Numerical Organizing Centers

Abstract

This first of two new chapters (16 and 17) for the Y2K edition tries to succinctly relate twenty years’ “experimental mathematics” about two- and three-dimensional rotors, following up the new parts of Chapter 9’s Section C. It stresses unsolved problems that await the reader’s attention. As noted in 1978’s Chapter 10 (Section C: Transition to Bestiary), Chapters 11 through 23 are less about concepts than about the specific experimental systems by which our many-forked tree of guesswork is pruned toward reality. This chapter combines mathematical concepts with numerical experiments that test conjectures about the solutions of equations, largely by providing counterexamples to steer thinking away from minefields. Equations in this area do not exactly represent physical mechanisms; to that extent they are only metaphors for the laboratory phenomena. These metaphors are awkward analytically, so to find their implications we do a fair amount of intuitive guessing and numerical simulating. The numerical experiments are in some respects as effective as laboratory experiments for disillusioning ourselves from mental mirages. In this mode we learn principally from failures: failures to produce a quantitatively workable incarnation of some hypothesis, or failures of the quantitative incarnation to do what a simpler analytical model does. And we learn from unexpected encounters with alternative solutions.

“If you can make a computer say it, it must be right,” enthused Simplicio. But Saviati scoffed, “That is too easy; I would rather say if you can’t, it probably isn’t right.”

— From a dreamed version of Galileo’s Dialogues