Humanity’s Strangest Language: On the Joys of Translating Math

I once asked an auditorium of undergraduates to recall their earliest memories of math. One of them shared a scene so peculiar—and yet so universal—that it burrowed deep into my unconscious, to the point where it has begun to feel like a memory of my own.

At age five, this student was assigned worksheets of addition problems. Trouble was, she didn’t know how to read the funny symbols on the pages, the 2’s and +’s and such. No one had ever taught her. Too intimidated to ask, she found a work-around, memorizing each sum not as a fact about numbers, but as an arbitrary rule about shapes.

For example, 8 + 1 = 9 was not the statement that 9 is one more than 8, but a coded set of instructions: if you are shown two stacked circles (8), followed by a cross (+), a vertical line (1), and a pair of horizontal lines (=), then you must fill in the blank space provided by writing a circle with a downward-curling tail (9). With painstaking diligence, she taught herself dozens of rules like this, each as baroque and pointless as the last. It was mathematics by way of Kafka.

Few people learn 8 + 1 = 9 this way. But sooner or later, almost every math student suffers a similar sense of confusion and resorts to similarly desperate work-arounds. Whether in preschool, middle school, or grad school, befuddlement eventually descends, and everyone seemed to say, terribly and perhaps fatally wrong, with the way that we teach math.

What, exactly, was wrong? Ah, that’s where the consensus dissolved. I’ve spent the last 15 years trying to puzzle it out.

One common complaint is a lack of “real-world applications.” Math is too abstract, too obscure, too far up its own ivory tower. As the timeless refrain goes: “When am I going to use this?” Many textbook authors take this grievance to heart. For example, they’ll turn a question about a quadratic (boring!) into one about a company whose revenue is, in defiance of all rhyme and reason, a quadratic (so real, so practical!). Other educators reject the premise of the “real-world” complaint. No one asks when they’ll “use” music or literature, do they? So why not follow the wisdom of Albert Einstein, and embrace math as “the poetry of logical ideas”?

No matter how we respond to the “real-world” concern, I suspect we’re taking it too literally. When students ask for usefulness, they don’t mean a sense of practicality. They mean a sense of purpose. “When will I use this?” means something like “What are we doing here?” or “Why does this stuff matter?” or “What does it all mean?”

They’re not saying: “Describe the distant date when these exercises will benefit my bank account.” Nor: “Explain the unlikely way in which these exercises might benefit my soul.” The question is more like: “Tell me, here and now, what exactly are these exercises?”

Math is more than a collection of ideas. It’s a specialized way of talking about those ideas. What the students are asking for, without realizing it, is help learning humanity’s strangest language.

So what does it mean to say that math is a language?

Math begins with numbers. Though numbers and words differ in a few notable ways, both are systems for labeling the world. Numbers, like words, let us reduce a complex experience (say, a lakeside stroll) to something far simpler. In the case of words, a description (“There were lots of expensive dogs”); in the case of numbers, a quantity (“3 miles”).

After numbers come calculations. Calculations generate new numbers from old numbers, which is to say, new knowledge from old mathematics becomes, in the words of mathematician David Hilbert, a game of “meaningless marks on paper.”

We’ve all been there. There’s a symbol you don’t recognize, a step you can’t follow, so you ask what the mess of marks is supposed to mean. The reply is a stream of gibberish. So you ask what that means. The reply is a torrent of nonsense. This continues, frustration building on all sides, until at last you nod, smile, and say, “Oh, yes. Thank you. That clears everything up.” Then, with all sense of meaning thwarted, you set about the grueling work of memorizing which shapes to write in what order.

Math, we like to say, is a language. (A “universal language,” even.) But if a language brings people together, then why does math make us feel so alone?

I am a professional apologist for mathematics. I use “apologist” both in the classical sense (an advocate; a proponent; an expounder of a worldview) and in the modern sense (someone doing public relations for a widely despised client). What brought me to this career—the reason I became a math teacher—was the vague and overblown conviction that mathematics needed my help. Something was wrong, knowledge. For example, if our 3-mile lake is a rough circle, then I can calculate the distance across the lake to be approximately 1 mile.

So far, so good. But then comes algebra.

Algebra, like literature or philosophy, takes a step back from the everyday world. We leave behind particular numbers (177) and particular calculations (177 ÷ 3) to study the nature of calculation itself. Algebra opens up new possibilities: streamlining computations, rearranging steps, comparing approaches, and so on. This requires a rich grammar, with a distinctive system of noun phrases and a small stable of workhorse verbs. Most notably, specific numbers like 3 give way to abstract placeholders like x. This leap of faith from the concrete 3 to the general x marks the dawn of a whole new language— and for many people, the dusk of understanding.

This little book has a lofty aim: to teach you the language of math. We’ll build from the abstract nouns of number to the active verbs of calculation to the nuanced grammar of algebra. A few cartoon-illustrated pages cannot teach the entire language, of course, but I hope they can give you a running start.

What I am proposing is a little out of the ordinary. When mathematicians write for a general audience, we tend to celebrate the subject’s ideas and applications, not the language in which they are expressed. Often we abandon the language altogether, translating the equations (as best we can) into English prose.

This book takes a more brambly and less trammeled path. It is not literature in translation, but an attempt to animate the beautiful and austere language that makes such literature possible.

A classic riddle asks whether math was discovered or invented. Is math out there in the fabric of nature? Or is it a tool we created for the sake of examining nature? Is mathematics the atom or the microscope?

My reply, of course, is both. Mathematics is an invention wrapped around a discovery; it is a house built around a tree. The house is a language so cleverly crafted that it feels like a work of nature. The tree is a discovery so magical in its architecture that it feels like a work of design. Mathematics is the atom and the microscope, unified so seamlessly it can be hard to tell where discovery ends and invention begins.

This intertwining of invention and discovery, of language and idea, is one reason math can be so hard to learn. To grasp the ideas, you must first learn the language, but the language makes no sense except as an expression of the ideas.

It was never my plan to make a career as a math apologist. If I have been guided toward math, then I was not like a Greek hero whom the gods usher to his fate but more like a confused tourist whom the locals usher out of traffic.

Still, here in the tree house, watching the light through the leaves, I can’t help wishing that everyone could stand right where I’m standing. I hope this little book can bring you here.

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From Math for English Majors: A Human Take on the Universal Language by Ben Orlin. Copyright © 2024. Available from Black Dog & Leventhal, an imprint of Hachette Book Group, Inc.