On Tue, Jul 24, 2007 at 12:39:06PM -0600, Peter Lissaman wrote: > Geometry has no place in mathematics. Mathematics cannot be explained > graphically -- all math proofs must be for blind men, as me tutor used to > say.
I vehemently disagree with this comment. Consider the theorem that the determinant of the product of two matrices is the product of the two determinants. This can be understood geometrically in a trice, as a determinant is simply the ratio of the changed hypervolumes undergo when passed through a linear map (for 2 dimensional hypervolumes, substitute "area", for 3D substitute "volume"). Sign captures whether the volume has undergone a mirror transformation. Obviously applying two linear maps one after the other leads to the desired composition rule. However, to show this theorem algebraicly requires at least a page of algebra, and it is not clear one hasn't made a mistake. One would never get to the theorem in the first place without the geometrical intuition. However, the algebra is needed to ensure one isn't mislead by intuition. I have met mathematicians one cannot talk to in geometry. They are a pain to work with. -- ---------------------------------------------------------------------------- A/Prof Russell Standish Phone 0425 253119 (mobile) Mathematics UNSW SYDNEY 2052 [EMAIL PROTECTED] Australia http://www.hpcoders.com.au ---------------------------------------------------------------------------- ============================================================ 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
