The Economics of Proof

Proof in mathematics (really in logic) is a kind of transaction. For any theorem which says that A implies B, the proof involves a series of transactions in which I exchange the hypotheses contained in A for the conclusions contained in B.

This has occurred to me for two reasons. One is that I've been thinking about economics a bit abstractly lately, in both philosophical and theological terms. The other reason is that my adviser happens to talk about mathematics in these terms quite often.

"Nothing in life is free," she says. Continuing on, she explains to our class that "if you want higher order integrability, you have to give up differentiability," referring to her judicious application of the Sobolev embedding theorem.

I spent countless hours this week doing one calculation that was difficult precisely because it wasn't obvious what exactly I had to trade for the end result. That's where creativity comes into the picture. Sometimes just taking inventory is half the battle: seeing what you already have in a different way is the kind of insight that math demands.

(For the mathematically inclined or anyone who's curious, I was calculating the density function of the exit time of Brownian motion by taking the Fourier transform of its characteristic function using complex contour integral methods. And yes, there's something satisfying about being able to string together that many technical terms in one sentence.)

But it occurs to me that mathematics is not unique in this way. This is exactly what an economy is all about. Goods and services are merely being exchanged--in some sense nothing is created or destroyed. Yet in another sense something is being created.

What's being created in a free market economy? Before I've called it meaning, or value. "Value" is probably a bit misleading, since it has so many prior connotations. "Meaning" is closer to what I'm talking about, but it's tricky to sort out the connotations of that word, also.

It might be a little easier to ask, what's being created in mathematical or logical proof? Knowledge? Yes, and certainly people agree knowledge is valuable. But valuable to whom? Is it valuable to all people when I prove existence and uniqueness of solutions to some partial differential equations? Is knowledge gained in one corner of mathematics valuable for the whole world?

Most academics would love to have convincing answers to this question, since often we're depending on the government for research funding. I don't think the answer is straightforward, and in fact I'm a little more hesitant than most academics to say that all knowledge gained is valuable.

I've just said mathematics is about exchange. It doesn't make sense to carry out some random series of exchanges and call that valuable to humankind. Part of what makes mathematics real work is that there's a result in mind, and I must figure out what I can give in exchange for that result. Presumably that result would actually be desired by people other than myself.

Perhaps this philosophical attitude has led me into a segment of mathematics that is more "applied." But on the other hand, I really do think that "application" is all relative. When I explained my research to a friend once, I started by illustrating that it could be applied to, say, stabilizing a helicopter. After I was finished, he asked, "So, what applications are there for this kind of thing?" I suppose if you don't care about helicopters, none at all!

The point for me is that people matter. So does the way people treat this world that we've been given. We humans have no ability to create or destroy matter; we are not gods. But we have the nearly god-like ability to create solutions to problems, and to create systems that work.

Where does that creative power come from? It comes precisely from the ability to take inventory of what we already possess, but in a new way. We have to figure out what we have that can be exchanged for what we want. This is no small task! Otherwise, mathematics would be easy.

And so it becomes clearer why "free" market does not mean easy success. In some ways, realizing the link between logical proof and free exchange in an economy makes me worry for our economy, for the simple reason that efficient exchange is not an easy task. It requires education, it requires discipline, and even then it doesn't always turn out so well.

But now I realize that I've traded one hour of my time for the opportunity to organize a few of my scattered thoughts, and unfortunately, I don't have any time left at the moment to exchange! This will be an interesting idea to return to later.