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<p class="MsoNormal"><span style="font-size:12.0pt;font-family:"Arial",sans-serif;color:#1F497D">Dear Louis, Pedro and All –
<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:12.0pt;font-family:"Arial",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:12.0pt;font-family:"Arial",sans-serif">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.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:12.0pt;font-family:"Arial",sans-serif"><o:p> </o:p></span></p>
<p class="MsoListParagraph" style="margin-left:54.0pt;text-indent:-36.0pt;mso-list:l0 level1 lfo3">
<![if !supportLists]><span style="font-size:12.0pt;font-family:"Arial",sans-serif"><span style="mso-list:Ignore">(i)<span style="font:7.0pt "Times New Roman"">
</span></span></span><![endif]><span style="font-size:12.0pt;font-family:"Arial",sans-serif">The bulk of Adam Svahn’s work co-authored with Mikhail Prokopenko (S & P) is already published and can be found here
<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:21.3pt"><span style="font-size:12.0pt;font-family:"Arial",sans-serif;color:black;border:none windowtext 1.0pt;padding:0cm;background:white"><a href="https://urldefense.com/v3/__https://doi.org/10.1162/artl_a_00370__;!!D9dNQwwGXtA!SRUQftsCdcVWuZ2nZSWbqdrZ5y2JlvuA5tZSuub4EMCuGaKiepU2hKD5nGCkDAAoeA7EI9f1d2PdQOiFULg$"><span style="color:black">https://doi.org/10.1162/artl_a_00370</span></a>
and is relevant to our discussion. While we are still in the dark about </span>
<span style="font-family:"Arial",sans-serif">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.</span><span style="font-size:12.0pt;font-family:"Arial",sans-serif;color:black;border:none windowtext 1.0pt;padding:0cm;background:white"> S & P share objectives as those that I have given
</span><span style="font-family:"Arial",sans-serif">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.</span><span style="font-size:12.0pt;font-family:"Arial",sans-serif"><o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:21.3pt"><span style="font-family:"Arial",sans-serif"><o:p> </o:p></span></p>
<p class="MsoNormal" style="margin-left:21.3pt"><span style="font-size:12.0pt;font-family:"Arial",sans-serif;color:black;border:none windowtext 1.0pt;padding:0cm;background:white">S& R give an interesting and plausible account of how RNA-push down automata
</span><span style="font-family:"Arial",sans-serif">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 2<sup>nd</sup> stack to reflect on codes and make changes to them.
</span><span style="font-size:12.0pt;font-family:"Arial",sans-serif"><o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:12.0pt;font-family:"Arial",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:12.0pt;font-family:"Arial",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoListParagraph" style="margin-left:54.0pt;text-indent:-36.0pt;mso-list:l0 level1 lfo3">
<![if !supportLists]><span style="font-size:12.0pt;font-family:"Arial",sans-serif;color:#1F497D"><span style="mso-list:Ignore">(ii)<span style="font:7.0pt "Times New Roman"">
</span></span></span><![endif]><span style="font-size:12.0pt;font-family:"Arial",sans-serif;color:#1F497D">Section 5 of S&P relates to undecidability as fixed points of negation functions
<span style="background:yellow;mso-highlight:yellow">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.</span>
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.
