<div dir="ltr">
<p class="gmail-p1" style="margin:0px;font-variant-numeric:normal;font-variant-east-asian:normal;font-stretch:normal;font-size:12px;line-height:normal;font-family:Helvetica"><span class="gmail-s1" style="font-kerning:none">Dear Folks,</span></p><p class="gmail-p1" style="margin:0px;font-variant-numeric:normal;font-variant-east-asian:normal;font-stretch:normal;font-size:12px;line-height:normal;font-family:Helvetica"><span class="gmail-s1" style="font-kerning:none">It is worth pointing out (as Lawvere has indicated long ago) that the construction of fixed points, the Goedelian construction and the Cantor diagonalization process are all instances of </span></p>
<p class="gmail-p1" style="margin:0px;font-variant-numeric:normal;font-variant-east-asian:normal;font-stretch:normal;font-size:12px;line-height:normal;font-family:Helvetica"><span class="gmail-s1" style="font-kerning:none">a natural construction in Cartesian closed categories. </span></p><p class="gmail-p1" style="margin:0px;font-variant-numeric:normal;font-variant-east-asian:normal;font-stretch:normal;font-size:12px;line-height:normal;font-family:Helvetica"><span class="gmail-s1" style="font-kerning:none">Lawvere's remarks are relevant to the present discussion and can be understood in many ways.</span></p><p class="gmail-p1" style="margin:0px;font-variant-numeric:normal;font-variant-east-asian:normal;font-stretch:normal;font-size:12px;line-height:normal;font-family:Helvetica"><span class="gmail-s1" style="font-kerning:none">I talked about this in previous emails without a directt reference to Lawvere.</span></p><p class="gmail-p1" style="margin:0px;font-variant-numeric:normal;font-variant-east-asian:normal;font-stretch:normal;font-size:12px;line-height:normal;font-family:Helvetica"><span class="gmail-s1" style="font-kerning:none">Here is a copy of his paper.</span></p><p class="gmail-p1" style="margin:0px;font-variant-numeric:normal;font-variant-east-asian:normal;font-stretch:normal;font-size:12px;line-height:normal;font-family:Helvetica"><span class="gmail-s1" style="font-kerning:none">I have more to say about this, but will just send the paper in this email.</span></p><p class="gmail-p1" style="margin:0px;font-variant-numeric:normal;font-variant-east-asian:normal;font-stretch:normal;font-size:12px;line-height:normal;font-family:Helvetica"><span class="gmail-s1" style="font-kerning:none">Best,</span></p><p class="gmail-p1" style="margin:0px;font-variant-numeric:normal;font-variant-east-asian:normal;font-stretch:normal;font-size:12px;line-height:normal;font-family:Helvetica"><span class="gmail-s1" style="font-kerning:none">Lou Kauffman</span></p></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">On Fri, Dec 23, 2022 at 2:31 AM guillaume.bonfante <<a href="mailto:guillaume.bonfante@loria.fr">guillaume.bonfante@loria.fr</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">
<div>
Dear all, <br>
<br>
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. <br>
<br>
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. <br>
<br>
All the best,<br>
<br>
Guillaume<br>
<br>
<br>
<div>Le 05/12/2022 à 00:25, Louis Kauffman a
écrit :<br>
</div>
<blockquote type="cite">
Dear Sheri,
<div>It would indeed be very helpful to have a zoom
conversation about these themes.</div>
<div>Please let me know when you would be available to
have it.</div>
<div>We could start with an hour meeting and discussion
and then perhaps extend to a second meeting with some
presentations.</div>
<div>Included below is a slide show of mine that is a bit
cryptic but does summarize some points of view.</div>
<div>I have downloaded the papers you indicated in your
email and will read them now.</div>
<div>Very best,</div>
<div>Lou </div>
<br>
<fieldset></fieldset>
<div><br>
</div>
<div><br>
<div>
<blockquote type="cite">
<div>On Dec 5, 2022, at 2:53 PM, Markose, Sheri
<<a href="mailto:scher@essex.ac.uk" target="_blank">scher@essex.ac.uk</a>>
wrote:</div>
<br>
<div>
<div style="font-family:Helvetica;font-size:12px;font-style:normal;font-variant:normal;font-weight:normal;letter-spacing:normal;line-height:normal;text-align:start;text-indent:0px;text-transform:none;white-space:normal;word-spacing:0px">
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:12pt;font-family:Arial,sans-serif;color:rgb(31,73,125)">Dear Louis, Pedro and All –<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:12pt;font-family:Arial,sans-serif;color:rgb(31,73,125)"> </span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:12pt;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.<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:12pt;font-family:Arial,sans-serif"> </span></div>
