Reflections

The Pleasure of Calculus

By Robert Silverberg

 

I have written some four hundred of these columns, I suppose, since I replaced the late Isaac Asimov in this slot in 1994, and attentive readers will be aware that they have covered a very wide range of topics indeed. A lot of them have been about science fiction, of course, this being a science fiction magazine, but over the years I have also dealt with astronomy, biology, geology, genetics, philology, paleontology, and a great many other scientific subjects.

So it is reasonable to assume that a column headed “The Pleasures of Calculus” is going to deal with the vast intellectual delight I have derived from using calculus in the course of my mental activities. You would be wrong. I know exactly two things about calculus. One is that it was invented by Gottfried Leibniz in the seventeenth century (though Sir Isaac Newton, working independently, developed a form of calculus about the same time. Newton found that he needed calculus for his scientific work, and since it didn’t yet exist, he invented it. But Leibniz’s notation is considered the more exact form of it.)

The second thing is that that bit of information about Leibniz and Newton, one of the vast accumulation of not very useful items of data that have come to clutter my mind over the past ninety years, is absolutely all that I know about calculus.

Mathematics was never an important part of my education. (Anyone paying even a little attention could see that from a very early age I was heading for a literary future.) I didn’t find arithmetic much of a challenge, of course—I could handle two plus two and five minus four with the greatest of ease; I quickly learned my multiplication tables, and in the course of time I mastered such more abstruse techniques as long division.

When I got along to higher education, I was confronted first with algebra, which struck me as utterly of no value to my future self, but I did get the knack of that x-minus-y stuff pretty quickly. Next came geometry, the only practical use for which that remains with me is my awareness that a straight line is the shortest distance between two points, something that has saved me any number of years, in the aggregate, in a long life of crossing from one side of the street to another.

Then came trigonometry, that esoteric thing of cosines, tangents, and secants and such—I stared at it, and it stared back at me, neither of us appreciating the other, but there was one thing about trigonometry that did attract me: the beautiful Miss Weinstein, my teacher, who was about 27 and absolutely gorgeous. I think every boy in the class was in love with her. Certainly I was. I paid close attention to every word the lovely Miss Weinstein uttered, and somehow I absorbed trigonometry along the way, and, a little to my surprise, got a remarkably high mark in that class.

But calculus—ah, calculus—I was now in my final year of high school, already sending short stories off to the science fiction magazines (and getting them back with sad rapidity), and, since calculus was an optional subject and I was clearly heading in a non-mathematical direction, I opted to skip it. And thus it became, and has remained everlastingly, a blank space in my knowledge.

Someone else, quite surprisingly, who got nowhere with calculus was Isaac Asimov, one of the most intelligent human beings ever to walk the Earth. Isaac did write a book called The Realm of Algebra, another called Realm of Numbers, and a third called Quick and Easy Math, but Asimov’s Realm of Calculus, no, there is no such book. He did find differential calculus relatively easy, but when he came to the other kind of calculus, integral calculus, he wrote, “I came up against a barrier. I realized that this was as far as I could go, and to this day I have never successfully gone beyond it in any but the most superficial way.”

It was rare for him to admit any sort of defeat for that remarkable mind. Somehow, though, Isaac made his way through the world without any command of calculus. I do take comfort in that.

The great editor John W. Campbell, Jr., another person of extraordinary intelligence, also reported having only a fair understanding of calculus. When he needed knowledge of trigonometry for his application to the Massachusetts Institute of Technology, Campbell taught himself trig overnight by skimming the textbook, and passed the exam. But calculus, no, it never came easily to him.

I do know people for whom calculus is not any sort of vast mystery, but a matter of ordinary and frequent use. There are, indeed, whole families that apparently have a genetic predisposition toward being at home in the world of calculus. The Benfords are a good example. Gregory and James Benford, twin brothers, are both physicists. Greg, Professor Emeritus of Physics at the University of California, has written significant science fiction also, and Jim was an executive at the California corporation Physics International, then founded his own company, Microwave Sciences, and has produced many scientific papers on plasma physics and similar subjects. (He wrote some science fiction stories as well.)

