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 half-way 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.
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.
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.
correct. The existence of parallels can be proved but not their uniqueness. There is a plane geometry with infinitely many parallels which satisfies the other primitive axioms. 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.
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.
Getting back to the nugget in the gold mine: what specifically will this new logic accomplish that the old logic is unable to?
Well, ideally I would like this logic to prove if a sentence is true or false by merely analyzing the meaning of the words. For example, take the sentence "statements must be types of data." How do you prove that? You prove it by stating what statements, types and data mean. So if statements mean x and type means y and data means z then you state what x y and z mean. You keep doing this until you reach the indefinable words. At that point you look for a contradiction and if there is none and if the contrary sentence "statements are not types of data" has a contradiction then you have one instance of your definitions being correct. At that point you try to make the definitions output the correct truth value for as many sentences as possible. It is easiest to define words if they are mathematical objects so we start with math. Once we have some success with math we move over to metaphysics since that is where the real gold is. However, you can't build this logic unless you have experience with artificial intelligence so I'm putting this project on hold until I know more about AI.
Not sure about the axioms but it seems that it would follow from a statement that the corresponding angles of congruent triangles are equal.
I think that adverb 'merely' is going to trip you up. If logic could have been reduced to the mechanical analysis of the meanings of words in a sentence, it would have been done so by now. Certainly, some aspects of the study of the law would be rendered less opaque if such were true. 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
I agree with SteamKing that "merely analyzing" will be very difficult, since words often have multiple meanings, sometimes even with opposite meanings. 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.