Long ago, when I was a freshman in engineering school, there was a required course in mechanical drawing. “You had better learn this skill,” the instructor said, “because all engineers start their careers at the drafting table.”
This was an ominous beginning to my education, but as it turned out, he was wrong. Neither I nor, I suspect, any of my classmates began our careers at the drafting table.
These days, engineers aren’t routinely taught drawing, but they spend a lot of time learning another skill that may be similarly unnecessary: mathematics. I confess this thought hadn’t occurred to me until recently, when a friend who teaches at a leading university made an off-hand comment. “Is it possible,” he suggested, “that the era of mathematics in electrical engineering is coming to an end?”
When I asked him about this disturbing idea, he said that he had only been trying to be provocative and that his graduate students were now writing theses that were more mathematical than ever. I felt reassured that the mathematical basis of engineering is strong. But still, I wonder to what extent—and for how long—today’s undergraduate engineering students will be using classical mathematics as their careers unfold.
There are several trends that might suggest a diminishing role for mathematics in engineering work. First, there is the rise of software engineering as a separate discipline. It just doesn’t take as much math to write an operating system as it does to design a printed circuit board. Programming is rigidly structured and, at the same time, an evolving art form—neither of which is especially amenable to mathematical analysis.
Another trend veering us away from classical math is the increasing dependence on programs such as Matlab and Maple. The pencil-and-paper calculations with which we evaluated the relative performance of variations in design are now more easily made by simulation software packages—which, with their vast libraries of prepackaged functions and data, are often more powerful. A purist might ask: Is using Matlab doing math? And of course, the answer is that sometimes it is, and sometimes it isn’t.
A third trend is the growing importance of a class of problems termed “wicked,” which involve social, political, economic, and undefined or unknown issues that make the application of mathematics very difficult. The world is seemingly full of such frustrating but important problems.
These trends notwithstanding, we should recognize the role of mathematics in the discovery of fundamental properties and truth. Maxwell’s equations—which are inscribed in marble in the foyer of the National Academy of Engineering—foretold the possibility of radio. It took about half a century for those radios to reach Shannon’s limit—described by his equation for channel capacity—but at least we knew where we were headed.
Theoretical physicists have explained through math the workings of the universe and even predicted the existence of previously unknown fundamental particles. The iconic image I carry in my mind is of Einstein at a blackboard that’s covered with tensor-filled equations. It is remarkable that one person scribbling math can uncover such secrets. It is as if the universe itself understands and obeys the mathematics that we humans invented.
There have been many philosophical discussions through the years about this wonderful power of math. In a famous 1960 paper entitled “The Unreasonable Effectiveness of Mathematics in the Natural Sciences,” the physicist Eugene Wigner wrote, “The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift [that] we neither understand nor deserve.” In a 1980 paper with a similar title, the computer science pioneer Richard Hamming tried to answer the question, “How can it be that simple mathematics suffices to predict so much?”
This “unreasonable effectiveness” of mathematics will continue to be at the heart of engineering, but perhaps the way we use math will change. Still, it’s hard to imagine Einstein running simulations on his laptop.