[FOM] Euclid's axioms

[Alasdair Urquhart]

I find myself to a large extent
in agreement with Fred Richman's
reply, but there are still places
where I disagree.  

Let's revert to Matthew Frank's original
remark:

"Matthew Frank pointed out to me that Euclid's axioms have a
model consisting only of the constructible points."

Now, taken literally, this is true, in the sense that
all of Euclid's Postulates and Common notions are
verified in the model of the constructible reals
(constructible in the sense of classical geometry,
that is).  But on the other hand, it is clear that
the classical geometers freely assumed the existence
of certain magnitudes whose existence is "obvious."
Take, for example, the first proposition in Archimedes's
"Measurement of a Circle".  It reads:

"The area of any circle is equal to a right-angled triangle
in which one of the sides about the right angle is equal
to the radius and the other to the circumference of
the circle."

I don't detect any doubt in Archimedes's statement
that there is such a right-angled triangle.
On the other hand, he obviously doesn't supply a means
of construction, but uses the method of exhaustion.

It's been known for over a century that Euclid's 
postulates are inadequate.  The question is how
they should be best supplemented to remedy their
deficiencies.  The standard solution to the problems
we've been discussing is the Principle of Continuity
(see, for example, Heath's edition of Euclid, 
Volume 1, p. 234).  

Fred Richman finds this solution unsatisfactory, and
I agree.  I would prefer to say that Euclid's practice
is better represented by a model consisting of 
relatively constuctible points.  That is to say, 
we assume the existence of certain "obvious"
magnitudes, and proceed to construct other magnitudes
based on these.  The idea is closely related to the
classical notion of analysis (as opposed to synthesis).

If one takes this model of classical geometrical practice,
then I think that it is not true that the existence of
a 20 degree angle is undecidable.  On the contrary,
it would have been one of these obvious and unproblematic
magnitudes for the Greek geometers.

I recommend the late Wilbur Knorr's beautiful book
"The Ancient Tradition of Geometric Problems"
to anybody who is interested in these questions.

-- Alasdair