If you have access to a library (e.g. any college library) that stocks the excellent quarterly magazine MATHEMATICAL INTELLIGENCER, check out a curious little essay by Eric Grunwald in the current (Spring ‘05) issue. The title of the piece is “Evolution and Design Inside and Outside Mathematics.”
Grunwald develops a long dichotomy — a sort of yin-yang schema — based on the difference in thinking pattersn between geometers and algebraists.
This is actually a line of thought that goes way back, at least as far as Wm Rowan Hamilton’s “Algebra as the Science of Pure Time” (1837). The starting point is the idea that geometry is about space, while algebra is about time.
Sir Michael Atiyah did a good piece on the same theme in the MAA Monthly back in 2000. Anyway, Grunwald stretches the thing out to breaking point (and beyond, in my opinion), enlisting all sorts of couplets in his
dichotomy: Newton/Leibniz, Poincare/Hilbert, socialism/capitalism, design/evolution, regular verbs/irregular verbs.
I found that with very little mental effort I could extend the list
indefinitely: Art/Music, Einstein/Mach, prescriptive (lexicography)/descriptive, etc. Somewhere at the end of this road, I feel sure, are our old friends Augustine and Pelagius…
I don’t buy Grunwald’s whole argument, but it’s a nice conversation piece.
If you like this kind of thing.