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<div class="moz-forward-container">-------- Mensaje reenviado
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<th valign="BASELINE" nowrap="nowrap" align="RIGHT">Asunto:
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<td>Re: [Fis] A new discussion session; Self-Other in
Biology</td>
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<th valign="BASELINE" nowrap="nowrap" align="RIGHT">Fecha: </th>
<td>Fri, 23 Dec 2022 09:31:53 +0100</td>
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<th valign="BASELINE" nowrap="nowrap" align="RIGHT">De: </th>
<td>guillaume.bonfante <a class="moz-txt-link-rfc2396E" href="mailto:guillaume.bonfante@loria.fr"><guillaume.bonfante@loria.fr></a></td>
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<th valign="BASELINE" nowrap="nowrap" align="RIGHT">Para: </th>
<td>Louis Kauffman <a class="moz-txt-link-rfc2396E" href="mailto:loukau@gmail.com"><loukau@gmail.com></a>, Markose, Sheri
<a class="moz-txt-link-rfc2396E" href="mailto:scher@essex.ac.uk"><scher@essex.ac.uk></a></td>
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<th valign="BASELINE" nowrap="nowrap" align="RIGHT">CC: </th>
<td>Pedro C. Marijuán <a class="moz-txt-link-rfc2396E" href="mailto:pedroc.marijuan@gmail.com"><pedroc.marijuan@gmail.com></a>, fis
<a class="moz-txt-link-rfc2396E" href="mailto:fis@listas.unizar.es"><fis@listas.unizar.es></a>,
<a class="moz-txt-link-abbreviated" href="mailto:guillaume.bonfante@mines-nancy.univ-lorraine.fr">guillaume.bonfante@mines-nancy.univ-lorraine.fr</a>
<a class="moz-txt-link-rfc2396E" href="mailto:guillaume.bonfante@mines-nancy.univ-lorraine.fr"><guillaume.bonfante@mines-nancy.univ-lorraine.fr></a>,
<a class="moz-txt-link-abbreviated" href="mailto:mikhail.prokopenko@sydney.edu.au">mikhail.prokopenko@sydney.edu.au</a>
<a class="moz-txt-link-rfc2396E" href="mailto:mikhail.prokopenko@sydney.edu.au"><mikhail.prokopenko@sydney.edu.au></a>, Neil Gershenfeld
<a class="moz-txt-link-rfc2396E" href="mailto:neil.gershenfeld@cba.mit.edu"><neil.gershenfeld@cba.mit.edu></a>, Koonin, Eugene
(NIH/NLM/NCBI) [E] <a class="moz-txt-link-rfc2396E" href="mailto:koonin@ncbi.nlm.nih.gov"><koonin@ncbi.nlm.nih.gov></a>, Oron
Shagrir <a class="moz-txt-link-rfc2396E" href="mailto:oron.shagrir@gmail.com"><oron.shagrir@gmail.com></a>,
<a class="moz-txt-link-abbreviated" href="mailto:Noson@sci.brooklyn.cuny.edu">Noson@sci.brooklyn.cuny.edu</a>
<a class="moz-txt-link-rfc2396E" href="mailto:Noson@sci.brooklyn.cuny.edu"><Noson@sci.brooklyn.cuny.edu></a>, Friston, Karl
<a class="moz-txt-link-rfc2396E" href="mailto:k.friston@ucl.ac.uk"><k.friston@ucl.ac.uk></a>, John Mattick
<a class="moz-txt-link-rfc2396E" href="mailto:j.mattick@unsw.edu.au"><j.mattick@unsw.edu.au></a></td>
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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 class="moz-cite-prefix">Le 05/12/2022 à 00:25, Louis Kauffman
a écrit :<br>
</div>
<blockquote type="cite"
cite="mid:E67E9DDD-D216-42CF-B2BF-E4FD9FB2EC02@gmail.com">
<meta http-equiv="Content-Type" content="text/html;
charset=UTF-8">
Dear Sheri,
<div class="">It would indeed be very helpful to have a zoom
conversation about these themes.</div>
<div class="">Please let me know when you would be available to
have it.</div>
<div class="">We could start with an hour meeting and discussion
and then perhaps extend to a second meeting with some
presentations.</div>
<div class="">Included below is a slide show of mine that is a
bit cryptic but does summarize some points of view.</div>
<div class="">I have downloaded the papers you indicated in your
email and will read them now.</div>
<div class="">Very best,</div>
<div class="">Lou </div>
<br>
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<div>
<blockquote type="cite" class="">
<div class="">On Dec 5, 2022, at 2:53 PM, Markose, Sheri
<<a href="mailto:scher@essex.ac.uk"
class="moz-txt-link-freetext" moz-do-not-send="true">scher@essex.ac.uk</a>>
wrote:</div>
<br class="Apple-interchange-newline">
<div class="">
<div class="WordSection1" style="page: WordSection1;
font-family: Helvetica; font-size: 12px; font-style:
normal; font-variant: normal; font-weight: normal;
letter-spacing: normal; line-height: normal; orphans:
auto; text-align: start; text-indent: 0px;
text-transform: none; white-space: normal; widows:
auto; word-spacing: 0px; -webkit-text-stroke-width:
0px;">
<div style="margin: 0cm; font-size: 11pt; font-family:
Calibri, sans-serif;" class=""><span
style="font-size: 12pt; font-family: Arial,
sans-serif; color: rgb(31, 73, 125);" class="">Dear
Louis, Pedro and All –<o:p class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt; font-family:
Calibri, sans-serif;" class=""><span
style="font-size: 12pt; font-family: Arial,
sans-serif; color: rgb(31, 73, 125);" class=""> </span></div>
<div style="margin: 0cm; font-size: 11pt; font-family:
Calibri, sans-serif;" class=""><span
style="font-size: 12pt; font-family: Arial,
sans-serif;" class="">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 class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt; font-family:
Calibri, sans-serif;" class=""><span
style="font-size: 12pt; font-family: Arial,
sans-serif;" class=""> </span></div>
<div style="margin: 0cm 0cm 0cm 54pt; font-size: 11pt;
font-family: Calibri, sans-serif; text-indent:
-36pt;" class=""><span style="font-size: 12pt;
font-family: Arial, sans-serif;" class=""><span
class="">(i)<span style="font-style: normal;
font-variant: normal; font-weight: normal;
font-size: 7pt; line-height: normal;
font-family: 'Times New Roman';" class=""> <span
class="Apple-converted-space"> </span></span></span></span><span
style="font-size: 12pt; font-family: Arial,
sans-serif;" class="">The bulk of Adam Svahn’s
work co-authored with Mikhail Prokopenko (S &
P) is already published and can be found here <o:p
class=""></o:p></span></div>
<div style="margin: 0cm 0cm 0cm 21.3pt; font-size:
