[FOM] Foundational Challenge

[Louis H Kauffman]

Dear Jose M.,
Lets take knot diagrams as an example. A given knot diagram D is a 4-valent plane graph with extra structure. It is convenient to condsider the set of  all diagrams S(D) obtained from D by the Reidemeister moves and to prove that a certain knot polynomial will take the same value on all of them. While one may focus on the diagram or the graph, nevertheless one needs the set of all the diagrams and the 
understanding that S(D) is a countable set and other matters. There is no felicitous way to handle knot diagrams and the associated tensor categories and functors of quantum topology without using set theory.
The whole enterprise is embedded in set theory with special languages of diagrams and compositionality that are convenient for both theory and calculation. This is what we expect of set theory - a broad basis that can handle special inventions as well. One more point. The diagram D can be seen as a projection of a curve that is set-theoretically embedded in three dimensional space. This relationship is of key importance for 
working with the knot theory and relating the diagrammatic parts of it with the classical, topological and geometrical parts. The set theory allows one to work with the subject as a whole. These remarks apply to the categories for quantum theory and the tensor diagrams as well.
Very best,
Lou Kauffman

> On Jan 17, 2020, at 12:20 PM, José Manuel Rodríguez Caballero <josephcmac at gmail.com> wrote:
> 
> Tim wrote:
> So again, I don't agree with M. Katz that ZFC as a universal
> foundation for all mathematics is not credible, if we understand that
> claim rightly.  The claim isn't that for every subfield X of mathematics,
> we must explicitly use raw set theory for the "foundations of X" and
> eschew any defined notions.  The claim, rather, is that set theory is
> still the most convincing candidate when it comes to Generous Arena,
> Shared Standard, and Metamathematical Corral for mathematics considered as
> a whole 
> 
> There is a contemporary tendency in mathematics, physics and computer science of substituting formulae by the so-called graphical calculus [1, 2, 3, 4], both in the statements of the theorems and in the proofs. In this approach, the sets are not fundamental, neither from a formal point of view nor from an intuitive point of view. The main focus is on the composition of processes and the fundamental intuition comes from topology, especially from knot theory. For this reason, this new tendency is known as compositionality [5]. Compositionality cannot be reduced to the foundation of X, because X is aimed to be everything, in mathematics and outside mathematics, e.g., physicist, biology, social sciences, computer science, etc. It is not unreasonable to predict that compositionality may become someday a new Generous Arena, Shared Standard, and Metamathematical Corral for mathematics as a whole.
> 
> As evidence that compositionality is already part of the current mainstream scientific activity, I would like to share the following typical fragment from an announce of Postdoctoral and Ph.D. positions in Edinburgh for a project about Quantum Theory (notice that both Category Theory and Causality, which are closely related to compositionality, are considered as important for this field):
> 
> Applicants must have or be about to receive a degree in Computer
> Science, Mathematics, or Physics, with a background in one or more of
> the following areas:
> * Quantum computing
> * Category theory
> * Programming languages
> * Causality
> * Concurrency  
> 
> Kind regards,
> Jose M.
> 
> [1] Penrose, Roger. Applications of negative dimensional tensors. Combinatorial mathematics and its applications 1 (1971): 221-244.
> [2] Kauffman, Louis H. Introduction to quantum topology. Quantum topology. 1993. 1-77.
> [3] Coecke, Bob, and Aleks Kissinger. Picturing quantum processes. Cambridge University Press, 2017.
> [4] Blinn, Jim, Using Tensor Diagrams to Represent and Solve Geometric Problems. 2002
> URL = https://www.microsoft.com/en-us/research/wp-content/uploads/2002/01/UsingTensorDiagrams.pdf <https://www.microsoft.com/en-us/research/wp-content/uploads/2002/01/UsingTensorDiagrams.pdf>
> [5] Compositionality
> URL = https://compositionality-journal.org/about/ <https://compositionality-journal.org/about/>
> 
> 
> 
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