Jim also is the co-author of a definitive text on microwave technology, and I have peered into it, understanding nothing at all, finding it full of arcane calculus symbology. Greg’s numerous scientific papers in the field of astrophysics also bristle with calculus stuff. Alvaro Zinos-Amaro, who studied theoretical physics as an undergraduate, works for Verizon, and writes science fiction on the side (he and I collaborated on a book a few years ago), tells this story of Greg Benford and calculus: “One day I walked into Greg Benford’s kitchen and found he’d scribbled down some integrals (derivatives and integrals being two main branches of calculus) on a scruffy piece of paper. I asked him what he was working on. He explained the scenario, and I was then able to follow his reasoning by going down the set of equations. I think that may have been his equivalent of you jotting down some of your story ideas and possible titles on used envelopes.”

And this is what Alvaro had to say about his personal experience with differential calculus: “I remember I’d read Asimov’s comments on his own difficulties with the subject so I approached it with some nervousness, because I knew he was a genius. But it helped me to realize that Asimov had other limitations too, as we all do. . . .I also knew other people who weren’t as bright as Asimov who were good at calculus. I suppose I started to think of it as learning a language, one more abstract than the typical ones we use to communicate in everyday life, but nevertheless one that could convey important mathematical messages. I’ve used some of the notions of calculus in my job, but I’ve not had to write down a lot of equations because the calculus needed for finance-type problems and data analysis is now mostly available in canned software. But while not directly applicable to what I do in terms of data analysis and data science, it’s informed my worldview. The world is change, after all, and calculus helps to quantify the relationship of changing variables with elegance and incredible power.”

Jim Benford’s son Dominic is also an astrophysicist, who works in the NASA space program as a program scientist for the Nancy Grace Roman space telescope. Calculus comes as second nature to him.

And so it continues through the generations. Dominic’s daughter Audrey is studying calculus in college—“I should think it will be useful to me in any career path I pursue,” she writes. “The study of math involves problem-solving skills that I think are applicable in any career.” Her brother Harrison, who is a brilliant pianist, learned calculus in high school, though he tells me that he does not make use of it these days while continuing his musical studies.

Then there is the Australian branch of the family. Jim’s daughter Vanessa, who moved to Melbourne some years back and is a banker, says, “I did study calculus in high school but have never really used it since.” She finds statistics and standard economic formulas more useful to her than calculus in her banking roles. But her son Jack, who is going to be yet another Benford astrophysicist, has studied calculus, as has his sister Anna.

And what, you may ask, actually is calculus? You may ask, but you won’t get the answer from me, because, although, as my extensive list of nonfiction books testifies, I have some Asimovian abilities as an explainer myself, I don’t try to explain subjects I know nothing about. (Likewise Isaac, though as noted above he wrote a number of books on mathematical subjects, wisely did not venture to discuss calculus, after admitting his lack of success with it.) If you happen to have any Benfords nearby, ask one of them. (They are scattered across our land from coast to coast, with that second group in Australia.) Dominic Benford summarizes calculus this way: “Calculus is certainly an important tool for many things, in physics and astronomy, certainly, and it also underlies many engineering and scientific disciplines.”

For a more detailed explanation of calculus, I commend you to the extensive Wikipedia article on the subject, which provides such a thorough account of it that it left me glassy-eyed within the first few paragraphs. May you have better luck with it.

Everybody has a different set of skills. One friend of mine can read Babylonian cuneiform; another flies his own plane; a third is an expert on indigenous African art, and can identify the tribal origin of a statuette at a glance; a fourth plays the cello.

I can do none of these things, any more than I can comprehend a chunk of calculus. Fortunately for me, I did know how to write science fiction stories that other people would want to read. And so I get along through life without ever knowing the pleasures of calculus. Different folks, different strokes.