TODO: intro
The word "Theory" means at least two things. In the case of an "economic theory", it lets you make predictions about things that are uncertain now, but that will be certain later. These are predictions like: "the price of milk will increase tomrrow". In the case of a "mathematical theory," it refers to a set of consistent statements about things that do not change in time. When you say that triangles have three sides, you're not predicting that if you meet a triangle in the future, it will have three sides. Instead you're clarifying that you mean by the words "triangle" and "side". Today I want to argue that we would have better economic theories if we created mathematical theories specifically to clarify the words that the economic theories use. In geometry, these end up being words like "point" and "line". In the mathematical theories I am proposing we will clarify the meaning of words like "debt" and "value".
In order to not become overloaded with "theory", you should know that the second kind of theory I've described here (the mathematical one) is more specifically known as a "Formal System". Formal systems have parts, and before I describe how those parts might come together so tha they can be useful for economic theories, we need to understand each one. They are:
- Undefined Term
- Axiom
- Theorem
In the late 1800's a smart guy named David Hilbert noticed that you can define a point as the spot where two (non-parallel) lines intersect, and you can define a line as the thing defined by two (non-coincident) points, but that things get weird if you try do define one without the other. One of his contributions to geometry was the undefined term. He proposed that we leave "line" and "point" undefined, and then just demonstrate how those words relate to one another. These demonstrations are called axioms. The idea is that as long as we both accept the axioms which clarify the term "line", there is no need to bother with defining "line" without the word "point" and "point" without the word "line". If we reason from those axioms and in those terms (so long as we agree on which logic rules are OK to use), we'll come up with new statements that make sense in conjunction with what we started with.
The new statements are called theorems. If you start with the undefined terms:
- point
- line
And add the axioms:
- For any two points there exists a line that goes through them
- For any two nonparalell lines there exists a point where they intersect
- Three non-colinear points exist
Then you can reason your way theorems, like:
- For each line, there is a point not on it
The whole thing (undefined terms, axioms, and all the theorems that follow) is called a formal system. By the way, we say call the reasoning that gets you from axioms to a theorem a "proof". This word also has multiple meanings. If I told you that the price of milk would increase tomorrow, and you asked me to "prove it," I could wait for a day, and then show you the price of milk. This might count as proof in some conversational sort of way, but not in the formal systems sense. I'm making a big deal of the difference because it would be easy to skim this paper and leave with the idea that I can use formal systems to discover something about the future price of milk. I can't. What might be useful, though, is to sart with a formal system, and then use it as a foundation for building one or more theories that do try to make predictions.
One benefit of such a move is that it would give us a way to classify the predictive economic theories. Just like a symphony in C-major is not the same as a symphony C-sharp-major, so too is a predictive economic theory on four undefined terms not the same as a peredictive economic theory on five undefined terms. Later on I'll show how it's possible to create such things, but more importantly, I want to demonstrate that it would be a good idea to do so.
The leap from "possible" to "good idea" goes like this: Presumably economists and policy decision makers are something more than rich men building justifications for the status quo. Presumably, they want to help people q