[Fis] Limits of Formal Systems

Tue Feb 6 01:28:48 CET 2024

Hi Carlos,
 thanks for this nice introduction. Good topic.
Please excuse my short provoking comments, which are directed towards the
FIS community.
I am simply trying to generate some food for thought.

On Mon, Feb 5, 2024 at 4:56 PM Carlos Gershenson <cgershen at gmail.com> wrote:

> 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.
>

Hilbert failed and this should be remembered for generations to come and
not make the same mistake.
Therefore, I like Felix Klein's program, which is much more realistic and
achievable.

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).
>
> Digital computers have enabled us to study concepts and phenomena
>

– Rule based systems - like us :-)

for which we did not have the proper tools beforehand, as they process much
> more information than the one our limited brains
>

Limited brains. This is perhaps what we, modern (humble) human beings,
think of ourselves. However, if you read what Rudolf Steiner wrote 100+
years ago about the Atlantes and their epistemic nature, - consider it as a
pure fiction, like the Star Wars saga - you will see that there are
possible other human races with very large scale memory who do to apply the
reasoning we are used to, but solve their problems on the base of past
experience and retrieval of analogous situations from their memory -
similar to analogous computing, something that we have recently delegated
to our AI systems recently, although they tried mimicking human behavior in
the past (alias "expert systems") in the Minsky's, Asimov's and Clarke's
(HAL) era.

can manipulate. These include intelligence, life,
>

- but we have set different rules for artificial intelligence and
artificial life, right? -

> and complexity.
>
> 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.
>
> 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.
>

– but this is still the same paradigm – rules & laws: human beings are
constructive and creative beings, so they obviously need recipes, and
rulers as well. It must be human nature that has got used to limiting
itself. It is not mathematics and computation as such.

> 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).
>

That’s called second order logic.

> 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.
>
> Our entire human science has been pre-set by our senses, so we have
continued to sense in the same manner, ub the direction of our senses even
after having developed equipment like telescopes, microscopes and jet
engines.

> 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.
>

Yes, going in this direction is part of the solution. But there is more.

> 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.
>

That’s truly so: the real problem of our human nature and rule-based
reasoning we cannot escape from. We always need causes and effects to
reduce complexity. At the same time, we have programmed our digital
computers and AI systems to do exactly the opposite: memorization and
pattern matching of huge data sets.

> 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]
>
> 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.
>

Yes, but are those the only modes of operation we can imagine?

> So maybe we are focussing on theoretical limits, while we should be
> concerned more with practical limits.
>
> As you can see, I have many more questions than answers, so I would be
> very interested in what everyone thinks about these topics.
>
> 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.
>

In fact, this is what I asked Lou Kauffman in the previous exchange on
Stu’s topic.

But again, axioms and rules. Is this the only way that mathematics as a way
of making distinctions and reasoning about them can be created?

Could we not do more?

> We can define a "multi-axiom geometry", so that we can switch between
> different versions of the 5th axiom for different purposes.
>

Yes, this could be practical in computer graphics, where different kinds of
spaces are appropriate to render different kinds of images and perspectives
of them, e.g. in multidimensional spaces.

> 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?
>

Yes.

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.
>

But this is still the same rule-based system view of Homo sapiens sapiens.

As you see, I just added more questions to those of Carlos.

Enjoy the discussion!

Best,

Plamen

>
> [1] Michael Arbib, The Metaphorical Brain 2. Neural Networks and Beyond
> (1989)
> [2] Stuart Kauffman, Andrea Roli. Is the Emergence of Life an Expected
> Phase Transition in the Evolving Universe?
> https://urldefense.com/v3/__https://arxiv.org/abs/2401.09514v1__;!!D9dNQwwGXtA!X-B-lRemcVytl0ED2UXtRdfkzqFiAkWY9yQEWSLMtJxtytC5YcaGbLaX-K5QFwGhHuShSp1ifVjtuis9nP1eT0F0EHI2$ 
> <https://urldefense.com/v3/__https://arxiv.org/abs/2401.09514v1__;!!D9dNQwwGXtA!Q9Wf2QzNb33Rbcm_rxf9I_P4EziZ3qwzNM9drNcS2M856SZcvJx6al-U8ZnYt5Fj0OfDWnNsNDd2RoZgOmc$>
>
>
> Carlos Gershenson
> SUNY Empire Innovation Professor
> Department of Systems Science and Industrial Engineering
> Thomas J. Watson College of Engineering and Applied Science
> State University of New York at Binghamton
> Binghamton, New York 13902 USA
> https://urldefense.com/v3/__https://tendrel.binghamton.edu__;!!D9dNQwwGXtA!X-B-lRemcVytl0ED2UXtRdfkzqFiAkWY9yQEWSLMtJxtytC5YcaGbLaX-K5QFwGhHuShSp1ifVjtuis9nP1eT-O3-Pik$ 
> <https://urldefense.com/v3/__https://tendrel.binghamton.edu__;!!D9dNQwwGXtA!Q9Wf2QzNb33Rbcm_rxf9I_P4EziZ3qwzNM9drNcS2M856SZcvJx6al-U8ZnYt5Fj0OfDWnNsNDd2yTKmSVg$>
>
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