[Fis] Limits of Formal Systems -- PARADIGM SHIFT(S) IN KNOWLEDGE PRODUCTION AND USE

Thu Feb 8 09:06:14 CET 2024

Dear All,

Thank you for this timely and important discussion that continues the theme of the paradigm shift we started with the New Year Lecture. It seems to me that we are already navigating the turbulent waters of several paradigm shifts in the production and use of knowledge.

Here is my list of concurrent shifts one can observe:

·        MULTIDISCIPLINARY approaches

·        MULTI-LEVEL ANALYSIS

·        THE ROLE OF THE OBSERVER – declaring an observer's perspective in the given context

·        DYNAMICAL approaches

·        AGENTIAL view of life. Living agents navigating „white-water“ (dynamic, largely unpredictable) environments („Wicked problems“)
Cells as collectives of active parts that „dance“ together to produce common goals, based on choreography constrained by chemistry and physics.

·        LOGIC (such as logic in reality-LIR and other non-standard logics)

·        INFORMATION AND COMPUTATION as conceptual glues

o   COMPUTING & algorithms beyond Turing model of computation. Non-algorithmic computation

o   INFORMATION beyond Shannon's model of communication (connects to meaning, that is the role of the observer and agential view of life)

·        New MATHEMATICAL approaches

  *   In the PRODUCTION of knowledge – SHIFT FROM MANUFACURE-STYLE BASED ON HUMAN INTELLIGENCE-ONLY to the extensive use of other, ARTIFICIAL INTELLIGENCES for tasks like data collection, verification, systematization, literature searches, writing, communication, etc.

The list goes on.

All the best,
Gordana

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From: Fis <fis-bounces at listas.unizar.es> on behalf of "Pedro C. Marijuán" <pedroc.marijuan at gmail.com>
Date: Wednesday, 7 February 2024 at 20:59
To: "fis at listas.unizar.es" <fis at listas.unizar.es>
Subject: Re: [Fis] Limits of Formal Systems

Dear All,

Thanks to Carlos for his kickoff text and indeed to Karl and Lou for their elegant duet . And also to Plamen.

The historical road to logics and formal systems has been drafted, somehow, in these interventions and in the previous NY Lecture--at least cursorily. Some mentions of our naming capabilities and biological limits have also been made by the above parties. It is in this aspect that I will briefly enter.

I worked in the 90s in some neuroscience themes with Ken Collins ("El Cerebro Dual", book we published in Spanish). One of the themes then emerging was the role of cerebellum in "higher functions". Also related to logics? My own contention after the work during those years (sorry to be insufficiently updated) is that language logic could be a byproduct of the way our cerebellum contributes to organize the myriad combinatory muscular modules and percept complexes involved in movement. I mean to achieve effective closure of an action, a congruence with the ongoing perception (and associated memories) has to be achieved. For a trivial grasping of, say, a mug, dozens and dozens of  muscles activation/inhibition processes, in a fiendish combinatorics of high precision changing terribly fast, has to be orchestrated. "True" efficient motion when a closure is achieved, versus "False" or failed motion when sensory-motor constellations do not match properly. Well, if we go to language, the muscles of phonation (particularly tongue and vocal cords) are supporting  the physical part of linguistic combinatorics, necessarily coupled with percepts and ad hoc memories related to the involved occurrences. Thus, there should also be a (logical) closure in the way our concepts or "cognits" (as Joaquin Fuster put) are verbally organized, so to achieve not only phonatory or lexical congruence but also an efficient adaptive matching with previous conceptual/perceptual experiences. Otherwise language would have never evolved.

I have always had trouble with logic. I think it is needed, beyond computation and formal systems, for communication & explanatory purposes, for making acceptable the new pieces of knowledge; but it would stand far away from the obtention part: the creative, visionary, intuition leaps that lead to real knowledge novelty. Nevertheless, the fascinating thing, tribute to human ingenuity, is the amazing historical development grounded in logics and formal systems that has been able to approach most physical events with uncanny precision. But concerning the bio-logics specifically, it is for me something not well solved yet. Joseph Brenner, a senior philosopher in FIS list, has developed a Logic in Reality, LIR, that tries to transcend formal logic limitations. I would be curious of hear from him in the present juncture.

Best wishes,
--Pedro

PS. Several new parties have joined FIS list these days. They are very welcome to express their views, always taking into account the general limitation of 2 messages per week (except the presenter), or sometimes 3 messages (for urged parties). The counting of messages follows the international business week (Monday to Sunday)... In any event, it is quite nice receiving fresh minds in our discussion community!

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El 05/02/2024 a las 16:43, Carlos Gershenson escribió:
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).

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.

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. 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.

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]

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.

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. 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.

[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!UvTDqp9f46c317buFOwM97xyNQUTyxOP5JY585DPS5QZwXFS33z0ccz40UJBtIU1VKjHJT5kQwN0fKvvX2_uYfI34_yqArR7$  <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
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