<o:p></o:p></span></p>
<p class="MsoListParagraph" style="margin-left:54.0pt"><span style="font-size:12.0pt;font-family:"Arial",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoListParagraph" style="margin-left:54.0pt"><span style="font-size:12.0pt;font-family:"Arial",sans-serif">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. <o:p></o:p></span></p>
<p class="MsoListParagraph" style="margin-left:54.0pt"><span style="font-size:12.0pt;font-family:"Arial",sans-serif;color:#1F497D"> <o:p></o:p></span></p>
<p class="MsoListParagraph" style="margin-left:54.0pt;text-indent:-36.0pt;mso-list:l0 level1 lfo3">
<![if !supportLists]><span style="font-size:12.0pt;font-family:"Arial",sans-serif;color:#1F497D"><span style="mso-list:Ignore">(iii)<span style="font:7.0pt "Times New Roman"">
</span></span></span><![endif]><span style="font-size:12.0pt;font-family:"Arial",sans-serif">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
<span style="color:#1F497D">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
</span></span><span style="font-size:12.0pt;font-family:"Arial",sans-serif">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<br>
neuroimmunology." By Aurora Krauset al. In, eLife 2021;10:e66706. DOI: </span><span style="font-size:12.0pt;font-family:"Arial",sans-serif"><a href="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!SRUQftsCdcVWuZ2nZSWbqdrZ5y2JlvuA5tZSuub4EMCuGaKiepU2hKD5nGCkDAAoeA7EI9f1d2PdD0BU0Q8$">https://doi.org/10.7554/eLife.66706</a></span><span style="font-size:12.0pt;font-family:"Arial",sans-serif">.<br>
<br>
</span><span style="font-size:12.0pt;font-family:"Arial",sans-serif;color:#1F497D"><o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:12.0pt;font-family:"Arial",sans-serif;color:#1F497D"> If you recall how the Recursive Fixed Point Theorem which starts with a mirror mapping between online self-assembly program execution,
</span><i><span style="font-size:12.0pt;font-family:Symbol">f</span></i><i><sub><span style="font-size:12.0pt">g</span></sub></i><i><span style="font-size:12.0pt">(g)</span></i><span style="font-size:12.0pt">)</span>
<span style="font-size:12.0pt;font-family:"Arial",sans-serif;color:#1F497D"> that have halted and create somatic identity and the
<i>offline</i> record of the <br>
same in a <span style="background:yellow;mso-highlight:yellow">
2- </span> <span style="background:yellow;mso-highlight:yellow">place</span> function
</span><span style="font-size:12.0pt;font-family:Symbol;color:#1F497D">s</span><span style="font-size:12.0pt;font-family:"Arial",sans-serif;color:#1F497D"> (g, g) in the MHC receptors creating the Thymic self. I say the first g from the left in
</span><span style="font-size:12.0pt;font-family:Symbol;color:#1F497D">s</span><span style="font-size:12.0pt;font-family:"Arial",sans-serif;color:#1F497D"> (g, g) and changes thereof relate to what happens to self and the second is self’s record of what
<br>
the other has done to self. So if the 2<sup>nd</sup> entry is different from the first entry in
</span><span style="font-size:12.0pt;font-family:Symbol;color:#1F497D">s</span><span style="font-size:12.0pt;font-family:"Arial",sans-serif;color:#1F497D"> (g, g) it is off diagonal etc. The non-self hostile other is a projection of self g ‘gene codes’ and
those <i>f ¬ !</i> which are reactive to the self g-<br>
codes denoted as g¬. As we know an astronomic number of potential indexes g¬ are generated by the RAG genes.
<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:12.0pt;font-family:"Arial",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:12.0pt;font-family:"Arial",sans-serif;color:#1F497D"> 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 <br>
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
<br>
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
<i>f ¬ !</i> requires <br>
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. <br>
<br>
<o:p></o:p></span></p>
<p class="MsoListParagraph" style="margin-left:54.0pt;text-indent:-36.0pt;mso-list:l0 level1 lfo3">
<![if !supportLists]><span style="font-size:12.0pt;font-family:"Arial",sans-serif;color:#1F497D"><span style="mso-list:Ignore">(iv)<span style="font:7.0pt "Times New Roman"">
</span></span></span><![endif]><span style="font-size:12.0pt;font-family:"Arial",sans-serif;color:#1F497D">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.<o:p></o:p></span></p>
<p class="MsoListParagraph" style="margin-left:54.0pt"><span style="font-size:12.0pt;font-family:"Arial",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal" style="margin-left:54.0pt"><span style="font-size:12.0pt;font-family:"Arial",sans-serif;color:#1F497D">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 ….<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:54.0pt"><span style="font-size:12.0pt;font-family:"Arial",sans-serif;color:#1F497D">Ditto for many others who I hope to be in touch with soon.
