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In Euclid's elements, some of the theorems (e.g. SAA congruence) can be proven using the parallel postulate, much easier than without it. But it seems that Euclid has intentionally avoided using it, when possible.

  1. Am I right?

  2. What is the reason behind this choice?

user2321323
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    Possible duplicate of Why were geometers dissatisfied with the parallel postulate? Presumably because he disliked postulating it and hoped someone would prove it, hence established as much as possible without it in preparation. – Conifold Oct 23 '17 at 23:03
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    See the very similar post why-did-the-ancients-hate-the-parallel-postulate. – Mauro ALLEGRANZA Oct 24 '17 at 13:42
  • How do you prove SSA congruence using Euclid's fifth postulate, and more easily than Elements I, 26? – Edward Porcella Oct 24 '17 at 16:01
  • SAA, not SSA!
    1. Prove that sum of the angles of any triangle is 180 degrees. 2. Use 1 to reduce SAA to ASA. QED!
     – user2321323 Oct 25 '17 at 07:20
  • Thanks, SSA was a typo. So Euclid would have had to delay SAA until after I, 32-- the sum of angles in a triangle is two right angles. I suspect he wanted to wrap up triangle congruence before moving on to parallelograms and quadrature of rectilinear figures. Even so, he achieves the latter not in Bk 1 (the great I, 47 is kind of a consolation prize), but only at the end of Bk 2. But can you say exactly how you use I, 32 to reduce SAA to ASA? – Edward Porcella Oct 25 '17 at 16:27
  • Ignore my question just preceding; clearly I, 32 with ASA proves SAA at once. Moreover, since Euclid's proof of SAA does not use ASA, either one of them easily proves the other with the aid of I, 32. But since Euclid needed the technique of I, 26 to prove at least one of the two, and it served equally well for both, he made them a pair. I can see the sense in his doing it this way. Do you see other, more striking, instances of Euclid apparently avoiding the use of his fifth postulate? – Edward Porcella Oct 25 '17 at 18:47
  • Avoiding unnecessary assumptions makes a stronger theory, logically. Extreme cases: Make SAA an axiom; then its proof is trivial. To make a proof easier, whenever a step seems too difficult to reason out, make it a hypothesis. This is done quite often in the undergraduate curriculum, at least in the US, which does not try to construct mathematics from a minimal set of first principles. By contrast, Euclid seemed interested in clarifying, one might assert despite the blemishes, the relation of geometrical properties to their starting points. – Michael E2 Nov 22 '17 at 15:50

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If he avoided use of that postulate where he saw that it is possible to do like that he did it most likely in the spirit of proving results with minimal assumptions, so that result proven in such a way will hold for some axiom systems different than the one used by him in his books, and different in the sense that those alternative axiom systems will contain some of his axioms but not all, so, if he proved that something is true without the usage of parallel postulate then he proved also that it is true with the usage of parallel postulate, so he most likely did it all in the spirit of trying to be as general as possible, and aware of the fact that other axioms systems and geometries are also possible.

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    Is there any evidence that Euclid or other mathematicians of his age have ever thought of any other axiomatic system? You're suggestion seems so "modern" to me. –  Oct 23 '17 at 18:52
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    @Behzad Haha, yes, it looks modern, but "ancient" and "modern" mathematicians, especially "big ones", are not much different as a people (modulo global changes in society), so the spirit is almost the same, there is no reason to think that Euclid did not see that other axiomatic approaches are possible. –  Oct 23 '17 at 18:57
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    If Euclid and other ancient mathematicians thought of other axiom systems, why did it take another 2000 years before non-Euclidean geometries were developed? –  Oct 23 '17 at 19:51
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    @Quidit: the vast majority of ancient western scientific literature was deliberately and permanently destroyed more than 1000 years ago. So the claim that it took "another 2000 years" to develop a non-Euclidean geometry is already making a statement about history whose truth is unknown. (I suspect Archimedes would have just said "yes, obviously there are non-Euclidean geometries" and would then have drawn you a picture of the Poincare disc.) For all we know Diophantus could have made what we now call Fermat's conjecture in a lost work. So the inference implied by your comment is vacuous. –  Oct 23 '17 at 20:15
  • From my philosophy classes, my understanding was not an awareness of other axiom systems, but a general opinion that the fifth postulate stood out from the rest as simply not being "obviously" true. –  Oct 23 '17 at 21:13
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I cannot answer with confidence as I havent extensively studied his work. I can think of a number of possible reasons, however.

  1. Maybe he didnt think of that solution. Genius or not, no one sees everything.

  2. Maybe that solution required a theorem he had yet to prove, and he thought it unnecessary to go back and change it. Its easy to judge in hindsight bias when you already know the theorem.

    • Maybe unbeknownst to you, the theorem you would have to invoke on some level requires the theorem you would like to prove with it, in order to itself be proven, and it would be circular reasoning to do it your way.

