[FOM] Understanding Euclid

Vaughan Pratt pratt at cs.stanford.edu
Sat Dec 13 00:31:13 EST 2008

--Original question answered--

After mulling over the very helpful input from everybody for a few days, 
I concluded that my original question (was Euclid considering spherical 
geometry in Book I?) had been answered definitively.

Answer:  No.  Euclid did not intend to accommodate spherical geometry in 
Book I or he would not have allowed Proposition 16, that an exterior 
angle of a triangle exceeds each of its two opposite interior angles. 
He did not notice that his postulates allowed spherical geometry, and so 
failed to realize that in that model of his axioms the relevant median 
(the line segment BE in the first figure at 
http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI16.html ) 
could be greater than a right angle.

In that case vertex F completing ABC to the parallelogram ABCF passes 
through (namely below in the figure) the plane of the great circle 
formed by BCD, simultaneously with its antipode reaching B.  This 
results in the line segment CF suddenly and very violently (from the 
standpoint of the diagram) flipping around 180 degrees at the instant of 
passage of F through BCD (thinking of F being pushed out continuously 
until EF equals BE rather than all at once) and splitting BCG instead of 

An easily visualized such triangle is the thick octant, consisting of 
two points on the Equator separated by a right angle, together with the 
North Pole, thickened by pulling each vertex distance epsilon away from 
the centroid of the triangle.  All internal angles are slightly over a 
right angle while all external angles are slightly under.

An equivalent coordinatization making the North Pole the centroid of the 
octant (for better symmetry) places the vertices of the triangle on 
three equally spaced meridians (i.e. 120 degrees apart), all at latitude 
epsilon less than asin(sqrt(1/3)), 35 degrees works fine.

--New question--

This answer generated for me a slew of questions, starting with the 

An interesting feature of this failure of I.16 is that this breakdown of 
the argument depended on the surface continuing to curve around to the 
back of the sphere well outside the triangle.  Taking the thick octant 
as a case in point, if one were to chop it and a neighborhood out of the 
sphere and continue its boundary with a different curvature aimed at 
making the octant one of a host of bumps in a large dimpled but 
otherwise flat sheet, the breakdown shouldn't be able to happen in the 
way it does for the sphere because there is no longer a back of the 
sphere for EF to go around; instead EF goes off to the upper right 
consistently with the diagram, rolling up hill and down but always away 
from B and remaining in a more or less flat (but dimpled) plane on average.

(The reason for making it multiple bumps rather than one isolated bump 
is that two points on opposite sides of the bump equally and 
sufficiently far removed from it will clearly find it shorter to go 
round the bump than over its top, giving the same ambiguity as arises 
for antipodes of the sphere.  Maybe this still happens with multiple 
bumps but I find this much harder to see.)

However the theorem remains false in this model because the thick-octant 
bump is a counterexample.  Why is it false?  The basic backwards flip of 
CF shouldn't happen.  Is it false because even a dimpled surface is 
problematic, and the only smooth surface, even without homogeneity, that 
has no pairs of points having two geodesics go through them is the 
hyperbolic plane (including the Euclidean plane as its limiting 
degenerate case)?  That's still not convincing because even if it does 
the flip is nowhere near as bad as for the sphere.  Because of this I 
have trouble understanding how I.32 can be true even taking I.16 as a 
postulate!  Could Euclid be smuggling yet more stuff in somewhere to get 
from I.16 to I.32?

I have several more questions, which divide into model-theoretic, 
concerning possible models of both the postulates and the propositions, 
and proof-theoretic, concerning the domain-independent aspects of 
Euclid's apparent reasoning.  I'll ask some of the former in this post 
and defer the others to later in the interests of not making this 
already overlong message (for which my apologies) impossibly long.

--Master question--

My model-theoretic questions all fall out as special cases of the 
following master question for the model theory of Book I.

What is a model of Euclidean geometry?

More precisely, what surfaces does Euclidean geometry describe?  (We can 
take two-dimensionality for granted.)  And of those, which are 
reasonable in some sense?

One might judge the reasonableness of a space by some combination of the 
following criteria (P2 denotes Postulate 2).