<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:54.0pt"><span style="font-size:12.0pt;font-family:"Arial",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal" style="margin-left:54.0pt"><span style="font-size:12.0pt;font-family:"Arial",sans-serif;color:#1F497D">Many thanks again for the great comments.<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:54.0pt"><span style="font-size:12.0pt;font-family:"Arial",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal" style="margin-left:54.0pt"><span style="font-size:12.0pt;font-family:"Arial",sans-serif;color:#1F497D">All best<o:p></o:p></span></p>
<p class="MsoNormal" style="margin-left:54.0pt"><span style="font-size:12.0pt;font-family:"Arial",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal" style="margin-left:54.0pt"><span style="font-size:12.0pt;font-family:"Arial",sans-serif;color:#1F497D">Sheri
<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:12.0pt;font-family:"Arial",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:12.0pt;font-family:"Arial",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:12.0pt;font-family:"Arial",sans-serif;color:#1F497D"> <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:12.0pt;font-family:"Arial",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:12.0pt;font-family:"Arial",sans-serif;color:#1F497D">
<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
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<p class="MsoNormal"><b><span lang="EN-US" style="mso-fareast-language:EN-GB">From:</span></b><span lang="EN-US" style="mso-fareast-language:EN-GB"> Pedro C. Marijuán <pedroc.marijuan@gmail.com>
<br>
<b>Sent:</b> 01 December 2022 13:03<br>
<b>To:</b> Markose, Sheri <scher@essex.ac.uk>; Louis Kauffman <loukau@gmail.com><br>
<b>Cc:</b> fis <fis@listas.unizar.es>; guillaume.bonfante@mines-nancy.univ-lorraine.fr<br>
<b>Subject:</b> Re: [Fis] A new discussion session<o:p></o:p></span></p>
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<p class="MsoNormal"><o:p> </o:p></p>
<div>
<p class="MsoNormal">Dear Sheri, Lou, and all discussants,<span style="mso-fareast-language:EN-GB"><o:p></o:p></span></p>
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<p class="MsoNormal"><o:p> </o:p></p>
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<div>
<p class="MsoNormal">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: <o:p></o:p></p>
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<p class="MsoNormal"><o:p> </o:p></p>
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<div>
<p class="MsoNormal">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<br>
neuroimmunology." By Aurora Krauset al. In, eLife 2021;10:e66706. DOI: <a href="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!SRUQftsCdcVWuZ2nZSWbqdrZ5y2JlvuA5tZSuub4EMCuGaKiepU2hKD5nGCkDAAoeA7EI9f1d2PdD0BU0Q8$">
https://doi.org/10.7554/eLife.66706</a><span style="font-size:10.0pt;font-family:"Times New Roman",serif">.</span> 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...<o:p></o:p></p>
</div>
<div>
<p class="MsoNormal"><o:p> </o:p></p>
</div>
<div>
<p class="MsoNormal">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 <u>indirect products of protein translation</u>. 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?? <o:p></o:p></p>
</div>
<div>
<p class="MsoNormal"><o:p> </o:p></p>
</div>
<div>
<p class="MsoNormal">Well, Sheri is far more acknowledged with all this stuff. And perhaps Lou can say something about the formal distinguishability of 10-12 aa.<o:p></o:p></p>
</div>
<div>
<p class="MsoNormal">Best--Pedro<o:p></o:p></p>
</div>
<div>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal"><b><span lang="EN-US">From:</span></b><span lang="EN-US"> Louis Kauffman
</span><a href="mailto:loukau@gmail.com"><span lang="EN-US">loukau@gmail.com</span></a><span lang="EN-US">
<br>
<b>Sent:</b> 15 November 2022 23:02<br>
<b>To:</b> Markose, Sheri </span><a href="mailto:scher@essex.ac.uk"><span lang="EN-US">scher@essex.ac.uk</span></a><span lang="EN-US"><br>
<b>Cc:</b> "Pedro C. Marijuán" </span><a href="mailto:pedroc.marijuan@gmail.com"><span lang="EN-US">pedroc.marijuan@gmail.com</span></a><span lang="EN-US">; fis
</span><a href="mailto:fis@listas.unizar.es"><span lang="EN-US">fis@listas.unizar.es</span></a><span lang="EN-US">;
</span><a href="mailto:guillaume.bonfante@mines-nancy.univ-lorraine.fr"><span lang="EN-US">guillaume.bonfante@mines-nancy.univ-lorraine.fr</span></a><span lang="EN-US"><br>
<b>Subject:</b> Re: [Fis] A new discussion session<o:p></o:p></span></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">Sheri,<span style="mso-fareast-language:EN-GB"><o:p></o:p></span></p>
<p class="MsoNormal">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.<o:p></o:p></p>
<p class="MsoNormal">My letter was quite long, and it is possible to not get to the second half.<o:p></o:p></p>
<p class="MsoNormal">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.<o:p></o:p></p>
<p class="MsoNormal">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.<o:p></o:p></p>
<p class="MsoNormal">Best,<o:p></o:p></p>
<p class="MsoNormal">Lou K.<o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
