The usual narrative suggests that the new mathematical tools of modern finance were like the wings that Daedalus gave Icarus. The people who put these tools to work soared too high and crashed.

In two posts, here and here, Tim Johnson notes that two government investigations (one in the UK, the other in the US) tell a different tale. People in finance used math to hide what they were doing.

One of the premises I used to take for granted was that an argument presented using math would be more precise than the corresponding argument presented using words. Under this model, words from natural language are more flexible than math. They let us refer to concepts we do not yet fully understand. They are like rough prototypes. Then as our understanding grows, we use math to give words more precise definitions and meanings. The canonical example of this process is the evolution of the concept of force.

To understand the motion of the planets, Newton and Leibniz both tried to characterize the concept that Leibniz called vis viva or vital force. Leibniz used mv^2 to define this concept. Based on an analysis of the motion of the planets, he conjectured that this quantity would be subject to a conservation law. However, this turned out to be a result that did not generalize. (For example, mv^2 is not conserved when two moving bodies with different masses collide and stick together.) So it was relabeled kinetic energy. This compound name signals that mv^2 it is part of total energy, which is the quantity that is conserved. Newton saw that the deeper concept of force came from the definition F=ma, which implied the conservation of momentum. Momentum is conserved more generally. Even with friction, momentum in a closed system does not “run down.”

Like Leibniz, I thought the behavior I observed in one special case would generalize. I assumed that because I was trying to use math to reason more precisely and to communicate more clearly, everyone would use it the same way. I knew that math, like words, could be used to confuse a reader, but I assumed that all of us who used math operated in a reputational equilibrium where obfuscating would be costly. I expected that in this equilibrium, we would see only the use of math to clarify and lend precision.

Unfortunately, I was wrong even about the equilibrium in the academic world, where mathiness is in fact used to obfuscate. In the world of for-profit finance, the return to obfuscation in communication with regulators is much higher, so there is every reason to expect that mathiness would be used liberally, particularly in mandated disclosures.

In response, regulators might want to adopt something like the two-tiered approach I’d also recommend for academic communication. Readers should take any exposition that involves the use of math seriously only if:

(1) The math is presented with the utmost clarity and simplicity.
(2) The words that describe the math are precise and accurate.

We should expect that there will be mistakes in math, just as there are mistakes in computer code. We should also expect some inaccuracies in the verbal claims about what the math says. A small number of errors of either type should not be a cause for alarm, particularly if the math is presented transparently so that readers can check the math itself and check whether it aligns with the words. In contrast, either opaque math or ambiguous verbal statements about the math should be grounds for suspicion. If someone uses math the way that Newton did, the math will be clear and the words will be precise. If someone is incapable of using math this way, they should not be allowed to use it at all.

Mathiness–exposition characterized by a systematic divergence between what the words say and what the math implies–should be rejected outright.