Teaching Myself Calculus at Sixty-Five

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.

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