Robert J. MacG. Dawson wrote:
...fatally flawed. (Why do the circles in Euc.I.1 intersect? None of his axioms assert that any pair of circles whatsoever do so.)
In axiomatic mathematics, everything that is not forbidden is permitted. Circles can climb trees and drape themselves over branches in Dali's axiomatization of geometry, not because he says they can but because he is not the strict disciplinarian that Euclid is. Euclid insists that his circles shape up or else. This creates many problems for the circles but few problems for mathematicians as their managers. Dali runs a looser ship, which lets the circles lead a less structured life while creating more problems for mathematicians. This is a win-win situation: the circles end up with fewer neuroses while the mathematicians thrive, problems being their lifeblood. Thank god for Lobachevsky and the others we can't remember because Tom Lehrer didn't. Vaughan Pratt