C1. Flat (zero Gaussian curvature, it's ok to roll up architectural 
plans for a pyramid when stowing them in the overhead bin because the 
Gaussian curvature is zero in that situation).

C2. Continuous (every Cauchy sequence has a limit).

C3. Orientable (clockwise and counterclockwise are distinct notions).

C4. Unbounded (infinite in extent) (entailed by C1 and P2)

C5. Homogeneous (a maze of points all alike).

C6. Boundaryless (not a half-plane and no holes) (entailed by C5 and P2).

--Three refinements of the master question--

These criteria allow the following refinements of the master question.

R1.  Which of these criteria might Euclid have had in mind?

R2.  Taking a model of Euclid's axioms as any surface for which the 
propositions of Book I hold, which criteria are satisfied by every such 
surface, and what logical relationships then hold between those criteria?

R3.  As for R2 with "postulates" in place of "propositions."

Refinement R1 is intrinsically speculative, while R2 and R3 replace the 
speculative element with more mathematically defined questions.

Although R1 is a nice question I don't know enough about pre-Euclidean 
mathematics, or Euclid's environment, to have any thoughts on it myself, 
but would be very interested to hear from those who know something about it.

Regarding the subject line of "understanding Euclid," R2 is really the 
right question because we depend on the propositions to clarify and fill 
in the gaps in Euclid's postulates making the latter a poor indicator of 
his intentions.  The answers to R2 should be a proper superset of those 
to R3.

By Proposition 32, that triangles contain exactly two right angles, 
there is no doubt that Euclid intended total flatness.  So Euclid's 
propositions, if not postulates, would appear to axiomatize the plane in 
the sense of zero Gaussian curvature if not e.g. continuity.

R3 is more in the nature of the game genies play when you make three 
wishes and they take them literally (don't you hate when that happens, 
like when the genie is your computer, or your two-year-old?).  In this 
case Euclid made the postulates and we're now proposing to take them 
literally.  As such it is more like pure mathematics than a serious 
effort to understand Euclid: a quite different question, more or less 
interesting than R2 depending on whether one is more interested in 
mathematics or mathematicians respectively.  I primed the inter-criteria 
relationships pump with two easy ones, namely C1 -> C4 and C5 -> C6, 
both solely on account of P2, to illustrate what I have in mind here.

--Consideration of each criterion--

If nonzero curvature is deemed unreasonable then I would be inclined to 
stop there and not pursue the remaining criteria.  Since nonflat models 
of Euclid's postulates have long been of interest and therefore are 
presumably not entirely unreasonable, let's push on.

Continuity doesn't seem high on people's list given that I've never 
heard anyone complain about iterated quadratic irrationals, the closure 
of the rationals under square root, as not being a model of Euclid's 
axioms on that ground.  This may be the result of wanting to consider 
Euclidean geometry a first order theory, which Tarski if not Hilbert 
adhered to.  (That ZF is first order should be an orthogonal issue.)

In the absence of continuity, how do you prove that a spherical model of 
Euclid's postulates contains any antipodal pairs of points?  Every point 
has its antipode when you work with degrees, or with ideal metres 
(40,000,000 of which will take you round a spherical Earth of radius of 
curvature that of the (nonspherical) WGS-84 ellipsoid at Lèves, Alsace 
or any other point at latitude 48.46791 degrees, north or south), or 
with cosines of angles instead of the angles themselves so as to retain 
the simplicity of iterated quadratic irrationals in relating sides and 
angles of spherical triangles, because cos(pi) = -1.  But if you make 
life computationally hard for yourself by using radians instead of 
cosines, taking as the basis for a coordinate system the vertices of an 
equilateral triangle one radian on each side, then it's not at all clear 
to me whether any great circles ever intersect in 2 points, nor whether 
they always intersect in 1 (though clearly infinitely many do).  (There 
exist antipodal points iff there exist great circles intersecting in two 
points iff there exist circles with center that are great circles.)

The absence of antipodal points has its good features and its bad.  On 
the good side, geodesics intersect in at most one point.  On the bad 
side, can we be sure that geodesics always intersect even once, let 
alone twice?  And how can we live with the thought of two geodesics 
coming closer to each other than any positive epsilon without actually 
intersecting?  Did any ancient Greek geometer contemplate such a 
travesty of common sense?