  3. To avoid unnecessary assumption. As Antoine pointed out, the whole point of his work is to minimize assumption.

  4. Maybe he foresaw that if one postulate were proven wrong at a later date, it would undermine your "simpler proof", whereas taking an alternate approach prevented this possibility.
  5. Maybe he didnt have the utmost confidence in that postulate.
  6. Maybe he wanted to keep things as generalizable as possible so that it could be applicable to a different set of axioms and the like.
  7. Maybe he wanted to exercise his genius instead of taking the "easy path".
  8. Maybe he wanted to demonstrate the approach/thinking/technique for later scholars benefit, rather than just arrive at a solution.
CogitoErgoCogitoSum
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  • Just about your seventh reason: please see my comment for @Antoine. –  Oct 23 '17 at 18:55
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    My reasons are explicitly numbered. Why are you being deliberately vague? You couldnt have glanced up to find the number? Make comments, dont cite them from elsewhere. I dont much care to circumnavigate the internet to find a comment you cant be bothered to make. Why should I be bothered to respond? –  Oct 23 '17 at 19:50
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    Maybe Euclid HAD thought about other axioms. He created them, after all, he should have been able to recognize the fact that he could have written anything down. Logical self-consistency was essential to his work. He wanted to capture reality as he knew it, which might be why he didnt discuss alternatives. If he had thought of other axiomatic systems, he might have said they were absurdities. He could have - and maybe he did - derive other systems he concluded didnt reflect reality, and rejected them. To say he didnt think of it at all seems even more absurd to me, a man of his genius. –  Oct 23 '17 at 19:54
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    It seems perfectly natural to me that if youre going to invent math, making up your own axioms, youre going to be taken down a multitude of paths that either lead to contradictions or lead to conclusions that seem unreasonable. Do you think he should have published these "mistakes"? –  Oct 23 '17 at 20:00
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    I'm surprised and shocked how an unintended small mistake can makes someone so annoyed. I honestly think that wasn't any big deal. –  Oct 23 '17 at 20:01
  • I don't think Euclid has been "inventing" mathematics. It sound so modern to me - something relating to recent centuries. He explicitly talks about geometric objects in the same sense as the corresponding real world objects. He considers "lines" not as an abstract entity but as the same thing every person outside mathematics thinks of. –  Oct 23 '17 at 20:06
  • My apologies for my slight over-reaction. I find this site in general to be annoying. I find the community by and large to be unhelpful, uncooperative, pretentious, and my guard is up every time I come here. As someone who came to this math forum to contribute, I can give you the benefit of the doubt and make the assumption that you appreciate and pay attention to the details, care for precision, exactness, etc. It just begs the question why you would make such an easy-to-avoid mistake unless you simply didnt care or werent qualified to contribute in the first place. – CogitoErgoCogitoSum Oct 24 '17 at 19:01
  • Math has always been invented. This is an age-old unresolved debate that reaches well into the realm of philosophy. Invented, discovered, its all the same thing and yet isnt. You dont think Euclid was being abstract when he talked about geometric figures, idealized shapes, parallel lines intersecting "at infinity"? The geometry he envisioned and made proofs on, published on, was the geometry that he could visualize, the geometry that made sense in the real world, and not a fantasy alternative that only makes sense in the higher abstractions/generalizations of today. – CogitoErgoCogitoSum Oct 24 '17 at 19:11
  • But again, my prior argument still holds. He may conceived of these alternatives or been lead into them with poorly chosen axioms, and ultimately disregarded them as nonsense. – CogitoErgoCogitoSum Oct 24 '17 at 19:13
  • His work is just the final product of years of exploration and effort, that hold two properties: logical self-consistency AND real-world relationship. He would most likely have disregarded anything else for one reason or another. Its perfectly reasonable to believe that when you are deciding on the rules of a game youre going to realize its contradictions or absurdities at some point, and then youre going to change the rules. – CogitoErgoCogitoSum Oct 24 '17 at 19:20
  • Can you back up any of this answer? Currently, it's simply conjecture. – HDE 226868 Oct 26 '17 at 04:49
  • As stated in the first line. Whats your point HDE? Arguing just to argue? Ive made my opinion of the arrogance and pretentiousness of you people clear and here you are proving my point. Read what I wrote before you judge it. – CogitoErgoCogitoSum Nov 25 '17 at 19:24
  • It was a request, @Cogito, not just a question. And please, be nice to other users using this thread, including some of the ones above. – HDE 226868 Nov 27 '17 at 01:32
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Euclid does not call on his fifth postulate until $I, 29$, where he cannot do without it. It is not needed until the treatment of parallels, which begins at $I, 27$. The last of the triangle congruence theorems is $I, 26$. Euclid had some dramatic sense: it would be premature to bring postulate five onstage needlessly, and just moments before the scene that really requires it.

Edward Porcella
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Yes. Somebody calls "Absolute geometry" the set of propositions that are provable without the $5$-th postulate.

IMHO this is an because it seems less obvious than the others, at least as we learnt at school the version "In a plane given a line and a point outside the line, there exists one and only one line passing through the given point and parallel to the given line".

Which is honestly quite hard to understand because it implies an impossible infinite producing of the parallel line to "verify" it does not intersect the given one.

But Euclid does not use these words! He says, very smartly here

"Let the following be postulated: [...]

  1. That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles."

This is much easier to digest: at least the producing stops at a certain point.

Hope it is useful

Raffaele
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  • -1: IMHO, your opinion is utterly devoid of value as it based on no knowledge whatsoever of the extensive criticism that Euclid's Elements have received from commentators since shortly after the books were written. You can't do useful history of mathematics by speculation. –  Oct 23 '17 at 19:17
  • ... so, looking no further than https://en.wikipedia.org/wiki/Parallel_postulate for some actual information rather than your opinions, how can you argue that Proclus in the 5th century and Omar Khayyam in the 11th century are "relatively recent" sources concerned with a "legendary" problem. –  Oct 23 '17 at 19:39