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<div class="moz-cite-prefix"><i>(This message has general interest,
and eliminating its attached docs it fits well into the list.
Interested parties in these attachments may ask Lou
directly--Pedro)</i></div>
<div class="moz-cite-prefix"><br>
</div>
<div class="moz-cite-prefix">El 13/02/2024 a las 9:05, Louis
Kauffman escribió:<br>
</div>
<p>Dear Folks,</p>
<p>This message was too long to reach fis. I cannot reduce it and
keep the content. So I am sending it for a private discussion.</p>
<p>I am interested in your notion Stu of jury rigging. You are
interested in structures that have a global interrelationship and
are not very likely to be deduced from some axioms although they
may indeed occur in processes constrained by some axioms. I am
interested in structures that can occur under such constraints.
This is what mathematics is often about for me. For example we
have the Borommean Rings in topology (And of course a long history
from the Borommean family to Ballentine been ads). Yellow
surrounds Blue surrounds Red surrounds Yellow...</p>
<p>Below I show you an example of generalized Borommean rings. They
are a kind of topological autocatalytic structure. Remove any ring
and the whole thing comes undone, but together they are
topologically linked. Now we topologists do not deduce these
things. We invent them or discover them. And if you were to look
around you could make more of them.Once we know about them we can
talk learnedly about the SET OF ALL BOROMMEAN TYPE RINGS. It can
be a pastime to to try to find things like this. And then we enter
a host of mathematical problems such as — how do you prove from
topological axioms that these rings are indeed linked? That is
something that can be deduced if you are clever enough. So we go
back and forth between invention/discovery and deductive
challenges. The prolixity of topological jury rigging is huge and
I imagine that <br>
</p>
<p>If you wonder how we might make proofs about such linkedness,
this paper will give you an introduction. High school students can
understand the techniques and in fact I wrote this paper with
Devika and Claudia when they were inhigh school in Chicago. But
lets go over to biology and in particular to recombination and
knotting, this kind of topology suitably generalized will be
important in biology in the future.</p>
<p>DNA molecules can be knotted and this was proved by the molecular
biologists Cozzarelli, Spengler and Stasiak by coating DNA with
protein and using electron microscopy. This allowed one to SEE the
weaving and to map the DNA via electron microscopy to the
diagrammatic formal systems of the knot theory. You can draw a
knot diagram corresponding to the self-entangled DNA. Then it is
possible to use topological analysis consisting as I said of
invention/discovery coupled with deduction to find about about the
topology of this micro molecular world. Representing the DNA as a
diagram, we can describe recombination as indicated below.</p>
<p>It is a GUESS that the replacement is a right-handed crossover as
shown above. How can one tell if this guess is right? (See the
articles above.) Experiments in processive recombination can be
performed. If you did it with this form recombination the diagrams
tell us that the results will be like this. A specific “spectrum”
of knot types should emerge. One can look for these with the
electron microscopy and in many cases the form of the
recombination can be found by this topological analysis.</p>
<p>
Now you should think about this. I have described one example of
how people have managed to map the unruly biology to a formal
diagrammatic topological system and how this could be used to
understand and even to find out about basically unobservable
processes in the biology (the form of the recombination). The
mathematical formalism into which things are mapped contains all
sorts of possibilities of its own for improbable and not deducible
structures. And the formalism admits to deductive analysis if we
are clever enough to find the ways to do it. There is a lot of
this kind of topological work going on in molecular biology and in
the study of protein folding. I see this as an exemplar of how in
the future it will be possible to flexibly interface biological,
topological and mathematical problems. I</p>
<p>In relation to the theme of the limits of formal systems, we see
here that diagrammatic formal systems allow a very wide
interaction among these different modes of working and if you were
to look at the practices of topologists, graph theorists,
combinatorialists, category theorists etc you would see that the
sort of strait-laced limitations of the era of Russelian and
Goedelian worries has been enfolded in a growing structure of
expanding and interrelated formal systems that in their prolixity
escape some of the problems of incompleteness. (But not all. See
my previous email).</p>
<div class="">Best,</div>
<p>Lou</p>
<p>On Feb 12, 2024, at 1:16 PM, Louis Kauffman <<a
href="mailto:loukau@gmail.com" class="" moz-do-not-send="true">loukau@gmail.com</a>>
wrote:</p>
<blockquote type="cite"
cite="mid:9178EF46-3997-4131-8B8D-283BE170F1CC@gmail.com">
<div class="">
<div class="">
<blockquote type="cite" class="">
<div class="">Dear Folks,<br class="">
<div style="word-wrap: break-word; -webkit-nbsp-mode:
space; -webkit-line-break: after-white-space;" class="">
<div class="">Please examine the Kleene argument that
you cannot list all algorithms (that halt) because you
can form</div>
<div class="">F(n) = F_{n}(n) + 1.</div>
<div class="">This is exactly Cantor diagonal transposed
to algorithms.</div>
<div class="">It is the core of incompleteness.</div>
<div class="">It is what it is.</div>
<div class="">You cannot sweep it under the rug.</div>
<div class=""><br class="">
</div>
<div class="">This argument fits into many different
formal systems and once you place it there,</div>
<div class="">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 class="">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 class="">a contradiction from the above.</div>
<div class=""><br class="">
</div>
<div class="">Thus (Turing) the halting problem is
undecideable in a wide class of formal systems.</div>
<div class=""><br class="">
</div>
<div class="">This part of the limitations of formal
systems is just what it is.</div>
<div class=""><br class="">
</div>
<div class="">There is something else however and I
