https://www.newyorker.com/culture/culture-desk/teaching-myself-calculus-at-sixty-five Skip to main content The New Yorker * Newsletter To revisit this article, select My Account, then View saved stories Close Alert Sign In Search * News * Books & Culture * Fiction & Poetry * Humor & Cartoons * Magazine * Puzzles & Games * Video * Podcasts * Archive * Goings On * Shop Open Navigation Menu To revisit this article, visit My Profile, then View saved stories Close Alert The New Yorker New Yorker Favorites * Why Nouns Slow Us Down * Hemingway's Mysterious Style * My Three Fathers * The Hockey Sister Culture Desk Teaching Myself Calculus at Sixty-Five I was never a good math student, but I was determined to penetrate the mysteries of mathematics. [wilkinson-] By Alec Wilkinson July 8, 2022 * * * * * Save this story for later. Child looking perplexed while looking at a chalkboard with a subtraction problem written on it reading 10 cents minus 5... As in childhood, I was more or less serially defeated by the practice of mathematics. Whereas some people are tone-deaf, I came to wonder if I was math-deaf.Photograph by William Gottlieb / Getty * * * * * Save this story for later. Hoping to make myself smarter and then write a book about it, I began studying mathematics when I was sixty-five, which was five years ago. As a boy, I had been kicked off the math train at the algebra station, so I decided to start there and then learn geometry and calculus--three of the disciplines that the eighteenth century called pure mathematics. I passed algebra and geometry in high school by cheating, which is not a good life lesson for an adolescent, but I had never taken calculus. I didn't even know what it was. It had always seemed less a subject than a destination, a private place where the bright girls and boys shared secrets. For two years, I spent my days studying things that children study. I was returning to childhood not to recover something but to try to do things differently from the way I had done them, to try to do better and see where that led. When I would hit the shoals, I would hear a voice saying, "There is no point to this. You failed the first time, and you will fail this time, too. Trust me. I know you." After a time, my studies began to occupy two channels. One channel involved trying to learn algebra, geometry, and calculus, and the other channel involved the things they introduced me to and led me to think about. While it was humbling to be made aware that what I know is nothing compared with what I don't know, this was also enlivening for me. I am done doing mathematics, so far as I was able to, but the thinking about it and the questions it raises are ongoing. What did I learn? Among other things, that, despite mathematics being the most explicit artifact that civilization has produced, it has also provoked many speculations that are not capable of being settled. Even those figures occupying the most exalted positions in regard to these speculations can't settle them. A lifetime doesn't seem sufficient to the task. What else? That mathematics is both real and not real. Like novelists and musicians, mathematicians produce thought objects that have no presence in the physical world. (Anna Karenina is no more actual than a thought about Anna Karenina.) Like other artists, mathematicians also have the run of a world that others hardly or only rarely visit. For mathematicians, though, this territory has more rules than it does for others. Also, what is different for mathematicians is that all of them agree about the contents of that world, so far as they are acquainted with them, and all mathematicians see the same objects within it, even though the objects are notional. No one's version, so long as it is accurate, is more correct than someone else's. Parts of this world are densely inhabited, and parts are hardly settled. Parts have been visited by only a few people, and parts are unknown, like the dark places on a medieval map. The known parts are ephemeral, but also concrete for being true, and more reliable and everlasting than any object in the physical world. Two people who do not share a language or understand a word the other is saying can do mathematics with each other, silently, like a meditation. An imaginary world's being infallible is very strange. This spectral quality is bewildering, even to mathematicians. The mathematician John Conway once said, "It's quite astonishing, and I still don't understand it, despite having been a mathematician all my life. How can things be there without actually being there?" Some things I had to learn were so challenging for me that I felt lost, bewildered, and stupid. I couldn't walk away from these feelings, because they walked with me in the guise of a gloomy companion, an apparition I could shake only by working harder and, even then, often only temporarily. There were times when I felt I had declared an ambition I wasn't equipped to achieve, but I kept going. I was inspired, partly, by indignation and bruised feelings. I had it in for mathematics, for what I recalled of its self-satisfaction, its smugness, and its imperiousness. It had abused me, and I felt aggrieved. I was returning, with a half century's wisdom, to knock the smile off math's face. As in childhood, I was more or less serially defeated by the practice of mathematics. Whereas some people are tone-deaf, I came to wonder if I was math-deaf. I found pleasure, though, this time around, in reading about mathematics and thinking about the world that it introduced me to. Even visiting that world as a tourist was broadening, and changed how I think. What I wrote, a little to my surprise, is a metaphysical travel book about an imaginary landscape. The title is "A Divine Language: Learning Algebra, Geometry, and Calculus at the Edge of Old Age." Mathematics, it turns out, is suffused with mysteries, and, though I could understand only simple ones, I was intrigued. The simplest one is where numbers come from. They don't typically appear in creation stories. In "Chinese Myths," Anne Birrell writes that the Coiled Antiquity myth, which belongs to "a minority ethnic group of south-western China," describes "how numbers were created" and "provides the etiological myth