My colleague Timothy Gowers is very close to finishing a project of really immense ambition: *the Princeton Companion to Mathematics*, a gigantic book which aims to be a panorama of all of contemporary mathematics, presented at an undergraduate or even interested-amateur level. He has jokingly suggested that a good alternate title would be *Mathematics: A Very Long Introduction. * Some of the book consists of expository articles on the subfields in math — things you might take a course in, like analytic number theory, probability, or partial differential equations. Others treat notable theorems (Mostow Rigidity, Hilbert’s Nullstellensatz), notable mathematicians (charmingly alphabetized by first name), and notable applications to other fields. And some of the articles — to my mind the most ambitious of all — attempt to give some sense the nature of the mathematical project to outsiders. (“The general goals of mathematical research,” “The language and grammar of mathematics.”) The editors have made many sample articles available online (userid: Guest, pwd: PCM) — I encourage people to have a look! In particular, if you are wondering what I do all day, you can read my article on “Arithmetic Geometry.” If you want to start from the beginning of things, try Gowers himself on “Some Fundamental Mathematical Definitions” or “The Language and Grammar of Mathematics.” For an applied article, try Madhu Sudan’s “Reliable Transmission of Information.” Or if you just want inspiration, see Sir Michael Atiyah on “Advice to a Young Mathematician.”

Important biographical notes: Tim has a Fields Medal and a blog.

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It’s a notable trope of mathematical titles that all too many of them, however arcane their subject matter or the treatment of it, are labeled “Introduction to” or “Elementary” or “Basic” or some combination of these. It’s as if books and articles can only provide the written law, and for the meat of the subject you need to go directly to the oral tradition.