[FOM] Euclid's axioms

[richman]

Original Message From Alasdair Urquhart <urquhart at cs.toronto.edu>

>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.

I don't think you need continuity (completeness) to justify "exhaustion". The 
theory of proportions goes through without completeness. No limits are 
involved in these kinds of arguments, just sufficiently close approximations 
to contradict the assumption that the proportions are different. You do need 
an Archimedean axiom to rule out infinitesimals.

>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.

Is the assumption that the area of a circle is well defined any more 
unjustified than that the length of a line is well defined? I assume you mean 
that it needs to be justified, not simply that Euclid didn't justify it.

Area seems pretty unproblematic to me. Of course the issue is always to 
compare the proportion of two areas to some other proportion (possibly of two 
other areas, as in the proposition you quote). You can't construct two squares 
whose proportion is the same as a circle is to the square on its diameter, but 
I don't think that means that the area of a circle is not well defined.

--Fred