<div dir="ltr">Dear Carlo, as an economist I agree with you when you say: «Information is the degree of being different from what is expected». Information is giving body to something or someone or taking shape from something or someone. My Trans-information is a change or transmutation of form that occurs over time. Thus «The discrepancies that present themselves as an artifact of the counting system are in their essence information». Such counting, calculating, computing is nothing but a matter of quantum reality - that's why my economics is quantum. This is the comment I send you with affection, also because you say what you do and do what you say.<br>A hug,<br>Francis<br><div><p class="MsoNormal" style="margin:0cm;text-align:justify;text-indent:14.2pt;line-height:normal;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:12pt;font-family:"Times New Roman",serif">Caro
Carlo, sono da economista d’accordo con Te quando dici: «L’informazione è il
grado di essere diverso da quanto previsto». L’informazione è un dare corpo a
qualcosa o qualcuno o il prendere corpo da qualcosa o qualcuno. La mia "Tras-informazione" è un cambiamento o una <b>trasmutazione</b> di forma che
avviene nel tempo. Quindi «Le discordanze che si presentano come un arte-fatto
del sistema di conteggio sono nella loro essenza l’informazione». Tale
conteggio, calcolo, computazione non è altro che una questione di realtà
quantica: ecco perché la mia economia è quantistica. Questo è il commento che
Ti mando con affetto, anche perché dici quel che fai e fai quel che dici.</span></p>
<p class="MsoNormal" style="margin:0cm;text-align:justify;text-indent:14.2pt;line-height:normal;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:12pt;font-family:"Times New Roman",serif">Un
abbraccio,</span></p>
<p class="MsoNormal" style="margin:0cm;text-align:justify;text-indent:14.2pt;line-height:normal;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:12pt;font-family:"Times New Roman",serif">Francesco</span></p>
<p class="MsoNormal" style="margin:0cm;text-align:justify;text-indent:14.2pt;line-height:normal;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:12pt;font-family:"Times New Roman",serif"> </span></p>
<p class="MsoNormal" style="margin:0cm;text-align:justify;text-indent:14.2pt;line-height:normal;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:12pt;font-family:"Times New Roman",serif"> </span></p></div></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">Il giorno ven 2 dic 2022 alle ore 13:22 Karl Javorszky <<a href="mailto:karl.javorszky@gmail.com" target="_blank">karl.javorszky@gmail.com</a>> ha scritto:<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 dir="ltr"><div dir="ltr"><div dir="ltr"><p class="MsoNormal"><span lang="HU">Another new
session within same frames of reference<span></span></span></p>
<p class="MsoNormal"><span lang="HU"> </span></p>
<p class="MsoNormal"><span lang="HU">Dear Friends,<span></span></span></p>
<p class="MsoNormal"><span lang="HU"> </span></p>
<p class="MsoNormal"><span lang="HU">There was an
interesting one-day conference online </span>just a few days ago on Zoom re Information
and Philosophy. <span></span></p>
<p class="MsoNormal">There are many noteworthy efforts to gain a deep
understanding of the meaning of the word ‘information’. Yet, they all remain –
alas! – within the same system of references. The cultural taboos and their
positive form, cultural agreements give us a standard frame of references which
is not easy to discard, leave behind as world views of a naïve person.<span></span></p>
<p class="MsoNormal">In clinial psychology, one is routinely used to discarding
the common framework of what is correct and reasonable, while one tries to
understand one specific client’s perspectives of the world. One has to reframe
the context so that it makes sense (at least for the client).<span></span></p>
<p class="MsoNormal"><span> </span></p>
<p class="MsoNormal">Two main points:<span></span></p>
<p>1.<span style="font-variant-numeric:normal;font-variant-east-asian:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span>Discard the unified, seamlessly fitting concept
