...and I'll recommend books that no one will ever read.
There's this perennial discussion whether math is "real" or not. My personal opinion is that it isn't. Are you "really" out when you get hit with a ball in dodge ball? Does something about you change when you get hit? Not really. If the recess bell rang right then, your outness and your friends' inness, would suddenly disappear. The property of "being in" or "being out" exists only within the game. So you're "really" out as much as the game is real, which I don't think is a whole lot.
Math is the same way. Is it a "real" truth that every odd-dimensional square matrix with real entries has at least one real eigenvalue, but some even-dimensional square matrices with real entries have no real eigenvalues? Well, it's a real truth in this game of mathematics.
Why are dodge-ball properties less real than say biological properties or physical properties? A wise man once said, "Reality is that which does not go away when when you stop believing in it." If we stopped believing in physics or biology, they would still exist. Nobody would know about their existence, just like nobody knew their existence for thousands of years before science came around. Dodge ball, on the other hand, exists only as long as we believe in it.
And math is the same.
If my blog had any readers other than those who have heard this lecture from me before, they would doubtlessly be clamoring, "What do you mean? Two and two would still be four, even if nobody knew it!"
To which I respond, would the polynomials still form a ring if nobody knew it? Would Gaussian integers still have unique factorizations (up to multiplication by a unit) if nobody knew it? Would "P implies Q" be equivalent to "not P or Q" if nobody knew it? Would e^(ix)=cos(x)+isin(x) if nobody knew about it? More simply, would (-2)*(-2) still equal positive four if nobody knew it?
The book Negative Math: How Mathematical Rules Can be Positively Bent, by Alberto A. Martínez, discusses this question at length. His answer is pretty much "No." Supposedly his book uses only basic algebra and a minimum of mathematical symbols and is easily accessible to people with rudimentary mathematical knowledge. I read it. I enjoyed it. It opened my eyes. But I don't think anyone reading this blog would enjoy it. So instead of telling you all to read it, I'll just tell you one of the central points of the book: If we wanted to, we could have made the rules of mathematics different than they are. Very different. It would have made math more suitable for some purposes and less suitable for others. Our current version is pretty damn good, and we should have no regrets. But things could have been very different.
To make matters worse, I believe City College's copy of this book is missing, even though the catalog keeps on telling me to "check [the] shelf." So I can't even get my mathematically inclined friends to read the book. Yes, life is tough.