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Thursday, June 22, 2006

Old correspondence

My old laptop returned to life today. It had refused to boot up a while ago, and so was left to gather dust on a shelf. While I was passing this morning I gave its ON button a small tap and with that it sprang into action. So I've had the chance to look into my correspondence of the period 1999-2001. It's intriguing to see nascent ideas, some since turned into papers, others since abandoned.

At the time, I had contacted several mathematicians for the writing of my chapter on groupoids, including Saunders Mac Lane, who told me that they didn't feature at all in the early days of category theory. I also corresponded with the late George Mackey. During my time as a PhD student I had loved the story of maths he had told about in The scope and history of commutative and noncommutative harmonic analysis, so I asked him how he saw groupoids fitting into the picture. His reply included the remark:
At the moment I am occupied with developing some recent ideas I have had on a possible extensive development of my methods to apply to a much larger part of mathematics and produce more unification. I will explain more fully when I have made a bit more progress in seeking the proper formulation.
He spoke of some manuscripts he had written along these lines. Someone could do us a great service by digging these out.

I was also interested in diagrammatic reasoning at the time, so contacted Todd Trimble about his category theoretic reconstruction of the American philosopher Charles Peirce's existential graphs. He said:
One thing I would have emphasized, had I addressed your group, is Lawvere's revolutionary insight that the connectives and quantifiers in logic are controlled by *adjoint functors*. I think this is the key to further progress in geometrizing logic: higher-dimensional adjunctions are intimately connected with Morse theory, esp. the calculus of cancelling and rearranging critical points of Morse functions. (I don't think Gerry Brady and I fully connected the Beta graphs with this geometric aspect of adjunctions -- it ought to be done.) It is interesting to me that Peirce perceived, at a pre-formal level, the structure of connectives via adjunctions.

This connection of logic and singularity theory really needs exploring.

To end this short stroll down memory lane, I had forgotten that I had developed an interest in Dudley Shapere, a philosopher of science. He seemed to my mind to be asking the kinds of question about science that I wanted to ask about mathematics. I had included this in an e-mail:

Shapere suggests that one should be answering the following questions:

(1) What considerations (or, better, types of considerations, if such types can be found) lead scientists to regard a body of information as a body of information - that is, as constituting a unified subject matter or domain tobe examined or dealt with?

(2) How is description of the items of the domain achieved and modified at sophisticated stages of scientific development?

(3) What sort of inadequacies, leading to the need for further work, are found in the bodies of information, and what are the grounds for considering these to be inadequacies or problems requiring further research? (Included here are questions not only regarding the generation of scientific problems about domains, but also scientific priorities - the questions of importance of the problems and of the "readiness" of science to deal with them.)

(4) What considerations lead to the generation of specific lines of research, and what are the reasons (or types of reasons) for considering some lines of research to be more promising than others in the attempt to resolve problems about the domain?

(5) What are the reasons for expecting (sometimes to the extent of demanding) that answers of certain sorts, having certain characteristics, besought for those problems?

(6) What are the reasons (or types of reasons) for accepting a certain solution of a scientific problem regarding a domain as adequate. (Shapere1984, 277-8).

He reckons that only the last has been seriously examined by philosophers of science.

These still strike me as very good questions.


lauret said...

"higher-dimensional adjunctions are intimately connected with Morse theory, esp. the calculus of cancelling and rearranging critical points of Morse functions"

This is an extremely interesting indeed, could anyone give some additional explanation or sources?

June 23, 2006 12:03 PM  
david said...

Any of Baez's stuff on the Tangle Hypothesis is relevant, like Week 121. Then there's this by Cheng and Gurski.

June 23, 2006 12:25 PM  
david said...

See also Higher-Dimensional Algebra IV: 2-Tangles should be able to compress the definition of `monoidal 2-category with
duals' using more of the language of 2-category theory. Doing so will shed more light
on the still mysterious general notion of `n-category with duals'. It bodes well that the
triangulator and its swallowtail coherence law have already been observed by Street
in his study of adjunctions between 2-categories [27]. In general, we expect a close
relation between the theory of n-categories with duals and the theory of adjunctions
between n-categories. This has already been noted in work on 2-Hilbert spaces [1, 22],
and the patterns found here should continue for higher n-Hilbert spaces. (p. 61)

June 23, 2006 12:31 PM  

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