11pt; font-family: Calibri, sans-serif;" class=""><span
style="font-size: 12pt; font-family: Arial,
sans-serif; border: 1pt none windowtext; padding:
0cm; background-color: white;" class=""><a href="https://urldefense.com/v3/__https://doi.org/10.1162/artl_a_00370__;!!D9dNQwwGXtA!R8lchYgfzUtRK_UvjczZjG4odrY4nCT90ap2vplSNdF7nmHYAjY30NMIkzAIZPRsIn5AxYjgQKn_X6p4BTG0vMTREdnb$" style="color: blue; text-decoration: underline;" class="" moz-do-not-send="true"><span style=""
class="">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
class="Apple-converted-space"> </span></span><span
style="font-family: Arial, sans-serif;" class="">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;" class=""><span
class="Apple-converted-space"> </span>S & P
share objectives as those that I have given<span
class="Apple-converted-space"> </span></span><span
style="font-family: Arial, sans-serif;" class="">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;" class=""><o:p class=""></o:p></span></div>
<div style="margin: 0cm 0cm 0cm 21.3pt; font-size:
11pt; font-family: Calibri, sans-serif;" class=""><span
style="font-family: Arial, sans-serif;" class=""> </span></div>
<div style="margin: 0cm 0cm 0cm 21.3pt; font-size:
11pt; font-family: Calibri, sans-serif;" class=""><span
style="font-size: 12pt; font-family: Arial,
sans-serif; border: 1pt none windowtext; padding:
0cm; background-color: white;" class="">S& R
give an interesting and plausible account of how
RNA-push down automata<span
class="Apple-converted-space"> </span></span><span
style="font-family: Arial, sans-serif;" class="">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
class="">nd</sup>stack to reflect on codes and
make changes to them.</span><span
style="font-size: 12pt; font-family: Arial,
sans-serif;" class=""><o:p class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt; font-family:
Calibri, sans-serif;" class=""><span
style="font-size: 12pt; font-family: Arial,
sans-serif; color: rgb(31, 73, 125);" class=""> </span></div>
<div style="margin: 0cm; font-size: 11pt; font-family:
Calibri, sans-serif;" class=""><span
style="font-size: 12pt; font-family: Arial,
sans-serif; color: rgb(31, 73, 125);" class=""> </span></div>
<div style="margin: 0cm 0cm 0cm 54pt; font-size: 11pt;
font-family: Calibri, sans-serif; text-indent:
-36pt;" class=""><span style="font-size: 12pt;
font-family: Arial, sans-serif; color: rgb(31, 73,
125);" class=""><span class="">(ii)<span
style="font-style: normal; font-variant:
normal; font-weight: normal; font-size: 7pt;
line-height: normal; font-family: 'Times New
Roman';" class=""> <span
class="Apple-converted-space"> </span></span></span></span><span
style="font-size: 12pt; font-family: Arial,
sans-serif; color: rgb(31, 73, 125);" class="">Section
5 of S&P relates to undecidability as fixed
points of negation functions<span
class="Apple-converted-space"> </span><span
style="background-color: yellow;
background-position: initial initial;
background-repeat: initial initial;" class="">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
class=""></o:p></span></div>
<div style="margin: 0cm 0cm 0cm 54pt; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 12pt; font-family: Arial,
sans-serif; color: rgb(31, 73, 125);" class=""> </span></div>
<div style="margin: 0cm 0cm 0cm 54pt; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 12pt; font-family: Arial,
sans-serif;" class="">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 class="Apple-converted-space"> </span><o:p
class=""></o:p></span></div>
<div style="margin: 0cm 0cm 0cm 54pt; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 12pt; font-family: Arial,
sans-serif; color: rgb(31, 73, 125);" class=""> <o:p
class=""></o:p></span></div>
<div style="margin: 0cm 0cm 0cm 54pt; font-size: 11pt;
font-family: Calibri, sans-serif; text-indent:
-36pt;" class=""><span style="font-size: 12pt;
font-family: Arial, sans-serif; color: rgb(31, 73,
125);" class=""><span class="">(iii)<span
style="font-style: normal; font-variant:
normal; font-weight: normal; font-size: 7pt;
line-height: normal; font-family: 'Times New
Roman';" class=""> <span
class="Apple-converted-space"> </span></span></span></span><span
style="font-size: 12pt; font-family: Arial,
sans-serif;" class="">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
class="Apple-converted-space"> </span><span
style="color: rgb(31, 73, 125);" class="">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 class="Apple-converted-space"> </span></span></span><span
style="font-size: 12pt; font-family: Arial,
sans-serif;" class="">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
class="">
neuroimmunology." By Aurora Krauset al. In, eLife
2021;10:e66706. DOI:<span
class="Apple-converted-space"> </span></span><span
style="font-size: 12pt; font-family: Arial,
sans-serif;" class=""><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!R8lchYgfzUtRK_UvjczZjG4odrY4nCT90ap2vplSNdF7nmHYAjY30NMIkzAIZPRsIn5AxYjgQKn_X6p4BTG0vN18TkB3$" style="color: blue; text-decoration: underline;" class="" moz-do-not-send="true">https://doi.org/10.7554/eLife.66706</a></span><span
style="font-size: 12pt; font-family: Arial,
sans-serif;" class="">.<br class="">
<br class="">
</span><span style="font-size: 12pt; font-family:
Arial, sans-serif; color: rgb(31, 73, 125);"
class=""><o:p class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt; font-family:
Calibri, sans-serif;" class=""><span
style="font-size: 12pt; font-family: Arial,
sans-serif; color: rgb(31, 73, 125);" class="">
If you recall how the Recursive Fixed Point
Theorem which starts with a mirror mapping between
online self-assembly program execution,<span
class="Apple-converted-space"> </span></span><i
class=""><span style="font-size: 12pt;
font-family: Symbol;" class="">f</span></i><i
class=""><sub class=""><span style="font-size:
12pt;" class="">g</span></sub></i><i class=""><span
style="font-size: 12pt;" class="">(g)</span></i><span
