Observations of a PhD Student: “Intuition” vs Mathematical Theory
A PhD student explores the use of math in graduate level economics courses.
By CobPD
As I mentioned in the last post of this series, mainstream economics is almost pure math. It’s very easy to forget that I’m actually in an economics course when the slides and lectures are in purely mathematical notation. However, every once in a while, the professor will step away from math to explain what we’re observing in a more qualitative, real sense, or a student will ask a question about the economic meaning of some particular assumption or result. When they do, they almost always call this analysis “intuition”. Logical analysis about real things in the world is “intuition”. Math is the actual theory and the proof.
I can sort of understand where this viewpoint comes from. Intuition is typically only a vague idea without precise underpinnings. Rigorous definition-based math can show those ideas in a much more explicit and concrete procedure, so it’s a certain proof.
However, in order to do all this math in the first place, they have to make a myriad of assumptions which introduce their own vagueness and uncertainties. For example, in math class we derived LeChatner’s Principle which states that a firm’s stock of capital will be more responsive to changes in wage rates in the long run (LR) than in the short run (SR), assuming that SR capital stock is already approximately equal to the LR capital stock.
There should already be a red flag here. What is the meaning between the short run and the long run when we’re playing around with equations of instantaneous change? Taking this distinction for granted, how could we possibly know if the SR capital stock is approximately close to the LR capital stock? More fundamentally, SR capital stock being equal to the LR capital stock implies an economy of no change (the unrealistic evenly rotating economy), so how could there even be a concept of a change in the capital stock in response to a change in wage rates? And finally, the model is assuming extreme homogeneity and continuity. The capital stock is made up of smooze and the wage rate is the single price of smooze_2. This completely ignores the differing degrees of specificity of various capital goods. This will inherently change each of the goods’ responsiveness to wage rates. Very few goods are truly continuous in nature. Most capital goods cannot be divided into an arbitrarily small unit but must be purchased as a whole. And even when a good is physically continuous, humans always act only in discrete terms.
All of these problems, however, may be marked up to “approximation error.” Academic economists will often admit that their models are unrealistic in a pure sense. They create simplified models in order to develop a better understanding and to have approximate applications. So as long as the inputs are a reasonable approximation of the real causes, it’s likely still worthwhile. But this defense seems to give up the entire argument for using mathematics in theory to begin with. Isn't math supposed to bring precision and certainty to our proofs? Instead, we’re forced to do all this hand-waving and pretending when it comes to fitting our economic concepts into the mathematical mold. However, the problem with the mathematization of theory runs much deeper than approximation error.
To return to LeChatner’s Principle, the proof, which is just a constrained optimization problem, runs into a problem where we need to take the derivative of a function which itself depends on another function. This is not an ordinary chain-rule differentiation. However, the problem is solved with a clever incorporation of the Envelope Theorem which allows us to pretend that the function is not dependent on another function, and to just treat the function as another given parameter. What does this mean in real terms? Absolutely nothing. But thank goodness the production function is twice differentiable so that we can apply the Envelope Theorem! The entire proof relies on a math trick which is only allowed because we simply assumed the requirements for the Envelope Theorem to be true.
This is the more fundamental problem with the mathematization of theory. This proof does not fall short of reality simply due to the accumulation of approximation errors. The proof, in fact, has no meaning in reality at all because it relies on an assumption which was precisely designed simply for the sake of forcing a mathematical framework. A lot of the theorems and proofs I’ve come across so far in my studies have had similar errors.
Another problem that pervades mainstream economic theory is that equations necessarily imply instantaneous and simultaneous actions. Whether the model is in discrete time or continuous time, an equation implies that people respond to changes in an infinitely-small unit of time. While it may be argued that this is more of an approximation error, Austrians who have studied the concepts of time and knowledge should understand how fundamental this is to economics. Instantaneous and simultaneous reactions between individuals have no meaning at all in the realm of real human action, and thus is a fundamental problem with mathematical theorizing.
So the theoretical proofs are no longer relevant to “intuition” anymore. The proofs are a completely different, self-contained analysis. It’s self-contained because they force the assumptions to be mathematical from the outset, and from there it’s just a matter of finding creative math tricks. Whenever we force facts of reality into our own preferred mathematical framework, it’s no surprise that we can come up with counterintuitive mathematical results.
These theorems and models are used on the grounds that they bring precision to economic discussions. Intuition, so it goes, is helpful for thinking about economics on the surface, but mathematics is necessary as a sure proof. However, as I argued in this post, the mathematical proofs do not bring certainty to our subject, they bring only a false confidence.
Fortunately, there is still a way for us to have precision and exactness in our economic discussions. The intuition-math paradigm completely overlooks the fact that verbal analysis can be just as formalized in a rigorously-defined system with concrete procedures. Rothbard’s “Man, Economy, and State” is a great example. And this should come as no surprise since math is just logic. If the purpose of the math is to concretize the definitions and procedures that we use in discussion, we can still get that through logical argumentation as the Austrians do, without having to make our analysis meaningless by forcing our assumptions into a mathematical framework.
The mainstream approach to economics takes intuition as vaguely useful guidelines while math is the real theory. I take the opposite view. Systematic logical analysis is the real theory of economics, while mathematics is more of an art that can sometimes be useful in moderate applications.
Other Articles Like This One
The Best Books for Learning Economics
How Economics Became a Mathematical Science
Discover New Opportunities
Click here for the Austrian Economics Discord Server.
Click here for the Austrian Economics Discord YouTube Channel.
Well said.