Training skeptics

This semester my teaching experience has been much different than last year. Whereas last year I taught a "throw-away" calculus course designed to fulfill a requirement for certain majors (but not designed to prepare students for any mathematics beyond calculus), this semester I'm teaching an honors course in multivariable calculus specifically designed to train students for higher level mathematics. Hence one of the essential parts of the course is teaching students to write proofs.

There is a certain level at which young mathematicians regard "proofs" almost as a specific area of math. They've learned algebra, trigonometry, calculus, and now they're learning "proofs." If they continue on in mathematics, they can't continue to think this way for long. Proof is not a part of mathematics. Proof is mathematics.

On the list of ways in which we teach mathematics badly in American schools, I would certainly add that we teach students that writing is not a part of mathematics. Writing is part of everything. If you can't write it, you can't communicate it. Perhaps part of the problem is that most people (even those who teach mathematics in high school and below) don't even realize mathematics needs to be communicated. Why would you need to convince someone that something is true in mathematics? Hasn't it all been figured out already? I kid you not--I would not be surprised if half the people I meet don't realize there's still being research done in mathematics. As in there's still math we don't know!

Since we teach students to do math without actually writing sentences, we also implicitly teach them that math is nothing more than a series of arcane symbolic manipulations that magically results in an answer. Many students will simply never grasp the idea that mathematics is really a set of questions that real people have asked out of genuine curiosity, and have answered purely through deductive reasoning. No magic necessary.

My students are a little sharper than that, but many of them still have a rough transition to make. They know how to get the answer, but they're totally new to actually writing mathematics. Thus if they were asked to prove that if x is a number other than 1, then 1 + x + x^2 + ... + x^n = (x^{n+1} - 1)/(x-1), they might give me an argument that looks like this:

1 + x + x^2 + ... + x^n = (x^{n+1} - 1)/(x-1)
(x-1)*(1 + x + x^2 + ... + x^n) = (x-1)*(x^{n+1} - 1)/(x-1)
(x + x^2 + ... + x^{n+1}) - (1 + x + x^2 + ... + x^n) = x^{n+1} - 1
x^{n+1} - 1 = x^{n+1} - 1

writing a check mark beside the last line. It seems like a legitimate way to argue, because the symbolic manipulations are exactly what they've been trained to do all through their previous math education, particularly when they're trying to solve equations. And the symbolic manipulations generate something that's true! So it must be right.

There is a sense in which this argument is correct. If the student were to indicate that each line is logically equivalent to the line before it, then I suppose the argument would work. But it still wouldn't feel like good style. Everyone who is trained in higher mathematics (or in logic) understands this basic fact about a good argument: you don't start with what you're trying to prove. You start with your hypothesis, then argue step by step to the conclusion.

A reasonable argument of the proposition I just mentioned might simply reverse the lines of the poorer argument I gave, but that would be rather clunky. A good argument would actually use words! It would be much easier and clearer to simply write the following:

Observe that (x-1)*(1 + x + x^2 + ... + x^n) = (x + x^2 + ... + x^{n+1}) - (1 + x + x^2 + ... + x^n) = x^{n+1} - 1. Now divide both sides of the equation by (x-1) to obtain the desired conclusion. QED

Here's what I tell my students: try to write as if you're trying to convince the most skeptical person in the world. Every line you write has to be 100% convincing. Starting with what you're trying to prove will never satisfy a skeptic, because a skeptic knows that you can get anything to be true if you just start by assuming it to be true. For instance, let's say I want to prove that -1 = 1. Well, my argument would be rather short and sweet:

-1 = 1
(-1)*(-1) = 1*1
1 = 1 (check mark!)

Since what I got in the end is clearly true, the argument must work, right? Well, of course not, because -1 does not equal 1. But that's basically how students will argue when they're first starting to write proofs in mathematics. They've never had to convince a skeptic before. All they had to do was convince their teachers, who really just wanted to see a bunch of symbolic manipulations that magically resulted in an answer. In other words, all they had to do before was computations. Now they actually have to write arguments.

