<html><head><meta http-equiv="content-type" content="text/html; charset=utf-8"></head><body style="overflow-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;">Lou, Fis, Carlos: Perfect Lou. This is also the point that Roger Penrose makes. Our own thinking can BE formal, but need not be formal. My argument along with Roli is that jury rigging is not deductive. Yet we humans can do multistep rigging very well. How do we do it? There is no Local clue at any step that one is on the way to a solution.<div><br></div><div><br></div><div>Is a counter argument that in a game with local legal moves, such as Chess or Go, hence a formally searchable space of strategies, one can have a good MID GAME marker of ultimate success? I do not think this is a counter argument. Go programs <i>do</i> create just such mid game markers of “probability of winning”. I see no way to do that for an N step jury rig. The famous 19 step jury rig in the Apollo near disaster may be a case. </div><div><br></div><div>And in the evolving biosphere, we do not know what is IN the non-deducible adjacent possible construction of the evolving biosphere or global economy. We do not deduce the evolution of the global economy. We invent it. </div><div><br></div><div>I suspect Descartes asked the wrong question. "Can I be wrong in what I believe to be true?" as a set of proposition, true or false. There is no “doing in the world” in this question. Living organisms are non-equilibrium self reproducing chemical reaction networks and have always had to interact with the world in order to “eat” and survive. Life has always had to search for old known, and novel new ways to get on with managing to live. We recognize old already classified “affordances” and somehow “cognize” for the first time novel opportunities. How do we Cognize for the first time? Teppo Fellin says, “The world LIGHT UP with relevant opportunities” He is correct. How does the world Light up? This includes how did Goldbach come to make his conjecture?</div><div><br></div><div>Best to all,</div><div><br></div><div>Stu</div><div><br></div><div><br></div><div><br></div><div><div><br><blockquote type="cite"><div>On Feb 12, 2024, at 12:16 PM, Louis Kauffman <loukau@gmail.com> wrote:</div><br class="Apple-interchange-newline"><div><meta http-equiv="Content-Type" content="text/html charset=utf-8"><meta http-equiv="Content-Type" content="text/html charset=utf-8"><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;">Dear Folks,<br><div>Please examine the Kleene argument that you cannot list all algorithms (that halt) because you can form</div><div>F(n) = F_{n}(n) + 1.</div><div>This is exactly Cantor diagonal transposed to algorithms.</div><div>It is the core of incompleteness.</div><div>It is what it is.</div><div>You cannot sweep it under the rug.</div><div><br></div><div>This argument fits into many different formal systems and once you place it there,</div><div>then the fact that all algorithms in a given formal system (not necessarily halting) can be listed (it is routine to check if some text is an algorithm, not routine to see if it halts), </div><div>shows that there can be no way in the formal system to decide whether algorithms halt. If you could do that, you could list all the halting algorithms and run into</div><div>a contradiction from the above.</div><div><br></div><div>Thus (Turing) the halting problem is undecideable in a wide class of formal systems.</div><div><br></div><div>This part of the limitations of formal systems is just what it is.</div><div><br></div><div>There is something else however and I would like to illustrate it with the Goldbach problem.</div><div>Try writing even numbers > 4 as a sum of two odd primes.