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<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>
<p></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">
<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>
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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>
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<br>
<br>
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