The Mathematician's Apology, if I recall, is one of the essays mentioned regularly in the qualitative understanding of Mathematics. While rhetorically some parts of the essay are quite beautiful, the whole thing doesn't really enlighten the everyday reader on what Mathematics actually is. It attempts to make Mathematics analagous to art, but does not' make full use of the comparison to achieve any technical understandings. That said, this line from the essay is beautiful: "The proof is by reductio ad absurdum, and reductio ad absurdum,
which Euclid loved so much, is one of a mathematician’s
finest weapons5
. It is a far finer gambit than any chess gambit: a
chess player may offer the sacrifice of a pawn or even a piece, but
a mathematician offers the game.
" I have not read Lockhart's Lament to the end, but the thesis it seems to be hinting at is that schools kill appreciation of education all too easily. Which raises a hell of a lot more questions for society, let alone mathematics.