style="font-size: 12pt;" class="">)</span><span
class="Apple-converted-space"> </span><span
style="font-size: 12pt; font-family: Arial,
sans-serif; color: rgb(31, 73, 125);" class=""> that
have halted and create somatic identity and the<span
class="Apple-converted-space"> </span><i
class="">offline</i><span
class="Apple-converted-space"> </span>record of
the<span class="Apple-converted-space"> </span><br
class="">
same in a<span
class="Apple-converted-space"> </span><span
style="background-color: yellow;
background-position: initial initial;
background-repeat: initial initial;" class="">2-<span
class="Apple-converted-space"> </span></span> <span
style="background-color: yellow;
background-position: initial initial;
background-repeat: initial initial;" class="">place</span>
function <span class="Apple-converted-space"> </span></span><span
style="font-size: 12pt; font-family: Symbol;
color: rgb(31, 73, 125);" class="">s</span><span
style="font-size: 12pt; font-family: Arial,
sans-serif; color: rgb(31, 73, 125);" class=""><span
class="Apple-converted-space"> </span>(g, g) in
the MHC receptors creating the Thymic self. I say
the first g from the left in<span
class="Apple-converted-space"> </span></span><span
style="font-size: 12pt; font-family: Symbol;
color: rgb(31, 73, 125);" class="">s</span><span
style="font-size: 12pt; font-family: Arial,
sans-serif; color: rgb(31, 73, 125);" class=""><span
class="Apple-converted-space"> </span>(g, g)
and changes thereof relate to what happens to
self and the second is self’s record of what<span
class="Apple-converted-space"> </span><br
class="">
the other has done to self. So if
the 2<sup class="">nd</sup><span
class="Apple-converted-space"> </span>entry is
different from the first entry in<span
class="Apple-converted-space"> </span></span><span
style="font-size: 12pt; font-family: Symbol;
color: rgb(31, 73, 125);" class="">s</span><span
style="font-size: 12pt; font-family: Arial,
sans-serif; color: rgb(31, 73, 125);" class=""><span
class="Apple-converted-space"> </span>(g, g) it
is off diagonal etc. The non-self hostile other
is a projection of self g ‘gene codes’ and those <span
class="Apple-converted-space"> </span><i
class="">f ¬ !</i> which are reactive to the
self g-<br class="">
codes denoted as g¬. As we know
an astronomic number of potential indexes g¬ are
generated by the RAG genes. <o:p class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt; font-family:
Calibri, sans-serif;" class=""><span
style="font-size: 12pt; font-family: Arial,
sans-serif; color: rgb(31, 73, 125);" class=""> </span></div>
<div style="margin: 0cm; font-size: 11pt; font-family:
Calibri, sans-serif;" class=""><span
style="font-size: 12pt; font-family: Arial,
sans-serif; color: rgb(31, 73, 125);" class="">
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
class="Apple-converted-space"> </span><br
class="">
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
class="Apple-converted-space"> </span><br
class="">
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
class="Apple-converted-space"> </span><i
class="">f ¬ !</i> requires<span
class="Apple-converted-space"> </span><br
class="">
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 class="">
<br class="">
<o:p class=""></o:p></span></div>
<div style="margin: 0cm 0cm 0cm 54pt; font-size: 11pt;
font-family: Calibri, sans-serif; text-indent:
-36pt;" class=""><span style="font-size: 12pt;
font-family: Arial, sans-serif; color: rgb(31, 73,
125);" class=""><span class="">(iv)<span
style="font-style: normal; font-variant:
normal; font-weight: normal; font-size: 7pt;
line-height: normal; font-family: 'Times New
Roman';" class=""> <span
class="Apple-converted-space"> </span></span></span></span><span
style="font-size: 12pt; font-family: Arial,
sans-serif; color: rgb(31, 73, 125);" class="">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 class=""></o:p></span></div>
<div style="margin: 0cm 0cm 0cm 54pt; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 12pt; font-family: Arial,
sans-serif; color: rgb(31, 73, 125);" class=""> </span></div>
<div style="margin: 0cm 0cm 0cm 54pt; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 12pt; font-family: Arial,
sans-serif; color: rgb(31, 73, 125);" class="">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 class=""></o:p></span></div>
<div style="margin: 0cm 0cm 0cm 54pt; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 12pt; font-family: Arial,
sans-serif; color: rgb(31, 73, 125);" class="">Ditto
for many others who I hope to be in touch with
soon. <o:p class=""></o:p></span></div>
<div style="margin: 0cm 0cm 0cm 54pt; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 12pt; font-family: Arial,
sans-serif; color: rgb(31, 73, 125);" class=""> </span></div>
<div style="margin: 0cm 0cm 0cm 54pt; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 12pt; font-family: Arial,
sans-serif; color: rgb(31, 73, 125);" class="">Many
thanks again for the great comments.<o:p class=""></o:p></span></div>
<div style="margin: 0cm 0cm 0cm 54pt; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 12pt; font-family: Arial,
sans-serif; color: rgb(31, 73, 125);" class=""> </span></div>
<div style="margin: 0cm 0cm 0cm 54pt; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 12pt; font-family: Arial,
sans-serif; color: rgb(31, 73, 125);" class="">All
best<o:p class=""></o:p></span></div>
<div style="margin: 0cm 0cm 0cm 54pt; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 12pt; font-family: Arial,
sans-serif; color: rgb(31, 73, 125);" class=""> </span></div>
<div style="margin: 0cm 0cm 0cm 54pt; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 12pt; font-family: Arial,
sans-serif; color: rgb(31, 73, 125);" class="">Sheri <o:p
class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt; font-family:
Calibri, sans-serif;" class=""><span
style="font-size: 12pt; font-family: Arial,
sans-serif; color: rgb(31, 73, 125);" class=""> </span></div>
