Engineers, Math, and Invention
By no stretch of the imagination am I a mathematician or an engineer. But nearly every January, I spend much of a week at the Joint Mathematics Meetings, the annual collection of lectures, seminars (and job interviews) of academic mathematicians. One obvious thought that came out of the latest meeting in San Antonio is just what is math education contributing these days to the non-mathematicians, many of whom run tech companies.
Are engineers mathematicians at heart? In the minds of many in the public engineers, especially those doing computer-related work, are indistinguishable from mathematicians. But many math teachers, especially those offering advanced courses in both high school and college, are more focused on theory than the practical. ((The concern here is not with controversy over the Common Core Standard, which is generally involved in earlier subjects.))
Mathematical tools. Once upon a time, when I was in school, the only tools you used were a slide rule and, maybe in college, a mechanical desktop calculator. One interesting effect is you learned a lot of calculating methods in high school courses, such as computing the square root of a number or using a trig table to find the cosine of an angle between those in the list.
The universal availability of electronic calculators, especially the Texas Instrument calculators, eliminated the need for anyone to understand these techniques. They could compute faster, but would lose some depth of knowledge.
To get some expert opinion on the difference in the belief of mathematics and engineering, I had the fortune of consulting my sons. Jonathan Wildstrom, a research computer engineer at IBM’s Watson Research Center, and D. Jacob (Jake) Wildstrom, a mathematics professor at the University of Louisville.
Calculation precision. One crucial difference in style is the precision of calculation. Mathematicians usually want an exact value, even if that means an expression full of square (or higher) roots and e’s. An engineer generally prefers a calculation that gives the result that is needed, which is both easier and more useful.
“In computer science and software engineering, for the bulk of things close, is usually enough,” says Jon. “In most applications, the existing algorithms are ‘good enough’. There are tricks of the trade—memoization is one, dynamic programming another—that help improve algorithms without needing mathematical support.” An example is calculating the Fibonacci numbers. ((Numbers in the infinite series 1,1,2,3,5,8… that are often needed in computations.)) Mathematicians proudly use a somewhat complicated formula that can compute the nth term in the series. For an engineer or a computer program, it will do well to just run the series of n terms. It’s not elegant, but it’s efficient and practical. For very large numbers, there’s a fast term that will give a close approximation.
“I think mathematicians tend to see approximation as an interesting challenge,” says Jake. “But unlike those in the engineering or scientific domain, we don’t have a well-defined notions of ‘good enough.’”
Engineers’ work focuses on the efficiency of their techniques. “Algorithms and tricks to access specific memory areas quickly can be important,” says Jon. “But in all these algorithms, multiplication is frowned upon and division is flat out forbidden because of the performance implications of even a single division.”
“It seems to me mathematicians do revel in the arcaneness for its own sake,” says Jake. “Even a mundane problem such as ‘how do we approximate this closely?’ ends up swaddled in layers of abstraction once mathematicians are through with it. We may exaggerate beyond the point where it is necessarily helpful or instructive to those who aren’t planning to be mathematicians.”
Practical separation. The separation of mathematicians from the more practical fields is a relatively recent development, no earlier than the middle of the nineteenth century. From Aristotle to Leonhard Euler in the eighteenth century, the best mathematicians were often put to work to apply their skills to military needs. More recently, serious mathematicians have typically found themselves limiting their involvement to theoretical physics, where those from Hermann Minkowski and Henri Poincare to Richard Feynman and Edward Witten have been leaders in both math and physics. ((As a matter of fact, a lot of interest in pure mathematics recently has become of much greater interest to biologists and biochemists. ))
But modern theoretical physics, like math, avoids obsessions with the practical or even understandable. “I’ve definitely thought of the mathematician/engineer divide as one going back philosophically to the long-standing difference between Plato’s ideal schools and Aristotle’s empirical schools,” says Jake. “Mathematicians invariably drank from a well of ideals.” Not surprising, or impractical, that their approach often offers something far from what engineers want or need.