Calculus Can Be Fun!

I have taught calculus for decades, but David Berlinski’s book, A Tour of the Calculus, made me look at the subject in entirely new and entertaining ways. He starts at the beginning: points and lines, and slowly builds up the mathematical foundation upon which calculus rests. Along the way, Berlinski shares his wonder and love of mathematics. Concepts I have taken for granted for years suddenly appeared to me in a new light.

Berlinski’s sometimes florid style could be off-putting to some readers, but I found it kind of charming:

If the calculus comes to vibrant life in celestial mechanics, as it surely does, then this is evidence that the stars in the sheltering sky have a secret mathematical identity, an aspect of themselves that like some tremulous night flower they reveal only when the mathematician whispers. It is in the world of things and places, times and troubles and dense turbid processes, that mathematics is not so much applied as illustrated.

So, I enjoyed his over-the-top delight in unfolding the miracle of the calculus to his audience. He also has a wicked sense of humor – throughout the book, his tongue is firmly in cheek:

Students who need not be persuaded that gender studies are good for something often ask innocently whether analytic geometry is good for anything.

He gives the most original explanation of what a function is that I’ve ever seen:

A function, those thousand bright and brittle textbooks say, is a rule that assigns to each element in a set A a unique element in a set B. On the left are the elements in A, on the right, the elements in B. The function acts to pick one in A and assign it uniquely to one in B. This definition is current in the mathematicians’ lounge, where the mathematicians gather after class, and where it is always four on a gray Friday afternoon, the rain just beginning to streak the sooty windows. The image of a function thus evoked suggests one of those ghastly preadolescent dances in which sullen boys are lined up along one side of the ineffaceably smelly gymnasium and preening girls along the other, an energetic social science teacher seizing one of the hideously embarrassed boys and, after dragging him by the lapels of his stiff sports jacket, depositing him in front of a pleased but stout and red-faced young girl: Gregory, you dance with Jessica here. The homely tableau succeeds in spite of itself. The sets A and B are represented by boys on the one hand, girls on the other, and the function itself by the Czar’s dancing mistress, mysteriously transposed to suburban Teaneck, New Jersey, and acting energetically to pick a boy and assign him to a girl.

High school textbooks aren’t spared his wit:

The examples offered by elementary algebra are often uninspiring if only because no one wishes really to know which numbers correspond to the unknowns, the unknowns in word problems referring always to a strangely meditative farmer standing forlornly on that illustrated textbook hill of his, wondering in a way that suggests nothing of the power of mathematics how many turnips he might grow if he had two tons of fertilizer.

Along the way, we sit in on some of his classes as he tries to get his bored and confused students to share his enthusiasm for the Intermediate Value Theorem, among other topics. Berlinski really shines when he provides beautiful little portraits of the mathematicians who discovered and developed the math behind the calculus: Newton, Leibnitz, Cauchy, Euler, Dedekind, Lagrange, Riemann. I gained a new appreciation for the difficulties the concept of a limit presented to rigorous mathematicians. I especially enjoyed Berlinski’s tribute to Reimann, who died tragically young:

He was in his temperament a geometer, in his affiliations a Platonist, in his soul a visionary; he saw through appearances to a world less voluptuous and less complex than the real world, but more ordered, harmonious, stable, and beautiful. … Alone among the mathematicians of the nineteenth century, he saw what he needed to see before ever he acquired the symbolic apparatus with which to express his vision; his certainty about each of his discoveries was richly merited, but exotic and spooky.

Even if you didn’t like math as a student, you will be entertained by Berlinski’s presentation of it here. He doesn’t assume the reader knows anything, and he carefully explains every new concept that he introduces. There are excellent illustrations throughout, clear and easy to understand. By the end of the book, you will have a firm grasp of both differentiation and integration, and the Fundamental Theorem of Calculus that weds them.

I’ve read many books on math that are written for the layperson, and Berlinski is definitely one of the most approachable. He joins Ian Stewart and Steven Strogatz as writers who never talk down to their audience, but manage to kindle interest in a subject that too often strikes fear into people’s hearts. Take Berlinski’s Tour of The Calculus, and see why

The calculus is the story this world first told itself as it became the modern world.