I'm currently re-reading (and re-relishing) mathematician William Byers' "How Mathematicians Think" -- for me, the richest, most interesting mathematical read in my entire library (though I recognize it won't be everyone's 'cup-o-tea' --- if you have little interest in the philosophical underpinnings of math, or if talk of Godel and Cantor tends to send you screaming out of the room, tearing your hair out by the roots.... well, then, this may not be a volume for you!)... but it is a great and accessible, thought-provoking treatment of ideas/issues underlying mathematical thinking (centering not around logic, but around ambiguity, contradiction, paradox, uncertainty, and creativity as being at the core of math).
But I'm also re-reading and refreshing myself on Byers because he has a new book out, "The Blind Spot: Science and the Crisis of Uncertainty," wherein he tackles many of the same concerns once again.
I can't imagine that his new book will surpass or even equal "How Mathematicians Think," but still looking forward to it. Additionally, it has a Facebook page devoted to it here:
Below, a passage from the older work, "How Mathematicians Think":
"Those who describe mathematics as an exercise in pure logic are blind to the living core of mathematics -- the mathematical idea -- that one could call the fundamental principle of mathematics. Everything else, logical structure included, is secondary...
"Now the mathematician can sense the presence of an idea even when the idea has not yet emerged... It occurs when you are looking at a certain mathematical situation and it occurs to you that 'something is going on here.' The data that you are observing are not random, there is some coherence, some pattern, and some reason for the pattern. Something systematic is going on, but at the time you are not aware of what it might be. This is a tangible feeling...
"The feeling that 'something is going on here' can even be brought on by a single fact, a single number. A case in point happened in 1978, when my colleague John McKay noticed that 196884 = 196883+1. What, one might ask, is so important about the fact that some specific integer is one larger than its predecessor? The answer is that these are not just any two numbers. They are significant mathematical constants that are found in two different areas of mathematics. The first arises in the context of the mathematical theory of modular forms. The second arises in the context of the irreducible representations of a finite simple group called the Monster. McKay intuitively realized that the relationship between these two constants could not be a coincidence, and his observation started a line of mathematical inquiry that led to a series of conjectures that go by the name, 'monstrous moonshine.' The main conjecture in this theory was finally proved by Fields Medal winner Richard E. Borcherds. Thus the initial observation plus the recognition that such an unusual coincidence must have some deep mathematical significance led to the development of a whole area of significant mathematical research...
"McKay noticed that there was something going on... Understanding what is going on is an ongoing process -- the very heart of mathematics."