<p class="MsoNormal">##########<o:p></o:p></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif">It is a very interesting question whether such encoding or such multiple relationships to context occur in biology. Here are some remarks.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif"> <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif">1. In biology is is NORMALLY the case that certain key structures have multiple interpretations and uses in various contexts.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif">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.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif"> <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif">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?<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif"> <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif">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.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif">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.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif"> <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif">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.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif">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.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif">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.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif"> <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif">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:<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif">PX: X can be printed.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif">~PX: X cannot be printed.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif">PRX: XX can be printed.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif">~PRX: XX cannot be printed.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif">Here X is any string of the symbols {~,P,R}.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif">Thus PR~~P means that XX can be printed where X = ~~P. Thus PR~~P means that ~~P~~P can be printed.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif">By printed we mean on one press of the button on the Machine, a string of characters is printed.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif">IT IS ASSUMED THAT THE SMULLYAN MACHINE ALWAYS TELLS THE TRUTH WHEN IT PRINTS A MEANINGFUL STATEMENT.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif">Then we have the<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif"> <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif">Theorem. There are meaningful true strings that the Smullyan Machine cannot print.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif"> <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif">This is a non-numerical analog of the Goedel Theorem. And the string that cannot be printed is G = ~PR~PR.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif">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.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif">If the machine were to print G, it would lie. And the machine does not lie.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif">Therefore G is unprintable.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif">But this is what G says.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif">So we have established the truth of G and proved the Theorem.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif"> <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif">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.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif">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.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif"> <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif">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.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif">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.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif"> <o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif">8. I will stop here. The relationships with biology are very worth discussing.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif">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.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif"> ###############<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif">Best,<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif">Lou Kauffman<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.0pt;font-family:"Arial",sans-serif"><br>
<br>
</span><o:p></o:p></p>
<p class="MsoNormal"><o:p> </o:p></p>
</div>
<div>
<p class="MsoNormal">El 15/11/2022 a las 21:19, Markose, Sheri escribió:<o:p></o:p></p>
</div>
<blockquote style="margin-top:5.0pt;margin-bottom:5.0pt">
<p class="MsoPlainText">Dear Louis, dear Colleagues - <o:p></o:p></p>
<p class="MsoPlainText"> <o:p></o:p></p>
<p class="MsoPlainText">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 ?
<o:p></o:p></p>
<p class="MsoPlainText"> <o:p></o:p></p>
<p class="MsoPlainText">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.
<o:p></o:p></p>
<p class="MsoPlainText"> <o:p></o:p></p>
<p class="MsoPlainText">1. I have found the following statement by Joel Hamkins ( :
<a href="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!SRUQftsCdcVWuZ2nZSWbqdrZ5y2JlvuA5tZSuub4EMCuGaKiepU2hKD5nGCkDAAoeA7EI9f1d2Pdeq-yEFE$">
<span style="font-size:11.0pt;font-family:"Calibri",sans-serif">http://jdh.hamkins.org/wp-content/uploads/A-review-of-several-fixed-point-theorems-1.pdf</span></a> ) 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.
<o:p></o:p></p>
<p class="MsoPlainText"> <o:p></o:p></p>
<p class="MsoNormal">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 <i>g builds the </i>machine that runs g. <o:p></o:p></p>
<p class="MsoPlainText"> <o:p></o:p></p>
<p class="MsoPlainText"> <o:p></o:p></p>
<p class="MsoPlainText">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).
