On Wed, Nov 28, 2012 at 3:49 AM, Peter Duveen <email@example.com> wrote: > I'm in Chandy's camp on this one. Mathematics takes common sense notions > of number and expands upon them. That is why I have often attempted to find > a common sense premise, which I often called "self-evident," such as that > the area of a rectangle is proportional to the length of its side, the > other side remaining constant, to demonstrate other properties, such as the > l x w formula. If we loose sight of the common-sense foundations of our > subject, we breed the confusion that is oft associated with the study of > mathematics.
This is also true of triangles, that if you extend only the base keeping the other side fixed, the area increases proportionally. Twice the base, twice the area.
So is a triangle a good model of l x w? Well, the angle is another variable. Once we see it's not 90 degrees we think "it could be anything" whereas a rectangle has fixed angles (by definition).
If we fix the edges to be at 90 degrees, so only a right triangle is considered, then we could *define* e1 x e2 = area = area of triangle and get away with it. It'd be a logical and consistent move. Instead, the convention is to *double* that area (of the triangle) and go with the rectangle uniquely defined by e1 and e2.
Rather than say "it has to be that way" (area has to be based on square and/or rectangular units) we have a moment to say "all math is ethno math" i.e. we have made decisions, as a culture, at various forks in the road. Other maths are out there, if we go back and make different turns. Some mathematicians explore these possibilities.