of the whole (which is imagined to be made up of identical elementary units)<span></span></p>
<p>2.<span style="font-variant-numeric:normal;font-variant-east-asian:normal;font-stretch:normal;font-size:7pt;line-height:normal;font-family:"Times New Roman"">
</span>Assume the existence of a crack along the whole,
which crack is serrated and can locally be overcome, but will not cease to
exist, because the two parts that one wants to fit together are by their nature
different. One cannot fit a lid on a cup if the diameters are slightly deviant,
whichever method one tries to trick one into fit with the other.<span></span></p>
<p class="MsoNormal"><span> </span></p>
<p class="MsoNormal">Doing the same but in stereo<span></span></p>
<p class="MsoNormal">One needs agreement that (while using <i>one </i>linear enumeration the problem appears not to exist) using <i>two </i>enumerations reveals inbuilt
controversies among the relations that underly <i>a+b=c. </i>We can put to use the diverse variants of the slight overall
discongruence which exists as a logical fact, to be read out of the properties
of natural numbers.<span></span></p>
<p class="MsoNormal">We need to agree that the world consists of two parts. The
parts <i>a, b </i>do not fit seamlessly into
an entity <i>c. </i>The extent of the
non-fit can be made visible by means of planar geometry. While <i>a+b=c </i>is true in a linear context, <i>pos(a) + pos(b) {<,=,>} pos(c)</i>
shows many, diverse variants of being not true, which variants each have a
definite numeric extent.<span></span></p>
<p class="MsoNormal">Treating the logical primitives in the context of planar
geometry, their interrelations give rise to many concepts the meaning of which
we have created by relating measurements of Nature. We find mathematical pictures
of natural constants.<span></span></p>
<p class="MsoNormal"><span> </span></p>
<p class="MsoNormal">Information is the extent of being otherwise than expected<span></span></p>
<p class="MsoNormal">Sorting and ordering a grand dozen of tin soldiers one will
observe typical patterns of movements while the exchanges of places take place.
These patterns are assemblies of numeric values. They can be brought into a web
of mutual expectations. This is a conceptually complicated but technically simple
data base management problem. <span></span></p>
<p class="MsoNormal"><span> </span></p>
<p class="MsoNormal">Change the Frame (coulisses, conceptual landscape, system of
what is normal)<span></span></p>
<p class="MsoNormal">Imagine a generally dualistic world. The two parts coexist
and transcend each other. The diameter of the proton is the result of numeric interdependencies
in a system of values which is the proton’s own system of references. The
diameter of the neutron is the result of numeric interdependencies in a system
of values which is the neutron’s own system of references. For our perception
we can establish two interacting systems of references when observing movement
patterns of realizations of <i>(a,b). </i>Having <i>two </i>counting systems in use,
we can exactly point out those snippets that are generated when our brain
assembles <i>one </i>picture out of the <i>two </i>sets of data coming from the
eyes. The discongruences that come as an artefact of the counting system are in
their essence the information. <span></span></p>
<p class="MsoNormal"><span> </span></p>
<p class="MsoNormal">Looking forward
your comments<span></span></p>
<p class="MsoNormal">Karl<span></span></p>
<p class="MsoNormal"><span> </span></p></div></div></div><br><div class="gmail_quote"><div dir="ltr" class="gmail_attr">Am Do., 1. Dez. 2022 um 14:03 Uhr schrieb Pedro C. Marijuán <<a href="mailto:pedroc.marijuan@gmail.com" target="_blank">pedroc.marijuan@gmail.com</a>>:<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>
<div>Dear Sheri, Lou, and all discussants,</div>
<div><br>
</div>
<div>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: <br>
</div>
<div><br>
</div>
<div>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 role="presentation">
neuroimmunology." By Aurora Krauset al. In, eLife 2021;10:e66706.
