Philosophy of Mathematics: An Introduction to a World of Proofs and Pictures

I've always had trouble with the idea that mathematicians discover things, as opposed to inventing them. You see, if you discover something, the implication is that that something is, in some sense, out there. But where would mathematical entities reside, if not inside human brains and thought processes? I must say, reading this book has if not changed my mind at least made me seriously question my positions - which is really what you want from any good book. Brown's treatment is relatively accessible, but of course you will be in for a good amount of philosophy, and some not so easily digestible math. Still, the attentive reader can get the gist of the arguments without having to follow every proof presented by the author. I am a little less convinced, though still intrigued, by Brown's claim that pictures can - in some circumstances - do the work of formal proofs. Then again, that notion does appeal to my generally pluralistic attitude about methods of inquiry, and it does fit very well with the author's overall contention that mathematics is - surprisingly - a lot more like the natural sciences than one might think at first. Of course, all of this leaves completely unanswered the underlying question of the ontological status of mathematical objects. Oh well, can't get everything out of a single book.