[Fis] Linking mathematical Indices in the natural sciences. was: Re: Links to YouTube videos of Dec 15 ZOOM session & reminder of meeting on Wed, Jan 19th 2022

[Louis Kauffman]

Dear Folks,
I am just popping in on this discussion to add a couple of comments about the Rota article.
He is discussing the fact that multiple formalizations of given mathematical topics add to the richness of their understanding.
If you take the formalist position that a mathematical topic is DEFINED by its axiomatization, this attitude seems problematical,
But if you take the attitude (Wittgenstein II) that the meaning of the mathematics is in its use, that is the meaning is  in the skein of relationships with mathematical use, applications and 
a variety of languages, then it is not surprising that to “one” mathematical topic there may be a variety of axiomatizations and a variety of contexts in which it may be understood.
Rota’s remarks are very well thought out and apply to many situations. 

You can look at the mathematical side of things, by thinking back to your childhood when you learned arithmetic.
(I am so ancient that we had no electronic calculators then, but we did have adding machines. I was fascinated by the machine in my father’s clothing store. Pull a lever and add!)
You did not learn arithmetic on the basis of axioms. But you learned that it made sense. Its results corresponding with your abilities to count and work with number signs.
It was language and activity that was part of the larger language and activity that was your life and school life at that time. You came to understand relations between arithmetic and 
geometric thinking (slicing a pie and fractions and all that). You understood what you were doing in the consistency of those actions.

Axiomatization, if you have seen it, for arithmetic requires that the person know algebra. You had to learn algebra before you could axiomatize.
When you think about it, you see that formal axiomatization of a mathematical topic is the LAST thing that happens. You do not start there, and it may be helpful to do it a different way.
Rota concludes that the mathematics is not captured by a single axiomatization. I agree. But it means that mathematics, like any other human activity is part of our language and relations.
All such activity involves both syntax and semantics. We can see how meaning does arise from choices of syntax. And we can see how choices of syntax arise from our meanings and our
uses. Meaning and Syntax are two poles of our activity that, like the poles of a magnet, are inseparable. But they are also malleable in relation to our dialogues.

I believe that all of these thoughts inform the discussions we have here about information where, once again, there are poles of meaning and syntax so strongly debated.
Very best,
Lou Kauffman


> On Jan 9, 2022, at 4:05 AM, Loet Leydesdorff <loet at LEYDESDORFF.NET> wrote:
> 
> Dear Jerry,
> 
> I donwloaded nd read "Syntax, Semantics, and the Problem of Identity of Mathematical Items”, but I find it difficult to relate the discussion. I guess that I am not the only one needing help. 
> 
> For example: Can you, please explain 
>> "the deep tensions between the Rosen - Husserl - C S Peirce philosophies of mathematics."?
> It seems to me that all three of them used a definition of information which includes meaning. 
> After Shannon (1948), one can distinguish between information and meaningful information. 
> 
> In my opinion, one can study the dynamics of meaning independent of the dynamics of information, and then study the dynamics of meaningful information as an interaction term. 
> 
> [...]
> 
>> More specifically, I am referring to the Peircian necessity for indexing symbols prior to creating logical propositions, a consequence of semiosis that is not embraced by Husserl and certainly not addressed by Rosen’s categorical diagrams. 
> Because the latter two abstracted from the bioogy?
> 
>> The indexing of biological symbols denotes the prior indexing of chemical symbols.  The intimate entangling of the indexes of chemical and biologic symbols is a mathematical fact that connects the syntax and semantics of molecular biology. 
>> 
>> These indices are essential to connecting the permutation groups of atomic numbers to the physical space groups of X-ray crystallography.  These indices are also essential to connecting the electrical attractor/repeller formula of the BZ reaction to the physical attractor-repeller formula such as the Rossler chaotic attractor.  The former comes to equilibrium, the latter does not!
>> 
>> Gian-Carlo Rota (discrete MIT math professor/ philosopher) addresses this syntactical <—> semantic distinction in mathematics in  "Syntax, Semantics, and the Problem of Identity of Mathematical Items” of Chapter 12 of Indiscrete Mathematics (1997). A difficult read to be sure, but extremely relevant to the failures of some FISers to either prehend, apprehend or comprehend the relative propositions of information transmission. Furthermore, this chapter illuminates several foundational issues of physical complexity theory.  After a couple of decades, I just re-read this chapter last evening.  It remains a “thought -energizer” that gracefully transcends academic boundaries from a Husserialian comprehension of mathematics. :-) 
> Rota is not using this terminology; it is yours. I would particularly be interest in "academic boundaries from a Husserlian comprehension of mathematics"?
> How do you see this? Please, explain. Do not hesitate to be too long. We may understand easier when there is some redundancy.
> 
>> Cheers
>> 
>> Jerry 
> Thanks in advance. Best, Loet
>> 
> 
> _______________
>  <https://www.springer.com/gp/book/9783030599508>Loet Leydesdorff
> 
> "The Evolutionary Dynamics of Discusive Knowledge" <https://link.springer.com/book/10.1007/978-3-030-59951-5>(Open Access)
> Professor emeritus, University of Amsterdam 
> Amsterdam School of Communication Research (ASCoR)
> loet at leydesdorff.net  <mailto:loet at leydesdorff.net>; http://www.leydesdorff.net/ <http://www.leydesdorff.net/>
> http://scholar.google.com/citations?user=ych9gNYAAAAJ&hl=en <http://scholar.google.com/citations?user=ych9gNYAAAAJ&hl=en>
> ORCID: http://orcid.org/0000-0002-7835-3098 <http://orcid.org/0000-0002-7835-3098>;    
> 
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