On Prediction

In 1859, French astronomer Urbain Le Verrier reported a peculiarity to l'Académie des sciences. Poring over others’ published results (no small feat before the advent of online scientific databases), he discovered that reported transit times of the planet Mercury across the Sun deviated from theoretically calculated values. Though the error was small, it was consistent; over 100 Earth years, Le Verrier reported, Mercury’s precession would deviate by about 38 arcseconds. Whereas Mercury’s precession was consistently measured at about 5695 arcseconds per tropical century, calculations based on the titanic Newton’s law of gravitation predicted 5557 arcseconds over the same period.

Le Verrier was no stranger to anomalous astronomical data. In 1845, he had engaged in a lengthy mathematical study to understand small, systematic differences between predicted and observed values for the orbital dynamics of Uranus. On the last day of August, 1846, he announced publicly to the French Academy his proposed reason for the discrepancies: there was an as-yet undiscovered planet whose gravitational pull was distorting Uranus’s orbit. What was more, Le Verrier claimed, his calculations revealed its exact location. Look, he told astronomers, where my calculations point, and you will find an unknown planet. Indeed, the Berlin Observatory found Neptune within 1 degree of the predicted location.

So, when the Frenchman found similar systematic discrepancies in Mercury’s orbit, he felt familiar fervor to point the telescopes toward yet another planet, to be called Vulcan. And yet, neither the calculations nor the observatories revealed anything new in the skies. When Le Verrier died in 1877, remembered by Arago as “the man who discovered a planet with the point of his pen,” the Mercurian mystery remained.

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Ask a physician, “how will I die?” and surely you will be met with ridicule. Anyone can see that such questions are best left to a reputable seer. But more reasonable questions will still be left without answer: “is he going to make it?” we might ask of a loved one in the ICU, and still the answer will be framed in terms of optimism or doubt. Or wonder “will the antibiotics clear up my infection?” and an honest physician will say “probably.”

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There is always a chaos term in physics; a part of every equation that tends toward unpredictable results. Often, our experiments allow us to safely ignore this term. For example, nonlinearity can be tossed aside when calculating theoretical motion of a simple pendulum, and still our predictions will agree well with experiment.

In a deterministic system, the same inputs lead always to the same outputs. The exact same ball, dropped from the exact same height, onto the exact same ground, by the exact same hand, with the exact same wind, will bounce to exactly the same height every time it is dropped (modulo any quantum effects, which we will squarely ignore in this discussion). Of course, in real life, none of these things are every exactly the same. The ball deteriorates slightly in the time between the drops. The hand shifts a little and cannot drop from the same height twice. The wind has shifted a little and blown some sand onto the landing zone. But, the ball bounces to about the same height because the system is about the same: this is the description of an unchaotic system.

Chaos is the opposite. A small perturbation in initial conditions has a large impact on the final conditions. It would be absurd for our ball to bounce a hundred feet higher if the wind shifted one degree southeast. But this is exactly the behavior of a chaotic system. If the conditions are exactly the same, the system is still deterministic. Nothing is random in that sense. But, if the conditions shift even slightly, the result could be drastically different. This is exactly our experience in medicine. The chaos term so often dominates the equation and gives us no hope of ignorance.

The exceptions are the powerful interventions – see Fleming’s penicillin, Jenner’s smallpox vaccine. In modernity, HIV cocktails or GLP-1 receptor agonists are easily recognizable as transformative. In this sense, we are good at understanding the impact of definitive interventions, and as a result we have discovered a great number. We know why AIDS kills, we know why our drugs work, and we know that the drugs will help. But ask medicine to make a prediction, a specific one, and you will be met with blank stares and hedged bets. So often we hear the story: “they gave him six months, and yet here he is years later.” These anomalies are not in a fundamental sense different from the discrepancies in the precession of Mercury; we simply do not have all the information nor do we have the cognitive powers to be able to process it all into a prediction. A mostly-linear, unchaotic system such as the pendulum or the bouncing ball can be modeled and studied with a level of precision that allows us to make accurate predictions. But nonlinear, chaotic systems can seldom be accurately modeled even as the number of parameters balloons. We are left with statistics to make sense of the noise.

If we could model every single atom in a human body and all the physical interactions that take place between them, perhaps we could accurately answer to who lives and who dies in the ICU. If we could model every atom in the universe, we might likewise be able to predict the future of any person or place. But, since this problem is safely impossible by any measure, we can rest easy without any predictions of the sort.

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In March of 1879, Albert Einstein was born in Ulm, Germany. Perhaps best remembered as a patent clerk, he was something of an inventor himself. His General Theory of Relativity, which around 1915 provided the explanation for those missing arcseconds in Mercury’s precession, is still among the most successful physical theories ever proposed. Adding the relativistic correction, our predictions of orbital mechanics are exceptionally accurate.