</div><div><br></div><div>6 = 3 + 2</div><div>8 = 5 + 3</div><div>10 = 7 + 3 = 5 + 5</div><div>12 = 7 + 5</div><div>14 = 11 + 3 = 7 + 7</div><div>16 = 13 + 3 = 11 + 5 </div><div>18 = 13 + 5 = 11 + 7</div><div>20 = 17 + 3 = 13 + 7 </div><div>22 = 19 + 3 = 17 + 5 = 11 + 11</div><div>24 = 19 + 5 = 17 + 7 = 13 + 11</div><div>26 = 23 + 3 = 19 + 5 = 13 + 13</div><div>28 = 23 + 5 = 17 + 11 </div><div>30 = 23 + 7 = 19 + 11 = 17 + 13 </div><div>32 = 29 + 3 = 19 + 13 </div><div>…</div><div>These decomposition go up and down but the number of decompositions grows as the even numbers get larger.</div><div>There is some principle here that we are missing about how numbers get constructed. </div><div>The problem to prove that there is at least ONE way to write any even </div><div>number greater than 4 as a sum of two odd primes is completely open.</div><div>That is the Goldbach Conjecture.</div><div>It bet that 12 is the last time you get only one decomposition for an even number into two odd primes.</div><div><br></div><div>It is possible that by thinking about how numbers are made</div><div>we will find new principles by which to reason about them and hence new formal systems, unknown at this time.</div><div>Any number theorist worth his or her salt is going to think about this. This means that the number theorist is thinking outside of the </div><div>known formal systems, trying to find new and better ways to work. This is normal. We need to promote the fact that creative thinking may use formal systems, but is</div><div>not limited to using only systems that already exist.</div><div>Best,</div><div>Lou</div><div><br></div><div><br><div><blockquote type="cite"><div>On Feb 9, 2024, at 9:44 AM, eric werner <<a href="mailto:eric.werner@oarf.org">eric.werner@oarf.org</a>> wrote:</div><br class="Apple-interchange-newline"><div>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<div><p>Dear Carlos,<br>
</p><p>Notice you contradicted yourself:<br>
</p>
<div class="moz-cite-prefix">On 2/5/2024 4:43 PM, Carlos Gershenson
wrote:<br>
</div>
<blockquote type="cite" cite="mid:FB72F1AF-73F7-4618-8D73-268E76157648@gmail.com">It is
clear that models/descriptions will never be as rich as the
modeled/phenomena, and that is the way it should be. As Arbib
wrote, “a model that simply duplicates the brain is no more
illuminating than the brain itself”. [1]</blockquote><p>On the one hand, you state that a model/description will never be
as rich as the model/phenomena it describes. And, then in the next
sentence you quote Arbib who presupposes that there could be a
model that duplicates the brain. Duplicating the brain
presupposes that that model is as rich in structure and
information as the thing it models, namely the brain.</p><p>Of course, Arbib is wrong. If we did have model that duplicates
the brain, then given the model is something like an LLM residing
on my laptop, that model would not only be as rich but richer in
many ways than the brain it modeled. It would give us
unprecedented insight into the organization and function of its
architecture.</p><p>My point is that models are often richer than the object that is
modeled. They often have further dimensions that go beyond the
object modeled. This extra-dimensionality and richness enable us
to understand that object and utilize it.</p><p>To be fair, you do state that computers can go beyond axiomatic
systems, implying perhaps that they can model phenomena that
axiomatic systems cannot. But I am skeptical of what appears to
be your wish to throw all axiomatic systems together and then get
meaning out of such a hodgepodge. <br>
</p><p>I must admit I long for the beauty of mathematics and logic, the
crystalline world of truth, even if Goedel and other's seem to
have made a mess of it. </p><p>And yet if you actually read Goedel's proof, reading as I did
while a student of Kleene (who was a student of Goedel's at
Princeton) and later teaching it as I did to undergrads, that
proof is itself a thing of beauty even if a bit messy. </p><p>But Goedel did steal the core idea from Cantor's diagonal method.