Homogeneity would imply no boundaries.  However even in the absence of 
homogeneity, boundarylessness is a no-brainer consequence of Postulate 
2: a line normal to a boundary is blocked by it.  Hence we can safely 
assume no boundaries.

Orientability seems a very natural requirement.  One tends to lose track 
of the time when your analog watch runs backward at the same time as 
it's running forward, a consequence of nonorientability.  This happens 
on the Bloch sphere modeling electron spin, being the projective sphere, 
but physicists are too busy worrying about Schroedinger's cat (curious) 
and Schroedinger's fish (delicious) to find nonorientability 
disorienting.  The designers of the Acropolis surely would have rejected 
a nonorientable surface---how would you indicate the orientation of a 
spiral staircase?

On that basis therefore the projective sphere (antipodal points 
identified, with the metric adjusted accordingly) seems a non-starter 
for a model of Euclidean geometry suitable for the typical applications 
of geometry contemplated by Euclid's contemporaries, or for modern 
architects for that matter.


Summarizing thus far, we've suggested that discontinuities are of 
debatable reasonableness (and we have the associated question of whether 
any discontinuous sphere can model the postulates without introducing 
antipodal points).  We expressed a sufficiently strong distaste for 
nonorientable surfaces that those can be ruled out.

(One approach to dealing with antipodal points would be to strengthen 
the requirement in Postulate 1 that the two points be distinct to the 
further requirement that they also not be antipodal when using them to 
determine a line.  But that does nothing to prevent the counterexample 
in Proposition 16 (violent flipping); it's not so much the singularity 
at an antipodal pair itself as the discontinuity it creates in its 
neighborhood that makes the proof of Proposition 16 unsound.)

--Down to just boundedness and homogenity.

This leaves boundedness and homogeneity.  I haven't heard anyone say 
that Euclid's postulates imply his domain must be unbounded.  If that 
follows from any reasonable interpretation of the postulates that 
doesn't assume homogeneity (currently also on the table) I would very 
much like to see the proof.

There does seem to be a strong sense that Euclid meant his space to be 
homogeneous, but that's not at issue because we know furthermore that he 
meant it to be Euclidean space, in particular flat.  The question here 
is not what he meant, only what Euclid's postulates might reasonably be 
taken to mean when others decide what they should mean.

There are a great many useful nonhomogeneous yet reasonably regular 
surfaces in the world, gently dimpled plastic sheets for example, which 
are often encountered, though more vigorously dimpled sheets containing 
thick octants might be much less common.  How about the Fermi surface of 
a monolayer crystal?

If Euclid's postulates taken as they stand, without Propositions 16, 17, 
32, etc. shoring them up, have dimpled sheets as a model, that could be 
very interesting.

--Onset of misgivings about the organization of Book I--

After contemplating all these possibilities, I came away with a feeling 
similar to that of Proclus as cited by Alasdair Urquhart: "[Postulate 5] 
ought even to be struck out of the Postulates altogether; for it is a 
theorem involving many difficulties..."

What I object to is not really Postulate 5 itself, however patched, but 
the whole protocol that it is part of.  I don't believe Euclid has quite 
the right axioms for Euclidean geometry.  Postulate 3 is a major part of 
the problem, and needs to be replaced with a set of postulates better 
adapted to precisely capturing the Euclidean plane, without using any 
form of Postulate 5.  Postulates 1, 2, and 4 are ok but a bit feeble, 
which Euclid wishfully compensates for by making Postulate 3 Euclid's 
hammer.  The problem with the hammer is that it is too universal for the 
purposes of capturing the geometry of the flat plane: it tries to do too 
much with a single axiom yet doesn't accomplish enough, and Postulate 5 
is the wrong way to take up that slack.  On the other hand I don't 
believe the right axioms need to differ from Euclid's as much as 
Tarski's did.  I'll say more about this (in part in response to George 
McNulty's post mentioning Borsuk and Szmielew) in another message after 
my promised proof theory message.

Vaughan Pratt

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