Most people think “mathematics” is the same as “arithmetic”, which is not the case. And some think “mathematics” is the same as “computation”– again not the case. There are many very fine mathematicians who are not particularly accurate at computation. Of course you need arithmetic and computation in order to do mathematics, just as you need a piano to play piano music. But the piano is not the music.

Mathematics is concerned with understanding mathematical concepts, developing general principles (theorems) about how they work, and exploring the connections among them. What are “mathematical concepts”? One of them — which everyone is familiar with — is number. Of course everyone learns counting, addition and multiplication, but there is much, much more to understand.

There are, to begin with, the distinctions between integers, rational numbers, real numbers, algebraic numbers, and transcendental numbers. There are questions concerned with the distribution of prime numbers and other special numbers. There are geometric representations of numbers, and deep questions about infinitely many numbers. Very early on it becomes clear that the real numbers are not adequate, so that we have to extend our number system to the complex numbers; and then we have to understand how those work. A simple question, with a more interesting answer than you might think, is: how might one find a square root of a complex number?

As a second example, some people are familiar with the concept of functions. There is an enormous amount to be known about functions. Concepts such as continuity, differentiability, integrability, computability, approximation, analyticity, monotonicity, bounded variation and hundreds of others have been exhaustively studied.

Every one of these concepts is of practical use in physics, engineering and many other disciplines. Yet every one of them is also an interesting and valid idea which connects in both obvious and subtle ways with all the others to form an intricate system. While all these systems are of immense practical use, each can stand on its own without reference to any usefulness it may have, and may be as beautiful as wonderful music or a gorgeous painting.

Almost all mathematics initially arises from real world problems. As some examples:

- Counting, rational numbers, negative numbers and arithmetic operations are a natural result of commerce.
- Early explorations in geometry and trigonometry (along with irrational numbers) arose out of the need to measure distances indirectly (such as in surveying), navigation, and astronomy.
- Development of calculus was propelled by studies in ballistics (how do you aim this cannon?) and astronomy (motions of the planets).

Once a concept has been introduced, mathematicians may study it for its own sake and learn much more about it. That in turn is almost always of practical use sooner or later. So on the one hand mathematics is in a constant give and take relationship with physical, biological and social sciences, and is inextricably entangled with them. And on the other hand all the concepts that have developed from these sources stand on their own without reference to outside sciences. The connections among them and the proofs used to reveal those connections can be very beautiful and elegant.

Finding those proofs, revealing those connections and building up the systems takes creativity — every bit as much as composing a symphony or painting a compelling picture.

To regard mathematics as an art as well as a science is natural and perfectly correct.