of the science of mathematics," but she does not give a source for this assertion, and I have not been able to find one, so I have to accept her word for it. Gods and protectors of numbers are rare. Plato says, in "Phaedrus," that he had heard that in Egypt there was a god named Theuth, who "invented number and calculation, geometry and astronomy, not to speak of draughts and dice," but that is the only other ancient reference to gods and the creation of numbers that I have been able to find, and it isn't clear that Plato didn't make this up. Babylonian, Indian, African, and Native American myths and traditions, so far as I can determine, are about other things than numbers. The people who wrote creation stories probably thought of numbers as practical objects, like the axe and the wheel, and didn't feel that they required a mythic explanation. A number would appear to be as simple as a letter--both are serial implements--but letters are literal and numbers have esoteric attributes. If I write the letter "A," it is the letter "A." It doesn't represent something, it is something. If I write "4," though, it isn't "4" in the sense that "A" is "A." "A" is concrete, the manifestation of a sound, but "4" is a symbol, a term denoting a collection. It has no scale or identity. It can be four cats or four galaxies. I can write "4," but I can't say that it is "4," at least not all the possible embodiments of "4." I can demonstrate "4" only obliquely, by gathering four things--"AAAA," for example. Numbers were invoked by counting, a form of organization. Letters changed speech from something ephemeral to something capable of being preserved, another form of organization. By means of addition or subtraction or some other mathematical operation, one number can deliver us to another, something letters can't do, though, unless you think that adding letters to one another to spell a word is similar, which it isn't. You can't divide a word by a word, or a letter by a letter. You can't have half a letter. Or the square root of a letter. Or 3.65 per cent of a letter. (Only in mathematics, I read somewhere, is "A/B" a sensible remark.) Numbers have two primary incarnations, positive and negative, but they also have hidden attributes, such as being prime. By agreement, we can change how words are spelled, but we can't change arithmetic. We can allow "theater" or "theatre," but with "5 + 7" we can't do anything to it at all. Someone who says that human beings created the operations of arithmetic cannot say that we created the results. Two objects and two objects are always four objects. We did not say, "We are adding two objects to two objects and, for the sake of clarity, deciding that the sum will usually be four objects." On all occasions, in all universes, the sum will be four, even if the term that denotes them is not "4." It is a property of their being, an inflexible trait. "Prayer," a poem by Ivan Turgenev begins, "Whatever a man prays for, he prays for a miracle. Every prayer reduces to this: 'Great God, grant that twice two be not four.' " Numbers are a mystery enfolded into ordinary life. They surround us the way radio waves and dark matter do, and, like hurricanes and white sharks and big cats, they suggest the edges of the inapprehensible. Numbers appear to be unambiguous in that, when I write a number, I can identify properties attached to it, but, except within the context of the logic that numbers embody, I can't say for sure why there are properties. Numbers did not initially provoke wonder or reverence. They did that later, partly because they invoked notions of infinity and therefore of God and, after that, because they appeared to be a language in which nature could be expressed. And because, on examination, numbers showed themselves to be complex in ways that had nothing to do with what we thought about them. Not being able to settle on an origin story for numbers means that there is no origin story for mathematics, either. It is a question that is always modern. New Yorker Favorites * The myth of whiteness in classical sculpture. * The many lives of a Princeton impostor. * Was the American Revolution a good idea? * Freud has been debunked again and again--and yet we can't give him up. * A juicy, unknown memoir about classic Hollywood. * A Personal History by Stephen King: after an accident, learning to write again. Sign up for our daily newsletter to receive the best stories from The New Yorker. [wilkinson-] Alec Wilkinson, a staff writer, is the author of ten books, including "The Protest Singer" and "The Ice Balloon." More:MathematicsGeometryAlgebraLearningEducation The New Yorker Recommends What our staff is reading, watching, and listening to each week. E-mail address [ ] Sign up By signing up, you agree to our User Agreement and Privacy Policy & Cookie Statement. Read More [undefined] News The Weakness of the Despot By David Remnick Illustration of abstract potrait of Maria Montessori made out of Montessori style toys Books The Miseducation of Maria Montessori By Jessica Winter women sitting on a chair in front of a beach, looking at a page from a book Magazine The Case Against the Trauma Plot By Parul Sehgal Supreme Court Justice Samuel Alito at Georgetown University Law Center. News Why There Are No Women in the Constitution By Jill Lepore The New Yorker Sections * News * Books & Culture * Fiction & Poetry * Humor & Cartoons * Magazine * Crossword * Video * Podcasts * Archive * Goings On More * Customer Care * Shop The New Yorker * Buy Covers and Cartoons * Conde Nast Store * Digital Access * Newsletters * Jigsaw Puzzle * RSS * Site Map * About * Careers * Contact * F.A.Q. * Media Kit * Press * Accessibility Help * Conde Nast Spotlight (c) 2022 Conde Nast. All rights reserved. Use of this site constitutes acceptance of our User Agreement and Privacy Policy and Cookie Statement and Your California Privacy Rights. The New Yorker may earn a portion of sales from products that are purchased through our site as part of our Affiliate Partnerships with retailers. The material on this site may not be reproduced, distributed, transmitted, cached or otherwise used, except with the prior written permission of Conde Nast. Ad Choices * * * * * Do Not Sell My Personal Info