On 3/19/08, David W. Cantrell <DWCantrell@sigmaxi.net> wrote:
... So did Jim's post reveal a "dirty secret of calculus"? I would say so. But the secret it reveals, IMO, is simply that the Riemann integral should be replaced by the gauge integral in introductory calculus courses.
A long time ago I'd heard of the Denjoy integral, but knew nothing of these later developments --- interesting stuff there! I'm no doubt broadening the discussion unacceptably at this point --- but we've all seen situations, in mathematics and elsewhere, in which the generally accepted way of doing something is apparently inferior to a little-known alternative, yet persists indefinitely. I've several times wondered if this behaviour can be modelled --- a science of "memetics", perhaps? To start with, it is necessary to identify the influences at work, then define how they interact, and build some kind of percolation process which needs to achieve a certain critical value. Ignorance, inertia, conflicting criteria, vested interests ... Umm --- maybe it's just called "politics"? Fred Lunnon