<div style="margin:0cm 0cm 0cm 54pt;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:12pt;font-family:Arial,sans-serif"><span>(i)<span> <span> </span></span></span></span><span style="font-size:12pt;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 <u></u><u></u></span></div>
<div style="margin:0cm 0cm 0cm 21.3pt;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:12pt;font-family:Arial,sans-serif;border:1pt none windowtext;padding:0cm;background-color:white"><a href="https://urldefense.com/v3/__https://doi.org/10.1162/artl_a_00370__;!!D9dNQwwGXtA!VWC9LTBHaSxeymx-5r0eHWMMDBrit4Ck9S5Tu7GLMTvgKzDwM0EGmZ214tQBczdWL4qaYl5QxkryRYqf$" style="color:blue;text-decoration:underline" target="_blank"><span>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></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:12pt;font-family:Arial,sans-serif;border:1pt none windowtext;padding:0cm;background-color:white"><span> </span>S
& P share objectives as those that I have given<span> </span></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:12pt;font-family:Arial,sans-serif"><u></u><u></u></span></div>
<div style="margin:0cm 0cm 0cm 21.3pt;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-family:Arial,sans-serif"> </span></div>
<div style="margin:0cm 0cm 0cm 21.3pt;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:12pt;font-family:Arial,sans-serif;border:1pt none windowtext;padding:0cm;background-color:white">S& R
give an interesting and plausible account of how
RNA-push down automata<span> </span></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:12pt;font-family:Arial,sans-serif"><u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:12pt;font-family:Arial,sans-serif;color:rgb(31,73,125)"> </span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:12pt;font-family:Arial,sans-serif;color:rgb(31,73,125)"> </span></div>
<div style="margin:0cm 0cm 0cm 54pt;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:12pt;font-family:Arial,sans-serif;color:rgb(31,73,125)"><span>(ii)<span style="font-style:normal;font-variant:normal;font-weight:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman""> <span> </span></span></span></span><span style="font-size:12pt;font-family:Arial,sans-serif;color:rgb(31,73,125)">Section
5 of S&P relates to undecidability as fixed
points of negation functions<span> </span><span style="background-color: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. <u></u><u></u></span></div>
<div style="margin:0cm 0cm 0cm 54pt;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:12pt;font-family:Arial,sans-serif;color:rgb(31,73,125)"> </span></div>
<div style="margin:0cm 0cm 0cm 54pt;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:12pt;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.<span> </span><u></u><u></u></span></div>
<div style="margin:0cm 0cm 0cm 54pt;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:12pt;font-family:Arial,sans-serif;color:rgb(31,73,125)"> <u></u><u></u></span></div>
<div style="margin:0cm 0cm 0cm 54pt;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:12pt;font-family:Arial,sans-serif;color:rgb(31,73,125)"><span>(iii)<span> <span> </span></span></span></span><span style="font-size:12pt;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> </span><span style="color:rgb(31,73,125)">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></span><span style="font-size:12pt;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></span><span style="font-size:12pt;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!VWC9LTBHaSxeymx-5r0eHWMMDBrit4Ck9S5Tu7GLMTvgKzDwM0EGmZ214tQBczdWL4qaYl5Qxu9jb2v9$" style="color:blue;text-decoration:underline" target="_blank">https://doi.org/10.7554/eLife.66706</a></span><span style="font-size:12pt;font-family:Arial,sans-serif">.<br>
<br>
</span><span style="font-size:12pt;font-family:Arial,sans-serif;color:rgb(31,73,125)"><u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:12pt;font-family:Arial,sans-serif;color:rgb(31,73,125)"> If you recall
how the Recursive Fixed Point Theorem which starts
with a mirror mapping between online self-assembly
program execution,<span> </span></span><i><span style="font-size:12pt;font-family:Symbol">f</span></i><i><sub><span style="font-size:12pt">g</span></sub></i><i><span style="font-size:12pt">(g)</span></i><span style="font-size:12pt">)</span><span> </span><span style="font-size:12pt;font-family:Arial,sans-serif;color:rgb(31,73,125)"> that
have halted and create somatic identity and the<span> </span><i>offline</i><span> </span>record of
the<span> </span><br>
same in a<span> </span><span style="background-color:yellow">2-<span> </span></span> <span style="background-color:yellow">place</span>
function <span> </span></span><span style="font-size:12pt;font-family:Symbol;color:rgb(31,73,125)">s</span><span style="font-size:12pt;font-family:Arial,sans-serif;color:rgb(31,73,125)"><span> </span>(g, g) in
