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Mathematics in Economics

Posted in economics by dingodonkey on September 1, 2009

I’m a “pure” mathematician by training, meaning that the math I studied was incredibly abstract and totally divorced from application outside the realm of ideas.  My job is the other side of that coin, essentially applied mathematician in engineering and science applications.  So when it comes to thinking mathematically about various fields, I have pretty good perspective.

Murray Rothbard rejected, as Austrian economists tend to do, what he called Mathematical Economics:

The mathematical method, like so many other fallacies, has entered and dominated present-day economic thought because of the pervading epistemology of positivism. Positivism is essentially an interpretation of the methodology of physics ballooned into a general theory of knowledge for all fields.

The reasoning runs like this: Physics is the only really successful science. The “social sciences” are backward because they cannot measure, predict exactly, etc. Therefore, they must adopt the method of physics in order to become successful. And one of the keystones of physics, of course, is the use of mathematics.

The positivists tend to separate the world into the truths of physics on the one hand and “poetry” on the other; hence their use of mathematics and their scorn for verbal economics as being “literary.”

If we grant Rothbard his terminology, it’s pretty easy to see what he’s driving at.  Mathematics (and what he really means is the “applied mathematics” style of math) is misapplied in conventional economics, in an effort to bolster it up with greater scientific rigor.  He goes on to explain that the very feature of physics that makes mathematical descriptions so fitting breaks down in the comparison to economics — economics is governed axiomatically by human behavior, by motivation, whereas physics is not.

But I don’t grant Rothbard his terminology.  Like most physicists, he has mathematics all wrong.  So-called “verbal economics”, and Austrian economics in particular, seems to me far more mathematical than the “mathematical economics” of the physicists.  Beginning with a set of axioms, and rules for interpreting logically and for defining concepts, conclusions in Austrian economics are in fact logically proven, mathematical proofs in verbal form.  Just like most “pure” math is.

In economics, … we know the cause, for human action, unlike the movement of stones, is motivated. Therefore, we may build economics on the basis of axioms — such as the existence of human action and the logical implications of action — which are originally known as true.

From these axioms we can deduce step by step, therefore, laws which are also known as true. And this knowledge is absolute rather than relative precisely because the original axioms are already known.

This is the nature of mathematical proof, not of demonstrated correlation between mathematics and physical reality (as in physics).  Even his discussion of the use of calculus, with its reliance on the infinitesimal, demonstrates the point — the infinitesimal in physics is just a convenient tool for simplifying the consideration of massive discrete systems.  In reality, as in math, the infinitesimal only applies to ideas.

It’s no wonder that as a mathematician I am so strongly attracted to the Austrian school.  I find their axioms intuitive.  I find their arguments compelling.  I find their conclusions reasonable.  I find their distrust of shoddy math reassuring.  I find their impact on my wallet convincing.