And, someday we may find that Cantor's method is flawed, in yet
another higher dimensional mathematical space. Which will bring
us back to the Greeks!<br>
</p><p>Thank you for your contribution, Carlos, and remember this is all
in good fun, <br>
</p><p>-Eric<br>
</p><p>****<br>
Dr. Eric Werner, FLS <br>
</p><p><br>
</p><div><br class="webkit-block-placeholder"></div>
<div class="moz-cite-prefix">On 2/5/2024 4:43 PM, Carlos Gershenson
wrote:<br>
</div>
<blockquote type="cite" cite="mid:FB72F1AF-73F7-4618-8D73-268E76157648@gmail.com">
<meta http-equiv="content-type" content="text/html; charset=UTF-8">
<div>In the 1920s, David Hilbert's program attempted to get rid
once and for all from the paradoxes in mathematics that had
arisen from the work of Cantor, Russell, and others. Even when
Hilbert’s PhD student — John von Neumann — was working avidly on
demonstrating that mathematics were complete, consistent, and
decidable, Kurt Gödel proved in the early 1930s that formal
systems are incomplete and inconsistent, while Alan Turing
proved in 1936 their undecidability (for which he proposed the
"Turing Machine", laying the theoretical basis for computer
science).</div>
<div><br>
</div>
<div>Digital computers have enabled us to study concepts and
phenomena for which we did not have the proper tools beforehand,
as they process much more information than the one our limited
brains can manipulate. These include intelligence, life, and
complexity.</div>
<div><br>
</div>
<div>Even when computers have served us greatly as "telescopes for
complexity", the limits of formal systems are becoming even more
evident, as we attempt to model and simulate complex phenomena
in all their richness, which implies emergence,
self-organization, downward causality, adaptation, multiple
scales, semantics, and more.</div>
<div><br>
</div>
<div>Can we go beyond the limits of formal systems? Well, we
actually do it somehow. It is natural to adapt to changing
circumstances, so we can say that our "axioms" are flexible.
Moreover, we are able to simulate this process in computers.
Similar to an interpreter or a compiler, we can define a formal
system where some aspects of it can be modified/adapted. And if
we need more adaptation, we can generalize the system so that a
constant becomes a variable (similar to oracles in Turing
Machines). Certainly, this has its limits, but our adaptation is
also limited: we cannot change our physics or our chemistry,
although we have changed our biology with culture and
technology.</div>
<div><br>
</div>
<div>Could it be that the problem lies not in the models we have,
but in the modeling itself? We tend to forget the difference
between our models and the modeled, between the map and the
territory, between epistemology and ontology; simply because our
language does not make a distinction between phenomena and our
perceptions of them. When we say "this system is
complex/alive/intelligent", we assume that these are inherent
properties of the phenomenon we describe, forgetting that the
moment we name anything, we are already simplifying and limiting
it. It is clear that models/descriptions will never be as rich
as the modeled/phenomena, and that is the way it should be. As
Arbib wrote, “a model that simply duplicates the brain is no
more illuminating than the brain itself”. [1]</div>
<div><br>
</div>
<div>Still, perhaps we're barking up the wrong tree. We also tend
to forget the difference between computability in theory
(Church-Turing's) and computability in practice (what digital
computers do). There are non-Turing-computable functions which
we can compute in practice, while there are Turing-computable
functions for which there is not enough time in the universe to
compute. So maybe we are focussing on theoretical limits, while
we should be concerned more with practical limits.</div>
<div><br>
</div>
<div>As you can see, I have many more questions than answers, so I
would be very interested in what everyone thinks about these
topics.</div>
<div><br>
</div>
<div>I'll just share some idea I've been playing with recently,
although it might be that it won't lead anywhere. For lack of a
better name, let's call them "multi-axiom systems". For example
in geometry, we know that if we change the 5th axiom (about
intersecting parallel lines), we can go from Euclidean to other
geometries. We can define a "multi-axiom geometry", so that we
can switch between different versions of the 5th axiom for
different purposes. In a similar way, we could define a
multi-axiom system that contains several different formal
systems. We know we cannot have all at once universal
computation and completeness and consistency. But then, in
first-order logic, we can have completeness and consistency. In
second-order logic we have universal computation but not
completeness. In paraconsistent logics we sacrifice consistency
but gain other properties. Then, if we consider a multi-axiom
system that includes all of these and perhaps more, in theory we
could have in the same system all these nice properties, but not
at the same time. Would that be useful? Of course, we would need
to find rules that would determine when to change the axioms.
