I was a math prodigy. But everything changed when I encountered an unfair problem more like the real world than mathematics.
Math had always come easily to me. I majored in math at Penn State, and everyone in the department knew I was a student with special abilities.
Math sort of shifts on you. In high school and first college courses, math is about problem solving: find x, determine the angle, compute the integral, get the answer. There are a lot of mathematical students who are good problem solvers. Many of them go on to become excellent engineers.
Once you get into junior and senior level classes and graduate classes, the objective changes. Your job is not to solve problems, it is to prove things. You are seeking to find general statements of truth. You must use reasoning and truths already proven to not only determine if if a statement is true, but to prove it with an airtight argument. Alternatively, you can show something is false by identifying a counterexample. Success is a proof or counterexample; failure is a suspicion that you can’t prove either way.
I was good at the proofs, too.
I wanted to train for the William Lowell Putnam exam. The Putnam exam is a nationwide, all-day exam that poses 12 problems. The best possible score on the Putnam exam was 120 — full credit for each of the 12 problems. The median score is 1 out of 120. The problems are that hard.
To train, I took an advanced problems class in which we received three problems per week. There were four students in the class. The whole class consisted of the students presenting their solutions to the three problems. The professor would present only if the students were unable to solve one of the problems.
These were hard problems, like the Putnam problems. They were often simple to describe, but maddening to prove (or disprove). You had to use creativity, guile, and a wide variety of mathematical techniques. I attacked them with every tool I had: converging series, reductio ad absurdum (assume the opposite and prove it leads to a contradiction), mathematical induction, and concepts from every branch of math in which I’d taken classes.
My goal was to prevent the professor from ever having to present an answer. A typical student in the class would solve one or two of the problems in a week. My goal was to solve all three. And even if took all my spare time for the whole week, I set out to do that.
And I succeeded. I felt invincible.
The next semester, I took the class again. But the original professor had retired. A new professor took over the class. I was the only student. We met in his office. And once again, I set out to master all three problems every week.
About halfway into the class, I met my match.
The professor had posed an equation about ordinary numbers that looked entirely routine. My job was to prove or disprove that the equation held true in all cases.
All week, I worked with the equation. I tried every approach I could think of. It was completely clear that the equation was true; I could find no evidence of any situation in which it was false. And yet, no matter what I tried, there was no angle that would yield a proof.
Finally, after struggling all week, I arrived, sleepless and red-eyed, in the professor’s office. “You have stumped me,” I said. “I know this is true, but I cannot prove it. Please tell me the answer.”
The professor said “The proof uses complex variables.”
This made no sense to me. The equation was about ordinary real numbers. Complex numbers, like the square root of -1, didn’t enter into the equation. It was as if you had been asked to construct a doorway, and found out that the key tool you would need was a hand mixer. Why would you use complex variables to solve an equation like this?
“I have not yet taken the complex variables class,” I told the professor.
“I know,” he said. “I wanted to see if there was a proof that did not use complex variables. No one has ever found one. If there was one, I figured you would find it.”
At that point I became very upset.
This was not fair. The result of his deception was that my streak was broken. I could no longer say I had solved all the problems in the advanced class.
This felt wrong.
First off, the professor knew the answer, but presented the problem in a way that was misleading, since it required a technique that had nothing to do with the original problem.
Second, he exploited the fact that I did not have the training to solve the problem for his own benefit — to try to unearth an unpublished proof.
And finally, he asked me to find something no one had found before. Everything I’d done before was on paths that others had made. He’d left me in the wilderness and told me to hack my way out.
I’m sure I seemed very agitated, but somehow, I didn’t blow up. You didn’t do that with a professor, even if you were just sitting in his office. I took the three problems for the following week and left, dejected.
I’m not sure why this incident has stuck with me so much that now, 40 years later, I want to tell the story. I think it is because, that day, I learned something a lot more important than math.
I think a lot of us go through life with the attitude I had as an undergraduate. We know the tools we have, and the problems we need to solve. We know that a solution exists. And we need to find it.
But that is not how the world works.
Sometimes the universe, or your boss, or your government withholds a key piece of information.
Sometimes the only path is to go into the wilderness rather than following the paths trod before.
Sometimes you work for weeks, only to find that there is no solution.
This feels unfair. Just like me on that day with the professor, you feel like you’ve been handed a raw deal, deceived, messed with.
But when the problem is how to get your marriage to work, or how to deal with your addiction or your child’s depression or a medical diagnosis or to fix global warming, there’s not only no set of rules, there is no guarantee that if you do the right thing and work hard, you will succeed. The self-help books are wrong. Confidence is not all you need. Some things are just impossible.
Not only that, but there’s no way to know if they are impossible or not when you set to work on them.
I remain a very confident person. I continue to succeed at most things I set my mind to. But thanks to that class 40 years ago, I know that the “rules” may not be rules, the tools I need may be completely unexpected, and the problem may not, in fact, be solvable.
The work is still worthwhile. The solutions are satisfying. But the work, in the end, is what makes life worth living, regardless of whether a solution exists — or whether life is fair in the end.