Euclid nowhere states an axiom (or postulate) of parallels

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.

Re: Euclid nowhere states an axiom (or postulate) of parallels

Perhaps I am wrong, but I believed that Playfair's Postulate is indeed logically equivalent to Euclid's Fifth Postulate given the first four of Euclid's postulates. That is, in Euclid's context, they are equivalent. Do you have a proof that they are not equivalent in that context?

Re: Euclid nowhere states an axiom (or postulate) of parallels

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**JeffM** Perhaps I am wrong, but I believed that Playfair's Postulate is indeed logically equivalent to Euclid's Fifth Postulate given the first four of Euclid's postulates. That is, in Euclid's context, they are equivalent. Do you have a proof that they are not equivalent in that context?

Yes, they are equivalent but I don't think that is what alan1000 means. Euclid's postulate is that if the sum of angles that two lines make with a transversal line is less than a straight angle (180 degrees) the lines intersect. That doesn't, directly, say anything about "parallel".

Re: Euclid nowhere states an axiom (or postulate) of parallels

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**HallsofIvy** Yes, they are equivalent but I don't think that is what alan1000 means. Euclid's postulate is that if the sum of angles that two lines make with a transversal line is less than a straight angle (180 degrees) the lines intersect. That doesn't, directly, say anything about "parallel".

I know that Euclid's Fifth Postulate is not directly about parallels, but if it is logically equivalent, in Euclid's context, to Playfair, I do not see what there is of substance to talk about. The concern always was whether it was a necessary postulate or a theorem. To the extent that philosophers and mathematicians believed it was an "embarrassment," they did so either for esthetic reasons or for the lack of a proof for what they assumed was a theorem, not because it involved, directly or indirectly, parallels. We now know that Euclid was correct to treat it (or its equivalent) as an axiom. It is in my view a classic case of a spurious problem: people simply assumed that the 5th postulate could be proved from the first four: they were wrong, and Euclid was right.

Re: Euclid nowhere states an axiom (or postulate) of parallels

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**JeffM** We now know that Euclid was correct to treat it (or its equivalent) as an axiom. It is in my view a classic case of a spurious problem: people simply assumed that the 5th postulate could be proved from the first four: they were wrong, and Euclid was right.

The OP asked for comments on the philosophical aspect of this question.

Prof. Howard Eves was one most important geometry teachers and historian of the last century.

Here is his take on the topic.

“Prior to its (non-Euclidean geometry) discovery it was believed that mathematics was concerned with finding unique and necessary about the real world, and the postulates and theorems of mathematics were essentially observed and derived laws of nature.…The discovery compelled mathematicians to adopt a new viewpoint of their subject.”

That was the loss of certainty. Some sixty years latter Gödel put the nail in certainty’s coffin.

Re: Euclid nowhere states an axiom (or postulate) of parallels

Plato

I do not disagree; it's partially why Gauss used empirical methods to verify that, within the accuracy of his instruments, the sum of the angles in a triangle were equal to the sum of two right angles. But I do not believe (although I am willing to be corrected) that the "problem of the parallels" for the ancients was worry about the existence of non-Euclidean geometries. As far as I know, and again I may be wrong, they were worried that they had not based what they believed to be a universal and empirically valid geometry on the most primitive (and perhaps most simple) axioms, and their concern centered on the fifth postulate. In fact, they were correct that their geometry was not rigorous as Hilbert (if I remember correctly) showed, but the problem was not with the fifth postulate.

I agree as well that the discovery of non-Euclidean geometries changed the nature of how mathematicians viewed their subject and that that change was very significant in terms of epistemology. But I simply do not read the OP as having raised any of that. I think he is mis-characterizing what concerned people before the 19th century about the fifth postulate, and if he is simply carping because people say that Euclid had a postulate about parallels, he is technically correct, but I do not see the philosophical interest in that because Euclid's postulate is logically equivalent, given Euclid's other postulates, to a postulate about parallels.

To put my point more succinctly, the concern that Euclid had erred in presenting his fifth postulate as a postulate rather than a theorem was a spurious problem. After almost 2500 years of wasted effort on that problem, it led to the discovery that Euclid had been correct all along, but that he and everyone else had overlooked a much broader mathematical world than either Euclid or his critics had imagined.

Re: Euclid nowhere states an axiom (or postulate) of parallels

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Originally Posted by

**Plato** The OP asked for comments on the philosophical aspect of this question.

Prof. Howard Eves was one most important geometry teachers and historian of the last century.

Here is his take on the topic.

“Prior to its (non-Euclidean geometry) discovery it was believed that mathematics was concerned with finding unique and necessary about the real world, and the postulates and theorems of mathematics were essentially observed and derived laws of nature.…The discovery compelled mathematicians to adopt a new viewpoint of their subject.”

That was the loss of certainty. Some sixty years latter Gödel put the nail in certainty’s coffin.

And I haven't been certain of **anything** since!

Re: Euclid nowhere states an axiom (or postulate) of parallels

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**HallsofIvy** And I haven't been certain of **anything** since!

The loss of certainty was an important step forward on the way of scientific progress. Actually, questioning everything and living with an open mind is critical for succeeding in modern scientific world and taking part in modern development.