Why Good Mathematicans Make Mistakes

So my girlfriend and I have started doing math again in our spare time (she's a biologist who's a good sport). The story begins when I discovered the awesome text Mathematical Masterpieces, which sent me on a search for it's prequel Mathematical Expeditions, which is geared towards lower-division undergrads so that I could read through it with Clare (any opportunity to do math with a partner is appreciated, since it's a lonely game in general). Both books talk about individual math topics through a math-historical perspective; each chapter has you reading original sources in chronological order in order to learn about, say, Fermat's Last Theorem or set theory. It's interesting stuff, and it doesn't shy away from real math like a lot of math history does. There desperately needs to be more historical context in the standard math education.

Anyways, the first chapter which Clare and I started reading talks about the development of hyperbolic geometry after generations of mathematicians failed to prove the independence of Euclid's "Parallel Postulate" from his other axioms. At the very beginning of Book I of Euclid's Elements, Euclid first defines a point, a line, and all the standard objects that Euclidean geometry is made of. I'll say more about this later. Then he gives five axioms for geometry. The first three basically say that all objects that should exist do: there's a line between every two points, a circle with any radius and center, and any finite line can be extended infinitely. Pretty non-controversial. The fourth one says that all right angles are equal, which sounds weird, but really just gives him the right to compare angles at all. The fifth one is a doozy (here I'm kind of paraphrasing):

If a line crosses two other lines, in such a way that the interior angles on the same side add to less than two right angles, than the two lines will converge on that side.

This is Euclid's Parallel Postulate, often just called Euclid's Postulate. (Here's the way I think of it. Take two lines, and two points, one on each of the lines. If at those points, the lines look like they're going to converge, than they will.) And that's it. Euclid then lists five "common notions", which are also axioms, but really for math in general. Stuff like equality is transitive, and the whole is greater than the part.

Anyways, from the moment that Euclid wrote it, this postulate was pretty controversial. Our earliest commentator mentions it, and tries to do away with it. On the one hand, it looks ugly. Turns out there are a lot of nicer statements that are equivalent. I was always taught it as this:

Given a line and a point off the line, there's only one line parallel to this line through the point.

Euclid almost certainly knew this equivalence, but phrased it in a way that he could work with more easily. It's his book.

Here's the more important issue: why is it even necessary? We all know what a straight line is, so assuming we have straight lines, we know that if they're coming towards each other, they're going to meet. We shouldn't need to take this an axiom, because it is inherent to the very notion of a straight line. This was the thought of pretty much every mathematician until the nineteenth century, and trying to prove it from the other postulates was a big deal, one of the great unsolved problems. Even Euclid lays out the Elements in such a way as to get as far as he can without using it, proving 26 theorems before he makes use of it.

From a modern perspective, the question is whether the fifth axiom is independent of the others. This was not the perspective of mathematicians prior to the discovery of hyperbolic geometry. The historical perspective was, this axiom must depend on the others, and we have to prove it. In Mathematical Expeditions, you read a source from the last major mathematician to believe in this dependence, Adrien-Marie Legendre. Legendre gave a number of proofs, and believed that each one was correct even when the flaws were pointed out to him. With each proof, he tried to make it simpler, make it the follow the same clear form of logic that Euclid himself used. Legendre is a smart guy; he solved some problems in number theory that Euler couldn't figure out, and he gave birth to the theory of elliptic functions, which makes him partly responsible for the solution to Fermat's Last Theorem. So why couldn't he or other mathematicians at the time acknowledge that they might be wrong about this?

The book answers this with a number of theories, and they're all pretty insightful. There's a cute historical argument that basically says that in an unstable world, mathematics, especially Euclid, was a torch-bearer for eternal, implacable stability, and the possibility of alternate geometries did not sit well. But I want to add a couple of my own theories, which I'm sure have been brought up many times before:

In modern math logic, one of the most fundamental ideas is the distinction between a syntactic model and a system of semantic logical statements that might be applied to that model. In our heads, we have a perfect visual model for Euclidean geometry. So did Euclid, and that's what he set out to work with. But Euclid wasn't working syntactically, he's working with a semantic system, and thus he needed to start somewhere. I get the sense after reading Legendre that even as late as 1792, this distinction was not well-known. I think he had serious problems with the notion that one would start with this awkward postulate and then prove how a straight line behaves, rather than the other way around. Reading one of his "proofs", you can see that he knows exactly where the flaw is, even adding a footnote with an explanation of why one should take his necessary assumption (which, too, was equivalent to the parallel postulate) as given. It's an assumption that one is perfectly willing to accept if one has the visual model in mind, but it's laughably easy to spot if one is working semantically. Smart mathematicians of then and now did not lay out axioms and figure out things to prove from them; they used intuition and insight, reverting to semantics when they need to convince others.

There's another more philosophical issue that is, I think, more to the point, and perhaps the real reason why Legendre was so adamant. That is that a 'straight line' is a perfectly intuitive notion that has no fundamental definition. I can see I'm going a little bit overboard for today, so I'll talk about this tomorrow.