<o:p></o:p></p>
<p class="MsoPlainText"> <o:p></o:p></p>
<p class="MsoPlainText"> <o:p></o:p></p>
<p class="MsoPlainText"><img border="0" width="896" height="240" style="width:9.3333in;height:2.5in" id="Picture_x0020_1" src="cid:image001.png@01D905CE.ACBBA200"><o:p></o:p></p>
<p class="MsoPlainText"> <o:p></o:p></p>
<p class="MsoPlainText">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. <o:p></o:p></p>
<p class="MsoPlainText"> <o:p></o:p></p>
<p class="MsoPlainText">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 <span style="font-size:12.0pt">g<sup>¬</sup></span> 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. <o:p></o:p></p>
<p class="MsoPlainText"> <o:p></o:p></p>
<p class="MsoPlainText">II. Once, (8) is in place, the Adaptive immune system has to identify novel negation software function
<i>f<sup>¬! </sup></i><sub> </sub>of non-self antigens which is an uncountable infinite possibilities. Hence the close to astronomic search with V(D) J of 10
<sup>20 </sup>– 10 <sup>30</sup> ) 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.
<o:p></o:p></p>
<p class="MsoPlainText"> <o:p></o:p></p>
<p class="MsoPlainText"> <o:p></o:p></p>
<p class="MsoPlainText">III. Roger Fixed Point assures us that the indexes of the fixed point for
<i>f<sup>¬! </sup></i><sub> </sub>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.
<o:p></o:p></p>
<p class="MsoPlainText"> <o:p></o:p></p>
<p class="MsoPlainText">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. <o:p></o:p></p>
<p class="MsoPlainText"> <o:p></o:p></p>
<p class="MsoPlainText">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. <o:p></o:p></p>
<p class="MsoPlainText"> <o:p></o:p></p>
<p class="MsoPlainText">So thankyou all again for your in depth comments and interest.<o:p></o:p></p>
<p class="MsoPlainText"> <o:p></o:p></p>
<p class="MsoPlainText">Best Regards<o:p></o:p></p>
<p class="MsoPlainText"> <o:p></o:p></p>
<p class="MsoPlainText">Sheri <o:p></o:p></p>
<p class="MsoPlainText"> <o:p></o:p></p>
<p class="MsoPlainText">-----Original Message-----<o:p></o:p></p>
<p class="MsoPlainText">From: Fis <<a href="mailto:fis-bounces@listas.unizar.es">fis-bounces@listas.unizar.es</a>> On Behalf Of Louis Kauffman<o:p></o:p></p>
<p class="MsoPlainText">Sent: 08 November 2022 00:13<o:p></o:p></p>
<p class="MsoPlainText">To: "Pedro C. Marijuán" <<a href="mailto:pedroc.marijuan@gmail.com">pedroc.marijuan@gmail.com</a>><o:p></o:p></p>
<p class="MsoPlainText">Cc: fis <<a href="mailto:fis@listas.unizar.es">fis@listas.unizar.es</a>><o:p></o:p></p>
<p class="MsoPlainText">Subject: Re: [Fis] A new discussion session<o:p></o:p></p>
<p class="MsoPlainText"> <o:p></o:p></p>
<p class="MsoPlainText">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.<o:p></o:p></p>
<p class="MsoPlainText"> <o:p></o:p></p>
<p class="MsoPlainText">Dear Pedro,<o:p></o:p></p>
<p class="MsoPlainText">Here are some comments about Goedel numbering and coding.<o:p></o:p></p>
<p class="MsoPlainText"> <o:p></o:p></p>
<p class="MsoPlainText">It is interesting to think about Goedel numbering in a biological context.<o:p></o:p></p>
<p class="MsoPlainText">Actually we are talking about how a given entity has semantics that can vary from context to context.<o:p></o:p></p>
<p class="MsoPlainText">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.<o:p></o:p></p>
<p class="MsoPlainText">1. F as a well formed formula in a formal system S.<o:p></o:p></p>
<p class="MsoPlainText">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.<o:p></o:p></p>
<p class="MsoPlainText">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.<o:p></o:p></p>
<p class="MsoPlainText">This is the key to the references and self-references of the Goedelian situations.<o:p></o:p></p>
<p class="MsoPlainText"> <o:p></o:p></p>
