Engineers, Math, and Invention

Slide rule

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.

Published by

Steve Wildstrom

Steve Wildstrom is veteran technology reporter, writer, and analyst based in the Washington, D.C. area. He created and wrote BusinessWeek’s Technology & You column for 15 years. Since leaving BusinessWeek in the fall of 2009, he has written his own blog, Wildstrom on Tech and has contributed to corporate blogs, including those of Cisco and AMD and also consults for major technology companies.

15 thoughts on “Engineers, Math, and Invention”

  1. One could write a several volume treatise on this subject, and still not scratch the surface. Thanks for bringing it forward.

    Mathematics and the sciences (yes, they are distinct) seek the truth. Engineering, in no way unimportant, are more like Applied Mathematics and Sciences. Still, all scientists and engineers are “mathematicians” to varying degrees. One asks of scientists “How good of a mathematician are they”. Faraday was a poor mathematician, Maxwell a great one. Both were profoundly significant contributors to science, revealing deeply profound truths.

    “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.”-SW

    In many ways, I see the contrast between science (and math) versus engineering to be like the contrast between education and training (also subtly distinct). Both training and education are incredibly vital to human endeavor. The difference is that training (engineering) teaches you to find answers, whereas education (math and science) teach you to ask questions worth answering.

    1. I mostly avoided talking about science because it gets complicated. Historically, physics was closely aligned to mathematics, but other sciences were more like engineers in their math view. Recently, however, the sciences, especially biology areas, are becoming much more allied with math. One of the good effects for mathematicians is that biologists with limited training in advanced math have been hiring mathematics to fill the gaps.

  2. Off topic, but I’m not digging the new look. Not a fan of WordPress either, the layout is buggy on mobile, but also on the desktop, there’s enough fiddly stuff with margins and alignment that the overall effect on the home page is that it looks messy. I don’t mind the article pages, nice and simple, but the body text is too light, pullquotes are too small. Still, it’s easy to complain, the effort is going in the right direction I think. Just some rough edges to work on.

    1. My biggest beef with the new look: the number of comments a post has recieved is not visible on the home page on mobile.

      1. There’s a few quirks, for sure. Most likely related to WordPress, and it can probably all be fixed. I’ve done some WordPress development, it’s a tangled ball of string in my opinion, a real mess of plugins, not worth the effort to get it working right.

        1. Yep, still working on some refinements with live content that didn’t show up dev server content. I agree with the wordpress stuff but since we have boot strapped this whole operation our budget for these type things is limited. 🙂

          1. WordPress can be decent enough, given enough time massaging it. Good luck as you work through it!

      2. That’s funny, I never could see that on Mobile unless I requested desktop view (which I always do, but it doesn’t always go there).

        1. Actually, It seems to not show up at 1024×768 regardless of device type. Reducing the font size on firefox a couple notches (ctrl – ) makes the number of comments show up on the home page at that resolution. Haven’t tried that trick on my ipad yet.

  3. This reminded me of an exchange I had with a professor at MIT while working on my (now ancient) BSEE. I was building a matrix that would weigh analog signals, and was dismayed to find that I could only obtain 20% tolerance resistors. When I mentioned this to my professor, his response was “That’s OK. I only believe this nonsense to within a 20% margin.”

    Finally, as nice as the TI calculators were in the ’70s, the “real deal” was always an HP using RPN. To this day I cannot use an algebraic entry calculator without making mistakes – thank goodness for PCalc!

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