the MHC receptors creating the Thymic self. I say
the first g from the left in<span> </span></span><span style="font-size:12pt;font-family:Symbol;color:rgb(31,73,125)">s</span><span style="font-size:12pt;font-family:Arial,sans-serif;color:rgb(31,73,125)"><span> </span>(g, g) and
changes thereof relate to what happens to self and
the second is self’s record of what<span> </span><br>
the other has done to self. So if
the 2<sup>nd</sup><span> </span>entry is
different from the first entry in<span> </span></span><span style="font-size:12pt;font-family:Symbol;color:rgb(31,73,125)">s</span><span style="font-size:12pt;font-family:Arial,sans-serif;color:rgb(31,73,125)"><span> </span>(g, g) it is
off diagonal etc. The non-self hostile other is a
projection of self g ‘gene codes’ and those <span> </span><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. <u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:12pt;font-family:Arial,sans-serif;color:rgb(31,73,125)"> </span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:12pt;font-family:Arial,sans-serif;color:rgb(31,73,125)"> 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<span> </span><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<span> </span><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<span> </span><i>f
¬ !</i> requires<span> </span><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>
<u></u><u></u></span></div>
<div style="margin:0cm 0cm 0cm 54pt;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:12pt;font-family:Arial,sans-serif;color:rgb(31,73,125)"><span>(iv)<span style="font-style:normal;font-variant:normal;font-weight:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman""> <span> </span></span></span></span><span style="font-size:12pt;font-family:Arial,sans-serif;color:rgb(31,73,125)">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.<u></u><u></u></span></div>
<div style="margin:0cm 0cm 0cm 54pt;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:12pt;font-family:Arial,sans-serif;color:rgb(31,73,125)"> </span></div>
<div style="margin:0cm 0cm 0cm 54pt;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:12pt;font-family:Arial,sans-serif;color:rgb(31,73,125)">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 ….<u></u><u></u></span></div>
<div style="margin:0cm 0cm 0cm 54pt;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:12pt;font-family:Arial,sans-serif;color:rgb(31,73,125)">Ditto
for many others who I hope to be in touch with
soon. <u></u><u></u></span></div>
<div style="margin:0cm 0cm 0cm 54pt;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:12pt;font-family:Arial,sans-serif;color:rgb(31,73,125)"> </span></div>
<div style="margin:0cm 0cm 0cm 54pt;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:12pt;font-family:Arial,sans-serif;color:rgb(31,73,125)">Many
thanks again for the great comments.<u></u><u></u></span></div>
<div style="margin:0cm 0cm 0cm 54pt;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:12pt;font-family:Arial,sans-serif;color:rgb(31,73,125)"> </span></div>
<div style="margin:0cm 0cm 0cm 54pt;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:12pt;font-family:Arial,sans-serif;color:rgb(31,73,125)">All
best<u></u><u></u></span></div>
<div style="margin:0cm 0cm 0cm 54pt;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:12pt;font-family:Arial,sans-serif;color:rgb(31,73,125)"> </span></div>
<div style="margin:0cm 0cm 0cm 54pt;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:12pt;font-family:Arial,sans-serif;color:rgb(31,73,125)">Sheri <u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:12pt;font-family:Arial,sans-serif;color:rgb(31,73,125)"> </span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:12pt;font-family:Arial,sans-serif;color:rgb(31,73,125)"> </span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:12pt;font-family:Arial,sans-serif;color:rgb(31,73,125)"> <u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:12pt;font-family:Arial,sans-serif;color:rgb(31,73,125)"> </span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:12pt;font-family:Arial,sans-serif;color:rgb(31,73,125)"> <u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;font-family:Arial,sans-serif;color:rgb(31,73,125)"> </span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;font-family:Arial,sans-serif;color:rgb(31,73,125)"> </span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;font-family:Arial,sans-serif;color:rgb(31,73,125)"> </span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;font-family:Arial,sans-serif;color:rgb(31,73,125)"> </span></div>
<div>
<div style="border-style:solid none none;border-top-color:rgb(225,225,225);border-top-width:1pt;padding:3pt 0cm 0cm">
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><b><span lang="EN-US">From:</span></b><span lang="EN-US"><span> </span>Pedro C.
