[Fis] Limits of Formal Systems

Wed Feb 7 20:58:25 CET 2024

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!UA-sb7aiYwrOiL7jQMQjOa5k6O51iouBF124nlyH4vm0BM8NRD0rtAio_puHOcGvGL62vJZ2Yrg8dOqvK6BAr3L3ktq1$  
> <https://urldefense.com/v3/__https://arxiv.org/abs/2401.09514v1__;!!D9dNQwwGXtA!Q9Wf2QzNb33Rbcm_rxf9I_P4EziZ3qwzNM9drNcS2M856SZcvJx6al-U8ZnYt5Fj0OfDWnNsNDd2RoZgOmc$>
>
>
> Carlos Gershenson
> SUNY Empire Innovation Professor
> Department of Systems Science and Industrial Engineering
> Thomas J. Watson College of Engineering and Applied Science
> State University of New York at Binghamton
> Binghamton, New York 13902 USA
> https://urldefense.com/v3/__https://tendrel.binghamton.edu__;!!D9dNQwwGXtA!UA-sb7aiYwrOiL7jQMQjOa5k6O51iouBF124nlyH4vm0BM8NRD0rtAio_puHOcGvGL62vJZ2Yrg8dOqvK6BArx6O6FUH$  
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>
>
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