Why math?

The Austrian Economists and Dani Rodrik have been talking about the use of mathematical formalism in their field. I think Rodrik gets it right:

In other words, we use math not because we are smart, but because we are not smart enough.

The low level of mathematics that I am familiar with, calculus, linear algebra, statistics and probability, is an aid to clear thinking, not an enabler of scientific obscurantism. Most people of some intelligence can understand the logic behind calculus and linear algebra, and one doesn't need to derive proofs from first principles to obtain greater insight through formalism. When it comes to discourse there is often a criticism that mathematical formalism gives one the illusion of greater precision than is truly warranted. And that is certainly often so, but the bigger point is that it allows communication to be more precise so that a common algebraic currency exists to trace information transactions. In purely verbal arguments it is often easy to disavow one's argument by assigning a different weight to a prior assertion up the chain of arguments; e.g., "What I really meant was this...." A lack of notational precision often means that you never have to say you were wrong, you were always just misinterpreted. It's like throwing out an equation where the coefficients of the variables aren't stipulated, but have to be inferred. If your interlocutor boxes you into a corner you can evade an admission of defeat simply be asserting that they assigned the wrong values to the coefficients and were arguing against a straw man.

This doesn't mean that there is a problem with the excessive formalism. Certainly in some fields there is an issue, when it comes to the agreed upon consensus in a discipline a restatement of the verbal argument is often sufficient to set the terms of the discussion. A fancy mathematical display adds ego, not value. That being said, an abuse of mathematics does not mean that it lacks a place at the table. Good formalism can sharpen and clarify any disagreement so that the axiomatic differences can be highlighted. Ultimately it is an economical strategy because it saves a great deal of time that might be allocated toward futile verbal exchanges which might sail past each other because of axiomatic differences.

Note: Generally in my posts I don't go beyond the simplest of algebra. That being said, compare and contrast the compactness of writing the Hardy-Weinberg Equilibrium in equation form to describing it in several sentences where you actually convey the central insights.