<div style="margin: 0cm; font-size: 11pt; font-family:
Calibri, sans-serif;" class=""><span
style="font-size: 12pt; font-family: Arial,
sans-serif; color: rgb(31, 73, 125);" class=""> </span></div>
<div style="margin: 0cm; font-size: 11pt; font-family:
Calibri, sans-serif;" class=""><span
style="font-size: 12pt; font-family: Arial,
sans-serif; color: rgb(31, 73, 125);" class="">
<o:p class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt; font-family:
Calibri, sans-serif;" class=""><span
style="font-size: 12pt; font-family: Arial,
sans-serif; color: rgb(31, 73, 125);" class=""> </span></div>
<div style="margin: 0cm; font-size: 11pt; font-family:
Calibri, sans-serif;" class=""><span
style="font-size: 12pt; font-family: Arial,
sans-serif; color: rgb(31, 73, 125);" class=""> <o:p
class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt; font-family:
Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif; color: rgb(31, 73, 125);" class=""> </span></div>
<div style="margin: 0cm; font-size: 11pt; font-family:
Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif; color: rgb(31, 73, 125);" class=""> </span></div>
<div style="margin: 0cm; font-size: 11pt; font-family:
Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif; color: rgb(31, 73, 125);" class=""> </span></div>
<div style="margin: 0cm; font-size: 11pt; font-family:
Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif; color: rgb(31, 73, 125);" class=""> </span></div>
<div class="">
<div style="border-style: solid none none;
border-top-color: rgb(225, 225, 225);
border-top-width: 1pt; padding: 3pt 0cm 0cm;"
class="">
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><b
class=""><span class="" lang="EN-US">From:</span></b><span
class="" lang="EN-US"><span
class="Apple-converted-space"> </span>Pedro
C. Marijuán <<a
href="mailto:pedroc.marijuan@gmail.com"
style="color: blue; text-decoration:
underline;" class="moz-txt-link-freetext"
moz-do-not-send="true">pedroc.marijuan@gmail.com</a>><span
class="Apple-converted-space"> </span><br
class="">
<b class="">Sent:</b><span
class="Apple-converted-space"> </span>01
December 2022 13:03<br class="">
<b class="">To:</b><span
class="Apple-converted-space"> </span>Markose,
Sheri <<a href="mailto:scher@essex.ac.uk"
style="color: blue; text-decoration:
underline;" class="moz-txt-link-freetext"
moz-do-not-send="true">scher@essex.ac.uk</a>>;
Louis Kauffman <<a
href="mailto:loukau@gmail.com" style="color:
blue; text-decoration: underline;"
class="moz-txt-link-freetext"
moz-do-not-send="true">loukau@gmail.com</a>><br
class="">
<b class="">Cc:</b><span
class="Apple-converted-space"> </span>fis
<<a href="mailto:fis@listas.unizar.es"
style="color: blue; text-decoration:
underline;" class="moz-txt-link-freetext"
moz-do-not-send="true">fis@listas.unizar.es</a>>;<span
class="Apple-converted-space"> </span><a
href="mailto:guillaume.bonfante@mines-nancy.univ-lorraine.fr"
style="color: blue; text-decoration:
underline;" class="moz-txt-link-freetext"
moz-do-not-send="true">guillaume.bonfante@mines-nancy.univ-lorraine.fr</a><br
class="">
<b class="">Subject:</b><span
class="Apple-converted-space"> </span>Re:
[Fis] A new discussion session<o:p class=""></o:p></span></div>
</div>
</div>
<div style="margin: 0cm; font-size: 11pt; font-family:
Calibri, sans-serif;" class=""><o:p class=""> </o:p></div>
<div class="">
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class="">Dear
Sheri, Lou, and all discussants,<span class=""><o:p
class=""></o:p></span></div>
</div>
<div class="">
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><o:p
class=""> </o:p></div>
</div>
<div class="">
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class="">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 class="Apple-converted-space"> </span><o:p
class=""></o:p></div>
</div>
<div class="">
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><o:p
class=""> </o:p></div>
</div>
<div class="">
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class="">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 class="">
neuroimmunology." By Aurora Krauset al. In, eLife
2021;10:e66706. DOI:<span
class="Apple-converted-space"> </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!R8lchYgfzUtRK_UvjczZjG4odrY4nCT90ap2vplSNdF7nmHYAjY30NMIkzAIZPRsIn5AxYjgQKn_X6p4BTG0vN18TkB3$" style="color: blue; text-decoration: underline;" class="" moz-do-not-send="true">https://doi.org/10.7554/eLife.66706</a><span
style="font-size: 10pt; font-family: 'Times New
Roman', serif;" class="">.</span><span
class="Apple-converted-space"> </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
class=""></o:p></div>
</div>
<div class="">
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><o:p
class=""> </o:p></div>
</div>
<div class="">
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class="">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
class="Apple-converted-space"> </span><u
class="">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
class="Apple-converted-space"> </span><o:p
class=""></o:p></div>
</div>
<div class="">
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><o:p
class=""> </o:p></div>
</div>
<div class="">
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class="">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
class=""></o:p></div>
</div>
<div class="">
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class="">Best--Pedro<o:p
class=""></o:p></div>
</div>
<div class="">
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><o:p
class=""> </o:p></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><o:p
class=""> </o:p></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><o:p
class=""> </o:p></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif; color: rgb(31, 73, 125);" class=""> </span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><b
class=""><span class="" lang="EN-US">From:</span></b><span
class="" lang="EN-US"><span
class="Apple-converted-space"> </span>Louis