Just to relate this idea to last month's topic — as it was
motivated by Stu's and Andrea's paper [2] — if we want to model
evolution, we can have "normal" axioms at short timescales (and
thus predictability), but at longer (evolutionary) timescales,
we can shift axioms set, and then the "rules" of biological
systems could change, towards a new configuration where we can
use again "normal" axioms.</div>
<div><br>
</div>
<div><br>
</div>
<div><br>
</div>
<div>[1] Michael Arbib, The Metaphorical Brain 2. Neural Networks
and Beyond (1989)</div>
<div>[2] Stuart Kauffman, Andrea Roli. Is the Emergence of Life an
Expected Phase Transition in the Evolving Universe? <a href="https://urldefense.com/v3/__https://arxiv.org/abs/2401.09514v1__;!!D9dNQwwGXtA!Q9Wf2QzNb33Rbcm_rxf9I_P4EziZ3qwzNM9drNcS2M856SZcvJx6al-U8ZnYt5Fj0OfDWnNsNDd2RoZgOmc$" moz-do-not-send="true">https://arxiv.org/abs/2401.09514v1 </a></div>
<br>
<br>
<div>
<meta charset="UTF-8">
<div dir="auto" style="letter-spacing: normal; text-align: start; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px; -webkit-text-stroke-width: 0px; text-decoration: none; overflow-wrap: break-word; -webkit-nbsp-mode: space;">
<div dir="auto" style="text-align: start; text-indent: 0px; overflow-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;">
<div dir="auto" style="text-align: start; text-indent: 0px; overflow-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;">
<div dir="auto" style="text-align: start; text-indent: 0px; overflow-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;">
<div dir="auto" style="text-align: start; text-indent: 0px; overflow-wrap: break-word; -webkit-nbsp-mode: space; line-break: after-white-space;">
<div style="letter-spacing: normal; text-transform: none; white-space: normal; word-spacing: 0px; text-decoration: none; -webkit-text-stroke-width: 0px;">Carlos
Gershenson</div>
<div>SUNY Empire Innovation Professor <br>
Department of Systems Science and Industrial
Engineering<br>
<span class="Apple-converted-space">Thomas J. Watson
College of Engineering and Applied Science<br>
State University of New York at Binghamton<br>
Binghamton, New York 13902 </span>USA<br>
</div>
<div style="letter-spacing: normal; text-transform: none; white-space: normal; word-spacing: 0px; text-decoration: none; -webkit-text-stroke-width: 0px;"><a href="https://urldefense.com/v3/__https://tendrel.binghamton.edu__;!!D9dNQwwGXtA!Q9Wf2QzNb33Rbcm_rxf9I_P4EziZ3qwzNM9drNcS2M856SZcvJx6al-U8ZnYt5Fj0OfDWnNsNDd2yTKmSVg$" moz-do-not-send="true">https://tendrel.binghamton.edu</a></div>
</div>
</div>
</div>
</div>
</div>
</div>
<br>
<br>
<fieldset class="moz-mime-attachment-header"></fieldset>
<pre class="moz-quote-pre" wrap="">_______________________________________________
Fis mailing list
<a class="moz-txt-link-abbreviated" href="mailto:Fis@listas.unizar.es">Fis@listas.unizar.es</a>
<a class="moz-txt-link-freetext" href="http://listas.unizar.es/cgi-bin/mailman/listinfo/fis">http://listas.unizar.es/cgi-bin/mailman/listinfo/fis</a>
----------
INFORMACIÓN SOBRE PROTECCIÓN DE DATOS DE CARÁCTER PERSONAL
Ud. recibe este correo por pertenecer a una lista de correo gestionada por la Universidad de Zaragoza.
Puede encontrar toda la información sobre como tratamos sus datos en el siguiente enlace: <a class="moz-txt-link-freetext" href="https://sicuz.unizar.es/informacion-sobre-proteccion-de-datos-de-caracter-personal-en-listas">https://sicuz.unizar.es/informacion-sobre-proteccion-de-datos-de-caracter-personal-en-listas</a>
Recuerde que si está suscrito a una lista voluntaria Ud. puede darse de baja desde la propia aplicación en el momento en que lo desee.
