Long ago, when I was a freshman in engineering school, there was a required course in mechanical drawing. On the first day of this course the instructor began his lecture by saying, “You had better learn this skill, because all engineers start their careers on the drafting table.”
This was an ominous beginning to my engineering education, but as it turned out, he was wrong. Neither I nor, I suspect, any of my classmates, began our careers on the drafting table. This is not to say that the course wasn’t valuable, as the ability to visualize and manipulate shapes can be important in many engineering applications. However, and most fortunately, I never subsequently had the occasion to put an inked pen on vellum.
But today I am wondering: could a similar forecast be made now about mathematics? At a recent meeting 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 comment afterwards, 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. Although I believe that the basis of engineering upon mathematics is as strong as ever, it is worth discussing whether or not the majority of undergraduate engineering students of today will be using classical mathematics in their future careers.
There are several trends that might suggest less use of math today. First, there is the very large and growing role of software in engineering design and development. While I don’t suggest that software is not amenable to mathematics, I do believe that it is less easily adapted to math than has been the physical world. And in seeming contradiction, software is rigidly structured while at the same time being an evolving art form, neither of which is especially helpful to the usefulness of math.
Another trend that might cause less use of classical math is the increasing dependence on computer simulation and mathematical analysis programs such as Matlab and Maple. In years past we used pencil-and-paper math as our tool for system analysis to evaluate the relative performance of variations in design. Now that is more easily done using computer simulation, and with the growing library of pre-packaged functions, often more powerful. Still, a purist might ask: is using Matlab doing math? And of course, the answer is that sometimes it is, and sometimes it isn’t.
There is also the enhanced awareness today of a class of problems, termed wicked, that 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. In the foyer of the National Academy of Engineering Maxwell’s equations are inscribed in marble. They foretold the possibility of radio. Shannon’s equation for channel capacity told us that it was possible to transmit data at arbitrarily small error rates, but only up to a limit he called capacity. It took about a half century, but Shannon’s capacity was finally achieved.
Theoretical physicists have predicted the existence of previously-unknown fundamental particles and explained through math the workings of the universe. The iconic image that I carry in my mind is Einstein at a blackboard covered with tensor-filled equations. It is remarkable that one person scribbling math can uncover such secrets. It is if the universe itself understands and obeys the mathematics that we humans invented.
For quite a few years there have been philosophical discussions about this wonderful power of math. There is a famous paper published in 1960 by the physicist Eugene Wigner entitled, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” In this paper he writes: “The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.”
Wigner’s paper was elaborated upon and argued by the computer science pioneer Richard Hamming in 1980. Hamming asks and tries 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. This does not compute.