I am sure that many among you have read Euclid's Elements of Geometry.

If you have, you will have noticed that nowhere does he state an axiom (or postulate) of parallels. (That's why we call it Playfair's Postulate, not Euclid's Postulate).

He gives a definition of parallelism - but of course, a definition has only very limited argumentative force. He also states Postulate 5, which deals with convergent lines, and by implication, explicity excludes "parallel" lines from consideration.

Nevertheless, many mathematicians - including professional mathematicians, who ought to know better - treat Postulate5 as synomymous with the Parallel Postulate. That this is fallacious is quickly demonstated by pointing out that a negation of the Parallel Postulate does not entail a negation of Postulate 5.

The "Axiom of the parallels" was a philosophical embarrassment in ancient philosophy - that's why Euclid doesn't explicitly state it. And it explains why he waits until Theorem 17 before introducing any problem which explicitly presupposes it.

Ladies and gentlemen, if you reply, please do so from the perspective of mathematical philosophy, and not from accepted mathematical wisdom. This will save us all a lot of angst.