Subject: Re: [HM] Kant and non-Euclidean geometry
From: Gordon Fisher (gfisher@shentel.net)
Date: Mon Mar 13 2000 - 18:16:38 EST
Bill Oates wrote:
> 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?
>
Aboout the 180 degrees: When, in an elementary physics class, you
measure how far something falls in a given time, you won't get an
exact result according to Galileo's law of falling bodies, except by
way of an extraordinary coincidence. Some of the reasons for this are:
(1) physical measurements are subject to error due to the ways the
measuring apparatus and the measurer interact with the falling body,
and inevitable (hopefully small) imperfections in the measuring
apparatus (and the measurer);
(2) in an elementary physics class, you probably weren't letting
the body fall in a vacuum;
(3) from a theoretical point of view, more precise results can be
obtained by way of Newton's law of gravity. Of course, in the
case of a ball, say, falling in a vacuum, close to the surface of
the earth, the mass of the earth is so large compared with the
mass of the ball that the fact that the earth is attracted by the
ball (as well as the other way around) can be neglected, and I
suppose would take a fiendishly clever experimental apparatus to
detect at all, given (1) above.
In the case of the angles of a triangle measuring 180 degrees, you
will have problems with (1), since you will be measuring some
realization of an ideal triangle with some physical apparatus such
as a protractor. However, here what is in question, from a
theoretical point of view, is a question of scale again. There's
no doubt that for triangles the size of those on our earth, the
measurements you get for the sum of the angles, assuming the
measurements are carried out caefully, will be very close to 180
degrees. The question is, though, suppose the vertices of a
triangle are taken to lie in three different galaxies, or even on
three widely separated stars in our own galaxy, with the side
determined by rays of light or some other kind of electromagnetic
signal. It may be that one would find some other angle sum for a
triangle so constructed was consistently closer to, say 182 degrees.
If memory serves me well, this sort of experiment was proposed back
early in the 19th century, and I seem to remember that some such
experiments have actually been carried out in this form, but I
think without definitive results.
In any case, in general relativity theory, there have been experiments
related to this, such as the one involving light ways and Mercury and
the Sun. As to having trouble with (2) above, well -- there's all
that so-called "dark matter" floating around our universe, and maybe
light behaves in some odd way we haven't discovered yet, or maybe one
can draw and analogy between doing a falling body experiment in air
or some other medium vs doing one in a vacuum (i.e. a vessel on earth
from which the air has been evacuated with a pump), and doing a
triangle measuring experiment near the surface of our earth vs doing
one involving three galaxies.
As to measuring the curvature of spacetime, a 4-dimensional manifold,
one does this by generalizing the Gaussian method of measuring the
curvature of a 2-dimensional surface, using tensors. In line with my
discussion of Kant's thought, I would say we have something here we
can in no way perceive (barring unusual mutations), but can understand
by way of analogy and reason. We have moved, then, from what we
perceive to something we can understand and use, but for which
perception itself fails us.
Gordon Fisher gfisher@shentel.net
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