The Sympathy of Clocks

The essay

A fever in The Hague

In February 1665, Christiaan Huygens lay sick in bed in The Hague. At thirty-five he had already worked out that Saturn wears a ring, and his pendulum clock kept better time than any machine built before it. Two of those clocks hung from a single wooden beam in his room, prototypes for a trial at sea. A fever leaves a man little to do but look at things, so for days he watched them.

The pendulums had started out of step and ended in perfect opposition, each swinging left as the other swung right. When Huygens disturbed one, the pair found each other again inside half an hour. Moving the clocks to separate supports broke the spell. In a letter to his father he called it a sympathy of clocks and blamed vibrations in the beam too faint for the eye to catch. He reported it to the Royal Society that March and went back to optics.¹

Historians count this as the first recorded case of spontaneous synchronization, two coupled systems falling into order with nobody in charge. Complexity science still uses it as a founding example. Huygens knew more about precision machinery than anyone alive, and he wrote the whole thing off as a curiosity. The mathematics that explains his clocks took until 1975 to arrive.

The shelf was already stocked

By the 1660s most of what an AI course opens with sat in plain view. Hobbes wrote in Leviathan, in 1651, that reasoning is “nothing but reckoning,” the adding and subtracting of thoughts, and every educated reader in Europe had an opinion about the book. Pascal was nineteen when he built a calculating machine in 1642. A gambling dispute led Pascal and Fermat to probability theory in 1654, and Huygens published its first textbook three years later. In 1662 John Graunt, a London haberdasher, went through the city's weekly death registers looking for patterns and founded statistics in the process.

Leibniz came closest to putting the pieces together. In 1673 he showed the Royal Society his stepped reckoner, the first machine that could multiply. He spent much of his life on a universal symbolic language in which two philosophers could settle any dispute by sitting down and saying calculemus, let us calculate. In November 1676 he travelled to The Hague to spend several days talking with Spinoza, who had already folded the mind into nature and treated mental life as one more aspect of the substance that makes up matter.² You can follow the paper trail from that visit all the way to a transformer model.

Why the clock won

The century's own success slowed everything down. Calculus arrived in the 1670s and 1680s and cracked planetary orbits and the flight of a cannonball. Problems like these are linear or close to it. The whole behaves like the sum of its parts, and a patient person with pencil and paper can solve them. After a generation of such wins, Europe came to picture the universe as a clock, wound once and running on rules anyone could write down.

Much of nature behaves otherwise. Its parts feed back on one another and change each other's rules as they go. Huygens' two clocks were a system of that kind, which is why his own equations could not touch them. Nonlinear problems give way to iteration: compute a step, feed the answer back in, and repeat it a million times. Nobody in 1665 owned a machine with that kind of patience, so the problem sat.

Poincaré breaks the clockwork

In 1889 Henri Poincaré won a prize set by the King of Sweden for work on whether the solar system is stable. Even three bodies under gravity, he showed, admit no general formula, and in the tangle of their orbits he found behavior so intricate he declined to try drawing it.

In 1961 the meteorologist Edward Lorenz restarted a weather simulation from a printout and typed 0.506 where the machine had stored 0.506127. Within a couple of simulated months the new weather had nothing in common with the old. His 1963 paper on sensitive dependence on initial conditions founded modern chaos theory.³

A burst of results followed in the 1970s. Robert May showed in 1976 that the simplest population equation a biology student meets tips from steady numbers into full chaos as one parameter is nudged upward. Mitchell Feigenbaum found that the tipping follows the same ratio, 4.669 and change, in systems with nothing else in common. Mandelbrot named the geometry in 1975: fractals. That same year Yoshiki Kuramoto published the equation Huygens never had, a solvable model of coupled oscillators that shows the exact moment a crowd of independent tickers snaps into shared time.

What the clocks were trying to say

With Kuramoto's result in hand, the same machinery turns up everywhere. Along riverbanks in Thailand, thousands of fireflies flash in unison, each one obeying a rule short enough to fit in a sentence: if a neighbor flashes first, flash a little sooner next time. Pacemaker cells in the heart keep time the same way, and so do the neurons whose rhythms let you read this page.

Scientists call this emergence. Simple rules and dense coupling at the bottom produce capable behavior at the top, and nothing anywhere holds a blueprint of the whole. In 1943 McCulloch and Pitts showed that networks of idealized neurons can compute anything logic can express. Rosenblatt's perceptron was learning from examples by 1958, and backpropagation made deep networks trainable in the 1980s. By the 1990s the theory was in decent shape and still waiting, as Huygens had, for a machine patient enough to run it at scale.

The patience machine finally shows up

ENIAC, unveiled in 1946, did 357 multiplications a second with eighteen thousand vacuum tubes and drew as much power as a small neighborhood. The transistor came a year later. Integrated circuits followed in the late 1950s, a whole processor fit on one chip by 1971, and for fifty years the number of transistors on a chip doubled about every two years.

Video games supplied the next turn. Drawing a 3D scene sixty times a second means recalculating millions of points and pixels in parallel, so graphics cards grew into machines built to multiply large matrices fast. Nvidia opened that hardware to general computing in 2007. Five years later two Toronto students trained a deep neural network on a pair of consumer gaming cards and won the ImageNet vision competition by a margin nobody in the field had seen before.⁴

The transformer arrived in 2017. Under the vocabulary it is matrix multiplication arranged so that every token in a sequence can attend to every other, and graphics silicon runs that arrangement well. Architecture and hardware locked together the way Huygens' pendulums did. One modern AI chip now performs about two thousand trillion low-precision multiplications every second. Out of multiplication at that scale came the emergence this whole lineage had been pointing toward, systems that write and plan even though no single part of them understands anything.

Filing things under curiosity has a price

I open AI sessions with this story because of how long it took to end. The best-equipped mind in Europe watched something new happen on his own wall, gave it a lovely name and went back to problems he could solve. The observation was sound, and Huygens filed it under curiosity, where it stayed for three centuries.

Plenty of companies are running the 1665 experiment again right now. A working demonstration of machine intelligence hangs on the beam, people gather round and agree it is fascinating, and the pilot gets a lovely name. The three hundred years live in the distance between that moment and the day the work changes.