On Jan 23, 2012, at 5:38 PM, Owen Densmore wrote: > The obvious question is "what next"? I.e. if we look at complex numbers at > 2-tuples with a peculiar algebra, shouldn't we expect 3-tuples and more that > are needed for operations beyond polynomial equations?
The Fundamental Theorem of Algebra states that complex numbers suffice. But that only means if you all you need is to do is express the solutions of polynomial equations. Abel showed that they do not suffice to solve quintics. Trigonometric functions allow easy solution of cubic equations with real roots, and Ramanujan used theta functions extensively. Hamilton felt the need for quaternions, which are convenient for 3-D transformations. There are generalizations in many directions: hypergeometric functions, Hestenes geometric algebra, Carl pointed to Baez and octonions, which go on to Clifford Algebras. Penrose has long advocated spinors as fundamental. But conventional mathematical physics chose to generalize in the direction of linear operators and functional calculus. Carl said it nicely as > Suspect you are about to pop out of algebra and end up someplace else as > interesting. -Roger ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org