<p class="MsoPlainText">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.<o:p></o:p></p>
<p class="MsoPlainText">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.<o:p></o:p></p>
<p class="MsoPlainText">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.<o:p></o:p></p>
<p class="MsoPlainText">And S can decode a number to find the corresponding expression as well. It is assumed that S as a logical system, is consistent.<o:p></o:p></p>
<p class="MsoPlainText"> <o:p></o:p></p>
<p class="MsoPlainText">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.<o:p></o:p></p>
<p class="MsoPlainText"> <o:p></o:p></p>
<p class="MsoPlainText">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.<o:p></o:p></p>
<p class="MsoPlainText">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).<o:p></o:p></p>
<p class="MsoPlainText">Now suppose that<o:p></o:p></p>
<p class="MsoPlainText">g —> F(x).<o:p></o:p></p>
<p class="MsoPlainText">Then we can form F(g) and this new statement has a Goedel number. Let #g denote the Goedel number of F(g).<o:p></o:p></p>
<p class="MsoPlainText">#g —> F(g).<o:p></o:p></p>
<p class="MsoPlainText">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.<o:p></o:p></p>
<p class="MsoPlainText">Then we can consider<o:p></o:p></p>
<p class="MsoPlainText">F(#x) and it has a Goedel number<o:p></o:p></p>
<p class="MsoPlainText">h —> F(#x)<o:p></o:p></p>
<p class="MsoPlainText">And we can shift that to<o:p></o:p></p>
<p class="MsoPlainText">#h —> F(#h).<o:p></o:p></p>
<p class="MsoPlainText">This is the key point.<o:p></o:p></p>
<p class="MsoPlainText">Now we have constructed a number #h so that F(#h) discusses its own Goedel number.<o:p></o:p></p>
<p class="MsoPlainText"> <o:p></o:p></p>
<p class="MsoPlainText">This construction allows the proof of the Goedel Incompleteness Theorem via the sentence B(x) that states<o:p></o:p></p>
<p class="MsoPlainText">B(x) = “The statement with Goedel number x is provable in S.” (This can also be encoded in S.)<o:p></o:p></p>
<p class="MsoPlainText"> <o:p></o:p></p>
<p class="MsoPlainText">We then construct<o:p></o:p></p>
<p class="MsoPlainText">h—> ~B(#x)<o:p></o:p></p>
<p class="MsoPlainText">and<o:p></o:p></p>
<p class="MsoPlainText">#h —> ~B(#h)<o:p></o:p></p>
<p class="MsoPlainText">and obtain the statement<o:p></o:p></p>
<p class="MsoPlainText">G= ~B(#h).<o:p></o:p></p>
<p class="MsoPlainText">G states the unprovability of the Goedel decoding of #h.<o:p></o:p></p>
<p class="MsoPlainText">But the Goedel decoding of #h is the statement G itself.<o:p></o:p></p>
<p class="MsoPlainText">Thus G states its own unprovability.<o:p></o:p></p>
<p class="MsoPlainText">Therefore, S being consistent, cannot prove G.<o:p></o:p></p>
<p class="MsoPlainText"> <o:p></o:p></p>
<p class="MsoPlainText">By making these arguments we have have proved that G cannot be proved by S.<o:p></o:p></p>
<p class="MsoPlainText">Thus we have shown that G is in fact true.<o:p></o:p></p>
<p class="MsoPlainText">We have shown that there are true statements in number theory unprovable by system S..<o:p></o:p></p>
<p class="MsoPlainText">##########################<o:p></o:p></p>
<p class="MsoPlainText"> <o:p></o:p></p>
<p class="MsoPlainText">The above is a very concise summary of the proof of Goedel’s Incompleteness Theorem, using Goedel number encoding.<o:p></o:p></p>
<p class="MsoPlainText"> <o:p></o:p></p>
<p class="MsoPlainText">It is a very interesting question whether such encoding or such multiple relationships to context occur in biology. Here are some remarks.<o:p></o:p></p>
<p class="MsoPlainText"> <o:p></o:p></p>
<p class="MsoPlainText">1. In biology is is NORMALLY the case that certain key structures have multiple interpretations and uses in various contexts.<o:p></o:p></p>
