Studying Mathematics Thread

[polymath257]

A very, very nice post. I only have a slight quibble about the history.

Plato was a couple of centuries *before* Euclid. So the geometry that Plato wanted for entrance to his Academy was not that of Euclid, but more likely that of Theatetus.

Second, Euclid's Elements *did* include a considerable amount of number theory. For example, his proof that there are infinitely many primes is simple and direct and used today. He also gave a condition when an even number is perfect (later showed to be the only situation where an even number is perfect by Euler).

Given your discussion of the 5th Postulate, I was also surprised that you didn't mention the rise of non-Euclidean geometry and the effect it had on the formalization movement leading up to Russell.

All said, though, well done!

Looks like the forums ate my last attempt at a reply, hopefully this one fairs better.

I didn't mention the number theory in Elements because, while the number theory was also rigorously approached and there was a significant amount of it, it wasn't axiomatized the same way that geometry was.  And I didn't mention non-Euclidean geometry because I'd already decided to reserve the 5th Postulate's epic for a potential later post, and also because I didn't feel like spending half the post trying to explain spherical and hyperbolic geometry and how that differs from the Euclidean postulates.  And I attributed the Academy's obsession with geometry to Euclid because... .... well, that one was a major screwup on my part.  I've believed that one for years but apparently it's false and I never, well, never did the math on that.  Good catch.

Well, Euclid made a distinction between axioms (which were general statements) and postulates (which were specific to the area of study). He actually had 5 axioms and 5 postulates for his geometry. My impression is that he regarded the axioms as the foundation for number theory and that geometry needed the additional postulates to make it work.

I definitely understand why you left out non-Euclidean geometry. Another aspect is that Euclid had  a number of 'hidden assumptions' that weren't really brought out until the 19th century, having to do with 'betweenness' properties. He implicitly used that a line meeting one side of a triangle will also meet at least one of the other sides (maybe at the corner).

In any case, I look forward to your later post!