[Fis] Happy Chinese New Year & Concluding the Session

[Louis Kauffman]

Dear Folks,
Thanks for this great month of exchanges! I see no need to make a summary of the whole exploration. A bit of mathematics did come out in relation to this discussion. We talked about it at the Laws of Form meeting earlier today. I discovered a “boundary arithmetic” similar to 3-valued logic that could be used to find out simple (but hard otherwise to determine) topological information about knots and links. The arithmetic itself is an extension of Laws of Form where one has the marked state <> and the rules <> <> = <>  (calling) and <<>> =       (crossing). In crossing, one crosses from the marked state and arrives at the unmarked state. In this notation the unmarked state is indicated by a blank space. To this we add a name for the unmarked state E and another name O which is a state where < O > = O so that crossing from O returns to O.  O is neither marked nor is it unmarked. You can think of O as the boundary between the inside and the outside of a first distinction. In logic it would be the third value, neither true nor false, in a three valued logic. We also assume that markedness dominates the third value so that O <> = <> O = <>. Now to get a structure that unfolds into the well-known Luaksiewcz three valued logic, one takes OO = O. BUT for “my” boundary arithmetic I take OO =     . That is O interacts with itself to disappear! You can think of this as a boundary joining along another boundary line and the seam between them becomes invisible. You can also think of it as analogous to Odd + Odd = Even in the sense that the sum of two odd numbers is even.
So we have an arithmetic with the rules
<><>=<>
<<>> =   
OO=
<> O = <>
<O> = O.
It turns out that one can use this arithmetic to calculate whether certain woven networks (that occur in DNA recombination e.g.) have one loop (odd) or two loops (even). I won’t repeat this here, but more information will be sent about this that could be downloaded.

A logicians point about such arithmetics is that they do not satisfy the “Law of the Excluded Middle” which in this notation would be P<P> = <>.
Thus here we have O<O> = OO =    , so that it is unmarked rather than marked. And in the usual three valued logic we have O<O> = OO = O, and again it is 
not marked.

I have a question for Pedro. Is it ok to send the Laws of Form announcement for my Friday seminar to fis? If not, then fis people who want the announcement should write me an email, and I will put you on the list. I’ll talk about the boundary arithmetic a bit more next Friday and Karl Jaworsky will give a talk about his theory. 

Getting back to the theme, with the boundary arithmetic we see how a very abstract, somewhat philosophical “logical arithmetic” can give us topological network information. This is quite interesting to contemplate as it shows how very macroscopic situations can be understood by using simple mathematical patterning.
Here it is happening  in a mathematical/logical distinction domain. In our original biologic examples we saw how the mathematical domain interacted with the microscopic molecular biological domain in terms of both theory and experiment. All of this suggests further exploration of our patterned thoughts in relation to sensing and understanding natural physical and biological worlds.
Very best,
Lou 


> On Jan 31, 2025, at 3:22 PM, Pedro C. Marijuán <pedroc.marijuan at gmail.com> wrote:
> 
> Dear All,
> 
> As is tradition in this List, the beginning of Chinese New Year announces the coming end of the FIS New Year Lecture. 
> 
> Thanks are given to Lou for his impressive posting work and for the high quality of literary contents (some concluding comments to be added?). 
> 
> The related discussions may continue, but now there are no chairing privileges--the iron rule of 2 - 3 mssgs per week applies to everybody.
> 
> And so, let us also wish our Chinese Colleagues a very Happy New Year!
> 
> 新年快乐
> All the best,
> 
> --Pedro
> 
> _______________________________________________
> Fis mailing list
> Fis at listas.unizar.es
> http://listas.unizar.es/cgi-bin/mailman/listinfo/fis
> ----------
> INFORMACI�N SOBRE PROTECCI�N DE DATOS DE CAR�CTER PERSONAL
> 
> Ud. recibe este correo por pertenecer a una lista de correo gestionada por la Universidad de Zaragoza.
> Puede encontrar toda la informaci�n sobre como tratamos sus datos en el siguiente enlace: https://sicuz.unizar.es/informacion-sobre-proteccion-de-datos-de-caracter-personal-en-listas
> Recuerde que si est� suscrito a una lista voluntaria Ud. puede darse de baja desde la propia aplicaci�n en el momento en que lo desee.
> http://listas.unizar.es
> ----------

-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://listas.unizar.es/pipermail/fis/attachments/20250131/4bf09545/attachment-0001.html>