<a class="moz-txt-link-freetext" href="http://listas.unizar.es/">http://listas.unizar.es</a>
----------
</pre>
</blockquote>
<div id="DAB4FAD8-2DD7-40BB-A1B8-4E2AA1F9FDF2"><br><table style="border-top: 1px solid #D3D4DE;"><tbody><tr><td style="width: 55px; padding-top: 13px;"><a href="https://urldefense.com/v3/__http://www.avg.com/email-signature?utm_medium=email&utm_source=link&utm_campaign=sig-email&utm_content=emailclient__;!!D9dNQwwGXtA!S9usrOp9JhvW8UwJEq9dcFtq7FKcSNqNVPec2sX1BgZjKXwC0mnBUV_SDjdu1Sa_AS3w3prDwKITzaqB41fYZVU$" target="_blank"><img src="https://s-install.avcdn.net/ipm/preview/icons/icon-envelope-tick-green-avg-v1.png" alt="" width="46" height="29" style="width: 46px; height: 29px;"></a></td><td style="width: 470px; padding-top: 12px; color: #41424e; font-size: 13px; font-family: Arial, Helvetica, sans-serif; line-height: 18px;">Virus-free.<a href="https://urldefense.com/v3/__http://www.avg.com/email-signature?utm_medium=email&utm_source=link&utm_campaign=sig-email&utm_content=emailclient__;!!D9dNQwwGXtA!S9usrOp9JhvW8UwJEq9dcFtq7FKcSNqNVPec2sX1BgZjKXwC0mnBUV_SDjdu1Sa_AS3w3prDwKITzaqB41fYZVU$" target="_blank" style="color: #4453ea;">www.avg.com</a></td></tr></tbody></table><a href="x-msg://124/#DAB4FAD8-2DD7-40BB-A1B8-4E2AA1F9FDF2" width="1" height="1"> </a></div></div>
_______________________________________________<br>Fis mailing list<br><a href="mailto:Fis@listas.unizar.es">Fis@listas.unizar.es</a><br><a href="http://listas.unizar.es/cgi-bin/mailman/listinfo/fis">http://listas.unizar.es/cgi-bin/mailman/listinfo/fis</a><br>----------<br>INFORMACI�N SOBRE PROTECCI�N DE DATOS DE CAR�CTER PERSONAL<br><br>Ud. recibe este correo por pertenecer a una lista de correo gestionada por la Universidad de Zaragoza.<br>Puede encontrar toda la informaci�n sobre como tratamos sus datos en el siguiente enlace: https://sicuz.unizar.es/informacion-sobre-proteccion-de-datos-de-caracter-personal-en-listas<br>Recuerde que si est� suscrito a una lista voluntaria Ud. puede darse de baja desde la propia aplicaci�n en el momento en que lo desee.<br>http://listas.unizar.es<br>----------<br></div></blockquote></div><br></div></div>_______________________________________________<br>Fis mailing list<br>Fis@listas.unizar.es<br>http://listas.unizar.es/cgi-bin/mailman/listinfo/fis<br>----------<br>INFORMACI�N SOBRE PROTECCI�N DE DATOS DE CAR�CTER PERSONAL<br><br>Ud. recibe este correo por pertenecer a una lista de correo gestionada por la Universidad de Zaragoza.<br>Puede encontrar toda la informaci�n sobre como tratamos sus datos en el siguiente enlace: https://sicuz.unizar.es/informacion-sobre-proteccion-de-datos-de-caracter-personal-en-listas<br>Recuerde que si est� suscrito a una lista voluntaria Ud. puede darse de baja desde la propia aplicaci�n en el momento en que lo desee.<br>http://listas.unizar.es<br>----------<br></div></blockquote></div><br></div></body></html>