A computer in the shoe
In 1961 the mathematician Edward Thorp and the information theorist Claude Shannon, in the basement workshop of Shannon’s home, built a computer the size of a pack of cigarettes with twelve transistors. Thorp had conceived the idea in 1955: a roulette wheel is a physical machine, so perhaps its ball could be timed. The wearer had two switches under the toes: one started the computer, and the other timed the rotor and then the ball. The computer worked out where the ball would probably fall and played eight musical tones into an earpiece, one for each eighth of the wheel; the last tone heard named the part of the wheel to bet on. It was used when the ball still had three or four revolutions left. In Thorp’s own laboratory estimate, the player’s edge was about 44 percent on the best bets. They took it to Las Vegas in the summer of 1961. The predictions agreed with the estimates, but the hair-thin speaker wires kept breaking, and they never bet seriously. Thorp kept the method secret until 1966.
Roulette was tried again at the University of California, Santa Cruz, in the late 1970s by a group of physics graduate students led by Doyne Farmer and Norman Packard, who called themselves the Eudaemons. Their computer was built around a 6502 chip, with about three kilobytes of machine code that Farmer wrote by hand, and two people to work it: one timed the ball and wheel with a toe switch, the other, sitting at the table, felt the answer as buzzing under a shirt and placed the bets. (The dates are given as 1976 to 1978 in some accounts and as the late 1970s into the early 1980s in others.) Their reported expectation in the laboratory was again about 44 percent; Farmer’s own account says they achieved a 20 percent advantage over more than eleven trips to casinos in Las Vegas, Reno, and Tahoe, never playing high stakes. Hardware failures, sweat that made the actuators shock the bettor, and fear of casino security ended the project. Thomas Bass told the story in The Eudaemonic Pie (1985, titled The Newtonian Casino in Britain). In 1985 Nevada passed a law, signed on 30 May, that made it a crime to use a device designed to predict the outcome of a game; today it is a felony.
Does a roulette wheel contradict chaos? No, and that is the point. A rolling ball is a nonlinear system with sensitive dependence, and no one can say where it will land ten seconds from now. But a few revolutions before the ball drops, the uncertainty is small. In 2012 Michael Small and Chi Kong Tse published in the journal Chaos a model that needs only the starting position, speed, and slowing of the ball and wheel, and reported that, with timing by hand, an expected return of at least 18 percent was possible (a random bet on their wheel loses 2.7 percent). The number comes from 13 correct predictions in 22 trials, a small sample, and was never tried in a casino. The honest summary of both stories is that prediction is possible for a short time, and the skill lies in knowing how short.
Laws are not forecasts
The source course makes a related point about the laws of physics: knowing the law does not mean that you can forecast. Take the case that looks easiest, the motion of three bodies pulling on one another by gravity. The law, Newton’s, is exact. Yet in 1887 Heinrich Bruns showed that the problem has no further simple algebraic integrals beyond the ten classical ones, and Henri Poincaré extended that result to a larger class of integrals around 1889 to 1892. (Poincaré’s corrected prize essay of 1890, which Course 14 describes, uncovered the tangle that we now call chaos.) In 1912 the Finnish mathematician Karl Sundman gave a series that converges and solves the three-body problem for all time, but it needs an astronomical number of terms, estimated at 10 to the power 8,000,000 or more for useful precision, so it has never been used. Around 1990 Qiudong Wang extended Sundman’s result to any number of bodies. The correct statement is that there is no general closed-form solution of the kind that exists for two bodies, and that special solutions exist: Euler found one with the bodies in a line in 1767, Lagrange one with an equilateral triangle in 1772, and in 1993 Chris Moore found one in which three equal masses chase each other around a figure eight. In 2013 Milovan Šuvakov and Veljko Dmitrašinović found several more families by computer.
The source text mentions one more consequence, that the laws of science form layers: mechanics, then physics, chemistry, biology, society. Each level has regularities of its own. And it mentions the oldest philosophical pair, change and permanence. Heraclitus’s surviving words, in the fragment numbered B12, say that on those who step into rivers, other and other waters flow; the line that you cannot step twice into the same river is a later paraphrase. Parmenides, writing a poem in the early fifth century before the common era, argued that what is cannot come to be or pass away. Science uses both ideas: conservation laws and symmetries (Course 5 meets them in Noether’s theorem) for what stays, and dynamics for what changes. Nonlinear science adds a third piece: that even when what changes follows exact laws, the future can be out of reach.
