
Re: Is Analytic Geometry really Geometry?
Posted:
Jun 22, 2006 5:37 PM


On Thu, 22 Jun 2006 09:01:15 0400, Hatto von Aquitanien <abbot@AugiaDives.hre> wrote:
>Toni Lassila wrote: > >> On Thu, 22 Jun 2006 05:43:00 0400, Hatto von Aquitanien >> <abbot@AugiaDives.hre> wrote: >> >>>Toni Lassila wrote: >>> >>>> On Wed, 21 Jun 2006 23:01:39 0400, Hatto von Aquitanien >>>> <abbot@AugiaDives.hre> wrote: >>>> >>>>>Lee Rudolph wrote: >>>>> >>>>>> Hatto von Aquitanien <abbot@AugiaDives.hre> writes: >>>>>> >>>>>>>How can one discuss points and lines without an a priori sense of a >>>>>>>geometric continuum? >>>>>> >>>>>> Very simply. Gene may yet show you how. >>>>> >>>>>I don't believe it is possible to ever show such a thing. The continuum >>>>>is a priori, and can, therefore, not be excluded from reasoning. >>>> >>>> http://planetmath.org/encyclopedia/FinitePlane.html >>> >>>Explain that to me without relying on the ordering of symbols on the >>>page/screen, and without any reference to concepts of time such as the >>>word "repeatedly". >> >> Why? > >To demonstrate that you can "discuss points and lines without an a priori >sense of a geometric continuum?".
Finite geometries like many other special geometries do not fulfill Dedekind's Axiom, so there is no appeal to "geometric continuum" there.
>I'm quite capable of applying concepts from one area of thought to another. > >My contention is that our concept of >continuity is a priori. I believe intuitively that the idea of real number >continuity is neither in need of proof, nor does it admit of proof.
The idea that real numbers are "intuitive" has popped up here recently, but I think it's blatantly false. Natural numbers, integers, and to a certain extent rationals could certainly be said to be intuitive. Real numbers are the result of a rather unintuitive completion process that produces a lot of misunderstandings and crankiness.
>Furthermore, I believe we innately understand certain other geometric >principles without any need of instruction.
Maybe so, but that does not affect the nature of purely axiomatic mathematical geometries.

