A well kept secret?

[Ronnie Brown <ronnie.profbrown@btinternet.com>]

In reply to André :


What seems reasonable to do is analysis, namely what is behind the 
success of category theory and how is this success  related to the 
progress of mathematics.
Which implies asking questions of mathematics, some of which have been 
aired in this discussion list. In this way, it should be possible to 
avoid seeming partisan, but to ask serious questions, which should help 
to steer directions, or suggest new ones. Of course lots of great maths 
does not arise in this way, but by following one's nose, but that does 
not mean that such analysis of direction is unhelpful.

I know some argue that this excursion into what might be called the 
theory of knowledge, or into methodology,  seems unnecessary to some. In 
reply I sometimes point to remarks of Einstein on my web site
www.bangor.ac.uk/r.brown/einst.html
or more mundanely retort that normal activities normally require some 
meta discussion: if you want to go on a holiday, you do some planning, 
you don't just rush to the station and buy some tickets. I develop this 
theme in relation to the teaching of mathematics in an article
What should be the output of mathematical education?
on my popularisation and teaching page.

I gave a talk to school children on `How mathematics gets into knots' in 
the 1980s, and a teacher came up to me afterwards and said: `That is the 
first time in my mathematical career that anyone has used the word 
`analogy' in relation to mathematics.' Yet abstraction is about analogy, 
and very powerful it is too. This was part of the motivation behind the 
article
146. (with T. Porter) `Category Theory: an abstract setting for analogy 
and comparison', In: What is Category Theory? Advanced Studies in 
Mathematics and Logic, Polimetrica Publisher, Italy, (2006) 257-274. pdf

There is also interest in the question of how category theory comes to 
be successful, and more successful than, say,  the theory of monoids. 
This seems connected with the underlying geometric structure being a 
directed graph, i.e. allowing a `geography of interaction'. A category 
is also a partial algebraic structure, with domain of definition of the 
operation defined by a geometric condition. Is this enough to explain 
the success?

It is worth noting that the article
Atiyah, Michael, Mathematics in the 20th century, Bull. London Math. 
Soc., {34},  {2002}, 1--15,
suggests that important trends in the 20th century were:
                              higher dimensions, commutative to non 
commutative, local-to-global, and the unification of mathematics,
but does not include the words `category' or `groupoid', let alone 
`higher dimensional algebra'!

This kind of analysis needs to be presented to other scientists, and to 
the public, not only to mathematicians. There is a hunger for knowing 
what mathematics is really up to, in common language as far as possible, 
what new concepts, ideas, etc., and not just `we have solved Fermat's 
last theorem'.

If your analysis of what category theory should do suggests some gaps, 
then that is an opportunity for work!

Good luck

Ronnie Brown


Joyal wrote:

Category theory is a powerful mathematical language.
It is extremely good for organising, unifying and suggesting new directions of research.
It is probably the most important mathematical developpement of the 20th century.

But we cant say that publically.

André Joyal

  



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