[FOM] Understanding Euclid
William Tait
williamtait at mac.com
Sat Dec 6 12:39:38 EST 2008
On Dec 5, 2008, at 3:13 AM, Vaughan Pratt wrote:
> Bill,
>
> Derivable from what? Proposition 32? In my message I fingered
> Proposition 32 as a counterexample to spherical geometry along with
> Proposition 47. We're in agreement that Proposition 32 does not
> hold in
> spherical geometry.
>
> The difficulty I'm having is that I don't see how any of Euclid's five
> postulates could fail in spherical geometry. My point was that
> propositions such as 32 and 47, which clearly don't hold in spherical
> geometry, can't follow logically from postulates all of which hold in
> spherical geometry.
>
> Which one of Euclid's five postulates fails in spherical geometry?
The first postulate, as it is actually used (and likely as it was
intended by Euclid: Who knows?), fails. Namely, the straight line
connecting two distinct points will not be unique in spherical
geometry when the points are antipodal. Postulate 3, that there is a
circle with a given point as center and a given straight line segment
as radius, is true only if one admits a point to be a circle. (Let the
radius be half a great circle.)
What goes wrong with the proof of Proposition 32 is in the
construction of a parallel to a line from a point not on it. This
depends on the construction of a perpendicular to a straight line from
a point on it: Proposition 11. This depends on Proposition 1 which
assumes APPARENTLY WITHOUT PROOF that, if A and B are distinct points,
the circle with center A and radius AB intersects the circle with
center B and the same radius. This of course fails in spherical
geometry with A and B antipodal points.
If Euclid intended that the circle be a line (i.e. curve), one can
construe the definitions to imply something like Dedekind continuity
for lines (understanding the statement that the 'extremities' of lines
are points to mean that however you cut the line, it determines a
point). On this reading, Postulate 3 is false in spherical geometry
and Euclid's proof of Proposition 1 is OK.
So Postulates 1 and 3 COULD be wrong in spherical geometry, depending
on what Euclid meant. But taken literally in the version that we have,
I agree that they hold and that the proof of Proposition 1 and
therefore of Proposition 32 that is defective,.
Bill
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