Why Good Mathematicians Make Mistakes (Part 2):

In my last post, I described why I think Legendre was so determined to prove the Parallel Postulate from the other axioms. Legendre knew where the faults in his proofs were, but he believed they could be overcome if we allow some basic assumptions to be made about straight lines. Indeed, he wrote explicitly that the problems we have with this postulate are due to the "imperfection of the common language and to the difficulty of giving a good definition of a straight line."

What Legendre didn't know is that the difficulty is not particular to the "common language," but that there simply is no way to define a straight line (or for that matter, a line itself or a point) without making reference to another model, that is itself undefined. Generations of mathematicians before Legendre had tried to give a simple, self-contained definition, but the clearest of all is, in my opinion, the very first one we have, which predates Euclid.

In the Parmenides, Plato gives what is believed to be the common definition of contemporary geometers: a straight line is one in which the middle covers the ends. Of course, we cannot be sure about these things, but this is the only way anyone's been able to make sense of this:

If you, yes you, put your eye in the middle of a straight line, then you wouldn't be able to see anything of the line but the point where you put your eye.

Modern mathematicians should be up in arms at this point; giving a definition that refers to a previous model is one thing, but a definition that refers to reality, to a human being, is unconscionable. In fact, Euclid's revolutionary act was to remove the human from the divine realm of mathematics. So here's Euclid's stumper of a definition:

A straight line is one which lies evenly on the points on it.

If you're like me, then you should be wondering, what is this shit? I thought there must be a translation issue here. So I looked at the source (Did I mention I took two years of ancient Greek in college? Liberal arts schools are great.) and here's what I found. Sorry, fellow Greek scholars, but you're just going to have to accept a poor transliteration until I can figure out how to use Greek characters in Blogspot:

"ex isou tois eph' eautes semeiois keitai" (lies evenly on the points on itself)

The ambiguity is in the 'ex isou', which literally means something like 'in an equal way'. The phrase can either be taken with 'keitai' (lies evenly on the points on itself) or 'tois semeiois' (sits on its evenly spread out points). What's the difference? Here I turned to Thomas Heath's 1908 edition of Euclid, what may still be the standard source on interpreting Euclid in Greek, if only because both reading Euclid and learning ancient Greek have been in serious decline since education stopped being so aristocratic.

Anyways, he says that the first translation might mean that a straight line looks the same no matter which point you look at; you'd be hard pressed to tell the difference between picking one point or another, since the line 'around it' will look the same. The second translation depends on what you mean by 'evenly spread out points'. He might mean that a curve amounts to a kind of 'bunching' of points. But nobody really knows. Proclus, the earliest commentator on Euclid, gives the least convincing interpretation of all; that a straight line measures out the smallest distance between two points. (Here I guess 'ex isou' is taken to mean 'of equal measure? I don't know.) It's pretty obvious why Euclid is the only one who gives this definition, and Greeks afterwards go back to Plato's definition.

Ironically, Platonism may have been the reason that Euclid changed to this vague and pretty meaningless definition. If Euclid was a Platonist (he is believed to have studied at the Academy), then it makes sense that he would have attempted to remove all traces of the material world from his geometry. I've been spent more than a bit of time studying Plato, so I find this pretty fascinating. I've always held the suspicion that Plato had mathematics in mind when he invented 'forms'. If so, then it's pretty interesting that Euclidean geometry suffers from some of the same issues with Platonism as more general philosophy does:

Does the form of a straight line exist in the world of the forms? If so, then it would be pure in itself; it wouldn't make reference to anything else. But then it would be impossible to describe it.

There are bigger Platonic problems with lines than the fact that lines may not exist in nature (or that no one can draw them perfectly straight). Euclid's failure and the failure of so many mathematicians exemplify the fact that even the idea of a line isn't sensible unless we give it some kind of context, a place to live like a real two-dimensional vector space.

I've always thought of myself as a mathematical Platonist, but after thinking about these issues, I'm not so sure.

Next time: Rocket Knight Adventures!!