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Spivak's Calculus.

It's "just" calculus... but it's also everything else leading up to it.

It's a wonderful book, written in a very engaging style, and it shows you how mathematicians think and how they play. It shows you why we have proofs, why things go wrong, and all that had to happen before we came up with a definition of derivatives and integrals that we're happy with (and of course, all of the things we can do with our newfound definitions).

One of the biggest conceptual moments in my life was taking an intro class to complex analysis in high school (not through my school itself, this was at a nearby university that runs a weekend program for interested applicants). It's probably not a novel intellectual framework for anyone who's spent time learning math from a theoretical point of view, but the guy who taught it opened with an (admittedly ahistorical, but that wasn't the point) tour of the development of numbers, starting with the intuitive case of counting, moving on to algebra and the question of what type of number could satisfy an equation like x + 5 = 2, and so on. He wasn't taking any philosophical position re the question of discovery vs. invention, but merely inviting us to consider the particular case of operations over a set not being algebraically closed and what it looks like to extend that set to support algebraic closure in a rigorously defensible way. Reading Spivak's Calculus, with the way that it starts off its path of inquiry by showing that, beginning with the basis of a totally-ordered field, the axioms at hand don't suffice to demonstrate the existence of e.g. a number that satisfies x^2 = 2, made me feel right at home again. It's like a detective story.

Apostols Calculus was huge for me. Probably the main reason I majored in math.

It's also a beautiful book.

Yeah, great layout and typesetting. I have a copy mostly for this reason.

Second this. All Spivak's books are gems.

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