Subject: Re: [HM] Kant and non-Euclidean geometry
From: Bill Oates (boates1@home.com)
Date: Sun Mar 12 2000 - 22:59:00 EST
Dear Gordon
Thank you for careful presentation of the alternatives.
> Case 2. Suppose Kant is right about his Proposal 2. How
> could this come about? Suggested answer: It may be that as far
> as *perception* is concerned, "normal" humans do indeed *perceive*
> in Euclidean geometric terms, although they have, by exercise of
> their reason and imagination, made and learned other geometries
> which differ from Euclidean geometry as to the parallel postulate.
> This seems odd to me, since to me, from some standpoints,
> Euclid's parallel postulate seems counterintuitive when it comes
> to human perception. Ditto for his notion of "line" (maybe "point",
> too, but that's a separate matter). A "straight" line going "to
> infinity"? What's this "straight" business, at least not without
> some theory of curvature which itself builds on the notion of a
> Euclidean line (e.g., when you interpret differentiation as
> producing "tangent lines"? And how about a line which "goes to
> infinity"? Lots of people have thought in these terms, with great
> success in many respects, but who ever perceived such a thing?
>
> So I conclude that Kant was wrong about his Proposal 2. However,
> I conclude this not on the grounds that people found some other
> systems like Euclid's except for the parallel postulate, nor on
> the grounds that it is possible to think about our universe as a
> 3-sphere, spatially speaking.
So far I am not convinced by your argument against Kant's proposal 2. I
measure the angles in a plane triangle and I get 180. Perhaps my thinking
is warped by studying too much analytical geometry but usually I attempt
to understand spherical by looking at the surface of a sphere. I understand
the difference between the two because I use three dimensions to see two
different 2-dimensional surfaces. When we get to three dimensions, we are up
against a difficulty. How do we move to a fourth dimension and measure the
curvature of our space? To the extent that I envisage space as Newton does,
something that exists when all else has been abstracted, I think in
Euclidean terms. What of the Leibnizian version of space?
Bill
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