How far ahead can you see?
Every system has its own horizon. A roulette ball: seconds. The weather: the best forecasts have useful skill to about ten days, according to a 2019 study by Fuqing Zhang and colleagues, who estimated that cutting the error in the starting data tenfold would buy at most about five more days; thirty years before, useful forecasts reached about seven days. The inner solar system: Jacques Laskar found a Lyapunov time of about five million years, so that a 15-meter error in the Earth’s position becomes about 150 meters after 10 million years and about 150 million kilometers after 100 million. (Course 14 tells these stories.) The scale changes by a factor of ten million, and the shape of the lesson does not: forecasts are good, then not.
What about the social world? The most careful evidence comes from the psychologist Philip Tetlock. For about twenty years, from 1984, he collected forecasts from 284 experts on political and economic questions, and in 2005 reported that they barely, if at all, outperformed informed non-experts, and that neither beat simple statistical rules; those he called foxes, who know many small things, did better than hedgehogs, who know one big thing. (The popular line that experts were no better than a dart-throwing chimpanzee is a simplification of this.) From 2011 to 2015 a tournament run by the US intelligence community’s research agency (IARPA) pitted teams of forecasters against each other, and Tetlock’s Good Judgment Project won it. In a 2014 paper, Barbara Mellers and colleagues showed that training, putting people in teams, and picking the best performers each improved accuracy: the top two percent, the “superforecasters,” did best, and teams beat individuals working alone. Two cautions apply. The questions were short-range, open for 102 days on average, and Tetlock has said that accuracy falls toward chance for questions about five or more years ahead. And the claim that teams do better rests on that tournament and is still being tested.
Like causes, like effects?
The source course turns on a remark by James Clerk Maxwell, and it is worth reading carefully, because the remark is sharper than the popular butterfly. On 11 February 1873 Maxwell gave a talk to a club of Cambridge friends on determinism and free will, printed after his death in Campbell and Garnett’s Life of James Clerk Maxwell (1882). He wrote that it is a metaphysical doctrine that from the same antecedents follow the same consequents, and observed that this is no use in a world where the same antecedents never happen twice. What science uses instead is the maxim that from like antecedents follow like consequents. That, he said, holds only for stable cases, and unstable cases multiply rapidly as the number of variables grows. In the same essay he noted that “every existence above a certain rank has its singular points”: a balanced rock, a spark that kindles a forest. At such points a small cause has a large effect. They are isolated, he added, so that most human conduct stays predictable, and if physicists took more interest in singularities and instabilities the prejudice in favor of determinism might fade. In Matter and Motion (1876 or 1877, depending on the edition) he repeated the point with a railway switch: the “like causes” maxim is true only when small differences in the starting state produce small differences in the final state. This is the sensitivity to starting conditions of Course 14, stated about 90 years before Lorenz.
Two starts a hair apart
A ball starts at rest on a landscape, and a second ball starts 0.01 to its right (blue and red). On a single valley the balls roll to the bottom and the difference dies away: like causes, like effects. On a ridge between two valleys, where the first ball starts decides which side both roll to. Slide the start across the ridge.
Maxwell’s “singular points” are places like the top of the ridge. Away from the ridge the pair always ends together; near it, a difference of 0.01 sends them to opposite valleys.
Two lessons follow, and the lecture draws both. First, an effect that is complicated does not need a complicated cause: simple rules, as Course 14 shows, can give wild behavior. But it is not always so, and only a particular case can tell you which. Second, you cannot always trace a large effect back to a single small cause. If a thousand small disturbances could each have tipped the outcome, naming one as “the” cause is a story, not an explanation. That is the sound point behind the popular line about a butterfly and a tornado.
