Нассим Талеб - The Black Swan. The Impact of the Highly Improbable

To show how endemic the problem of misusing the Gaussian is, and how dangerous it can be, consider a (dull) book called Catastrophe by Judge Richard Posner, a prolific writer. Posner bemoans civil servants’ misunderstandings of randomness and recommends, among other things, that government policy makers learn statistics … from economists. Judge Posner appears to be trying to foment catastrophes. Yet, in spite of being one of those people who should spend more time reading and less time writing, he can be an insightful, deep, and original thinker; like many people, he just isn’t aware of the distinction between Mediocristan and Extremistan, and he believes that statistics is a “science,” never a fraud. If you run into him, please make him aware of these things.

QUÉTELET’S AVERAGE MONSTER

This monstrosity called the Gaussian bell curve is not Gauss’s doing. Although he worked on it, he was a mathematician dealing with a theoretical point, not making claims about the structure of reality like statistical-minded scientists. G. H. Hardy wrote in “A Mathematician’s Apology”:

The “real” mathematics of the “real” mathematicians, the mathematics of Fermat and Euler and Gauss and Abel and Riemann, is almost wholly “useless” (and this is as true of “applied” as of “pure” mathematics).

As I mentioned earlier, the bell curve was mainly the concoction of a gambler, Abraham de Moivre (1667–1754), a French Calvinist refugee who spent much of his life in London, though speaking heavily accented English. But it is Quételet, not Gauss, who counts as one of the most destructive fellows in the history of thought, as we will see next.

Adolphe Quételet (1796–1874) came up with the notion of a physically average human, l’homme moyen . There was nothing moyen about Quételet, “a man of great creative passions, a creative man full of energy.” He wrote poetry and even coauthored an opera. The basic problem with Quételet was that he was a mathematician, not an empirical scientist, but he did not know it. He found harmony in the bell curve.

The problem exists at two levels. Primo , Quételet had a normative idea, to make the world fit his average, in the sense that the average, to him, was the “normal.” It would be wonderful to be able to ignore the contribution of the unusual, the “nonnormal,” the Black Swan, to the total. But let us leave that dream for utopia.

Secondo , there was a serious associated empirical problem. Quételet saw bell curves everywhere. He was blinded by bell curves and, I have learned, again, once you get a bell curve in your head it is hard to get it out. Later, Frank Ysidro Edgeworth would refer to Quételesmus as the grave mistake of seeing bell curves everywhere.

Golden Mediocrity

Quételet provided a much needed product for the ideological appetites of his day. As he lived between 1796 and 1874, so consider the roster of his contemporaries: Saint-Simon (1760–1825), Pierre-Joseph Proudhon (1809–1865), and Karl Marx (1818–1883), each the source of a different version of socialism. Everyone in this post-Enlightenment moment was longing for the aurea mediocritas, the golden mean: in wealth, height, weight, and so on. This longing contains some element of wishful thinking mixed with a great deal of harmony and … Platonicity.

I always remember my father’s injunction that in medio stat virtus , “virtue lies in moderation.” Well, for a long time that was the ideal; mediocrity, in that sense, was even deemed golden. All-embracing mediocrity.

But Quételet took the idea to a different level. Collecting statistics, he started creating standards of “means.” Chest size, height, the weight of babies at birth, very little escaped his standards . Deviations from the norm, he found, became exponentially more rare as the magnitude of the deviation increased. Then, having conceived of this idea of the physical characteristics of l’homme moyen , Monsieur Quételet switched to social matters. L’homme moyen had his habits, his consumption, his methods.

Through his construct of l’homme moyen physique and l’homme moyen moral , the physically and morally average man, Quételet created a range of deviance from the average that positions all people either to the left or right of center and, truly, punishes those who find themselves occupying the extreme left or right of the statistical bell curve. They became abnormal . How this inspired Marx, who cites Quételet regarding this concept of an average or normal man, is obvious: “Societal deviations in terms of the distribution of wealth for example, must be minimized,” he wrote in Das Kapital .

One has to give some credit to the scientific establishment of Quételet’s day. They did not buy his arguments at once. The philosopher/mathematician/economist Augustin Cournot, for starters, did not believe that one could establish a standard human on purely quantitative grounds. Such a standard would be dependent on the attribute under consideration. A measurement in one province may differ from that in another province. Which one should be the standard? L’homme moyen would be a monster, said Cournot. I will explain his point as follows.

Assuming there is something desirable in being an average man, he must have an unspecified specialty in which he would be more gifted than other people—he cannot be average in everything. A pianist would be better on average at playing the piano, but worse than the norm at, say, horseback riding. A draftsman would have better drafting skills, and so on. The notion of a man deemed average is different from that of a man who is average in everything he does . In fact, an exactly average human would have to be half male and half female. Quételet completely missed that point.

God’s Error

A much more worrisome aspect of the discussion is that in Quételet’s day, the name of the Gaussian distribution was la loi des erreurs , the law of errors, since one of its earliest applications was the distribution of errors in astronomic measurements. Are you as worried as I am? Divergence from the mean (here the median as well) was treated precisely as an error! No wonder Marx fell for Quételet’s ideas.

This concept took off very quickly. The ought was confused with the is , and this with the imprimatur of science. The notion of the average man is steeped in the culture attending the birth of the European middle class, the nascent post-Napoleonic shopkeeper’s culture, chary of excessive wealth and intellectual brilliance. In fact, the dream of a society with compressed outcomes is assumed to correspond to the aspirations of a rational human being facing a genetic lottery. If you had to pick a society to be born into for your next life, but could not know which outcome awaited you, it is assumed you would probably take no gamble; you would like to belong to a society without divergent outcomes.

One entertaining effect of the glorification of mediocrity was the creation of a political party in France called Poujadism, composed initially of a grocery-store movement. It was the warm huddling together of the semi-favored hoping to see the rest of the universe compress itself into their rank—a case of non-proletarian revolution. It had a grocery-store-owner mentality, down to the employment of the mathematical tools. Did Gauss provide the mathematics for the shopkeepers?

Poincaré to the Rescue

Poincaré himself was quite suspicious of the Gaussian. I suspect that he felt queasy when it and similar approaches to modeling uncertainty were presented to him. Just consider that the Gaussian was initially meant to measure astronomic errors, and that Poincaré’s ideas of modeling celestial mechanics were fraught with a sense of deeper uncertainty.