The parallel axiom, Stillwell's "Reverse Mathematics"

[Hill]

TL;DR

How is this statement of the parallel axiom used in the following deduction?

Stillwell in the "Reverse Mathematics: Proofs from the Inside Out" states

Then, he deduces

My question is: where/how in this deduction the parallel axiom is used?

 

[mathwonk]

It does not appear to be used here. I can only suppose that he does not mean this latter fact actually follows from the parallel postulate, only that it follows from something. In fact it follows from Euclid's exterior angle theorem, and is stated and proved in Prop. 17, Book I, of the Elements, as the result that in any triangle, any two angles are together less than a straight angle. Consequently the two angles alpha, beta, which equal a straight angle, cannot form part of a triangle, so line l cannot meet line m. But it is hard to judge, based on an excerpt from a book I don't have access to.

 

[Hill]

It does not appear to be used here. I can only suppose that he does not mean this latter fact actually follows from the parallel postulate, only that it follows from something. In fact it follows from Euclid's exterior angle theorem, and is stated and proved in Prop. 17, Book I, of the Elements, as the result that in any triangle, any two angles are together less than a straight angle. Consequently the two angles alpha, beta, which equal a straight angle, cannot form part of a triangle, so line l cannot meet line m. But it is hard to judge, based on an excerpt from a book I don't have access to.

Thank you.
It appears that he in fact means that the latter fact follows from the parallel axiom, as immediately after that passage he says,

 

[mathwonk]

Well again, I am handicapped by not having the book. He visibly states that the result follows from the ASA principle, so that is what he is using in his argument. You might look and see how he proved that principle, e.g. whether he used the parallel axiom there.

 

[Hill]

Well again, I am handicapped by not having the book. He visibly states that the result follows from the ASA principle, so that is what he is using in his argument. You might look and see how he proved that principle, e.g. whether he used the parallel axiom there.

He does not use the parallel axiom there. Moreover, he says that ASA principle is proved without this axiom.

Here is all that he says about it (he calls other Euclidean axioms excluding the parallel one, "basic axioms"):

 

[mathwonk]

I can't judge a book's logical flow based on selected excerpts. But the ones you have reproduced do puzzle me. If Stillwell is following Euclid, he must know that the existence of parallel lines (Prop. I.27) is proved in Euclid without using the parallel postulate, but based instead on the exterior angle theorem (Prop.I.16). This uses also the SAS criterion (Prop. I.4), which is the one people usually criticize, since it uses the principle of superposition, which Euclid has not made a postulate. I also do not feel that Euclid actually states a postulate of uniqueness of a straight line joining two points. This does seem to be needed however, since in spherical geometry, the truth of Euclid's parallel postulate, and ASA, do not imply existence of parallel lines. Of course spherical lines joining two points are also not always unique (take the 2 points to be opposite poles), and are bounded in length.

Anyway, based on what you have shown me, I do not see anywhere that Euclid's (non) parallel postulate is used in Stillwell's argument for existence of parallel lines.

 

[Hill]

I can't judge a book's logical flow based on selected excerpts. But the ones you have reproduced do puzzle me. If Stillwell is following Euclid, he must know that the existence of parallel lines (Prop. I.27) is proved in Euclid without using the parallel postulate, but based instead on the exterior angle theorem (Prop.I.16). This uses also the SAS criterion (Prop. I.4), which is the one people usually criticize, since it uses the principle of superposition, which Euclid has not made a postulate. I also do not feel that Euclid actually states a postulate of uniqueness of a straight line joining two points. This does seem to be needed however, since in spherical geometry, the truth of Euclid's parallel postulate, and ASA, do not imply existence of parallel lines. Of course spherical lines joining two points are also not always unique (take the 2 points to be opposite poles).

I think, Stillwell made a mistake there. Nevertheless, this discussion helps me to obtain a deeper understanding. Thank you.

 

[mathwonk]

Having found and read Stillwell's account in full, I agree with you that he seems mistaken in his claim. I find that very puzzling, as he has a PhD in logic from MIT under Alonzo Church, and I wonder if I have understood him correctly.

 

[mathwonk]

But let's try to prove existence of parallels actually using Euclid's parallel postulate. let p be the point where n meets m in his diagram 1.2, and let q be the point where n meets l, and assume alpha +beta = π. If l meets m on the right side, say at x, then choose a point, say y, further out on l, and join p to y by a line k. Then at p, k makes an angle ypq greater than alpha, so the sum of ypq and beta is greater than π. I.e. the angles on the right side of n, cut by n on l and k, add to more than a straight angle. Hence the angles cut by n from k and l on the left side of n, add to less than π. Hence by actually using the parallel postulate this time, k meets l again on the left side of n. But since k also meets l on the right side of n at y, this contradicts the uniqueness of a line through two points.

To be even more picky about Stillwell's argument, I don't quite see how he argues that lines l and m must meet again on the left, assuming they meet on the right. Until he knows that, they don't form two triangles, and he can't apply ASA. He seems to need some superposition principle, such as Euclid used also without justification.

 

[Hill]

This seems to work.
I doubt a bit about this step: "Then at p, k makes an angle ypq greater than alpha." It is obvious if I make a naive sketch, but is it rigorous? (And you mean rather "greater than beta", I think.)