The average is not you
A third lesson concerns averages. Suppose a fair coin decides whether your money is multiplied by 1.5 or by 0.6, and you play repeatedly. The average of the two outcomes is 1.05, so the expected wealth grows 5 percent a round. But a heads and a tails, in either order, multiply your money by 1.5 × 0.6 = 0.9, a 10 percent loss, so the typical player, who sees about half of each, shrinks by about 5 percent a round, since the geometric mean is the square root of 0.9, or 0.949. After 100 rounds the expected wealth is 131 times the start, while the typical player has about half of one percent of it. The average is carried by a few very lucky players. The physicist Ole Peters has argued, in a 2019 paper in Nature Physics, that economics has too often confused such an ensemble average with what happens to a person over time (a coin-toss example like this one is in his 2016 paper with Murray Gell-Mann). Economists have objected, in 2020 in the same journal, that the remedies Peters proposes are unjustified. The arithmetic is not in dispute, and Edward Thorp’s own betting systems are descendants of John Kelly’s 1956 rule for maximizing the average logarithm of wealth, which is exactly the geometric mean.
A favorable bet that most players lose
Twenty thousand players start with $100. Each round a fair coin multiplies their money by 1.5 (heads) or 0.6 (tails). Slide to choose how many rounds they play. The red dashed line is the average that the arithmetic promises; the blue line is the typical (median) player.
Gray lines are 25 individual players. After 20 rounds the average is $265 and the typical player has about $35, and three out of four players are below $100.
The source course uses the idea about education. Statistically, graduates of famous universities earn more than others, but an average says little about a particular child: whether your child lands in the high or low part of the distribution, you do not know, and some children flourish elsewhere. Economists have tested a version of this. In 2002 Stacy Dale and Alan Krueger compared students who were admitted to selective colleges but chose less selective ones with those who attended the selective ones. After allowing for the schools they applied to, the two groups later earned about the same, with an exception: for students from disadvantaged families, attending the selective school paid off more. In a follow-up in 2014, using later earnings records, they found the same. The authors give caveats: the range of selectivity in the comparison was limited, and the findings do not mean that selectivity never matters. A 2023 study by Raj Chetty, David Deming, and John Friedman compared attending an Ivy-Plus college with attending a flagship public university, using the waitlist as a natural experiment, and found that it raised the chance of reaching the top one percent of earnings by half; the two results are about different questions, and neither proves what is best for your child.
How the method changed
The last teaching of the source course is about how science is done. For centuries philosophers argued about whether it is built by deduction, reasoning from principles, or induction, generalizing from observations. Francis Bacon’s Novum Organum (1620) pressed for induction by careful tables of cases and devised experiments, and the Royal Society, founded in London in 1660 with the motto Nullius in verba (“take nobody’s word for it”), made experiment a habit. Galileo showed both legs working: he rolled balls down slopes and also held that the book of nature is written in the language of mathematics. Friedrich Engels, in a note in Dialectics of Nature, said that induction and deduction belong together as necessarily as synthesis and analysis, and each should be used in its place; the standard Chinese translation of the sentence, 归纳和演绎，正如分析和综合一样，是必然相互联系着的, is familiar to many readers in China. The source course adds that mathematics itself is not purely deductive, since a conjecture comes first, and a conjecture is a leap.
Nonlinear science adds a third leg, which is simulation, the computer experiment. When equations cannot be solved on paper, one can run them and look. That is how the Fermi–Pasta–Ulam–Tsingou chain of Lesson 1 gave its surprise in 1953; how Edward Lorenz found sensitive dependence on a Royal McBee computer around 1960; how Mitchell Feigenbaum, with an HP-65 pocket calculator in 1975, found the number 4.669 that is the same for a whole family of rules (to three places, because of the calculator’s limits); and how Mandelbrot, at IBM, drew the pictures that made fractals visible. In each case an experiment on a machine came first and a proof, if any, came later. The worldview here is practical: for nonlinear systems, qualitative understanding and experiment often come before formulas.
Meeting the future
The lecture ends with a film. In Next (2007, directed by Lee Tamahori, loosely based on a 1954 short story by Philip K. Dick, “The Golden Man”) Nicolas Cage plays Cris Johnson, a Las Vegas stage magician who can see about two minutes into his own future. When he meets a woman, Liz Cooper, played by Jessica Biel, his vision reaches further, and the plot has him using it to stop a terrorist plot. The source course reads the plot as a parable: foresight grows when you are bound to someone you care about and face a complex world together, not alone. A film is fiction, and this one was a commercial disappointment. But the parable lines up with two things we saw. The magician sees the next two minutes, and the Santa Cruz roulette players saw the next few seconds, so each has a horizon. And the one real result on foresight and company is that, in forecasting tournaments, teams beat individuals.
