[FOM] Understanding Euclid
Vaughan Pratt
pratt at cs.stanford.edu
Wed Dec 3 18:15:17 EST 2008
Euclidean geometry is standardly understood as the geometry of the
plane, more generally of flat or uncurved space. To make this stick
however, Euclid's fifth postulate should be phrased as an equivalence:
two lines fail to meet if and only if a third line intersecting them
both meets them at the same angle (or any equivalent phrasing thereof,
e.g. that the interior angles on the same side of the cutting line sum
to two right angles, or that a point P and a line L determine a unique
line through P parallel to L).
Wording the postulate in this way then allows "parallel lines" to be
defined equivalently as lines that don't meet, or as lines that cut any
third line if at all in the same angle. Before contemplating weakening
the Parallel Postulate in any way, one should first decide which of
these one is going to take as the definition of "parallel," because the
respective predicates denoted by the two definitions diverge when it is
weakened. Here I'll follow the custom of defining parallel lines to be
lines that don't meet.
When the Parallel Postulate is omitted altogether, other geometries such
as hyperbolic and elliptic satisfy the first four postulates. In
particular those postulates are true on the sphere, for example the
Earth's surface so modeled.
Euclid required just the only-if direction of that equivalence, that a
line cutting parallel lines meet them at the same angle. His phrasing
more precisely was of the contrapositive, couched in terms of two
adjacent angles summing to less than two right angles, which one might
interpret constructively as the Triangle Postulate, that a base and two
such angles always determine a (not necessarily unique) triangle,
analogously to the other two odd-numbered postulates, that two points
determine both a line and a circle (in those cases uniquely). Yet
another phrasing is that *at most* one line can pass through a point P
parallel to a line L, also constructive but in this case partially and
uniquely instead of totally and nonuniquely, concomitant with point-line
and epi-monic duality.
In this weaker form the Parallel Postulate is true not only on the plane
but on the sphere when lines are understood to be great circles or
geodesics. Hence all of Euclid's postulates as originally stated are
true on the sphere.
The great search for a counterexample to the Parallel Postulate only
makes sense for Euclid's statement of it. Its phrasing as an
equivalence (or even just as the contrapositive, two lines cut by a
third at the same angle don't meet) admits the sphere as an obvious
counterexample worth not even brownie points at the level of
mathematical sophistication at which the search was conducted. The
searchers were clearly well aware of the soundness of Euclid's
postulates for the sphere. With regard to the preference for the
one-directional version of the Parallel Postulate, Euclid and the
searchers were surely on the same page.
I would welcome pointers to sources of information and opinion bearing
on either of the following questions.
1. All intuition about Euclidean space per se demands that the Parallel
Postulate be bidirectional, in order to rule out positive curvature as
well as negative when proving theorems true on the plane but false on
the sphere. Given this, why were Euclid and the searchers in agreement
about stating the Parallel Postulate in its weakened one-directional
form? Was Euclid covertly hoping that his theorems would all hold for
surveying projects of a scale where the earth's curvature was
significant, or did it merely not occur to him that he might need the
converse at some point?
2. Proposition 47 of Book 1 of the Elements states and proves
Pythagoras's theorem. This is clearly false on the sphere, which
contains an equilateral triangle all three of whose angles are right.
This is not the earliest counterexample: Proposition 32 shows that the
angles of a triangle sum to two right angles. Euclid is evidently not
playing by his own rules, and is appealing to unstated postulates. Does
he appeal somewhere to the converse of his fifth postulate? If so how
did he fail to notice he was doing so? This would make him an early
user of the Fallacy of the Converse, from P implies Q infer Q implies P,
and an embarrassingly visible one at that. Proclus's Commentary raises
only relatively finicky concerns, I don't know who first pointed out
this more glaring discrepancy but it's hard to imagine anyone seriously
working on the independence of the Parallel Postulate without being
aware of it. It was surely common knowledge by the time Hilbert
embarked on his program to make Euclid's system more rigorous by
formalizing it.
Vaughan Pratt
--I was raised to be rigorous, not formal.
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