Math

*Published in IEEE Spectrum Magazine, Sept
2007*

I was browsing some of the features of a popular computer program for doing
mathematics. Wow, I thought! What I would have given for this years ago!

But suddenly I was overcome with sadness. I don’t need this anymore, I
realized. In fact, it has been many years since I worked with “real”
mathematics. I just never really thought about that loss before. It was if my
profession had slipped away while I hadn’t been looking.

I commiserated with several engineering friends. Two of them
weren’t concerned at all. That’s what happens when you move along
in your engineering career, they said, and it doesn’t make you any less
of an engineer. The other, a researcher like me, shared my nostalgic pain. It
made him think of what he was, and is no more.

I wonder what percentage of engineers use advanced mathematics in their jobs,
and I wonder if that percentage is less now that computers have consumed so
much of our work. Has mathematics disappeared behind the screens of our monitors
like so many other things? And as more and more of engineering becomes the writing
of software, how much math is being used?

Yet mathematics is a way of thought that binds us to our profession. Maxwell’s equations are inscribed in the entrance foyer of the National Academy of Engineering as the very symbol of what we do. I look at them as the scripture of engineering – a concise and elegant description of the laws that govern electromagnetism. But I also wonder: how many engineers have actually used Maxwell’s equations in their work? Alas, I’ve never had the pleasure myself.

Our journals are still full of mathematics. If you want to publish and have your work inscribed in stone for eternity, you must code your work in mathematical symbolism. If you want to parade among the elite of the profession, you must cloak yourself in mathematics. That is the way it has always been. If now math is disappearing from our practice, I would be saddened.

I remember well the day when I was first introduced to imaginary numbers. It was in a high school algebra class. The teacher talked about the square root of a negative number. It didn’t actually exist, she said, so it was called imaginary. That bothered me a lot. If it didn’t exist, why give it a name and study it? Unfortunately, the teacher had no answers for those kinds of questions. It was just “there” and something that mathematicians seemed to be concerned about.

As in a lot of the mathematics that we’ve all studied, it’s only later that understanding comes about. We all have had the experience of learning mathematical principles before we had any idea what they were good for. Looking back on that day in high school algebra, what would I have told my young self in retrospect?

I can think of two pathways of explanation, although somehow I doubt that my young self would have been happy with either. The first is that mathematics is beautiful in itself, a study of consistent rules of logic that can be appreciated as an art form. It doesn’t have to be rationalized as immediately applicable to everyday problems.

The second explanation is that this square root of minus one
is actually useful (in problems that you don’t know about yet). It opens
the door to two-dimensional thinking – a dimension that gets you off the
line of real numbers. So, whether or not this imaginary number exists in your
world of arithmetic training, it’s useful. In real world problems, it
works.

I’m reminded of a famous saying in physics that often occurs to me. Dirac
is quoted as having said, “Shut up and calculate.” (This saying
is often attributed to Richard Feynman.) In physics it refers to the fact that
in quantum mechanics the Shroedinger wave equation often contradicts common
perception, yet always gets the right answers when applied. So don’t worry
about it; quit complaining and just calculate. Like using the square root of
minus one, it works.

Since that first day of imaginary numbers, I’ve come almost full circle. I learned to appreciate math, and I found imaginary numbers useful. But now I’m thinking that, though the appreciation remains, the usefulness to me has faded.

The more I think about this as I write, the sadder I get. I’m going to
go back and look at the features of that mathematics program again.

Robert Lucky