Kauffman<span class="Apple-converted-space"> </span></span><a
href="mailto:loukau@gmail.com" style="color:
blue; text-decoration: underline;" class=""
moz-do-not-send="true"><span class=""
lang="EN-US">loukau@gmail.com</span></a><span
class="" lang="EN-US"><span
class="Apple-converted-space"> </span><br
class="">
<b class="">Sent:</b><span
class="Apple-converted-space"> </span>15
November 2022 23:02<br class="">
<b class="">To:</b><span
class="Apple-converted-space"> </span>Markose,
Sheri<span class="Apple-converted-space"> </span></span><a
href="mailto:scher@essex.ac.uk" style="color:
blue; text-decoration: underline;" class=""
moz-do-not-send="true"><span class=""
lang="EN-US">scher@essex.ac.uk</span></a><span
class="" lang="EN-US"><br class="">
<b class="">Cc:</b><span
class="Apple-converted-space"> </span>"Pedro
C. Marijuán"<span class="Apple-converted-space"> </span></span><a
href="mailto:pedroc.marijuan@gmail.com"
style="color: blue; text-decoration: underline;"
class="" moz-do-not-send="true"><span class=""
lang="EN-US">pedroc.marijuan@gmail.com</span></a><span
class="" lang="EN-US">; fis<span
class="Apple-converted-space"> </span></span><a
href="mailto:fis@listas.unizar.es" style="color:
blue; text-decoration: underline;" class=""
moz-do-not-send="true"><span class=""
lang="EN-US">fis@listas.unizar.es</span></a><span
class="" lang="EN-US">;<span
class="Apple-converted-space"> </span></span><a
href="mailto:guillaume.bonfante@mines-nancy.univ-lorraine.fr"
style="color: blue; text-decoration: underline;"
class="" moz-do-not-send="true"><span class=""
lang="EN-US">guillaume.bonfante@mines-nancy.univ-lorraine.fr</span></a><span
class="" lang="EN-US"><br class="">
<b class="">Subject:</b><span
class="Apple-converted-space"> </span>Re:
[Fis] A new discussion session<o:p class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><o:p
class=""> </o:p></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class="">Sheri,<span
class=""><o:p class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class="">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
class=""></o:p></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class="">My
letter was quite long, and it is possible to not
get to the second half.<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class="">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 class=""></o:p></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class="">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 class=""></o:p></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class="">Best,<o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class="">Lou K.<o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><o:p
class=""> </o:p></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class="">##########<o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class="">It is a very interesting
question whether such encoding or such multiple
relationships to context occur in biology. Here
are some remarks.<o:p class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class=""> <o:p class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class="">1. In biology is is
NORMALLY the case that certain key structures
have multiple interpretations and uses in
various contexts.<o:p class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class="">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 class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class=""> <o:p class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class="">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 class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class=""> <o:p class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class="">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 class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class="">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
class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class=""> <o:p class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class="">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
class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class="">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
class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class="">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 class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class=""> <o:p class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class="">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 class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class="">PX: X can be printed.<o:p
class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class="">~PX: X cannot be printed.<o:p
class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class="">PRX: XX can be printed.<o:p
class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class="">~PRX: XX cannot be
printed.<o:p class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class="">Here X is any string of
the symbols {~,P,R}.<o:p class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class="">Thus PR~~P means that XX
can be printed where X = ~~P. Thus PR~~P means
that ~~P~~P can be printed.<o:p class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class="">By printed we mean on one
press of the button on the Machine, a string of
characters is printed.<o:p class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class="">IT IS ASSUMED THAT THE
SMULLYAN MACHINE ALWAYS TELLS THE TRUTH WHEN IT
PRINTS A MEANINGFUL STATEMENT.<o:p class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class="">Then we have the<o:p
class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class=""> <o:p class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class="">Theorem. There are
meaningful true strings that the Smullyan
Machine cannot print.<o:p class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class=""> <o:p class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class="">This is a non-numerical
