Can there be anything that gives a person more thrills in life?
What’s that? You say you can think of one or two possibilities.
Well, I may have to concede the point. But I’m hoping I can give a hint as to why someone might pursue my line of work.
I wasn’t always a mathematician, by the way. At one point, for too long a time in my career, I was an electrical engineer. I was a terrible electrical engineer. You’ve heard those wonderful stories about the scientist in his lab (or garage) developing circuits that would change the world. Apple comes to mind. I might be in the lab all right, but never did I change the world. In college I held the record for creating the largest short circuit in my school’s history.
Fortunately, as I was forming, so were computers and I found a niche in that area that allowed me to house and feed my family. And it was fun. But not fun enough. I didn’t have the knowledge to do things I wanted to do. I needed to learn mathematics, mathematics beyond what engineers learned.
One day, after seven long years of half-time school, full-time work, and double-time husband and father, I finally could say, “I am a mathematician.”
This perhaps is not the best phrase to utter at parties. If I’m speaking to someone and am asked what I do, the answer often is, “Oh.” And after a pause, “That was my worst subject.” And after another delay, “Well, it’s been nice meeting you, but I have to leave.” This is not all bad. When I’m in a discussion with someone who is boring me to tears, I make certain to mention my life’s work and once again I’m left to my own devices.
But when I’m with other mathematicians, no one thinks I’m strange. When one of us asks, “How many “n”s in innovation, everyone knows the person is really requesting how many come after the initial “i.” But a mathematician will answer, “three,” and the asker will respond, “Thank you.” I shudder at what you might be thinking.
So what is it that drives one to mathematics? Let me try to explain with the story of a now famous mathematician.
His name is Andrew Wiles. He proved Fermat’s Last Theorem. Most of us learned that 3 squared plus 4 squared is equal to 5 squared, that is, (3 x 3) plus (4 x 4) is equal to (5 x 5) (exponent is 2). There are lots of other triples of numbers (meaning positive integers such as 1, 2, 3, …) that satisfy this, like 5 and 12 and 13. But what if I asked for three numbers b, c, and d such that b cubed (exponent is 3) plus c cubed is equal to d cubed, that is, (b x b x b) plus (c x c x c) is equal to (d x d x d)? Pierre de Fermat, a giant in mathematical history, said there are no such numbers. In fact, he said there are no such numbers if the exponent is 4 or 5 or any integer bigger than 2, and he declared in a margin of a notebook he had a beautiful proof but didn’t have the space to include it there. Most doubt he did have a proof, because some of the best mathematical minds over centuries couldn’t prove it. Until Andrew Wiles.
Now in a world that’s going to pot, one might wonder why anyone would care. But, to a mathematician, his proof is a big deal, a really big deal. A deal big enough for a documentary to be produced showing the drama of its discovery. I have watched it many times and shown it to students. It depicts Wiles’ many year effort involving extremely advanced mathematics. He thought he had it proved, but then an error was found. Much more time went into discovering a fix. Finally, the proof was complete.
As I watched the story, I could feel the emotions sweeping Wiles. Great joy as he thought the proof done, intense misery at the discovery of the error, and ecstasy when the final proof was confirmed.
Most research mathematicians don’t deal with mathematics at the level of Wiles. But we still share the emotions.
To us it’s theorems, mathematical truths that have been proven by rigorous well-defined rules of argument, that we live by. What we do is think up some question no one knows the answer to. It’s the type of statement that has a yes or no answer. For example, if it’s Saturday today, will it be Saturday again in exactly seven days? Of course, in this case the answer is yes, so we have to find a question whose answer is unknown.
Then we try to figure out what the answer is. Usually we hope it’s yes. Suppose it is. Then think what has happened. You now know something no one else in the world does. It might not be important. It might not be of interest to another soul. But it’s all yours. I suspect there is just as much elation as Wiles experienced.
I don’t think it can be significantly different from what a musician feels when she’s composed a symphony or a mechanic when a confusing problem falls to his expertise. A mathematician in Math Is Murder describes the feeling.
Now imagine the joy evaporating in an instant when you discover an error in your argument. I remember one paper I submitted for publication with a colleague. It was in the refereeing process when I discovered an error. Through late night phone calls, my colleague and I worked to find a fix. Alas, we could not, because the theorem simply was not true. The answer to the question we had asked was “no,” not the “yes” we had originally thought, and we had to withdraw the paper.
I’ve been involved in the development of hundreds of theorems, perhaps as many as a thousand. Most of them I no longer can remember. But I have never forgotten the one time a paper had to be recalled.
But, despite the potential pitfalls, I still look for theorems I can prove. Because the thrill of success is impossible to describe.