One of the reasons I’ve always had difficulty with mathematical proofs is that they’re presented as final, finished works of art — or, to use a perhaps better metaphor, they’re presented as magic tricks, where at the beginning the magician says, “I’m going to do this one weird thing with the birds and the scarves. Grant me this one thing; you’ll see later on that it makes sense.”
So it is with proofs. In mathematical analysis you see all the time some construction like, “let ε equal some weird value”. You ask, “Why the hell would I set ε to that?” By the end of the proof, it turns out that this funny value of ε is exactly what you needed to make the proof complete in an aesthetically pleasing way. Or the proof will involve some specially constructed infinite series. And I haven’t had the intuition to construct these funny things on my own. It just takes practice, of course, but that’s another part of math phobia that many, many people — myself very much included — feel: “I’m just not good enough at math to get this, so practicing isn’t going to be worth it.” This is, of course, ultimately self-defeating. Nevertheless, many people feel the same way.
The Mathematical Experience tackles all of this head on. And I think it finally sealed the deal with me — finally established its greatness — with one chapter that tries to map out how mathematics is actually built. That is, when mathematicians are trying to find a proof of something that may or may not be provable, how do they go about doing it? They experiment. They play around. They come up with examples and counterexamples. And in one chapter, structured as a dialogue with their own students, Davis and Hersh play around with a small-scale mathematical hypothesis that nonetheless reveals a lot about how real mathematics is practiced.
The experiment starts with the observation that numbers ending in the digit 2 are divisible by 2. Are any other numbers like this? Well, it’s true of 5. Let’s call this property “magicness”. 2 and 5 are magic, for short. Are there any others? Is 4 magic? Quick counterexample: 14 ends in 4 but is not divisible by 4, so 4 is not magic. Perhaps only for the sake of completeness, let’s say that 1 is magic. Looks like 1, 2, and 5 are the only magic numbers less than 10. Now what if we look for all magic numbers, whether or not they’re less than 10? Davis and Hersh run through several beautiful pages, all in the form of a dialogue, wherein they chase this notion of magic numbers seemingly as far as it can be chased.
What’s most marvelous about this is that they’re doing mathematics as humans do it. This is one of the few times in my life when I’ve seen mathematical proofs that have felt like humans could have constructed them. To use a different metaphor: normally proofs feel to me, the mathematics student, as though I were a student in a sculpture class, plopped down in front of Michelangelo’s David. “Here,” says the teacher. “This is sculpture. Now you know.” I so rarely get a picture of mathematics the way the young Michelangelo presumably experienced sculpture: with a lot of false starts, a lot of tiny slabs of marble on the floor, a lot of chisels slammed to the floor in frustration, and the frequent feeling that he would never make a proper sculptor.
Of course Michelangelo just kept working at it. Here I’m reminded of something that Ira Glass mentioned when my partner and I saw him in Boston a while ago: you get into this business — Glass meant the business of making art, but it carries over virtually without change to mathematics or science or any other creative endeavor — because you have taste. You see a startling result in mathematics or a beautiful piece of writing, and you say, “I want to make one of those myself.” Since you’re smart, and since you have taste, you also see right away that the math or the writing that you’re currently able to produce is way down here (mime a hand down by your knees), while the thing you aspire to is way up here (reach far above your head). That gap infuriates you and often leaves you dejected.
The solution is, of course, to keep working to narrow that gap. Some people, maybe most people, give up before they’ve closed the gap. But the answer is to keep working at it.
Of course, then, the counterargument is: maybe you aren’t any good at it. Maybe you in fact don’t have it in you to ever close that gap. I knew from a pretty early age that I would never be as good at basketball as Michael Jordan, because of certain fundamental biological limitations: I wasn’t tall enough or muscular enough, and my eyesight has always made coordination difficult. Perhaps I was also missing the math gene.
I’m not a teacher; I don’t know how to reinforce the self-confidence of students who legitimately do have mathematical talent and convince them to keep striving. A couple approaches occur to me. One is to do what karate teachers in the U.S. do: give students steady recognition that they are advancing (white belt, blue belt, etc.). Another is to try different approaches to presenting proofs; among these approaches is the approach of play, or experimentation. Play and experimentation are where Davis and Hersh shine. The slogan might be “Real Mathematicians Play”.
In my experience mathematical pedagogy as she is actually practiced doesn’t feature much play. Davis and Hersh give any number of reasons for this, at least one of which hadn’t occurred to me: the professor himself or herself (and yes, I did have at least two female professors of mathematics) may be almost as scared of the material as you are. He or she may be keeping just barely ahead of it, and may be terrified to veer from the formally perfect proof that he or she has just read in the textbook. A willingness to experiment, to play, to end up at dead ends and then back into a solution, is risky. You might fail. You might show yourself to be nearly as intimidated as your students. You might lose the godlike status normally accorded to you by standing in front of the class. Whereas presenting the ironclad proof — the marble sculpture — and delivering it with utter confidence, is a certain path.
Davis and Hersh make mathematics into a human science. Of course it always has been, but I’ll be damned if I’ve seen much at all of that in my mathematical education. What I’ve always wanted out of mathematics is to be taught the intuition, which is one of the parts inside of the human that can find ways to an argument. Intuition is the thing which tells you, “I don’t really know the answer, but something tells me it’s over here.” It’s the thing that tells you, “What you’ve just said doesn’t sound like it could be right.” My experience of mathematical education has, seemingly, been long on formal proof and short on finding solutions in your gut that lead you to the formal proof. The Mathematical Experience suggests that I’m not alone on this, and that at least a couple mathematicians would like to restore the humanity to mathematics education. This book is a pure delight.