What Is Mathematics?
Alec Wilkinson March 2, 2021
A while ago, I got interested in mathematics, mostly because I had done so poorly at it in school. I’m being coy. I didn’t do poorly; I pretty much failed. I only passed by cheating. Anyway, I bought a copy of “Algebra for Dummies” to see whether I could improve, but it turned out that I didn’t like algebra as an adult any more than I had as a boy. Even so, I was determined to see whether I could understand why I hadn’t been able to learn it. Doing teen-age mathematics as an older person, though, was harder than I had expected it to be, and I’m not sure how long I could have kept going if I hadn’t become aware, mostly from reading books about mathematics and talking to mathematicians, that outside my overheated room at the Algebra Hotel, mathematics had a grandeur and reach that I hadn’t even suspected. I then spent more of my time trying to learn what I could of its qualities.
Mathematicians know what mathematics is but have difficulty saying it. I have heard: Mathematics is the craft of creating new knowledge from old, using deductive logic and abstraction. The theory of formal patterns. Mathematics is the study of quantity. A discipline that includes the natural numbers and plane and solid geometry. The science that draws necessary conclusions. Symbolic logic. The study of structures. The account we give of the timeless architecture of the cosmos. The poetry of logical ideas. Statements related by very strict rules of deduction. A means of seeking a deductive pathway from a set of axioms to a set of propositions or their denials. A science involving things you can’t see, whose presence is confined to the imagination. A proto-text whose existence is only postulated. A precise conceptual apparatus. The study of ideas that can be handled as if they were real things. The manipulation of the meaningless symbols of a first-order language according to explicit, syntactical rules. A field in which the properties and interactions of idealized objects are examined. The science of skillful operations with concepts and rules invented for the purpose. Conjectures, questions, intelligent guesses, and heuristic arguments about what is probably true. The longest continuous human thought. Laboriously constructed intuition. The thing that scientific ideas, as they grow toward perfection, become. An ideal reality. A story that has been written for thousands of years, is always being added to, and might never be finished. The largest coherent artifact that’s been built by civilization. Only a formal game. What mathematicians do, the way musicians do music.
Bertrand Russell said that mathematics, by its nature as an explorative art, is “the subject in which we never know what we are talking about, nor whether what we are saying is true.” Darwin tried studying mathematics with a tutor when he was nineteen and hated it, mainly from “not being able to see any meaning in the early steps in algebra.” He is supposed to have concluded that “a mathematician is a blind man in a dark room looking for a black cat which isn’t there.” In “Alice’s Adventures in Wonderland,” Lewis Carroll has the Mock Turtle say that the four operations of arithmetic (addition, subtraction, multiplication, and division) are ambition, distraction, uglification, and derision. A complicating circumstance is that mathematics, especially in its higher ranges, is hard to understand. It begins as simple, shared speech (everyone can count) and becomes specialized into dialects so arcane that some of them are spoken by only a few hundred people in the world. Other fields haven’t even been discovered yet.
No scripture is as old as mathematics is. All the other sciences are younger, most by thousands of years. More than history, mathematics is the record that humanity is keeping of itself. History can be revised or manipulated or erased or lost. Mathematics is permanent. A² + B² = C² was true before Pythagoras had his name attached to it, and will be true when the sun goes out and no one is left to think of it. It is true for any alien life that might think of it, and true whether they think of it or not. It cannot be changed. So long as there is a world with a horizontal and a vertical axis, a sky and a horizon, it is inviolable and as true as anything that can be thought.
Mathematicians live within a world that is essentially certain. The rest of us, even other scientists, live within one where what represents certainty is so-far-as-we-can-tell-this-result-occurs-almost-all-of-the-time. Because of mathematics’ insistence on proof, it can tell us, within the range of what it knows, what happens time after time.
As precise as mathematics is, it is also the most explicit language we have for the description of mysteries. Being the language of physics, it describes actual mysteries—things we can’t see clearly in the natural world but suspect are true and later confirm—and imaginary mysteries, things that exist only in the minds of mathematicians. A question is where these abstract mysteries exist, what their home range is. Some people would say that they reside in the human mind, that only the human mind has the capacity to conceive of what are called mathematical objects, meaning numbers and equations and formulas and so on—the whole glossary and apparatus of mathematics—and to bring these into being, and that such things arrive as they do because of the way our minds are structured. We are led to examine the world in a way that agrees with the tools that we have for examining it. (We see colors as we do, for example, because of how our brains are structured to receive the reflection of light from surfaces.) This is a minority view, held mainly by neuroscientists and a certain number of mathematicians disinclined toward speculation. The more widely held view is that no one knows where math resides. There is no mathematician/naturalist who can point somewhere and say, “That is where math comes from” or “Mathematics lives over there,” say, while maybe gesturing toward magnetic north and the Arctic, which I think would suit such a contrary and coldly specifying discipline.
The belief that mathematics exists somewhere else than within us, that it is discovered more than created, is called Platonism, after Plato’s belief in a non-spatiotemporal realm that is the region of the perfect forms of which the objects on earth are imperfect reproductions. By definition, the non-spatiotemporal realm is outside time and space. It is not the creation of any deity; it simply is. To say that it is eternal or that it has always existed is to make a temporal remark, which does not apply. It is the timeless nowhere that never has and never will exist anywhere but that nevertheless is. The physical world is temporal and declines; the non-spatiotemporal one is ideal and doesn’t.
A third point of view, historically and presently, for a small but not inconsequential number of mathematicians, is that the home of mathematics is in the mind of a higher being and that mathematicians are somehow engaged with Their thoughts. Georg Cantor, the creator of set theory—which in my childhood was taught as a part of the “new math”—said, “The highest perfection of God lies in the ability to create an infinite set, and its immense goodness leads Him to create it.” And the wildly inventive and self-taught mathematician Srinivasa Ramanujan, about whom the movie “The Man Who Knew Infinity” was made, in 2015, said, “An equation for me has no meaning unless it expresses a thought of God.”
In Book 7 of the Republic, Plato has Socrates say that mathematicians are people who dream that they are awake. I partly understand this, and I partly don’t.