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 beta, so the sum of ypq and alpha 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.)

 

[mathwonk]

yes, my drawing labeled the angles oppositely to Stillwell. I have (hopefully) corrected the notation in response to your observation.

As to rigor, such arguments require a rigorous discussion of sides of a line, and what happens when a line meets and hence crosses to the other side of a line. Euclid omitted such discussions, but I hope my argument is ok on those points.

EDIT: I.e. it seems all that is needed is that q and y are on opposite sides of m. So if l meets m at x, choose y on l and on the opposite side of m from q. Then segment yp is on the opposite side of m from segment qp, and x is between q and y. So angle xpq, i.e. beta, is interior to angle ypq, so ypq is greater than beta.

I.e. since q,x,y are collinear on line l, and p is off l, since x is between q and y, it follows that angle ypq is greater than angle xpq.

 

[Gavran]

I think, Stillwell made a mistake there.

The author is right. It is all about “if and only if”.
The parallel axiom includes “⇒” and it does not include “⇐”. “⇐” is included by using the uniqueness of the line through any two points as it is shown in the text.

 

[Hill]

The author is right. It is all about “if and only if”.
The parallel axiom includes “⇒” and it does not include “⇐”. “⇐” is included by using the uniqueness of the line through any two points as it is shown in the text.

The author is not right.
The author says that
"if $\alpha+\beta$ equals a straight angle then $l$ and $m$ do not meet"
follows from
"if $\alpha+\beta$ is less than a straight angle then $l$ and $m$ meet",
while in fact the former is independent of the latter.

 

[Gavran]

The author is not right.
The author says that
"if $\alpha+\beta$ equals a straight angle then $l$ and $m$ do not meet"
follows from
"if $\alpha+\beta$ is less than a straight angle then $l$ and $m$ meet",
while in fact the former is independent of the latter.

The author says, “It follows that if α+β equals two right angles (that is, a straight angle) then l and m do not meet.” and he keeps on with the next, “Because if they meet on one side (forming a triangle) they must meet…”.
It seems that l and m do not meet for two reasons. The first one is presented with “It follows …” and the second one is presented with “Because …” in the text.
Based on the parallel axiom, we have that “α+β is less than two right angles” is sufficient for “l and m meet”, but “α+β is less than two right angles” is not necessary for “l and m meet”. To provide necessity, the author includes an additional statement based on the uniqueness of the line through any two points.

 

[Hill]

The author says, “It follows that if α+β equals two right angles (that is, a straight angle) then l and m do not meet.” and he keeps on with the next, “Because if they meet on one side (forming a triangle) they must meet…”.
It seems that l and m do not meet for two reasons. The first one is presented with “It follows …” and the second one is presented with “Because …” in the text.
Based on the parallel axiom, we have that “α+β is less than two right angles” is sufficient for “l and m meet”, but “α+β is less than two right angles” is not necessary for “l and m meet”. To provide necessity, the author includes an additional statement based on the uniqueness of the line through any two points.

So, only what is presented with "Because..." is a reason for them not to meet. While what is presented with "it follows..." is NOT a reason for them not to meet.

 

[mathwonk]

Here is the situation as I understand it. Stillwell states Euclid's parallel postulate, then gives an argument meant to prove its converse, and then states that "thus" the parallel postulate implies its own converse. One would expect this to mean that his argument has used the parallel postulate in the proof of its converse.

However a close reading of Stillwell's argument shows that it does not use the parallel postulate at all. It refers only to the ASA principle, which is proved in Euclid, as Prop. 26, without using the parallel postulate. In fact Euclid's own proof of the converse of the parallel postulate is his Prop. 28, and it is well known that the parallel postulate is not needed for any of the first 28 propositions of Euclid (just read the proofs).

It is in fact true that the converse of the parallel postulate can be proved by an argument that does essentially use the parallel postulate, and one such argument is given in post # 9, but Stillwell does not give this argument.

I claim further that Stillwell's argument is not even complete, since it asserts, without proof, that if the two lines, m,l, meet on one side of the line n, (assuming alpha+beta = π) then they must also meet on the other side. He gives no argument for this claim at all. He cites ASA for the fact that they would then form on say the left side, a triangle congruent to the one on the right side. But ASA does not allow one to conclude anything, until one has two triangles to begin with, i.e. it does not allow the conclusion that three given lines meeting in two points at certain angles do meet in a third point and hence form a triangle, at least not without a further argument which I do not see.

One can however argue that the lines do also meet on the opposite side of n, if one uses the superposition principle that Euclid used for his Prop. 4, (SAS). I.e. if one can superpose one set of three lines on the other, then the third intersection point of the first triple must fall upon a third intersection point on the second triple. Also If, as Hilbert does, one takes Prop. 4 as an axiom, one can prove the superposition principle; but Stillwell makes no mention of any of this.

One can also proceed without using that principle, as in post #9. I.e., since the parallel postulate does give a criterion for two lines to meet, which however is not satisfied by lines l,m, the argument in post #9 was cooked up to use that criterion by forcing its hypothesis to be true, by constructing a new line k, and applying the parallel postulate to lines l and k.

Thus in my opinion, Stillwell's assertion, that the parallel postulate implies its own converse, is true. His argument for the truth of the converse however, not only does not demonstrate that fact, but is not even quite complete on its own terms since he gives no argument for why the lines l,m, meet again on the other side of n. It is of course conceivable that he had in mind some unstated argument that did use the parallel postulate, maybe along the lines of post #9, but, without introducing another line, I don't know what it would be.

 

[Hill]

it is of course conceivable that he had in mind some unstated argument

We don't know what he had in mind, but as a matter of fact, existence of parallel lines does not require the parallel axiom and what the parallel axiom implies is the uniqueness of the parallel line.