Posted by
Russell Standish on
Jul 25, 2007; 12:39am
URL: http://friam.383.s1.nabble.com/Intuition-geometry-computation-mathematics-tp524269p524270.html
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.
--
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A/Prof Russell Standish Phone 0425 253119 (mobile)
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