[FOM] Euclid's axioms

Alasdair Urquhart urquhart at cs.toronto.edu
Fri Nov 1 15:00:16 EST 2002

Fred Richman wrote:

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

Of course Hilbert's "Axiom of line completeness" does not
hold in this model, but then neither God nor Euclid
introduced that axiom into geometry."

There is certainly a sense in which this is true, but
at the same time, it's difficult to account for some
of the propositions in Euclid on this basis.  
For example, the propositions in Book XII that 
depend on the method of exhaustion, such as:

XII.2 "Circles are to one another as the square 
	on the diameters."

This uses the method of exhaustion, something we
would justify today by a continuity axiom.
In the case of Euclid, XII.2 is grounded
in the (unjustified) assumption that the area
of a circle is well defined, together with the use
of approximating polygons, the final step
being a reductio ad absurdum.  This is also
Archimedean standard practice.

-- Alasdair

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