Finally, here's a very different kind of applied mathematics. Structure and Interpretation of Classical Mechanics is basically an expository account of classical mechanics. But it takes a few unusual stances. First of all, the authors use "correct" notation for partial derivatives, avoiding the conceptual mess that the Leibnizian notation often brings with it. They do this, they claim, not only because it is right and the other notation is wrong (the authors quote a strong argument to this effect from Spivak's Calculus on Manifolds), but also because this notation avoids critical conceptual traps allowed by less formal notation. Second, the authors use the Scheme programming language to introduce a computational aspect to their exposition. Again, they do this not just because it allows them to make pictures of their dynamical systems, but because making the mathematics sufficiently precise to be attacked computationally is itself a valuable exercise that will help students understand it better. As they say, "Our requirement that our mathematical notations be explicit and precise enough that they can be interpreted automatically, as by a computer, is very effective in uncovering puns and flaws in reasoning." Third, they focus on understanding motion rather than on deriving equations, and finally, they include many recent results into the text. The book, which comes out of a course taught at MIT, is one that mathematicians may want to read, not just for the mathematical content, but also for the pedagogical ideas it reflects. Maybe we can learn something here.
Fernando Q. Gouvêa (firstname.lastname@example.org) is the editor of FOCUS and MAA Online. He teaches both "History of Mathematics" and "Number Theory", among others, at Colby College. He is a number theorist whose main research focus is on p-adic modular forms and Galois representations.