DOI: <a href="https://urldefense.com/v3/__https://doi.org/10.7554/eLife.66706__;!!D9dNQwwGXtA!SqizL3x3CcntbhLaIdrhBHmGMi7btD5ZwhfvU15L4xIxjE3QCOJv6ua8XvelEf-PtASzBqEnHi_-s40a8WSen8bu1LAR$" target="_blank">https://doi.org/10.7554/eLife.66706</a><span style="font-size:13.2836px;font-family:serif" role="presentation" dir="ltr">.</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...</div>
<div><br>
</div>
<div>And second, about the adaptive immune
system, it is where the ongoing mostly formal discussion would
apply (can we agree with that?). Then, it seems that the core of
this adaptive immune branch is the Major Histocompatibility
Complex molecule (MHC). This MHC molecules of two major classes
are highly complex (polygenic and polymorphic) and they are in
charge of presenting to lymphocyte T cells the protein fragments
churned out from the proteosomes inside cells (fragments of
variable lenght: 8-10 amino acids residues
for Class 1, mostly "self", and 13-18 amino acids residues for
Class 2, mostly "non self"). Then, the thymus is in charge of
deactivating the T cells loaded with self stuff. My point is that
the defense in front of the non-self is based on <u>indirect
products of protein translation</u>. This causes me some
uneasiness, as protein translation (see Youri's presentation
months ago) introduces a layer of extra complexity, not to speak
the processing via proteosomes. Further, with just 10 or 12 amino
acids can we faithfully ascertain algorithmic non-self
provenance?? <br>
</div>
<div><br>
</div>
<div>Well, Sheri is far more acknowledged
with all this stuff. And perhaps Lou can say something about the
formal distinguishability of 10-12 aa.<br>
</div>
<div>Best--Pedro<br>
</div>
<div><br>
</div>
<div>El 15/11/2022 a las 21:19, Markose,
Sheri escribió:<br>
</div>
<blockquote type="cite">
<div>
<p>Dear Louis, dear Colleagues - <u></u><u></u></p>
<p><u></u> <u></u></p>
<p>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></p>
<p><u></u> <u></u></p>
<p>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></p>
<p><u></u> <u></u></p>
<p>1. I have found the following statement
by Joel Hamkins ( :
<a href="https://urldefense.com/v3/__http://jdh.hamkins.org/wp-content/uploads/A-review-of-several-fixed-point-theorems-1.pdf__;!!D9dNQwwGXtA!SqizL3x3CcntbhLaIdrhBHmGMi7btD5ZwhfvU15L4xIxjE3QCOJv6ua8XvelEf-PtASzBqEnHi_-s40a8WSenw8_avhB$" 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>
) 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></p>
<p><u></u> <u></u></p>
<p class="MsoNormal">I choose Rogers Fixed Point Theorem of
(total) computable functions starting with the staple I have
already indicated Diag (g) (RHS of (8) below) is what Neil
Gerschenfeld calls ribosomal self-assembly machines in gene
expression where the program <i>g builds the </i>machine
that runs g. <u></u><u></u></p>
<p><u></u> <u></u></p>
<p><u></u> <u></u></p>
<p>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></p>
<p><u></u> <u></u></p>
<p><u></u> <u></u></p>
<p><img style="width: 9.3333in; height: 2.5in;" id="m_-303142184314655114m_-1490578781154761285m_1703372105328360732m_6698188255541615770Picture_x0020_1" src="cid:184d2c8c9a64cff311" width="896" height="240" border="0"><u></u><u></u></p>
<p><u></u> <u></u></p>
<p>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. <u></u><u></u></p>
<p><u></u> <u></u></p>
<p>I have suggested that the Adaptive
Immune System and the Mirror Neuron System have these
structures in (8). And the domain of self-halting machines as
in (8) are the Theorems of the system and a subset of Post
(1944) Creative Set. The non-Theorems have codes say <span style="font-size:12pt">g<sup>¬</sup></span> which cannot
halt in a formal system that is consistent. To my mind, the
embodiment via the physical self being self-assembled and an
offline record of this on LHS of (8) is what fuses syntax and
semantics. <u></u><u></u></p>
<p><u></u> <u></u></p>
<p>II. Once, (8) is in place, the Adaptive
immune system has to identify novel negation software function
<i>f<sup>¬! </sup></i><sub> </sub>of non-self antigens which
is an uncountable infinite possibilities. Hence the close to
astronomic search with V(D) J of 10
<sup>20 </sup>– 10 <sup>30</sup> ) of non-self antigens
that can hijack the self- assembly machines as recorded on
RHS of (8). Only from knowledge of self can the hostile
other, in the case of the AIS, be identified.
