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... 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" <pratt@cs.stanford.edu> To: <categories@mta.ca> 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...]