Marijuán <<a href="mailto:pedroc.marijuan@gmail.com" style="color:blue;text-decoration:underline" target="_blank">pedroc.marijuan@gmail.com</a>><span> </span><br>
<b>Sent:</b><span> </span>01
December 2022 13:03<br>
<b>To:</b><span> </span>Markose,
Sheri <<a href="mailto:scher@essex.ac.uk" style="color:blue;text-decoration:underline" target="_blank">scher@essex.ac.uk</a>>;
Louis Kauffman <<a href="mailto:loukau@gmail.com" style="color:blue;text-decoration:underline" target="_blank">loukau@gmail.com</a>><br>
<b>Cc:</b><span> </span>fis <<a href="mailto:fis@listas.unizar.es" style="color:blue;text-decoration:underline" target="_blank">fis@listas.unizar.es</a>>;<span> </span><a href="mailto:guillaume.bonfante@mines-nancy.univ-lorraine.fr" style="color:blue;text-decoration:underline" target="_blank">guillaume.bonfante@mines-nancy.univ-lorraine.fr</a><br>
<b>Subject:</b><span> </span>Re:
[Fis] A new discussion session<u></u><u></u></span></div>
</div>
</div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><u></u> <u></u></div>
<div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">Dear Sheri, Lou, and
all discussants,<span><u></u><u></u></span></div>
</div>
<div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><u></u> <u></u></div>
</div>
<div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">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:<span> </span><u></u><u></u></div>
</div>
<div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><u></u> <u></u></div>
</div>
<div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">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:<span> </span><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!VWC9LTBHaSxeymx-5r0eHWMMDBrit4Ck9S5Tu7GLMTvgKzDwM0EGmZ214tQBczdWL4qaYl5Qxu9jb2v9$" style="color:blue;text-decoration:underline" target="_blank">https://doi.org/10.7554/eLife.66706</a><span>.</span><span> </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...<u></u><u></u></div>
</div>
<div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><u></u> <u></u></div>
</div>
<div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">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<span> </span><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??<span> </span><u></u><u></u></div>
</div>
<div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><u></u> <u></u></div>
</div>
<div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">Well, Sheri is far
more acknowledged with all this stuff. And perhaps
Lou can say something about the formal
distinguishability of 10-12 aa.<u></u><u></u></div>
</div>
<div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">Best--Pedro<u></u><u></u></div>
</div>
<div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><u></u> <u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><u></u> <u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><u></u> <u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;font-family:Arial,sans-serif;color:rgb(31,73,125)"> </span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><b><span lang="EN-US">From:</span></b><span lang="EN-US"><span> </span>Louis
Kauffman<span> </span></span><a href="mailto:loukau@gmail.com" style="color:blue;text-decoration:underline" target="_blank"><span lang="EN-US">loukau@gmail.com</span></a><span lang="EN-US"><span> </span><br>
<b>Sent:</b><span> </span>15
November 2022 23:02<br>
<b>To:</b><span> </span>Markose,
Sheri<span> </span></span><a href="mailto:scher@essex.ac.uk" style="color:blue;text-decoration:underline" target="_blank"><span lang="EN-US">scher@essex.ac.uk</span></a><span lang="EN-US"><br>
<b>Cc:</b><span> </span>"Pedro C.
