNdF7nmHYAjY30NMIkzAIZPRsIn5AxYjgQKn_X6p4BTG0vHf5JlgG$
>> <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 sayg^¬ 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 at listas.unizar.es> On Behalf Of Louis Kauffman
>> Sent: 08 November 2022 00:13
>> To: "Pedro C. Marijuán" <pedroc.marijuan at gmail.com>
>> Cc: fis <fis at listas.unizar.es>
>> Subject: Re: [Fis] A new discussion session
>> CAUTION: This email was sent from outside the University of
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>> 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 at listas.unizar.es
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>> ----------
>> 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
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>> 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!R8lchYgfzUtRK_UvjczZjG4odrY4nCT90ap2vplSNdF7nmHYAjY30NMIkzAIZPRsIn5AxYjgQKn_X6p4BTG0vJLaKpZp$
>> <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
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>> 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$
>> <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!R8lchYgfzUtRK_UvjczZjG4odrY4nCT90ap2vplSNdF7nmHYAjY30NMIkzAIZPRsIn5AxYjgQKn_X6p4BTG0vJvGi7BP$
>> <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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