[Fis] Limits of Formal Systems

[Carlos Gershenson]

Dear All,

Thank you for sharing several interesting ideas.

Yesterday we had a seminar at Binghamton by Dr. Pedro Márquez-Zacarías, postdoc at the Santa Fe Institute:
"Lessons from Life Itself: Relational Models of Complexity and Self-Organization” https://urldefense.com/v3/__https://vimeo.com/910918333__;!!D9dNQwwGXtA!RvTqWpX2kvtCXXuvi0Lxx5vRJ8mEMzFFpxvIoq-RQa4JCQ_LpnEJLuOFL-9y0sUowwjSXTVIKJhZJXeYga8$  
Where he mentions several topics related to this discussion (and several interesting anecdotes). At the end I realized that in many cases we are trying to predict complex phenomena (evolution, life, societies, intelligence), but it seems that actually undecidability is not a problem, but a feature and a requirement to exhibit these levels of complexity (See Hernández-Orozco, S., Hernández-Quiroz, F., & Zenil, H. (2018). Undecidability and irreducibility conditions for open-ended evolution and emergence. Artificial Life, 24(1), 56–70. ).
Thus, it seems that the path forward should not be to try to patch classic tools to understand complexity, but “break” them even more, because only then we will be able to model complexity. This reminds me of infinitesimals: we did not have the proper tools to describe them, so they were basically ignored until Cantor. Then people initially ridiculed him, but after some time, speaking about the infinitely small or large stopped being a problem or taboo.

One example of this is a Paraconsistent Logic I proposed some time ago that extends Fuzzy Logic and is able to handle contradictions:
Gershenson, C. (1999). Modelling emotions with multidimensional logic. Proceedings of the 18th International Conference of the North American Fuzzy Information Processing Society (NAFIPS ’99), 42–46. https://urldefense.com/v3/__https://tendrel.binghamton.edu/unam/jlagunez/mdl/mdlemotions.html__;!!D9dNQwwGXtA!RvTqWpX2kvtCXXuvi0Lxx5vRJ8mEMzFFpxvIoq-RQa4JCQ_LpnEJLuOFL-9y0sUowwjSXTVIKJhZ_HDPbnA$  

And coincidentally, yesterday I found this note from 2019:
On the completeness of potential mathematics
Conjecture: under all possible axiomatic systems, all possible statements are valid.
i.e. for any statement, there exists an infinite set of axioms under which that statement is valid.
i.e. A mathematics (formal system) is not complete; but the (infinite) set of all possible mathematics (formal systems) is.
 
e.g. for all functions, one can find an axiomatic system where that function halts.
e.g. one can define an axiom: after T cycles, the machine will halt. Then all machines will halt in time <=T, and one can check the halting time by simply running the machine.
Many of these potential axioms will be trivial or not useful, that is why experience helps us select useful axioms. But these are restrictive, and thus incompleteness. Consequence of being finite. For all potential axioms, i.e. for the infinite set of all possible mathematics, there is completeness… 

Still Computational Irreducibility, because you will need more and more axioms to prove more and more random/complex statements/strings.    



What I want to say is that most probably we will have advances not by finding ways to extend formal systems beyond their limits, but by accepting them. In other words, we won’t find predictability, consistence, completeness, etc. as we would like. But we have to “abandon” them if we want to understand and engineer complexity. (I use quotes because we will still use them when we can, it is just that we need alternative tools for when these are not viable. Some tools have been proposed already for decades, but it seems that what we are lacking is a synthesis that brings them all together).


Glad to have more comments rolling.

Best wishes,
Carlos

-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://listas.unizar.es/pipermail/fis/attachments/20240208/1d8a7d83/attachment.html>