Marijuán"<span> </span></span><a href="mailto:pedroc.marijuan@gmail.com" style="color:blue;text-decoration:underline" target="_blank"><span lang="EN-US">pedroc.marijuan@gmail.com</span></a><span lang="EN-US">; fis<span> </span></span><a href="mailto:fis@listas.unizar.es" style="color:blue;text-decoration:underline" target="_blank"><span lang="EN-US">fis@listas.unizar.es</span></a><span lang="EN-US">;<span> </span></span><a href="mailto:guillaume.bonfante@mines-nancy.univ-lorraine.fr" style="color:blue;text-decoration:underline" target="_blank"><span lang="EN-US">guillaume.bonfante@mines-nancy.univ-lorraine.fr</span></a><span lang="EN-US"><br>
<b>Subject:</b><span> </span>Re: [Fis]
A new discussion session<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><u></u> <u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">Sheri,<span><u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">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.<u></u><u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">My letter was quite
long, and it is possible to not get to the second
half.<u></u><u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">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.<u></u><u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">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.<u></u><u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">Best,<u></u><u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">Lou K.<u></u><u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><u></u> <u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">##########<u></u><u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;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.<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;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.<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;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.<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;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?<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;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.<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;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.<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;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.<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;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.<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;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.<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;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:<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;font-family:Arial,sans-serif">PX: X can be printed.<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;font-family:Arial,sans-serif">~PX: X cannot be printed.<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;font-family:Arial,sans-serif">PRX: XX can be printed.<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;font-family:Arial,sans-serif">~PRX: XX cannot be printed.<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;font-family:Arial,sans-serif">Here X is any string of the
symbols {~,P,R}.<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;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.<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;font-family:Arial,sans-serif">By printed we mean on one
press of the button on the Machine, a string of
characters is printed.<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;font-family:Arial,sans-serif">IT IS ASSUMED THAT THE
SMULLYAN MACHINE ALWAYS TELLS THE TRUTH WHEN IT
PRINTS A MEANINGFUL STATEMENT.<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;font-family:Arial,sans-serif">Then we have the<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;font-family:Arial,sans-serif">Theorem. There are
meaningful true strings that the Smullyan Machine
cannot print.<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;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.<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;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.<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;font-family:Arial,sans-serif">If the machine were to print
G, it would lie. And the machine does not lie.<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;font-family:Arial,sans-serif">Therefore G is unprintable.<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;font-family:Arial,sans-serif">But this is what G says.<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;font-family:Arial,sans-serif">So we have established the
truth of G and proved the Theorem.<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;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.<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;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.<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;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.<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;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.<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;font-family:Arial,sans-serif">8. I will stop here. The
relationships with biology are very worth
discussing.<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;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.<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;font-family:Arial,sans-serif"> ###############<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;font-family:Arial,sans-serif">Best,<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;font-family:Arial,sans-serif">Lou Kauffman<u></u><u></u></span></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:10pt;font-family:Arial,sans-serif"><br>
<br>
</span><u></u><u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif"><u></u> <u></u></div>
</div>
<div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">El 15/11/2022 a las
21:19, Markose, Sheri escribió:<u></u><u></u></div>
</div>
<blockquote style="margin-top:5pt;margin-bottom:5pt">
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">Dear Louis, dear
Colleagues - <span> </span><u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">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 ? <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">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. <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">1. I have found the
following statement by Joel Hamkins ( :<span> </span><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!VWC9LTBHaSxeymx-5r0eHWMMDBrit4Ck9S5Tu7GLMTvgKzDwM0EGmZ214tQBczdWL4qaYl5Qxl8pLdy3$" style="color:blue;text-decoration:underline" target="_blank"><span style="font-size:11pt;font-family:Calibri,sans-serif">http://jdh.hamkins.org/wp-content/uploads/A-review-of-several-fixed-point-theorems-1.pdf</span></a><span> </span>) 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.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
<div style="margin:0cm;font-size:11pt;font-family:Calibri,sans-serif">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<span> </span><i>g
builds the<span> </span></i>machine
that runs g.<span> </span><u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">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).<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"><span id="m_3057625756610017441cid:image001.png@01D905CE.ACBBA200"><image001.png></span><u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">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. <span> </span><u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">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> </span><span style="font-size:12pt">g<sup>¬</sup></span><span> </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.<span> </span><u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">II. Once, (8) is in
place, the Adaptive immune system has to identify
novel negation software function<span> </span><i>f<sup>¬!<span> </span></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<span> </span></sup>– 10<span> </span><sup>30</sup><span> </span>) 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.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">III. Roger Fixed Point
assures us that the indexes of the fixed point for<span> </span><i>f<sup>¬!<span> </span></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.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">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.<span> </span><u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">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. <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">So thankyou all again
for your in depth comments and interest.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">Best Regards<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">Sheri<span> </span><u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">-----Original
Message-----<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">From: Fis <<a href="mailto:fis-bounces@listas.unizar.es" style="color:blue;text-decoration:underline" target="_blank">fis-bounces@listas.unizar.es</a>>