analog of the Goedel Theorem. And the string
that cannot be printed is G = ~PR~PR.<o:p
class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class="">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
class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class="">If the machine were to
print G, it would lie. And the machine does not
lie.<o:p class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class="">Therefore G is
unprintable.<o:p class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class="">But this is what G says.<o:p
class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class="">So we have established the
truth of G and proved the Theorem.<o:p class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class=""> <o:p class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class="">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 class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class="">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 class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class=""> <o:p class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class="">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
class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class="">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 class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class=""> <o:p class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class="">8. I will stop here. The
relationships with biology are very worth
discussing.<o:p class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class="">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 class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class=""> ###############<o:p
class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class="">Best,<o:p class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class="">Lou Kauffman<o:p class=""></o:p></span></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><span
style="font-size: 10pt; font-family: Arial,
sans-serif;" class=""><br class="">
<br class="">
</span><o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class=""><o:p
class=""> </o:p></div>
</div>
<div class="">
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class="">El
15/11/2022 a las 21:19, Markose, Sheri escribió:<o:p
class=""></o:p></div>
</div>
<blockquote style="margin-top: 5pt; margin-bottom:
5pt;" class="">
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">Dear
Louis, dear Colleagues - <span
class="Apple-converted-space"> </span><o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">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
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">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 class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">1. I
have found the following statement by Joel Hamkins
( :<span class="Apple-converted-space"> </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!R8lchYgfzUtRK_UvjczZjG4odrY4nCT90ap2vplSNdF7nmHYAjY30NMIkzAIZPRsIn5AxYjgQKn_X6p4BTG0vHnETe5i$" style="color: blue; text-decoration: underline;" class="" moz-do-not-send="true"><span
style="font-size: 11pt; font-family: Calibri,
sans-serif;" class="">http://jdh.hamkins.org/wp-content/uploads/A-review-of-several-fixed-point-theorems-1.pdf</span></a><span
class="Apple-converted-space"> </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.<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 11pt;
font-family: Calibri, sans-serif;" class="">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 class="Apple-converted-space"> </span><i
class="">g builds the<span
class="Apple-converted-space"> </span></i>machine
that runs g.<span class="Apple-converted-space"> </span><o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">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
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""><span
id="cid:image001.png@01D905CE.ACBBA200"><image001.png></span><o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">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 class="Apple-converted-space"> </span><o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">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 class="Apple-converted-space"> </span><span
style="font-size: 12pt;" class="">g<sup class="">¬</sup></span><span
class="Apple-converted-space"> </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 class="Apple-converted-space"> </span><o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">II.
Once, (8) is in place, the Adaptive immune system
has to identify novel negation software function<span
class="Apple-converted-space"> </span><i
class="">f<sup class="">¬!<span
class="Apple-converted-space"> </span></sup></i><sub
class=""> </sub>of non-self antigens which is an
uncountable infinite possibilities. Hence the
close to astronomic search with V(D) J of 10<sup
class="">20<span class="Apple-converted-space"> </span></sup>–
10<span class="Apple-converted-space"> </span><sup
class="">30</sup><span
class="Apple-converted-space"> </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.<o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">III.
Roger Fixed Point assures us that the indexes of
the fixed point for<span
class="Apple-converted-space"> </span><i
class="">f<sup class="">¬!<span
class="Apple-converted-space"> </span></sup></i><sub
class=""> </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 class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">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
class="Apple-converted-space"> </span><o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">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 class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">So
thankyou all again for your in depth comments and
interest.<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">Best
