There is a famous mathematical proof called The Jordan Curve Theorem. It's wrong. Camille Jordan came up with it at the end of the 19th century, and it bears his name, despite being inaccurate, because there is no justice in the world. There are plenty of proofs out there that are wrong, but this one is notable for proving something so very, very simple. Find out how hard it is to prove an obvious statement.

One of the frustrating things about academia is the fact that the simplest things are the hardest to define. For example, give me the definition for the word 'and.' And, for, the, and other simple words can take up pages in the dictionary. Likewise, straightforward mathematical concepts are massively difficult to prove or define. The Jordan Curve Theorem was a long, difficult proof that was an attempt to show that any finished curve, circle, oval, or squiggly loop, divides a plane into an inside and an outside. Visually, all you need to do that is a piece of paper, some markers, and your fingers to point — and possibly your face to accurately convey the word 'duh.' Math makes things a little more stringent.

Again the problem comes down to definition. We know that a circle is a simple, closed curve, but what about a square? How about a long, complex loop? Intuitively, we know what we mean, but how do we actually define the shape, which can be straight, angled, curved, complicated, as long as it doesn't have an end or cross over itself. Even tougher on math, how do you distinguish the inside from the outside? If you got as far as, "The inside is the part that's inside the curve," you got as far as many others did. It took Jordan years to do this . . . he thought.

In the end, the best way to tell inside from out is with a straight line across the curve to any given point. If the line you draw, from the edge of the plane to any given point in the curve, intersects with the line of the curve an even number of times, the point you picked is outside the curve. If it intersects an odd number of times, the point is inside the curve. Jordan figured that one out, but other parts of his proof were found to be incomplete, and it took another decade and a half until someone stepped up and proved that a circle has an inside and an outside once and for all. The hero was an American, which is strange, because we're usually good at taking credit for things. Sadly, he had the not-so-flowing name of Oswald Veblen, and so the Jordan Curve Theory kept its original, and erroneous, name to this day.

*Image: **Nevit Dilman*

## DISCUSSION

Yes, lots of simple things are hard to prove. The four color theorem is a good example. Any kid with a box of crayons will figure out that no more than four colors are needed to color the states or countries on a map so that no two adjacent areas (sharing a common side) are the same color. It's pretty obvious but people tried for over a hundred years to prove it mathematically and couldn't. Finally in 1976 two guys sort of proved it by exhaustively computing all possibilities.

Now if you want a real headache, read up on Gödel's Theorem. He showed that there can be true things in mathematics that are unprovable even in principle.