On Behalf Of Louis Kauffman<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">Sent: 08 November 2022
00:13<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">To: "Pedro C. Marijuán"
<<a href="mailto:pedroc.marijuan@gmail.com" style="color:blue;text-decoration:underline" target="_blank">pedroc.marijuan@gmail.com</a>><u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">Cc: fis <<a href="mailto:fis@listas.unizar.es" style="color:blue;text-decoration:underline" target="_blank">fis@listas.unizar.es</a>><u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">Subject: Re: [Fis] A
new discussion session<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">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.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">Dear Pedro,<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">Here are some comments
about Goedel numbering and coding.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">It is interesting to
think about Goedel numbering in a biological
context.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">Actually we are talking
about how a given entity has semantics that can vary
from context to context.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">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.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">1. F as a well formed
formula in a formal system S.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">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.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">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.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">This is the key to the
references and self-references of the Goedelian
situations.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">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.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">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.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">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.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">And S can decode a
number to find the corresponding expression as well.
It is assumed that S as a logical system, is
consistent.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">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.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">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.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">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).<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">Now suppose that<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">g —> F(x).<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">Then we can form F(g)
and this new statement has a Goedel number. Let #g
denote the Goedel number of F(g).<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">#g —> F(g).<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">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.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">Then we can consider<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">F(#x) and it has a
Goedel number<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">h —> F(#x)<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">And we can shift that
to<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">#h —> F(#h).<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">This is the key point.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">Now we have constructed
a number #h so that F(#h) discusses its own Goedel
number.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">This construction
allows the proof of the Goedel Incompleteness
Theorem via the sentence B(x) that states<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">B(x) = “The statement
with Goedel number x is provable in S.” (This can
also be encoded in S.)<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">We then construct<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">h—> ~B(#x)<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">and<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">#h —> ~B(#h)<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">and obtain the
statement<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">G= ~B(#h).<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">G states the
unprovability of the Goedel decoding of #h.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">But the Goedel decoding
of #h is the statement G itself.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">Thus G states its own
unprovability.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">Therefore, S being
consistent, cannot prove G.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">By making these
arguments we have have proved that G cannot be
proved by S.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">Thus we have shown that
G is in fact true.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">We have shown that
there are true statements in number theory
unprovable by system S..<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">##########################<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">The above is a very
concise summary of the proof of Goedel’s
Incompleteness Theorem, using Goedel number
encoding.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;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.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;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.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;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.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;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?<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;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.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;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.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;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.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;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.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;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.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;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:<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">PX: X can be printed.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">~PX: X cannot be
printed.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">PRX: XX can be printed.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">~PRX: XX cannot be
printed.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">Here X is any string of
the symbols {~,P,R}.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;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.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">By printed we mean on
one press of the button on the Machine, a string of
characters is printed.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">IT IS ASSUMED THAT THE
SMULLYAN MACHINE ALWAYS TELLS THE TRUTH WHEN IT
PRINTS A MEANINGFUL STATEMENT.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">Then we have the<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">Theorem. There are
meaningful true strings that the Smullyan Machine
cannot print.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;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.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;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.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">If the machine were to
print G, it would lie. And the machine does not lie.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">Therefore G is
unprintable.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">But this is what G
says.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">So we have established
the truth of G and proved the Theorem.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;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.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;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.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;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.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;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.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">8. I will stop here.
The relationships with biology are very worth
discussing.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;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.<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">Very best,<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif">Lou Kauffman<u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
<div style="margin:0cm;font-size:10pt;font-family:Arial,sans-serif"> <u></u><u></u></div>
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</blockquote>
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