Regards<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">Sheri<span
class="Apple-converted-space"> </span><o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">-----Original
Message-----<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">From:
Fis <<a
href="mailto:fis-bounces@listas.unizar.es"
style="color: blue; text-decoration: underline;"
class="moz-txt-link-freetext"
moz-do-not-send="true">fis-bounces@listas.unizar.es</a>>
On Behalf Of Louis Kauffman<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">Sent: 08
November 2022 00:13<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">To:
"Pedro C. Marijuán" <<a
href="mailto:pedroc.marijuan@gmail.com"
style="color: blue; text-decoration: underline;"
class="moz-txt-link-freetext"
moz-do-not-send="true">pedroc.marijuan@gmail.com</a>><o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">Cc: fis
<<a href="mailto:fis@listas.unizar.es"
style="color: blue; text-decoration: underline;"
class="moz-txt-link-freetext"
moz-do-not-send="true">fis@listas.unizar.es</a>><o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">Subject:
Re: [Fis] A new discussion session<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">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 class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">Dear
Pedro,<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">Here are
some comments about Goedel numbering and coding.<o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">It is
interesting to think about Goedel numbering in a
biological context.<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">Actually
we are talking about how a given entity has
semantics that can vary from context to context.<o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">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
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">1. F as
a well formed formula in a formal system S.<o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">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 class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">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 class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">This is
the key to the references and self-references of
the Goedelian situations.<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">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
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">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 class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">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 class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">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 class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">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 class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">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 class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">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 class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">Now
suppose that<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">g —>
F(x).<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">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
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">#g —>
F(g).<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">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 class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">Then we
can consider<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">F(#x)
and it has a Goedel number<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">h —>
F(#x)<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">And we
can shift that to<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">#h —>
F(#h).<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">This is
the key point.<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">Now we
have constructed a number #h so that F(#h)
discusses its own Goedel number.<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">This
construction allows the proof of the Goedel
Incompleteness Theorem via the sentence B(x) that
states<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">B(x) =
“The statement with Goedel number x is provable in
S.” (This can also be encoded in S.)<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">We then
construct<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">h—>
~B(#x)<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">and<o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">#h —>
~B(#h)<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">and
obtain the statement<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">G=
~B(#h).<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">G states
the unprovability of the Goedel decoding of #h.<o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">But the
Goedel decoding of #h is the statement G itself.<o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">Thus G
states its own unprovability.<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">Therefore,
S being consistent, cannot prove G.<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">By
making these arguments we have have proved that G
cannot be proved by S.<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">Thus we
have shown that G is in fact true.<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">We have
shown that there are true statements in number
theory unprovable by system S..<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">##########################<o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">The
above is a very concise summary of the proof of
Goedel’s Incompleteness Theorem, using Goedel
number encoding.<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">It is a
very interesting question whether such encoding or
