[FOM] Foundational Challenge

[Patrik Eklund]

Dear Louis,

When you say

> There is no felicitous way ...

I guess you actually say you have not been able to create one, or you 
haven't seen anybody else having created one. And if so, this does 
obviously not mean it is impossible to create one.

Let me give you another example, not related to your graphs, but what we 
nevertheless find as a "felicious way".

Look at signatures, i.e., sorts and operators. We usually, and in 
particular if we do not use categories, view them as sets of sorts and 
sets of operators. However, they can be arranged as objects in a 
monoidal category (see pages 219-221 in our Fuzzy Terms paper, 
https://www.sciencedirect.com/science/article/pii/S0165011413000997). 
This can be seen as an extensions to what Benabou did in the 1960s.

What has been said in previous FOM postings by others generally about 
"what we want to do with them", in this particular case is e.g. for the 
Goguen category to be viewed as a monoidal category, which enables 
algebraic many-valuedness to be attached with operators. This is an 
important step for the foundations of many-valued logic and as 
contribution to the long-standing debate between "fuzzy" and 
"probability". This "felicious way" of viewing signatures in a 
many-valued setting can be used also in very practical situations, like 
analysing information structures related with disease and functioning in 
health care.

What I say here is that we would never have seen this if we would have 
continued to view sorts and operators to reside in 'sets' only. So the 
meaning of

> The whole enterprise is embedded in set theory ...

is now very different as expected. Larry Paulson was a bit into this 
also when sharing his view on how category theory builds upon set 
theory. This is different, I guess, but nevertheless related to the 
"intertwining" of set theory and category theory.

Needless to say, tree automata and such things could also quite 
felicitously be handled similarly, and then we are a bit on the graph 
side, aren't we?

Best,

Patrik



On 2020-01-21 06:31, Louis H Kauffman wrote:
> 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
>> [5] Compositionality
>> URL = https://compositionality-journal.org/about/
>> 
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