[FOM] FOM Understanding Euclid
Vaughan Pratt
pratt at cs.stanford.edu
Sat Dec 6 05:35:53 EST 2008
Colin McLarty wrote:
> Euclid's postulate that a line segment can always be extended was
> understood to mean "extended to new points," i.e. a segment can always
> be extended without returning to itself. That is how he proves
> Proposition 16, the first of his propositions that fails in the sphere
> geometry: "In any triangle, if one of the sides is produced, then the
> exterior angle is greater than either of the interior and opposite
> angles."
Colin, thanks very much for that.
By "he proves," do you mean Euclid himself or those looking for some
justification for his reasoning?
Where does Euclid exploit the no-new-points premise in his proof?
How plausible is it, really, that this premise is what Euclid had in
mind at the time?
There are just too many missing facts and logical connections to make
this explanation plausible.
1. Unless I'm missing something Euclid says nothing about new points
anywhere.
2. He proves Proposition 16 by appealing to the diagram, which being
drawn in a medium satisfying the converse of Postulate 5 makes it
impossible to see how it could fail to follow from the postulates in the
absence of that converse. (I find this the most telling point of all.)
His apparent success with Proposition 16 may have led him to weaken
Postulate 5 because the other direction *did* follow from 1-4, so he
thought. The grand project over the following millennia of eliminating
Proposition 5 altogether may well have been simply one of somehow
persuading the other shoe to drop, along similar lines.
3. The no-new-points premise only rules out spherical geometry under
the assumption that spherical geometry is the only possible
counterexample to the converse of Postulate 5, making it a
model-theoretic argument. What evidence is there that Euclid intended
his system to be limited to the models of Postulates 1-5 with 5
bidirectional, plus spherical geometry?
Only slightly more plausible is that Euclid had just the plane and
spherical geometry in mind. But in that case surely he would have said
*something* by way of explanation of how Postulate 2 was supposed to
rule out spherical geometry. This post hoc explanation of Proposition
16, with nothing to back it up other than the absence of any other
explanations, might have satisfied those who bought it, but to my
thinking it sheds little if any light on what Euclid might have had in
mind here.
-----
A convenient way of organizing Euclid's postulates is according to
locality: which postulates take their user well outside the scope of
their data?
Postulate 1 is naturally interpreted as being about line segments rather
than lines, otherwise there would be no need for Postulate 2 concerning
extendibility of lines. With this very reasonable interpretation
Postulate 1 is entirely local.
Postulate 2 is therefore the first non-local postulate, allowing line
segments to be extended into foreign countries and beyond.
Postulate 3, two points determine a circle, is intrinsically local: the
resulting figure fits within twice the separation of the data.
The locality of Postulate 4, all right angles are congruent, is unclear
for want of any notion of locality of a right angle. On the one hand
one might argue that infinitesimal neighborhoods of right angles
suffice, on the other one might invoke Postulate 2 to argue that right
angles can grow to arbitrary size, which while not a problem for a
perfect sphere might be interpreted as ruling out all other
positive-curvature geometries. However the precise logic of how
Postulate 4 as stated would rule those out while leaving the sphere
intact as a model remains completely unclear to me. In any event I
don't see how any such interpretation of Postulate 4 could rule out
spherical geometry itself, the sphere being perfectly symmetrically and
therefore admitting no obvious (to me) counterexamples to Postulate 4.
Postulate 5 in the direction stated by Euclid is intrinsically nonlocal
by virtue of a given base and angles summing to less than 180 degrees
producing a third point arbitrarily far away. The (missing) converse of
Postulate 5 however is easily seen to be local, by consideration of
Proposition 32 as a suitable surrogate for the converse. Proposition 32
asserts that the angles of a triangle add up to 180 degrees, and this is
a completely local statement. Taking the earlier Proposition 16 instead
makes no difference, it being just as local; likewise for Proposition 47
(Pythagoras).
How is a nonlocal condition such as novelty of a point per Postulate 2
supposed to bear on a purely local postulate organized along the lines
of Propositions 16, 32, or 47? There you are with Toto somewhere out
near Arcturus and suddenly you stumble upon a triangle whose angles add
up to 181 degrees. How natural would it be to say, "Toto, I have a
feeling of deja vu." This might make sense if there were some reason
for the existence of "large" triangles in this sense to imply a return
to an old haunt, which arguably there is in the very special case of a
sphere itself. But this is model-theoretic reasoning based on
particular models, not logic, and the requisite thought processes seem
impossibly far from anything suggested by the Elements.
> I know that Johann Heinrich Lambert also took the postulate that way,
> much later, and I believe pretty much everyone did before the 19th
> century.
What was the alternative? To say that Euclid's argument was unsound?
Let those who have never argued unsoundly cast the first stone.
While I don't want to stone Euclid I do want to understand him.
Painting him as an infallible mystic expecting a high degree of
creativity from his students isn't a hugely satisfying insight into his
thought processes.
Vaughan
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