Subject: categories: Re: Demystifying the categorial approach

From: "Ronnie Brown" <[email-masked]>
Date: Tue, 4 Apr 2006 22:49:53 +0100

I have not been able to keep  up with all this correspondence, but would
like to take up Vaughan Pratt's counter to Reinhard Boerger's point saying:
`it can't be done'  which would be better as a question: `How should one do
it?'

The many books and films on maths show there is some kind of hunger to know
what this subject is about, but the books and films often avoid the subject
itself in order to emphasise the weirdness of their chief characters.

The Bangor approach has for years been `advanced mathematics from an
elementary viewpoint', and we have been trying this out on 13 year olds, and
the general public,  for the last 22 years. We have gained a lot from the
exercise, and have used the experience in talks to mathematicians, science
festivals and other scientists.

Here are some references.

136. (with T. Porter), `Category theory and higher dimensional algebra:
potential descriptive tools in neuroscience', Proceedings of the
International Conference on Theoretical Neurobiology, Delhi, February 2003,
edited by Nandini Singh, National Brain Research Centre, Conference
Proceedings 1 (2003) 80-92.

137. (with R.Paton and T. Porter), `Categorical language and hierarchical
models for cell systems', in Computation in Cells and Tissues - Perspectives
and Tools of Thought, Paton, R.; Bolouri, H.; Holcombe, M.; Parish, J.H.;
Tateson, R. (Eds.) Natural Computing Series, Springer Verlag. (2004)
289-303.

I presented the talk for 136. and it went well. One aim was to explain the
concept of colimit, as a way of putting structures together.  A scientist
from the Salk Inst said he kept on thinking about the ideas that night and
could not go to sleep!  This is a better reaction than I get generally from
algebraic topologists (pace Marta's comments)!

Here is another article:

146. (with T. Porter) `Category Theory: an abstract setting for analogy and
comparison', Advanced Studies in Mathematics and Logic (to appear) UWB Math
Preprint 05.10 .
http://www.informatics.bangor.ac.uk/public/mathematics/research/preprints/05/cathom05.html#05.10

If you examine our pages on www.popmath.org.uk, you will find discussion of
many topics, such as AIMS.

My whole approach to higher dimensional algebra since 1965 or so has been to
give expression to intuitions such as expressing a big object as a
composition of small objects, i.e. the algebraic inverse to subdivision, and
also explaining the notion of commutative cube.

There are deep ideas (e.g. knots) that can be explained to children, and so
to a general audience, and others for which  this is much more difficult.
There is a hunger among scientists and the general public to get some idea
of what is going on in mathematics, apart from solving some  famous but
weird  problems. The idea of `structure' is a good way to start, perhaps. I
have used Bill Lawvere's tag: The mathematical notion of space is for the
representation of motion, i.e. of change of data. How can space be
structured? How can we calculate and control ideas in that context? How
strange is space?

In 1987 or so,  I gave a lecture on knots to an audience from schools, 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.'  Think on it!

So I have made something of a song and dance on the theme of `analogy' in
the new edition of my book: and the notion of universal property as a way of
making analogies, to obtain understanding and calculation.

Finally, to see an argument about physics and maths, see an article

S. Novikov `The Second Half of the 20th Century and its
Conclusion: Crisis in the Physics and Mathematics Community in Russia and in
the West',   published in American  Math Society Translations,  (2) vol 212,
2004

which might rise some ire among some of us. There are also important points.
It is certainly an aspect of the debate, e.g. `7. Second half of the 20th
century: excessive formalization of mathematics.'
I was sent a pdf file of this. But I do not have a url for this English
translation.

Ronnie Brown
www.bangor.ac.uk/r.brown

----- Original Message -----
From: "Vaughan Pratt" <[email-masked]>
To: <[email-masked]>
Sent: Monday, April 03, 2006 4:09 PM
Subject: categories: Demystifying the categorial approach

> Reinhard Boerger wrote:
>  Of course, category
>> theoty usrather abstract and can hardly be explained to non-mathematians.
>
> Would this not be both true and false of any mathematical subject worth
> its salt?  Such a subject will include deep results and/or abstract
> concepts that are inaccessible even to many mathematicians, let alone
> nonmathematicians.  At the same time it should be possible to trace the
> chains of reasoning motivating those results and concepts back to
> origins that should be easy to motivate for the nonspecialist and/or
> nonmathematician.
>
> As a case in point, category theory can be motivated by presenting it as
> an approach to axiomatizing sets and functions (and more generally
> algebraic structures and homomorphisms as their structure-respecting
> functions, but one need not start there).  In that approach, instead of
> defining a function to be a binary relation of a certain form, one
> postulates functions as primitives in their own right, axiomatized by
> the laws of composition and the existence of identities.
>

[balance of quotation omitted...]