I have a great deal of uneasiness about the whole process of teaching good mathematical writing. I can tell them all day what's wrong with their proofs, and they might write little notes to themselves to try to figure out what I want to see on their homework. But that's the last thing I want them to learn. I don't want them to be able to convince me. I want them to be able to convince anyone who understands the symbols being used. There's no precise way to say this, but what I'm really going for is that they would be able to convince reason itself. Somehow they have to transcend the personal motives of finding acceptance from their teacher and getting good grades. They have to develop an innate desire to critique their own arguments, and write something about which they can confidently say, This is simply irrefutable.

Of course, mathematics is never really exactly like that in the real world of research. Mistakes are made. In the rush to get results published, sometimes mathematicians have indeed overlooked important details. But members of the mathematical community are constantly attempting to hold each other to that rigorous standard of irrefutable proof. This principle is emulated in all the sciences, yet mathematics has the privilege of working by pure logic. There is no "methods" section of a research paper in mathematics.

What an interesting sort of community this is, though. We're not a community of experimentalists, who argue over the analysis of data. We're not a community of theologians or philosophers, who argue over the meanings of words or doctrines. In fact, arguments between mathematicians are rarely over the actual content of what they study. We can argue over what the best method of proof is (though such arguments can be absurd), or we can argue over what we think the solution to a problem will be (though such arguments are rendered somewhat meaningless once the problem is actually solved). (There are also plenty of arguments between mathematicians mainly concerning their own egos; in that our community is not at all unique.)

If all the theologians or philosophers in the world were asked to reach a conclusive solution to a problem in one of these disciplines, they would only succeed in dividing into several camps. Yet do the same thing for all the mathematicians in the world, and all of them, working entirely independently from one another, can still reach the same solution. There will be no disagreement on the final result. Of all the other academic disciplines, the scientific community is most similar to this; yet the scientific community routinely has to correct past theories.

Does that make the mathematics community somehow ideal, which the whole world ought to emulate? Not at all. Mathematics progresses through skepticism; society in general does not. Part of skepticism is the ability to say, "I don't know." In the case of mathematics, we can say that for quite a long time--many famous problems have taken centuries to solve. It would be absurd to advocate the kind of rigor that mathematics demands as a way to unite all people. There are plenty of decisions in life that need to be made now. There are plenty of questions of profound importance which demand incomplete answers. I believe it was an insight of Friedrich Hayek that if we only accepted what we could justify on purely rational grounds, no human institutions could exist.

In short, mathematics is a luxury sport (like everything academic). That's not to say one must come from a wealthy family to do it. But it is certainly something that society could never have developed if it had never moved beyond the point of just doing enough to survive. More than that, it fundamentally arises out of a certain kind of dissatisfaction with mere acquaintance with the world around us. Mathematics can be characterized by a strange and relentless longing to see the inherent logical connections between ideas.

That is basically what I'm trying to train my students into. It should not be surprising if some of them never get it. If this innate desire is in you, you don't really need your teacher to justify it. On the other hand, if this desire isn't in you, then I wonder if you can ever really understand more than what it is that your teacher wants to see on your homework.

The point is that if a student wants to really do mathematics, he has to be skeptical of himself, first of all. He has to understand that it doesn't really matter whether or not he has the basic gist of a problem. He has to learn to expect a kind of precision of himself that can only be based on a relentless desire to see clearly the logical connections between ideas. If one is genuine about this, it really demands a great deal of humility. It demands the ability to say, I don't know until I have truly seen. And when one has seen clearly, then one has to humbly accept that there is no denying what one has seen. Yet while this is a humbling experience, it is simultaneously empowering, as it grants the ability to prove irrefutable claims.

Even as I teach students this, I am still learning it myself. It is an endless process. Yet I have to believe there is some inherent value in it. On the other hand, whether or not mathematics has value is hardly a mathematical question.