The last advice is about restraint. The lecture warns against “tinkering” with a living world without limit, and the cautionary examples are well known. In China, the Four Pests campaign that began in 1958 told people to kill rats, flies, mosquitoes, and sparrows. Sparrows eat grain, but also insects; by 1960 the party’s own health instruction said that sparrows should no longer be hunted, and named bedbugs as the fourth pest instead. Government figures claimed that about two billion sparrows were killed in 1958, which historians treat with caution. The Great Chinese Famine killed between 15 and 55 million people, by different estimates, and historians stress grain procurement, false output reports, and the diversion of labor; it is an oversimplification to say that the sparrow campaign caused it. A 2025 working paper (Frank and colleagues, National Bureau of Economic Research) estimates that, combined with procurement, it may have added nearly two million deaths, as a contributing factor and not the main one.
In the United States, a similar story runs in the other direction. The last wolves in Yellowstone National Park were killed by 1926. In 1995 and 1996 about thirty were brought from Canada, 14 in January 1995 and 17 in January 1996. Some ecologists reported a cascade: elk browsed less, young aspen and willow recovered, beavers returned. In a 2012 paper William Ripple and Robert Beschta found the browsing of the tallest young aspen had fallen from nearly all leaders in 1998 to under a quarter by 2010. Others dispute how far the cascade goes. A 2010 study found that aspen are not recovering despite the wolves, and a 2013 experiment showed that fewer browsing elk do not by themselves bring back willows when the hydrology has changed. A 2014 review concluded that the question “cannot lead to simple, precise, or definitive answers.” Yellowstone is not a closed experiment; the result depends on many things, and wolves are one.
Closer to home
The two stories share a lesson that fits both China and the United States: in a living system, an intervention starts a chain of effects that no one has fully mapped, and the sensible habit is to act in steps that can be undone, watch the result, and be ready to reverse. Neither story is proof that intervention is always a mistake: the wolves were restored on purpose, to repair an earlier intervention, and the Four Pests campaign also targeted flies and mosquitoes, which spread disease. They are cases of what Maxwell called singular points, where a small push has a large and uncertain effect. The Chinese phrase the source course uses for the right posture is 少折腾 (shǎo zhēteng), roughly “less fuss,” and the medical maxim for it is “first, do no harm.”
Another view
The honest question is what nonlinear science has actually delivered, and the answer is mixed. Delivered: ensemble weather forecasts, which turned chaos into probabilities and extended useful forecasts; chaotic mixing; nonlinear time-series methods with real successes in short-term prediction; the demonstration that chaos can be controlled in laboratory systems; numerical experiment as a method; and fractal geometry, which describes rough shapes over a measured range. Not delivered: forecasts beyond the horizons above; a proof that markets or hearts are chaotic; a single theory of complexity; or, for the most part, advice that follows from the mathematics. When a lecture says that nonlinear science teaches humility, cooperation, or restraint, those lessons are reasonable, and they were taught by many people long before. The mathematics supports a narrower set of claims: horizons exist and can be estimated, averages mislead for multiplicative growth, and small differences matter at singular points. The step from there to a way to live is interpretation.
The lecture also says that Einstein believed the world is simple. He believed that nature’s basic laws are mathematically simple, as he put it in a 1933 lecture at Oxford; he did not believe its phenomena are simple. The line “as simple as possible, but not simpler,” often quoted as Einstein’s, is a paraphrase traced to a 1950 newspaper article. And the claim that the twenty-first century belongs to biology and information instead of physics is an opinion, not a result.
A friend says: “The expected return on this bet is +5% per round, so everyone who plays long enough gets richer.” What is the flaw?
What to keep
- Laws are not forecasts. The laws of gravity are exact, and the three-body problem still has no general closed-form solution. Every system has a horizon: seconds for a roulette ball, about ten days for weather, millions of years for the solar system.
- Maxwell’s 1873 point is the sharp one: “like causes, like effects” holds in stable cases, and at singular points a small difference has a large effect that cannot be traced to one cause.
- Averages mislead for multiplied growth: a bet that gains 5 percent on average can leave most players poorer. What happens to a group is not what happens to you.
- The method gained a third leg, the computer experiment, and nonlinear science taught restraint with living systems: try small, watch, and be ready to reverse.
