jim stasheff wrote:
now for a mathematical subject a math proof is sometimes but not always necessary
There's a saying about Lefschetz that he "never wrote a valid proof, and never made a false conjecture". Now it's not an attitude that want to encourage, but if you have great mathematicians who are like that (and Lefschetz was not just a good mathematician, but a great mathematician, without whom a good deal of modern algebraic geometry would be unimaginable), then this ought to tell us something. What it tells us is, of course, not easy to formulate: it's an example that causes severe problems for almost every philosophy of mathematics that I know. But it ought to stop us saying things of the form "if we don't do category theory in such and such a way, then it won't be mathematics at all". (Of course we'll all keep saying this, because we all have a secret fear that, if we aren't really careful about what we do, the grown up mathematicians will kick sand in our face, but that's a psychological problem and not a mathematical problem.) -- Dr. Graham White Lecturer Department of Computer Science Queen Mary, University of London Mile End Road London E1 4NS http://www.dcs.qmul.ac.uk/~graham (+44)(020)7882 5242