would like to illustrate it with the Goldbach problem.</div>
<div class="">Try writing even numbers > 4 as a sum
of two odd primes.</div>
<div class=""><br class="">
</div>
<div class="">6 = 3 + 2</div>
<div class="">8 = 5 + 3</div>
<div class="">10 = 7 + 3 = 5 + 5</div>
<div class="">12 = 7 + 5</div>
<div class="">14 = 11 + 3 = 7 + 7</div>
<div class="">16 = 13 + 3 = 11 + 5 </div>
<div class="">18 = 13 + 5 = 11 + 7</div>
<div class="">20 = 17 + 3 = 13 + 7 </div>
<div class="">22 = 19 + 3 = 17 + 5 = 11 + 11</div>
<div class="">24 = 19 + 5 = 17 + 7 = 13 + 11</div>
<div class="">26 = 23 + 3 = 19 + 5 = 13 + 13</div>
<div class="">28 = 23 + 5 = 17 + 11 </div>
<div class="">30 = 23 + 7 = 19 + 11 = 17 + 13 </div>
<div class="">32 = 29 + 3 = 19 + 13 </div>
<div class="">…</div>
<div class="">These decomposition go up and down but the
number of decompositions grows as the even numbers get
larger.</div>
<div class="">There is some principle here that we are
missing about how numbers get constructed. </div>
<div class="">The problem to prove that there is at
least ONE way to write any even </div>
<div class="">number greater than 4 as a sum of two odd
primes is completely open.</div>
<div class="">That is the Goldbach Conjecture.</div>
<div class="">It bet that 12 is the last time you get
only one decomposition for an even number into two odd
primes.</div>
<div class=""><br class="">
</div>
<div class="">It is possible that by thinking about how
numbers are made</div>
<div class="">we will find new principles by which to
reason about them and hence new formal systems,
unknown at this time.</div>
<div class="">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 class="">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 class="">not limited to using only systems that
already exist.</div>
<div class="">Best,</div>
<div class="">Lou</div>
<div class=""><br class="">
</div>
<div class=""><br class="">
<div class="">
<blockquote type="cite" class="">
<div class="">On Feb 9, 2024, at 9:44 AM, eric
werner <<a href="mailto:eric.werner@oarf.org"
class="" moz-do-not-send="true">eric.werner@oarf.org</a>>
wrote:</div>
<br class="Apple-interchange-newline">
<div class="">
<meta http-equiv="Content-Type"
content="text/html; charset=UTF-8" class="">
<div class="">
<p class="">Dear Carlos,<br class="">
</p>
<p class="">Notice you contradicted yourself:<br
class="">
</p>
<div class="moz-cite-prefix">On 2/5/2024 4:43
PM, Carlos Gershenson wrote:<br class="">
</div>
<blockquote type="cite"
cite="mid:FB72F1AF-73F7-4618-8D73-268E76157648@gmail.com"
class="">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 class="">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 class="">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 class="">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 class="">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 class="">
</p>
<p class="">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 class="">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 class="">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 class="">
</p>
<p class="">Thank you for your contribution,
Carlos, and remember this is all in good
fun, <br class="">
</p>
<p class="">-Eric<br class="">
</p>
<p class="">****<br class="">
Dr. Eric Werner, FLS <br class="">
</p>
<p class=""><br class="">
</p>
<div class=""><br
class="webkit-block-placeholder">
</div>
<div class="moz-cite-prefix">On 2/5/2024 4:43
PM, Carlos Gershenson wrote:<br class="">
</div>
<blockquote type="cite"
cite="mid:FB72F1AF-73F7-4618-8D73-268E76157648@gmail.com"
class="">
<meta http-equiv="content-type"
content="text/html; charset=UTF-8"
class="">
<div class="">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 class=""><br class="">
</div>
<div class="">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 class=""><br class="">
</div>
<div class="">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 class=""><br class="">
</div>
<div class="">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 class=""><br class="">
</div>
<div class="">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 class=""><br class="">
</div>
<div class="">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 class=""><br class="">
</div>
<div class="">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 class=""><br class="">
</div>
<div class="">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 class=""><br class="">
</div>
<div class=""><br class="">
</div>
<div class=""><br class="">
</div>
<div class="">[1] Michael Arbib, The
Metaphorical Brain 2. Neural Networks and
Beyond (1989)</div>
<div class="">[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" class="">https://arxiv.org/abs/2401.09514v1 </a></div>
<br class="">
<br class="">
<div class="">
<meta charset="UTF-8" class="">
<div dir="auto" style="letter-spacing:
normal; text-align: start; text-indent:
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class="">
<div dir="auto" style="text-align:
start; text-indent: 0px;
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-webkit-nbsp-mode: space; line-break:
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<div dir="auto" style="text-align:
start; text-indent: 0px;
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class="">
<div dir="auto" style="text-align:
start; text-indent: 0px;
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class="">
<div dir="auto" style="text-align:
start; text-indent: 0px;
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class="">
<div style="letter-spacing:
normal; text-transform: none;
white-space: normal;
word-spacing: 0px;
text-decoration: none;
-webkit-text-stroke-width:
0px;" class="">Carlos
Gershenson</div>
<div class="">SUNY Empire
Innovation Professor <br
class="">
Department of Systems
Science and Industrial
Engineering<br class="">
<span
class="Apple-converted-space">Thomas
J. Watson College
of Engineering and
Applied Science<br class="">
State University of New York
at Binghamton<br class="">
Binghamton, New York 13902 </span>USA<br
class="">
</div>
<div style="letter-spacing:
normal; text-transform: none;
white-space: normal;
word-spacing: 0px;
text-decoration: none;
-webkit-text-stroke-width:
0px;" class=""><a
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moz-do-not-send="true"
class="">https://tendrel.binghamton.edu</a></div>
</div>
</div>
</div>
</div>
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<br class="">
<br class="">
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