<p class="MsoPlainText">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.<o:p></o:p></p>
<p class="MsoPlainText"> <o:p></o:p></p>
<p class="MsoPlainText">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?<o:p></o:p></p>
<p class="MsoPlainText"> <o:p></o:p></p>
<p class="MsoPlainText">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.<o:p></o:p></p>
<p class="MsoPlainText">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.<o:p></o:p></p>
<p class="MsoPlainText"> <o:p></o:p></p>
<p class="MsoPlainText">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.<o:p></o:p></p>
<p class="MsoPlainText">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.<o:p></o:p></p>
<p class="MsoPlainText">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.<o:p></o:p></p>
<p class="MsoPlainText"> <o:p></o:p></p>
<p class="MsoPlainText">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:<o:p></o:p></p>
<p class="MsoPlainText">PX: X can be printed.<o:p></o:p></p>
<p class="MsoPlainText">~PX: X cannot be printed.<o:p></o:p></p>
<p class="MsoPlainText">PRX: XX can be printed.<o:p></o:p></p>
<p class="MsoPlainText">~PRX: XX cannot be printed.<o:p></o:p></p>
<p class="MsoPlainText">Here X is any string of the symbols {~,P,R}.<o:p></o:p></p>
<p class="MsoPlainText">Thus PR~~P means that XX can be printed where X = ~~P. Thus PR~~P means that ~~P~~P can be printed.<o:p></o:p></p>
<p class="MsoPlainText">By printed we mean on one press of the button on the Machine, a string of characters is printed.<o:p></o:p></p>
<p class="MsoPlainText">IT IS ASSUMED THAT THE SMULLYAN MACHINE ALWAYS TELLS THE TRUTH WHEN IT PRINTS A MEANINGFUL STATEMENT.<o:p></o:p></p>
<p class="MsoPlainText">Then we have the<o:p></o:p></p>
<p class="MsoPlainText"> <o:p></o:p></p>
<p class="MsoPlainText">Theorem. There are meaningful true strings that the Smullyan Machine cannot print.<o:p></o:p></p>
<p class="MsoPlainText"> <o:p></o:p></p>
<p class="MsoPlainText">This is a non-numerical analog of the Goedel Theorem. And the string that cannot be printed is G = ~PR~PR.<o:p></o:p></p>
<p class="MsoPlainText">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.<o:p></o:p></p>
<p class="MsoPlainText">If the machine were to print G, it would lie. And the machine does not lie.<o:p></o:p></p>
<p class="MsoPlainText">Therefore G is unprintable.<o:p></o:p></p>
<p class="MsoPlainText">But this is what G says.<o:p></o:p></p>
<p class="MsoPlainText">So we have established the truth of G and proved the Theorem.<o:p></o:p></p>
<p class="MsoPlainText"> <o:p></o:p></p>
<p class="MsoPlainText">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.<o:p></o:p></p>
<p class="MsoPlainText">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.<o:p></o:p></p>
<p class="MsoPlainText"> <o:p></o:p></p>
<p class="MsoPlainText">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.<o:p></o:p></p>
<p class="MsoPlainText">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.<o:p></o:p></p>
<p class="MsoPlainText"> <o:p></o:p></p>
<p class="MsoPlainText">8. I will stop here. The relationships with biology are very worth discussing.<o:p></o:p></p>
<p class="MsoPlainText">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.<o:p></o:p></p>
<p class="MsoPlainText"> <o:p></o:p></p>
<p class="MsoPlainText">Very best,<o:p></o:p></p>
<p class="MsoPlainText">Lou Kauffman<o:p></o:p></p>
<p class="MsoPlainText"> <o:p></o:p></p>
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<p class="MsoPlainText"> <o:p></o:p></p>
<p class="MsoPlainText"> <o:p></o:p></p>
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<p class="MsoPlainText">INFORMACIN SOBRE PROTECCIN DE DATOS DE CARCTER PERSONAL<o:p></o:p></p>
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