[FOM] Understanding Euclid

[Michael J Barany]

I don't have a direct answer to your question, but I can point out that we
shouldn't be surprised that obvious fallacies or counter-examples may have
persisted for even thousands of years.  Lakatos's Proofs and Refutations is
full of such examples, including during the time when the parallel postulate
was most actively being challenged.  Counterexamples were proposed to Cauchy's
theorem that limits preserve integrals (without stating the uniform limit
condition) some half a century before they were widely accepted as genuine
counterexamples.  It wasn't that mathematicians were being misguided or
unthorough,... just that a lot of things seem more obvious looking backwards
than looking forwards.

Smiles,

Michael


> 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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