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June 26 - July 1, 2018
Three states of matter—solid, liquid, and gas—have long been known. An analogous distinction between three states of randomness—mild, slow, and wild— arises from the mathematics of fractal geometry.
Biologists know that studying disease helps to understand the healthy body. Physicists collide high-energy particles to understand ordinary matter. Meteorologists study hurricanes to forecast the local weather. And economists? Well, by comparison they are a curiously incurious lot.
A bank in which the research department thinks it has discovered something new and useful will not share it with anyone else. Being focused on profit, not knowledge, it is unlikely to fund fundamental research.
what is needed is a Project Apollo for economics—a sizeable, coordinated effort to advance human knowledge. We need to understand, in much closer fidelity to reality, how different kinds of prices move, how risk is measured and how money is made and lost. Without that knowledge, we are doomed to crashes, again and again.
INDEPENDENCE IS A GREAT VIRTUE.
As a scientist, all of my research has, in one way or another, veered between the two poles of human experience: deterministic systems of order and planning, and stochastic, or random, systems of irregularity and unpredictability.
Wild randomness is uncomfortable. Its mathematics is unfamiliar and in many cases remains to be developed. It looks difficult, often requiring elaborate computer simulations rather than a quick punch on a calculator. Unfortunately, the world has not been designed for the convenience of mathematicians. There is much in economics that is best described by this wilder, unpleasant form of randomness—perhaps because economics is about not just the physics of wheat, weather, and crop yields, but also the mercurial moods and unmeasurable anticipations of wheat farmers, traders, bakers, and consumers.
The judges included Henri Poincaré, one of the most celebrated mathematicians of all time. He was a genius whose restless energy had led him across virtually every field of mathematical inquiry and beyond: probability, function theory, topology, geometry, optics, and, above all, celestial mechanics. He was a widely read popularizer of math and science, and his collected columns fill several books read to this day.
He had a keen sense of the beautiful in mathematics. He once said: “A scientist worthy of the name, above all a mathematician, experiences in his work the same impression as an artist; his pleasure is as great and of the same nature.”
So in October 1970, Black and Scholes submitted a paper to the Journal of Political Economy. Rejection: Too specialized, the journal said. They tried another journal. Rejection: Too many papers competing for too little space in its pages. Black, himself, suspected the ivory-tower class system at work. He later grumbled, “One reason these journals didn’t take the paper seriously was my non-academic return address.” In the end, the paper was rewritten and published in the Journal of Political Economy—but only after two friends from the University of Chicago, Fama and Merton Miller, lobbied the
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Their collective brainpower, both carbon-and silicon-based, is astounding. As an industry, finance buys more computers than almost any other. It hires a huge proportion of the world’s newly minted math and economics graduates.
But the sensation of roughness had almost entirely been ignored by scientists. Euclid, the Greek geometer whose Elements is the world’s oldest treatise with near-modern mathematical reasoning, focused on its opposite, smoothness.
The key is spotting the regularity inside the irregular, the pattern in the formless. Contrary to popular opinion, mathematics is about simplifying life, not complicating it.
Fractal geometry is about spotting repeating patterns of this kind, analyzing them, quantifying them and manipulating them; it is a tool of both analysis and synthesis.
Like Cantor’s dust and Sierpinski’s gasket, its intent was to defy conventional mathematical notions. Its outline is truly monstrous: continuous but of infinite length; you could not draw a line that was tangent to it anywhere along its infinite length. This sort of mathematical anarchy annoyed many contemporaries, who were still pursuing ideals of continuity and order. A French mathematician, Charles Hermite, wrote in 1893 of “turning away in fear and horror from this lamentable plague of functions with no derivatives.”
Such is the power of fractals and chance working together: Simple rules build complex structures, and complex structures deconstruct into simple rules.
But I was seeking no advisers. In fact, I wrote the thesis without one, and persuaded one of the university’s bureaucrats to rubber-stamp it when it was completed.
Look at it another way, through the lens of what mathematicians call conditional probability. That is a fancy term for a straightforward concept: Given a starting condition, what is the probability that some event will happen?
In the physics I learned as a student, there is a clear barrier between the very large and the very small. At the very large scale of the cosmos, the relativistic space-time laws of Einstein apply. In the medium-scale world of our daily lives, Newtonian mechanics holds. And in the subatomic world of electrons and quarks, the entirely different laws of quantum mechanics apply. Three different regimes, three different scales, each one distinct from the last. The laws of physics do not scale.
Pictures can deceive as well as instruct. The brain highlights what it imagines as patterns; it disregards contradictory information. Human nature yearns to see order and hierarchy in the world. It will invent it where it cannot find it.
Models are important in science. They help us understand. If, on a computer, we can build a small-scale model of the global climate, of a planet’s orbit, or of an economy’s growth, we can test our knowledge. Models also help us act.
The future is shrouded in mist and doubt.
Consider Newton’s famous law of gravity: The force of attraction between two bodies depends on their distance. He needed just a few pen strokes to express that thought, mathematically. But from it, he showed why the planets move as they do, where comets fly—even how high the tides flow. Later generations elaborated, until we had rockets, satellites, and men in space. His was a very small seed of thought, from which a great forest of science and engineering has grown.
There is something in the human condition that abhors uncertainty, unevenness, unpredictability.
