What is math, and why should we use it in economics?

In my last post, I pointed out that the Nobel Prize-winning work of Lloyd Shapley and Al Roth, makes heavy use of mathematics, and indeed would be completely impossible without math. This, I said, is evidence against the idea that economics doesn't need (or shouldn't use) math.

But then some commenters asked me: What do you mean by "math"? And I thought that was an interesting question.

There is no "correct" definition of the word "math", any more than there is a correct definition of the word "art", or the word "love". There are many different definitions, all of which are drawn from similar connotations; in other words, people look at a bunch of things, say "This is math, and that is math", and then try to distill and formalize the similarities between the things that seem like math. For example, the definition I tended to like in college was called the "formalist" definition: 

"Mathematics is the manipulation of the symbols of a language according to explicit, syntactical rules."

Basically, this just means "math" = "logic". Philosophically, I'm fine with that. It's an expansive definition. But it's not very helpful when talking about economic methods, since it includes lots of stuff that people wouldn't normally call "math".

So what do I think is a useful definition? When it comes to scientific methodology, I think of "math" as basically being the same thing as "precision of meaning." This working definition is not a yes-or-no sort of thing; it's a sliding scale. Methods can be more math-y or less. 

So what do I mean by "precision of meaning"? Basically, something with a precise meaning has fewer alternative things that it could mean. For example, compare the two scientific propositions:

1. If you push something, it will push you back.

2. Momentum is conserved.

The second statement has a more precise meaning than the first. For example, the first statement could mean "If I push on something with a force of 5 Newtons, it will push on me with a force of 5 Newtons in the exact opposite direction that I pushed." Or, it could just as easily mean "If I push on something with a force of 5 Newtons, it will push on me with a force of anywhere between 1 to 1,000,000 Newtons, in a direction 15 degrees east of the direction I pushed." But the second statement can only mean the first of those two things, not the second. 

Therefore, I would say that the second statement is more mathematical than the first. Note that both of these statements are logical statements; for example, you can apply the rules of first-order logic to either statement to rule out the situation where I push something and it doesn't push me back at all. By the formalist definition, we can do "math" with either statement. But my "precision" definition makes a distinction between the two.

So by this definition, are probabilistic statements less mathy than deterministic ones? No, as long as they are explicit about the fact that they are probabilistic statements.

Are qualitative statements less mathy than quantitative statements? Not necessarily ("The sign of the first derivative is positive" is qualitative but is precise in its meaning), but in practice, this often tends to be the case. Quantitative statements must be precise, while qualitative statements may or may not be. This is just due to differences in the languages we use for expressing qualitative and quantitative statements. And this tendency is why people usually think math is about numbers and/or symbols that stand for numbers.

Incidentally, I noticed that Lloyd Shapley's definition of "math" is pretty similar to my own:

What, then, to raise the old question once more, is mathematics? The answer, it appears, is that any argument which is carried out with sufficient precision is mathematical, and the reason that your friends and ours cannot understand mathematics is not because they have no head for figures, but because they are unable [or unwilling, DRH] to achieve the degree of concentration required to follow a moderately involved sequence of inferences. This observation will hardly be news to those engaged in the teaching of mathematics, but it may not be so readily accepted by people outside of the profession. For them the foregoing may serve as a useful illustration.

So there you go. Great minds think alike...and mine occasionally happens to stumble to the same conclusions.

So why should we use math in economics? Well, I can think of a number of reasons:

1. We may want to make precise predictions about what will happen in a market.

2. We may want to make precise predictions about the conditions under which things will happen in a market.

3. Precise statements often help resolve debates, avoiding the phenomenon of "talking past each other".

4. Precise statements often lead to unintuitive but logically inescapable results.

5. It is usually easier to check sets of precise statements for logical inconsistencies.

I think all of these reasons are good reasons sometimes and bad reasons sometimes (note how imprecise of a statement that is!). I have no hard-and-fast rule about how much precision to use, and when. But I do know that if you tried to implement a Shapley-Roth matching algorithm without mathematically precise statements about what happens when, it would be hopeless. 

And I also know that in the blogosphere, many debates go on and on without being resolved, when both sides are really just talking past each other. Egos get bruised, grudges develop, and understanding is not advanced, even when the different sides' positions are not mutually incompatible or even that far off. That's why, when debates get really long and confusing, I think it's time to whip out the math, define terms, and get really precise. (By the way: In my experience, defining terms is really the critical piece of this. It's very very hard to make imprecise statements when all your words are precisely defined!)

So are there times when we should use less math in economics? Sure. Sometimes we understand a phenomenon so little that imprecise statements are more valuable than precise ones; precise formulations, if we believe them, give us the illusion of understanding, while imprecise statements, by pointing us in many directions at once, give us a menu of options for seeking the truth. And I also suspect (without proof) that some authors use excessive precision as a form of obscurantism, cloaking simple ideas in daunting reams of equations, or performing byzantine manipulations of simplistic assumptions, in order to deter outsiders from entering their hyper-specialized sub-field and criticizing their work. 

But these are cases in which the purpose of imprecision is to lead us to greater future truth. And that truth, if it is found, will certainly be expressed with great precision - i.e., if there is an economic theory that really works, it's going to use some math. The only time not to use math in econ is when we haven't found the right math yet.

And in practice, I find that a few of the people calling for less math in economics (You know who you are!) don't seem to have any such goal in mind. There are a few people out there who would rather econ stay imprecise forever - so that nobody will ever be proved wrong or right, and we can let a million flowers bloom, and everyone's scholarly opinion about the economy will be equally valid. Paul Krugman discusses these folks when he says:

[Some people] claim to reject neoclassical economics, but their alternative is not an alternative model but a lot of verbiage; they talk at the economy, and imagine that by so doing they achieve a higher level of sophistication and realism than economists who try to express their ideas in terms of little models. 

And they’re kidding themselves; all they’ve done is hide their implicit models and prejudices behind a dust cloud.

Agreed. Math is not always the most appropriate tool in economics. But the more real successes economics achieves, the more good math it will use.

Update: And here is a useful reminder that the things people call "math" don't always meet my definition...computer-generated gibberish was accepted for publication in a math journal. Gibberish, of course, has no precision of meaning at all.

Update 2: Alex Marsh has a good post that discusses the pitfalls of using math in economics. The main pitfall he identifies is that people start to believe in their own math because it's simple. Marsh is absolutely right. Making simplifications is a necessary evil, and when people do it, sometimes they forget - or decide not to believe - that the things they left out of the model still exist. Believing that your own oversimplifactions are the Laws of the Universe is easy, seductive, and deadly. Only empiricism - the relentless insistence on matching models to real-world data - can provide an effective check on this tendency.