Ugly Facts

Thomas Henry Huxley has this famous quote: "Science is organized common sense where many a beautiful theory was killed by an ugly fact." The analogue in mathematics is when a beautiful conjecture is killed by an ugly counterexample. I have been working for the last month on proving that V_n + V_n is pseudo-convex. Please don't worry about what that means; the important thing is that I was able to obsess over this problem. If I can show that kind of obsession in the future, I may yet have a successful career in mathematics...

Anyways, I thought I almost had a proof at least two or three times, but it kept getting away from me. And for good reason; it isn't true! I found a counterexample today. Mathematica gets the assist on this one. I was trying to write a program to test my algorithm for small cases, and what do you know, when n = 20 mathematica found the following vector, which was ultimately my demise:

(1,1,0,1,2,0,0,0,1,0)

Again, don't worry about why. I do love the fact that I can talk honestly about my research and it sounds so complex and difficult, but again, the important thing was I had written a program to try to weed out counterexamples. So far, I had been able to apply my method to deal with each one, but in this case, I had finally met my match. My algorithm didn't work, and in fact no algorithm could, since it turns out this little fella is in the convex hull of V_n + V_n without being in V_n itself.

Wow, I really can't stop with the math jargon. Well, now that I found this counterexample, I have some mixed emotions. I am a little upset that the result isn't true, as a proof of it would really make my thesis nice and complete. Plus, my advisor had always been skeptical, so it would have been nice to be able to prove him wrong. But I also feel some relief, in that now I know the answer. Plus, it's not like this counterexample was just sitting there waiting to be found. I am proud of the methods I used and the devotion I have had in searching for a proof/counterexample these last couple of weeks.

So there is this ugly fact, but it is my ugly fact. And that makes it pretty beautiful to me.