Answer by Alon Amit:
In the preface toTim Gowers quotes Bertrand Russell's definition of mathematics.
"Pure mathematics is the class of all propositions of the form "p implies q", where p and q are propositions containing one or more variables, the same in the two propositions, and neither p nor q contains any constants except logical constants."
Gowers then goes on to say that the Princeton Companion is about everything that Russell's definition leaves out.
Russell's definition is, in some ways, formally correct. Mathematics is the things we can prove, described in a language that lets us express what we regard as mathematical objects, properties and relations. To Russell, those objects were sets (and only sets), and this is indeed sufficient for much of modern mathematics.
However, this definition isn't particularly helpful in understanding what mathematicians actually do, what questions interest them, what answers they are finding and why anyone should care. This is my understanding of Gowers' comment: he created the Princeton Companion in order to provide at least a partial view into what modern, pure mathematics actually is in all the ways that we can't glean from Russell's definition.
Later, Gowers points out that defining "mathematics" is notoriously difficult, and the book doesn't attempt to come up with such a definition – instead, it shows what mathematics is by surveying some of its most important concepts, features, theorems and questions.
One of the best ways to get an idea of what mathematics is is to read The Princeton Companion, or at least browse through it, or at least read the Introduction. But it's better to just read it.
To provide a general idea, we can very crudely say that mathematics consists of the following areas:
- Algebra. The study of abstract structures like rings, fields and groups; to give and idea of what those are, think about the numbers you know: they can be added and multiplied, and there are various rules and properties ("0 added to something leaves it unchanged", "multiplication is associative"). Algebraic structures keep some or all of those rules intact but let us replace the numbers by anything we could dream up. Many beautiful structures emerge, with an uncanny ability to answer questions about ordinary numbers as well as the physical world.
- Analysis. The study of limiting processes leading to concepts like derivatives, integrals and differential equations. This, too, has been generalized enormously into abstract structures that somewhat resemble the algebraic structures I mentioned, except that they tend to be infinite-dimensional.
- Geometry and Topology. The study of shapes and forms, and the ways in which they can be similar to each other or different from each other. A circle and a square are different in some ways, but they are similar, too, in a way that a circle and a ball are not. Branches of geometry and topology study those shapes under various rules of allowed manipulations, in particular the smooth and nice shapes called manifolds.
- Number Theory. The study of the natural numbers 1, 2, 3,… is, amazingly, one of the deepest and hardest domains in mathematics, and it rests on virtually everything I mentioned earlier (algebra, analysis and – yes – geometry and topology).
- Combinatorics. The study of finite structures.
- Logic. The study of the rules of mathematical reasoning.
- Set Theory. The study of sets with no structure at all, which in many ways is the study of the foundations of all of mathematics.
There are many other domains of mathematics that overlap with these ones but extend them in a great variety of ways. It's a jungle out there, but a very pretty one.
Mathematics is also a language, in that it allows us to express ideas and notions that are hard or impossible to communicate otherwise. Music is a language in precisely this sense, too. Mathematics is far more than just a language, though. It is more than a way of communicating and expressing ideas. It has theorems, truths, proven facts about things. That is something that languages simply lack. Those theorems are expressed in mathematical language, but they aren't merely that language. This is why I feel that "mathematics is a language" doesn't quite capture what math is.