I have a paper with Loday on calculating the third homotopy group of a suspension of a K(G,1) 51. (with J.-L. LODAY), ``Van Kampen theorems for diagrams of spaces'', {\em Topology} 26 (1987) 311-334. in terms of a new tensor product of non abelian groups. It would not be a sensible task to rewrite the paper without category theory (considered as a unifying principle, as a mode for efficient calculation in certain algebraic structures, and as a supplier of new algebraic structures). To go further back (and higher, of course) Grothendieck's extension of the Riemann-Roch Theorem uses category theory explicitly. In the 1950's people were trying to give algebraic proofs of this theorem, when AG came up with an algebraic proof of a vast generalisation. Then there is the categorical background to the proof of the Weil conjectures .... But the question is misplaced - in the early part of the 20th century, I expect some would ask if set theory was really necessary for a confirmed problem solver! The history of maths shows that maths greatest contribution to science, culture and technology has been in terms of expressive power, to give a language for intuitions which enables exact description, calculation, deduction. (Exam question: discuss the last statement, with an emphasis on particular examples!) It also allows for the *formulation* of new problems, which perhaps cause old interests to lapse as people perceive there are more exciting things to do. It is a narrow view to regard `important maths' as necessarily that which solves well known problems, and so leave evaluation as akin to a sports league table: how old is the problem? who has worked on it? etc, etc. The progress of maths is much more complicated and interesting than that! It won't help the public image of maths if it is seen that mathematicians believe the most important aspect of their subject is a (to the public) bizarre set of problems which seem to interest no one else. However, it is important that these questions be asked, together with questions on modes for evaluating `good maths'. As AG remarked in a letter, maths was held back for centuries for lack of the `trivial' concept of zero! For more discussion, look at http://www.bangor.ac.uk/ma/CPM/cdbooklet/knots-m.html knots2.html So let the debate be broadened and continued! Ronnie Brown (will someone please forward this to the newsgroup?) PS I am trying to trace a combinatorics problem solved in the 1950s using categories of paths, and which was at the time held up as the sort category theory could not do! Maybe my memory is failing! But the question put on the newsgroup is an old war horse! -------- Original Message -------- Subject: categories: A question on: <newsgroup sci.math> Date: Mon, 19 Jun 2000 17:13:05 +1000 From: Ross Street <street@ics.mq.edu.au> To: "categories@mta.ca" <categories@mta.ca> I thought the CatNet would be interested in the following question which appeared on <newsgroup sci.math>. The letter was pointed out to me by a colleague at Macquarie. --Ross ************************************************* From: mathwft@math.canterbury.ac.nz (Bill Taylor) Newsgroups: sci.math Subject: Query about Category Theory. Date: 18 Jun 2000 05:36:30 GMT Organization: Department of Mathematics and Statistics, University of Canterbury, Christchurch, NewZealand Does anyone know of any cases where Category Theory (morphisms, functors & all that) has helped solve an unsolved problem? That is, a problem in some other branch of math, posed without reference to categorical ideas, and previously unsolved, that was first solved via Category T. I realize that CT provides a unifying framework for many seemingly disparate ideas in math, and that is a fine thing of course; but I was just wondering if it had this problem-solving capability. TIA for any helpful responses. ---------------------------------------------------------------------------- --- Bill Taylor W.Taylor@math.canterbury.ac.nz