<u></u><u></u></p>
<p><u></u> <u></u></p>
<p><u></u> <u></u></p>
<p>III. Roger Fixed Point assures us that
the indexes of the fixed point for
<i>f<sup>¬! </sup></i><sub> </sub>be generated. I have cced
Guillame Bonfante who I think was among the first (with
coauthors, 2006) to suggest how SRT can be used to identify
computer viruses. But they do not use the full force of
Self-Ref and Self -Rep and only implicitly use Post Creative
and Productive Sets. The index of the Godel Sentence for the
fixed point will endogenously lie outside of Post listable or
recusively enumerable set for Theorems and known non-Theorems.
<u></u><u></u></p>
<p><u></u> <u></u></p>
<p>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. <u></u><u></u></p>
<p><u></u> <u></u></p>
<p>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></p>
<p><u></u> <u></u></p>
<p>So thankyou all again for your in depth
comments and interest.<u></u><u></u></p>
<p><u></u> <u></u></p>
<p>Best Regards<u></u><u></u></p>
<p><u></u> <u></u></p>
<p>Sheri <u></u><u></u></p>
<p><u></u> <u></u></p>
<p><u></u><u></u></p>
<p>-----Original Message-----<u></u><u></u></p>
<p>From: Fis <<a href="mailto:fis-bounces@listas.unizar.es" target="_blank">fis-bounces@listas.unizar.es</a>>
On Behalf Of Louis Kauffman<u></u><u></u></p>
<p>Sent: 08 November 2022 00:13<u></u><u></u></p>
<p>To: "Pedro C. Marijuán" <<a href="mailto:pedroc.marijuan@gmail.com" target="_blank">pedroc.marijuan@gmail.com</a>><u></u><u></u></p>
<p>Cc: fis <<a href="mailto:fis@listas.unizar.es" target="_blank">fis@listas.unizar.es</a>><u></u><u></u></p>
<p>Subject: Re: [Fis] A new discussion
session<u></u><u></u></p>
<p><u></u> <u></u></p>
<p>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></p>
<p><u></u> <u></u></p>
<p>Dear Pedro,<u></u><u></u></p>
<p>Here are some comments about Goedel
numbering and coding.<u></u><u></u></p>
<p><u></u> <u></u></p>
<p>It is interesting to think about Goedel
numbering in a biological context.<u></u><u></u></p>
<p>Actually we are talking about how a
given entity has semantics that can vary from context to
context.<u></u><u></u></p>
<p>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></p>
<p>1. F as a well formed formula in a
formal system S.<u></u><u></u></p>
<p>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></p>
<p>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></p>
<p>This is the key to the references and
self-references of the Goedelian situations.<u></u><u></u></p>
<p><u></u> <u></u></p>
<p>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></p>
<p>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></p>
<p>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></p>
<p>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></p>
<p><u></u> <u></u></p>
<p>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></p>
<p><u></u> <u></u></p>
<p>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></p>
<p>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></p>
<p>Now suppose that<u></u><u></u></p>
<p>g —> F(x).<u></u><u></u></p>
<p>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></p>
<p>#g —> F(g).<u></u><u></u></p>
<p>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></p>
<p>Then we can consider<u></u><u></u></p>
<p>F(#x) and it has a Goedel number<u></u><u></u></p>
<p>h —> F(#x)<u></u><u></u></p>
<p>And we can shift that to<u></u><u></u></p>
<p>#h —> F(#h).<u></u><u></u></p>
<p>This is the key point.<u></u><u></u></p>
<p>Now we have constructed a number #h so
that F(#h) discusses its own Goedel number.<u></u><u></u></p>
<p><u></u> <u></u></p>
<p>This construction allows the proof of
the Goedel Incompleteness Theorem via the sentence B(x) that
states<u></u><u></u></p>
<p>B(x) = “The statement with Goedel
