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Proving Euclid's Propositions 
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#1
Nov913, 01:56 AM

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I'm working on a new logic which I hope will be better than the old logic and decided to use it to prove Euclid's first proposition and I was rather shocked that Euclid's explanation was not all that rigorous. I had a translation that was very close to the original, plus I can halfway read Ancient Greek and they had a side by side text. I'm not saying that Euclid is not a good mathematician I'm just saying that by today's standard's I'm not sure his proofs would pass muster. I was wondering if any mathematician has since come up with a more rigorous way of proving Euclid's propositions.



#2
Nov913, 11:18 AM

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There has been various commentary on the rigor in the Elements ever since it was first published. The book of Thomas Heath, "The thirteen books of Euclid's Elements", now in public domain, has extensive commentary. In fact, the commentary there and "filling the gaps" take (a lot) more volume than the original content.



#3
Nov913, 02:51 PM

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From what I heard, the fifth proposition, or the parallel proposition, is independent of the other propositions, and so it cannot be proven from the other propositions.



#4
Nov913, 03:36 PM

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Proving Euclid's Propositions
BTW: The parallel postulate is equivalent to the Pythagorean Theorem and to the law of similar triangles. So these also do not follow from the primitive axioms. 


#5
Nov913, 03:58 PM

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Propositions are the things that are proven from the axioms...
I have heard that historically the proof that an icoseles triangle has two equal angles has been thought of as dubious to the point that some people said that it was really Euclid's sixth axiom. 


#6
Nov913, 05:05 PM

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#7
Nov913, 10:55 PM

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Getting back to the nugget in the gold mine: what specifically will this new logic accomplish that the old logic is unable to?



#8
Nov1013, 04:51 AM

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#9
Nov1013, 06:50 PM

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#10
Nov1013, 07:17 PM

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Still, I think you should study 'semantics', which is a branch of linguistics dealing with the meanings of words (and which is no simple subject): http://en.wikipedia.org/wiki/Semantics 


#11
Nov1013, 08:45 PM

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Here's a simple example that I cooked up. You see me write "I read the book." Noticing that I am not reading a book, you conclude that my statement is false. However, the word I wrote is the past tense of "to read," so if I have ever read a book, my statement is actually true. 


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