Almost fifty years ago, when my student T. Y. Li and I wrote a math paper titled "Period 3 Implies Chaos", I could not predict the effect that title would have. Chaos. It would go on to have a life of its own, far beyond the mathematical proof contained in our short paper. Since then, the word has risen and fallen in popularity, while others, like complexity, have emerged. But the principles are the same: there are limits to what we can accurately predict. Many systems are sensitive to initial conditions. And above all: a fully deterministic system can still be unpredictable.

This reflection could have easily been titled “50,000 Years of Chaos.” Humans have always known that slight differences can have dramatic consequences: a boulder lands a meter away from a man’s sleeping head—a meter that contains an entire world. 1 1 Benjamin Franklin knew about sensitivity to initial conditions when he popularized the old piece For Want of a Nail. Expand Footnote Collapse Footnote Mathematicians merely put numbers to a principle that needed no introduction.

When novelists use Heisenberg’s uncertainty principle as an explanation for divorce, they are speaking by analogy. 2 2 Strictly speaking, Heisenberg’s principle only holds for subatomic particles. Expand Footnote Collapse Footnote Chaos, on the other hand, is both a formal mathematical discipline, and a fact about the contingency of our daily lives—we buy health and life insurance in order to manage it. A world that is shorn of chaos doesn’t look anything like ours. But chaos is also a phenomenon that applies to precise numerical quantities, whether those quantities are water molecules or animal populations. 3 3 It is ironic that “scale invariance” is a topic within chaos theory, and also applies to the theory itself. Expand Footnote Collapse Footnote The reason why it took so long to mathematically formalize, is because people wanted to find clean linear solutions to differential equations. But the vast majority of such equations are not easily solvable, being nonlinear and chaotic. As Stanislaw Ulam famously quipped, “to call the study of chaos ‘nonlinear science’ was like calling zoology ‘the study of non elephant animals’.” 4 4 This quote by Ulam was related in James Gleick’s book Chaos (1987), a book which helped popularize chaos for mainstream audiences. Expand Footnote Collapse Footnote

Most of the world that we see around us is chaotic, with linear solutions being the rare exceptions. A lot of great math has been done with such ideal linear constructions, but the real world is hairy with impurities and noise. To clarify, there is a difference between random noise and chaos, although it can often be difficult to distinguish one from the other. 5 5 I have always avoided studying situations that are both chaotic and random, so as not to confuse my readers about which is which. Although I encourage others to investigate combined systems that are more realistic. Expand Footnote Collapse Footnote Furthermore, not all systems are chaotic, sometimes it is possible to trim down a system so that chaos does not emerge. Water is chaotic as a gas, much less chaotic as a fluid, not chaotic as ice, all depending on a single parameter: temperature.

Chaos persists. Of course, the field already existed before it was named, a loose spiderweb stretching across disciplines. When we wrote our paper, Edward Lorenz had already written about his “strange attractors”—those generalized patterns of movement lacking any precise repetition. 6 6 Lorenz’s 1963 paper “Deterministic Nonperiodic Flow” motivated us to write “Period Three Implies Chaos.” In fact, one could read our paper as a commentary on Lorenz’s work. Expand Footnote Collapse Footnote But the antecedents of chaos stretch back in the scientific literature. In my opinion, James Clerk Maxwell (1831-1879) was likely the first person to understand chaos as sensitivity to initial conditions. For instance, take his writings about the behavior of gas molecules, where a simple collision results in an unpredictable direction of rebound due to initial starting conditions. Maxwell wrote that “small differences in the initial conditions produce very great ones in the final phenomena” and urged scientists to pursue “singularities and instabilities, rather than the continuities and stabilities of things.” 7 7 See my article with Brian Hunt. I had read about Maxwell’s ideas in high school and it made a lasting impression. Expand Footnote Collapse Footnote In 1890, Henri Poincaré (1854-1912) wrote about the three body problem, demonstrating that the orbits of three celestial bodies often behave chaotically and are difficult to predict.

Chaos says that we cannot predict the future with precision or certainty, especially when the future is distant. Given an entirely deterministic universe, we still can’t predict the temperature a year from now. 8 8 Even though we can say with confidence that July in Texas will be hot. The seasons are a macro pattern that is never the same in the details, in other words, weather a strange attractor. Expand Footnote Collapse Footnote This is because we can’t fully quantify our current state! Even if we measure an object’s location to three hundred decimal places of accuracy, its behavior will still diverge from our predictions. Each additional digit of precision extends our predictive ability, so that twenty decimal places of precision, rather than ten, doubles the length of time we can make accurate predictions. But it is immensely difficult to increase precision. Adding more decimal places of accuracy doesn’t do very much—any inaccuracy will be quickly magnified across time. The reason we can’t predict the future precisely is because we can’t measure the present precisely. 9 9 Maxwell reiterated LaPlace’s point in an essay delivered at Cambridge in 1873, “If we knew exactly the laws of nature and the situation of the universe at the initial moment. We could predict exactly the situation of that same universe at a succeeding moment. But even if it were the case that the natural laws had no longer any secret for us, we could still only know the initial situation approximately.” Once again, see my article with Brian Hunt. Expand Footnote Collapse Footnote LaPlace’s demon can’t even tie his own shoes.

