What Is Mathematics?

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.