
Re: Analytic vs. Synthetic Geometry
Posted:
Sep 5, 2013 10:55 PM


On Thursday, September 5, 2013 10:03:28 AM UTC4, Peter Percival wrote: > Lite Beta wrote: > > > > > > > > Also, R^n is the Cartesian product. So isn't it by default that the n numbers in R^n are cartesian coordinates? > > > > It seems to me that "R^n" is ambiguous, it might be nothing more than > > this set: {(x_1,...,x_n): x_i real, 1 <= i <= n}. Or it might be that > > set with structure on it. In the latter case, there are various > > possible structures: additive group, vector space, metric space,... >
So do you think cartesian coordinates are not more closely linked to R^n than other coordinate systems? The wikipedia page on Euclidean spaces mixes the two and reading it scrambled (or rescrambled) my brain.
How about this: If you have a constant function f from a subset S of R^2 into R, and you do the double integral of f*dA over S, that is interpreted as the volume of a box with base area S and height f. But doesn't this require us to consider R^2 as the cartesian coordinates (x,y) ?

