Couching them in the form of letters is a little awkward, but Ian Stewart's observations on mathematics and advice on becoming a mathematician are insightful and entertainingly presented. Samples:
"What is mathematics? In despair, some have proposed the definition 'Mathematics is what mathematicians do.' And what are mathematicians? 'People who do mathematics.' This argument is almost Platonic in its perfect circularity. But let me ask a similar question. What is a businessman? Someone who does business? Not quite. It is someone who sees opportunities for doing business when others might miss them. A mathematician is someone who sees opportunities for doing mathematics. I'm pretty sure that's right, and it pins down an important difference between mathematicians and everybody else." (p. 32)
"When you study any subject, the rate at which you can understand new material tends to accelerate the more you already know. You've learned the rules of the game, you've gotten good and playing it, so learning the next level is easier. At least it would be, except that at higher levels you set yourself higher standards. Math is like that. To perhaps an extreme degree, it builds new concepts on top of old ones. If math were a building, it would resemble a pyramid erected upside down. Built on a narrow base, the structure would tower into the clouds, each floor larger than the one below. The taller the building becomes, the more space there is to build more. That's perhaps a little too simple a description. There would be funny little excrescences protruding all over the place, twisting and turning; decorations like minarets and domes and gargoyles; stairways and secret passageways unexpectedly connecting distant rooms; diving boards suspended over dizzying voids. But the inverted pyramid would dominate. All subjects are like that to some extent, but their pyramids do not widen so rapidly, and new buildings are often put up beside existing ones. These subjects resemble cities, and if you don't like the building you are in, you can always move to another one and start afresh. Mathematics is all one thing, and moving house is not an option." (pp. 38-39)
--Stewart, Ian. Letters to a Young Mathematician. New York: Basic Books, 2006.
Link to MnCat Record