Think it over
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Sources and further reading
- Edward Thorp, “The invention of the first wearable computer,” Digest of Papers, Second International Symposium on Wearable Computers (1998), 4–8; Doyne Farmer, “Roulette,” doynefarmer.com; Thomas Bass, The Eudaemonic Pie, Houghton Mifflin, 1985; Nevada Revised Statutes 465.075 and 465.088.
- Michael Small and Chi Kong Tse, “Predicting the outcome of roulette,” Chaos 22 (2012), 033150; Peter Pennock, “The Eudaemonic Pie: a review,” AI Magazine 25 (2), 2004.
- Zoltan Musielak and Billy Quarles, “The three-body problem,” Reports on Progress in Physics 77 (2014), 065901; Karl Sundman, Acta Mathematica 36 (1912), 105–179; Qiudong Wang, “The global solution of the N-body problem,” Celestial Mechanics and Dynamical Astronomy 50 (1991), 73–88; June Barrow-Green, Archive for History of Exact Sciences 48 (1994), 107–131; Cristopher Moore, Physical Review Letters 70 (1993), 3675; Milovan Šuvakov and Veljko Dmitrašinović, Physical Review Letters 110 (2013), 114301.
- Fuqing Zhang and colleagues, “What is the predictability limit of midlatitude weather?” Journal of the Atmospheric Sciences 76 (2019), 1077–1091; Jacques Laskar, “Is the Solar System stable?” (arXiv 1209.5996).
- Philip Tetlock, Expert Political Judgment, Princeton University Press, 2005; Philip Tetlock and Dan Gardner, Superforecasting, Crown, 2015; Barbara Mellers and colleagues, “Psychological strategies for winning a geopolitical forecasting tournament,” Psychological Science 25 (2014), 1106–1115.
- Lewis Campbell and William Garnett, The Life of James Clerk Maxwell, Macmillan, 1882, chapter 14 (the 1873 essay); James Clerk Maxwell, Matter and Motion, 1876, article 19; Brian Hunt and James Yorke, “Maxwell on chaos,” Nonlinear Science Today 3 (1), 1993.
- Ole Peters, “The ergodicity problem in economics,” Nature Physics 15 (2019), 1216–1221; Ole Peters and Murray Gell-Mann, “Evaluating gambles using dynamics,” Chaos 26 (2016), 023103; Jason Doctor, Peter Wakker, and Tong Wang, Nature Physics 16 (2020), 1168; John Kelly, Bell System Technical Journal 35 (1956), 917–926.
- Stacy Dale and Alan Krueger, Quarterly Journal of Economics 117 (2002), 1491–1527 and Journal of Human Resources 49 (2014), 323–358; Raj Chetty, David Deming, and John Friedman, “Diversifying society’s leaders?” NBER working paper 31492 (2023).
- Francis Bacon, Novum Organum (1620); Royal Society, “History”; Friedrich Engels, Dialectics of Nature (English edition, Clemens Dutt, 1934); Marx and Engels Collected Works (Chinese edition), volume 20; Mitchell Feigenbaum, MacTutor biography and Journal of Statistical Physics 19 (1978), 25–52.
- Next (Paramount, 2007); Philip K. Dick, “The Golden Man,” If, April 1954.
- Central Committee of the Communist Party of China, instruction on health work (March 1960); Judith Shapiro, Mao’s War Against Nature, Cambridge University Press, 2001; Frank, Gong, and colleagues, NBER working paper 34087 (2025); “Great Chinese famine” and “Four Pests campaign” (reference entries, for the range of death estimates).
- William Ripple and Robert Beschta, Biological Conservation 145 (2012), 205–213; Matthew Kauffman, Jedediah Brodie, and Erik Jules, Ecology 91 (2010), 2742–2755; Kristin Marshall, N. Thompson Hobbs, and David Cooper, Proceedings of the Royal Society B 280 (2013), 20122977; Rolf Peterson and colleagues, Annual Review of Ecology, Evolution, and Systematics 45 (2014), 325–345.
- Stanford Encyclopedia of Philosophy, “Heraclitus” and “Parmenides”; Albert Einstein, Herbert Spencer Lecture, Oxford, 10 June 1933; Quote Investigator, “Everything should be made as simple as possible, but not simpler.”