such multiple relationships to context occur in
biology. Here are some remarks.<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">1. In
biology is is NORMALLY the case that certain key
structures have multiple interpretations and uses
in various contexts.<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">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 class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">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 class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">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 class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">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 class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">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
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">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
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">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 class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">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 class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">PX: X
can be printed.<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">~PX: X
cannot be printed.<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">PRX: XX
can be printed.<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">~PRX: XX
cannot be printed.<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">Here X
is any string of the symbols {~,P,R}.<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">Thus
PR~~P means that XX can be printed where X = ~~P.
Thus PR~~P means that ~~P~~P can be printed.<o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">By
printed we mean on one press of the button on the
Machine, a string of characters is printed.<o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">IT IS
ASSUMED THAT THE SMULLYAN MACHINE ALWAYS TELLS THE
TRUTH WHEN IT PRINTS A MEANINGFUL STATEMENT.<o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">Then we
have the<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">Theorem.
There are meaningful true strings that the
Smullyan Machine cannot print.<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">This is
a non-numerical analog of the Goedel Theorem. And
the string that cannot be printed is G = ~PR~PR.<o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">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
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">If the
machine were to print G, it would lie. And the
machine does not lie.<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">Therefore
G is unprintable.<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">But this
is what G says.<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">So we
have established the truth of G and proved the
Theorem.<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">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
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">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 class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">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
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">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 class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">8. I
will stop here. The relationships with biology are
very worth discussing.<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">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 class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">Very
best,<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">Lou
Kauffman<o:p class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class=""> <o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">_______________________________________________<o:p
class=""></o:p></div>
<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">Fis
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<div style="margin: 0cm; font-size: 10pt;
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<div style="margin: 0cm; font-size: 10pt;
font-family: Arial, sans-serif;" class="">Recuerde
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<div style="margin: 0cm; font-size: 10pt;
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class=""></o:p></div>
</blockquote>
<p class=""><o:p class=""> </o:p></p>
<div id="DAB4FAD8-2DD7-40BB-A1B8-4E2AA1F9FDF2"
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<div style="margin: 0cm; font-size: 11pt;
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<div style="margin: 0cm; font-size: 11pt;
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class=""><a href="https://urldefense.com/v3/__https://www.avast.com/sig-email?utm_medium=email&utm_source=link&utm_campaign=sig-email&utm_content=emailclient__;!!D9dNQwwGXtA!R8lchYgfzUtRK_UvjczZjG4odrY4nCT90ap2vplSNdF7nmHYAjY30NMIkzAIZPRsIn5AxYjgQKn_X6p4BTG0vNvQg6vb$" target="_blank" style="color: blue;
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line-height: 13.5pt;" class=""><span
style="font-size: 10pt; font-family:
Arial, sans-serif; color: rgb(65, 66,
78);" class="">Libre de virus.</span><a href="https://urldefense.com/v3/__https://www.avast.com/sig-email?utm_medium=email&utm_source=link&utm_campaign=sig-email&utm_content=emailclient__;!!D9dNQwwGXtA!R8lchYgfzUtRK_UvjczZjG4odrY4nCT90ap2vplSNdF7nmHYAjY30NMIkzAIZPRsIn5AxYjgQKn_X6p4BTG0vNvQg6vb$" target="_blank" style="color: blue;
text-decoration: underline;" class="" moz-do-not-send="true"><span
style="font-size: 10pt; font-family:
Arial, sans-serif; color: rgb(68, 83,
234);" class="">www.avast.com</span></a><span
style="font-size: 10pt; font-family:
Arial, sans-serif; color: rgb(65, 66,
78);" class=""><o:p class=""></o:p></span></div>
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