Despite our inability to precisely predict the future, there are people with remarkable forecasting abilities, so called “superforecasters.” 10 10 I tell young people that Superforecasting by Philip Tetlock is a must read. Learning to make well calibrated estimates based on available information is a crucial skill. Far more important than memorizing facts is learning how to use evidence to make testable hypotheses about the world. Expand Footnote Collapse Footnote These talented individuals are able to probabilistically predict the outcome of macroscale events much better than the average person. Perhaps surprisingly, predicting a coup in a foreign country is more tractable than weather forecasting. Superforecasters are likely taking advantage of strange attractors, those endlessly novel patterns that do have a recognizable shape. As Twain said “history may not repeat itself, but it does rhyme”—our historical moment bears similarities with other historical moments.

Can we predict the rise and fall of civilizations? There have always been those who have tried, without much success. We are not in the position of Asimov's psychohistorians—there is no exact science of collapse. Our civilization is far too chaotic for accurate prediction, I don't think we can make meaningful predictions beyond ten years. For instance, one of our great triumphs against global warming has been substituting natural gas for coal. I don’t know anyone who predicted we would find immense new reserves of natural gas. But we did.

That is not to say that long-term thinking is futile. There are specific dangers that it's reasonable to assume we'll confront unless we make course corrections. But these reasonable concerns are not derived from a mathematical function that takes “society” as its input, and outputs what will happen in the future. Making sure we don’t run out of natural resources is similar to the imperative to “not run out money before you die.” It’s a very simple prescription that can be surprisingly difficult to achieve—we should start working on that now.

There are three great limitation principles: Heisenberg’s uncertainty, Gödel’s incompleteness theorem, and chaos. All three of these theorems tell us what will forever be unknowable, what is beyond the limits of science. Counterintuitively, limitation theorems seem to be remarkably productive. While demonstrating the limits of formal systems and computing, Gödel and Turing were also setting technology free. The computer is a physical embodiment of the limits of thought.

Chaos also has its applications. Over the last 50 years, it’s been wonderful to see chaos come into its own as a scientific discipline that is both theoretical and practical. It’s proved useful in everything from electrical circuits to fluid dynamics. It is widely used for modeling the spread of diseases, proving instrumental during the COVID pandemic. Chaos has also had an enormous impact on cryptography, the digital economy being secured by simple functions that produce seemingly random outputs. The difficulty in distinguishing chaotic output from random noise works in our favor here.

Another way of framing chaos is that a simple set of rules can produce massive complexity. From a simple set of operations on a vast amount of data, the black box we call AI emerges. We understand the basic operations being performed when we train an LLM, and we like the output, but we don’t understand its actual functioning. It turns out that high level natural language can emerge from matrix multiplication (who would have thought?). Of course, we don’t understand how our own brains function, despite understanding the general laws that it operates under. And many of the flaws of AI mirror our own—when people are asked questions, the answers are often misremembered fabrications. We too are chaotic functions produced by evolution.

With AI, we’ve once again shown that it’s easier to create chaos than it is to reverse engineer it. Unfortunately, this is a dangerous state of affairs. AI is almost an exact corollary of weather forecasting. Because we don’t understand how current AI works now, we will not understand what it will do in the future. I’m hoping that we can understand AI faster and better than we’ve understood the weather. As Nick Bostrom has said, “self-improving AI will be the last invention that man ever makes”, because we’ll either perish from it, or it will produce all further inventions.

Technological development can sometimes progress incrementally, but it can also surge and leap forward. After the first airplane flew in 1903, it was only 66 years before we landed on the moon, but a Saturn V rocket is not just a really good airplane. When it comes to preventing catastrophes it’s a good idea to focus on unprecedented, and therefore unpredictable, technologies. The stakes have never been higher—humans can now make themselves extinct. We are stuck with two bad options: we can focus on tractable but trivial predictions; or we can focus on difficult, but important problems. 11 11 Working on tractable and trivial problems is like the old metaphor about looking for a lost key under a lamppost because that’s where the light is. Expand Footnote Collapse Footnote

People are often blinded by rosy visions and don't see possible disasters. Just because things have gone well in the past does not mean that they’re guaranteed to go well in the future. Nothing is promised. I've always told my kids "figure out what's the worst that can happen, and see that it doesn't"—the most successful people are good at plan B.

Today, my computer writes better poetry than I do, but it’s not doing my job quite yet. AIs are being used as automated proof checkers, but they aren’t constructing the proofs. 12 12 Proofs that can be checked by AIs have to be very clearly structured and cannot make logical inferences that are too large. If the proof uses a novel method, current AIs will also have problems recognizing it. This will likely change in the future. Expand Footnote Collapse Footnote There is still a large amount of creativity that goes into the production of a novel proof, even when the problem is relatively simple or involves just connecting the dots. 13 13 I tell students, pick a simple problem. As you work on it, it will become harder than you expected. If you pick an obviously hard problem, it is likely to become impossible to solve. Expand Footnote Collapse Footnote Li and I did not invent chaos. We saw connections between everyday life and mathematics and made the connection in a way people could understand. And we picked a good name.

About the author

James Yorke

James Yorke is a mathematician at the University of Maryland.