A Divine Language: Learning Algebra, Geometry, and Calculus at the Edge of Old Age

From a Wall Street Journal review by Siobhan Roberts of the book by Alec Wilkinson titled “A Divine Language: Learning Algebra, Geometry, and Calculus at the Edge of Old Age”:

When the New Yorker’s Alec Wilkinson ran into his colleague Calvin Trillin at a party and mentioned that he was working on a book about mathematics, Mr. Trillin asked, “For or against?”

Well, it’s complicated.

In “A Divine Language: Learning Algebra, Geometry, and Calculus at the Edge of Old Age,” Mr. Wilkinson sets out to right a wrong he suffered as a student half a century ago. He confesses that, but for cheating, he would’ve flunked mathematics in high school. So he embarks, at the age of 65, on an intellectually rollicking remedial quest, undertaken in the spirit of some 11th-hour self-improvement.

He soon learns, however, that his “math allergy” hasn’t waned much with time. Despite a fairly steady state of high math anxiety (he probes research on this bona fide condition, as well as other scientific tangents), Mr. Wilkinson proceeds with Beckettian perseverance and entitled petulance. “A lot of things I did in algebra I did in the spirit of exposing its irrationality, that is, destructively,” he says. “I had a bad attitude. I was a type of math punk.”

He recruits two professors of mathematics (his niece Amie Wilkinson at the University of Chicago and Deane Yang at New York University) for guidance, if not teaching proper—a tutor was against his rules. Instead, he plays the autodidact and, along the way, struggles through failure and frustration. But this, at least—as he gathers from his experts along the way, and many a mathematician would agree—is a genuine mathematical experience.

Mr. Wilkinson gives his journey the quintessential New-Yorker-super-duper-feature treatment. While he struggles with learning math, his capacious curiosity leads him to roam far and wide through the discipline’s history and philosophy. For starters, the question, “What is mathematics?” generates an illuminating riff, with answers as varied as “the theory of formal patterns,” “the poetry of logical ideas,” “the longest continuous human thought” and “only a formal game.” In this context, Mr. Wilkinson is entirely in his element, telling tales with his easily acquired, elegant erudition—especially when he ventures out to meet mathematicians.

Then he returns to The Learning and The Suffering. He suspects that math, “a species of fancy grifter,” is out to get him, trip him up; the equations seem “rigged.” Sometimes the messenger is to blame—that is, the textbooks, which he reviews with an amusing haughtiness: “When I had trouble with one textbook, I found another, but nearly all of the books were poorly written,” he complains. “In addition to leaving things out, they were careless about language, their sentences were disorderly, their thinking was frequently slipshod, and their tone was often cheerfully and irrationally impatient.”

Mr. Wilkinson might have fared a bit better had he been in possession of “Algebra the Beautiful: An Ode to Math’s Least-Loved Subject” by G. Arnell Williams, a professor of mathematics at San Juan College in New Mexico. Mr. Williams announces at the outset that his teaching is motivated by his students’ skepticism, and their emotional and conceptual struggles.

Before the revolution of symbolic algebra—with unknowns and equations conveyed by the likes of x, y and z (and often the Latin or Greek alphabet)—there was rhetorical algebra, with problems and solutions expressed by words or abbreviations of language. The medieval Indians on the subcontinent devised a color scheme to handle multiple unknowns. As the Indian mathematician Bhāskara II explained in the 1100s: One unknown “is the color black, another blue, yellow, and red. [Colors] beginning with these have been imagined by the best of teachers to be the designations of the measures of the unknowns in order to accomplish their calculation.”

“Algebra the Beautiful” is a rich paean within a workbook (of sorts), or possibly a workbook within a paean. On a number of occasions, the educational philosopher John Dewey and his book “Art as Experience” are invoked, as Mr. Williams articulates and affirms his goal, admittedly “lofty, illusory”—yet attainable, he believes: teaching algebra through fulfilling experiences and unifying ideas.

His book contains plenty of word problems and their ilk, of the sort taught in the average algebra class. (For instance: A plane travels 450 m.p.h. How far does it travel if . . . ?) In Mr. Williams’s hands, however, these algebraic entities are fetchingly deconstructed as “numerical symphonies” and “quantitative cocktails.” If you’re up for some frustration and struggle, the active pulling apart of the variables, unknowns and parameters does facilitate satisfying “Aha!” moments.

But why bother to teach algebra at all, he asks, when most people are unlikely to ever employ its contrived and often unrealistic word problems in everyday life? Mr. Williams aims to convey algebra’s transformative power—whether dealing with unknowns in a word problem or in a scientific formula—to reveal patterns, relationships, analogies and metaphors, variations and commonalities from one scenario to the next.

He likens mathematics to language, and writes: “Languages, in general, give us this wide-ranging ability to describe lots of objects and ideas with a relatively small glossary of words. Taking these words, then, in combination to form sentences—language expressions—gives us the breathtaking ability to describe nearly everything that we experience in life or are able to think about in the world around us. We seek the same in the world of numerical variations.”

Mr. Wilkinson has an “Aha!” moment of sorts when he sees a kindred connection. “I was staring at a page of equations, and it was borne in on me that mathematics is a language and an equation is a sentence. . . . I was pleased with this insight, which struck me as deep and even lyrical. It cheered me in light of the discouragements I had encountered and reinforced my sense of purpose.”

He gives cameos along the way to a pantheon of literary comrades, including Gustave Flaubert and Ernest Hemingway, Edna St. Vincent Millay and Anne Carson, together with the usual suspects of science—Kurt Gödel, George Pólya, Bertrand Russell, Eugene Wigner. Alas, there’s no Emmy Noether. (Lamentably, too, he offers no index.)

Mr. Wilkinson is always good company when he takes on the big ideas. He becomes enamored of prime numbers: Palindromic primes are the same forward and backward; naughty primes are comprised mostly of zeros; holey primes contain only numbers with holes; sexy primes are six numbers apart; beastly primes have “666” in the middle. He contemplates the infinite: Borrowing from the German mathematician and physicist Hermann Weyl, he recounts that “the goal of mathematics is the symbolic comprehension of the infinite with human, that is finite, means.” Mr. Wilkinson adds: “This makes the most precise of sciences a way of approaching the most unknowable part of our existence.”

And he delves into the debate about whether mathematics was invented or discovered. “Once I encountered Platonism”—which, in various shapes and forms, contends the latter—“I was a goner.” He was entranced. “Mathematics mostly rebuffed me, since I could perform it usually only badly, but thinking about mathematics I could do, because anyone can.”

Siobhan Roberts is the author of “Genius at Play: The Curious Mind of John Horton Conway.”

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