This is how Cédric Villani puts it [Birth of a theorem, p.38]:

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Ask a mathematician the average of a bunch of numbers, and he will likely respond: what average? Geometric, arithmetic, truncated, weighted (with what weight?)… There are an infinity of averages. 350 years ago, when mathematical probabilities were being invented, marking the beginning of formal economics, this question possibly only had one answer. But since then we have refined our concepts, asked new questions, changed entirely the context. What’s the average return on this investment? Average… what average?

In the new context, let’s say in pricing financial derivatives or in macroeconomic policy, the old answers are invalid — their context was betting insignificant stakes in a friendly game of dice. A dollar, maybe — not a million, not a house, not the oil price, not GDP. Given the youth of probability theory, our concepts are bound to continue evolving.

Mathematical truth is concept-dependent

This brings to mind “Proofs and refutations,” the classic text by Imre Lakatos. Since you ask: yes, the title alludes to Popper’s “Conjectures and refutations“, and yes, Lakatos meant to imply that mathematics is just as tentative, just as malleable and alive, as science. Subject to never-ending revision and creative extension.

Lakatos discusses the dynamics of mathematical concept formation, of proof and disproof. He uses the example of Euler’s polyhedral formula V-E+F=2 that relates the numbers of vertices V, edges E, and Faces F of a polyhedron (a cube or pyramid or diamond). When Euler first wrote down his formula in 1758 (or 1750), it was “correct” but as we learned more about polyhedra, the truth of the statement went through numerous revisions. It was proved to be true, proved to be false, true, false…

A tetrahedron satisfies Euler’s original formula, V-E+F=2.

No error had been committed — instead, our understanding of polyhedra changed. We asked different questions. The meanings of words changed, arguments ensued, were settled, compromises found. For example, no one initially considered that a polyhedron could have holes, or exist in 12 dimensions. Euler imagined pyramids and cubes and so on — the question whether holes are allowed had not been asked. The implicit assumption was that there are no holes, but why not? When holes were introduced, it turned out that Euler’s formula was false, or was it true in the special case of no-hole polyhedra? Sure enough, once that was settled some smart Alec wanted to know about 12 dimensions…

A picture-frame-type polyhedron does not satisfy Euler’s original formula. I’ll leave the counting of vertices, edges, and faces to you (the Euler characteristic is zero).

Lakatos’ work — as much as Gödel’s? — limits the epistemological status of mathematical proof. Something has been proven to be true. So what? The proven statement has meaning only within its context of agreed-upon mathematical concepts (and we haven’t even started to translate to physical reality!). A mathematical proof, as Reuben Hersh argues convincingly in “Experiencing mathematics,” is an argument that convinces your mathematician colleagues. For the alert reader: has he proved this?

Which norm is right depends on what it’s used for

Back to those norms Cédric Villani talks about, and back to economics. Formal economics, in one way or another, is about evaluating gambles, and that’s precisely like comparing the pluviometry of Brest with that of Bordeaux — when comparing distributions, or other high-dimensional objects (vectors, functions), we must ask: what do we care about? Then select a mathematical object that reflects that. It may be a norm — perhaps the maximum, L^\infty. For instance, I have to inhale at least about once a minute, so the L^\infty norm of the intervals between opportunities to take a  breath is quite important to me.

How do we set this up? Consider the factors r_t by which your wealth changes on each day t. Assuming multiplicative dynamics, these will be instances of a stationary random variable. We’ll think of them as the elements of a vector. We assume that they can’t be negative, r_t\geq 0 — that doesn’t mean you can’t lose money, it just means that your wealth can’t become negative if it’s positive right now (real wealth can do that).

Maximizing the expectation value of your wealth is equivalent to maximizing the L^1 norm of this vector. But that’s just not an interesting norm — as Villani says: “there are many, many others.”

Our claim is that economic agents care about how they will do over time if they follow some behavioral protocol. That has nothing to do with expectation values or the L^1 norm. As a first approximation, I’m interested in the time-average growth rate. To compute that, I must find out the dynamics of repetition of my gamble. If the dynamic is multiplicative, then the relevant quantity is the geometric mean of the elements of our vector (aside: because multiplicative dynamics is often quite a good model, the Kelly criterion is so important). The geometric mean is the limit of the generalized p-mean (in our case the L^p norm divided by the number of days) as p\to 0.

Here’s a challenge: pick some actual norm and ask yourself what dynamic it implies. Come again? I mean: under what dynamic does the maximization of some norm imply optimal long-term growth. The answer goes beyond Kelly, Whitworth (who arrived at Kelly’s famous 1956 result in 1872), and beyond the model of geometric Brownian motion. A trivial example (the L^1 norm) is given in “Evaluating gambles…”.

It’s been fun writing this post. It will be useful beyond hedonism if it helps erode the notion that mathematical knowledge can be applied mechanically without understanding. Let me know if you work out the dynamics corresponding to some norm other than L^1.


p.s.: many thanks to Dan Crisan who held up “Birth of a theorem” at Cédric Villani’s Imperial College Mathematics Colloquium the other week. I’m half-way through. It’s a different and very valuable approach to popular science writing. Highly recommended.

p.p.s.: Finished the book now. Even more recommended!

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