July 17th, 2017, 04:55 AM  #1 
Member Joined: Apr 2017 From: Canada Posts: 32 Thanks: 2  Euclid's Fifth Postulate
Even after reading about Euclid's fifth postulate, I can't understand why it couldn't have been proven for Euclidean space. All of the geometers that lived between Euclid's time and Lobachevsky's worked only within the parameters of euclidean space (in which the postulate holds). So why could the postulate not be proven for Euclidean space only?

July 17th, 2017, 05:09 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,237 Thanks: 2412 Math Focus: Mainly analysis and algebra 
It can't be proven because it is independent of the other axions. If you choose to enforce the fifth postulate, your space is forced to be Euclidean. If you choose to enforce a different fifth postulate, your space is forced to be nonEuclidean. Flat space is called Euclidean because Euclid chose to enforce the fifth postulate. 
July 17th, 2017, 05:41 AM  #3  
Senior Member Joined: Oct 2009 Posts: 275 Thanks: 92  Quote:
Sure, you can invent other definitions of what it means that space is Euclidean. People throughout history have done that and proved those definitions to be exactly equivalent to the parallal postulate. I think the most famous one is Playfair's axiom, but there are many many many others. The question really was (or should have been) whether there were any space besides Euclidean space.  
July 17th, 2017, 11:11 AM  #4 
Senior Member Joined: Jun 2015 From: England Posts: 768 Thanks: 223 
We tend to call postulates axioms these days. It is often said that Euclid is built on just five axioms. But these are valueless without the 23 definitions and 5 what he called Common Notions. 
July 17th, 2017, 01:52 PM  #5 
Senior Member Joined: Oct 2009 Posts: 275 Thanks: 92  Perhaps. But most of his definitions and common notions are 1) Incomplete 2) Too vague to be useful at all.

July 17th, 2017, 02:42 PM  #6  
Senior Member Joined: Jun 2015 From: England Posts: 768 Thanks: 223  Quote:
Definition 1 for instance has incredibly far reaching effects on the whole of mathematical physics. Proposition 1 was the first enunciation of the transitive relation and the basis for the zeroth law of thermodynamics. Not useful?  
July 17th, 2017, 03:16 PM  #7  
Senior Member Joined: Oct 2009 Posts: 275 Thanks: 92  Quote:
Surely you mean common notion 1? Yes, the common notions are somewhat more useful. Although I highly doubt Euclid had thermodynamics in mind haha. Still, the common notions are very incomplete. He very often uses common notions that he didn't state. Last edited by skipjack; July 17th, 2017 at 09:05 PM.  
July 17th, 2017, 07:28 PM  #8 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,237 Thanks: 2412 Math Focus: Mainly analysis and algebra 
It's very common that the inventor/discoverer of a mathematical idea has no idea of the practical uses to which it will eventually be put. It's also extremely common for the first attempt at formalising an idea to be incomplete and sometimes inaccurate. Are you going to have a go at Newton and Leibnitz for not having a solid theoretical basis for calculus? Or for not having path integrals in the complex plane and string theory in mind when they created it? The value of Euclid is not the accuracy of his work, but the formal nature of it and what he was trying to achieve. Last edited by v8archie; July 17th, 2017 at 07:32 PM. 
July 17th, 2017, 09:44 PM  #9  
Senior Member Joined: Oct 2009 Posts: 275 Thanks: 92  Quote:
Discussing "flaws" in Euclid has been done by mathematicians for centuries. Proclus and Zeno were among the first to discuss the elements, point out flaws and fix them up. This continued for centuries, as every geometry student was required (rightfully) to read Euclid. Finally, Hilberts and Tarski's formulization of geometry was the end of it. Still Euclid is worth reading even today, even though most mathematics students sadly do not read it and most high schools do not teach from it. Having a critical(!!!!!!) look at Euclid teaches a lot. I always love to read Euclid with interested students and then to discuss with them things that could be better. It's very much worth it. Although Newton's Principia was recommended reading at Cambridge until well in the 19th century, the text aged much less well. I find it very difficult to read. It is almost completely outdated, even though there are many interesting modern looks at it. Still, I would not require a physics student to look at it, important as it was. Euclid is different. Yes, it aged somewhat in many places. But it is still extremely readable, and when reading it you really do get a feeling for its immense beauty revealed in its structure. Last edited by Micrm@ss; July 17th, 2017 at 10:01 PM.  
July 17th, 2017, 10:37 PM  #10 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,237 Thanks: 2412 Math Focus: Mainly analysis and algebra 
I just think your criticism is over the top. Many mathematical texts are difficult to understand if you don't already know what they are trying to say. Especially the prenotation ones. I think it was some of alKhwarizmi's writing I saw recently that sounded like utter gobbledygook. Such was the rhetorical tradition. In contrast "that which has no parts" is a reasonably good definition of a point from 1800 years earlier and without the benefit of any existing definitions to work from. Sure, it might not tell you much, but if you know what he means, it makes sense.


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