number x is provable in S.” (This can also be encoded in S.)<u></u><u></u></p>
<p><u></u> <u></u></p>
<p>We then construct<u></u><u></u></p>
<p>h—> ~B(#x)<u></u><u></u></p>
<p>and<u></u><u></u></p>
<p>#h —> ~B(#h)<u></u><u></u></p>
<p>and obtain the statement<u></u><u></u></p>
<p>G= ~B(#h).<u></u><u></u></p>
<p>G states the unprovability of the Goedel
decoding of #h.<u></u><u></u></p>
<p>But the Goedel decoding of #h is the
statement G itself.<u></u><u></u></p>
<p>Thus G states its own unprovability.<u></u><u></u></p>
<p>Therefore, S being consistent, cannot
prove G.<u></u><u></u></p>
<p><u></u> <u></u></p>
<p>By making these arguments we have have
proved that G cannot be proved by S.<u></u><u></u></p>
<p>Thus we have shown that G is in fact
true.<u></u><u></u></p>
<p>We have shown that there are true
statements in number theory unprovable by system S..<u></u><u></u></p>
<p>##########################<u></u><u></u></p>
<p><u></u> <u></u></p>
<p>The above is a very concise summary of
the proof of Goedel’s Incompleteness Theorem, using Goedel
number encoding.<u></u><u></u></p>
<p><u></u> <u></u></p>
<p>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></p>
<p><u></u> <u></u></p>
<p>1. In biology is is NORMALLY the case
that certain key structures have multiple interpretations and
uses in various contexts.<u></u><u></u></p>
<p>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></p>
<p><u></u> <u></u></p>
<p>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></p>
<p><u></u> <u></u></p>
<p>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></p>
<p>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></p>
<p><u></u> <u></u></p>
<p>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></p>
<p>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></p>
<p>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></p>
<p><u></u> <u></u></p>
<p>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></p>
<p>PX: X can be printed.<u></u><u></u></p>
<p>~PX: X cannot be printed.<u></u><u></u></p>
<p>PRX: XX can be printed.<u></u><u></u></p>
<p>~PRX: XX cannot be printed.<u></u><u></u></p>
<p>Here X is any string of the symbols
{~,P,R}.<u></u><u></u></p>
<p>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></p>
<p>By printed we mean on one press of the
button on the Machine, a string of characters is printed.<u></u><u></u></p>
<p>IT IS ASSUMED THAT THE SMULLYAN MACHINE
ALWAYS TELLS THE TRUTH WHEN IT PRINTS A MEANINGFUL STATEMENT.<u></u><u></u></p>
<p>Then we have the<u></u><u></u></p>
<p><u></u> <u></u></p>
<p>Theorem. There are meaningful true
strings that the Smullyan Machine cannot print.<u></u><u></u></p>
<p><u></u> <u></u></p>
<p>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></p>
<p>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></p>
<p>If the machine were to print G, it would
lie. And the machine does not lie.<u></u><u></u></p>
<p>Therefore G is unprintable.<u></u><u></u></p>
<p>But this is what G says.<u></u><u></u></p>
<p>So we have established the truth of G
and proved the Theorem.<u></u><u></u></p>
<p><u></u> <u></u></p>
<p>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></p>
<p>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></p>
<p><u></u> <u></u></p>
<p>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></p>
<p>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></p>
<p><u></u> <u></u></p>
<p>8. I will stop here. The relationships
with biology are very worth discussing.<u></u><u></u></p>
<p>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></p>
<p><u></u> <u></u></p>
<p>Very best,<u></u><u></u></p>
<p>Lou Kauffman<u></u><u></u></p>
<p><u></u> <u></u></p>
<p><u></u> <u></u></p>
<p><u></u> <u></u></p>
<p><u></u> <u></u></p>
<p><u></u> <u></u></p>
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