Scientists Are Mapping the Boundaries of What Is Knowable and Unknowable

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Apr 6, 2025

Math and computer science researchers have long known that some questions are fundamentally unanswerable. Now physicists are exploring how physical systems put hard limits on what we can predict.

The French scholar Pierre-Simon Laplace crisply articulated his expectation that the universe was fully knowable in 1814, asserting that a sufficiently clever “demon” could predict the entire future given a complete knowledge of the present.

His thought experiment marked the height of optimism about what physicists might forecast. Since then, reality has repeatedly humbled their ambitions to understand it.

One blow came in the early 1900s with the discovery of quantum mechanics.

Whenever quantum particles are not being measured, they inhabit a fundamentally fuzzy realm of possibilities.

They don’t have a precise position for a demon to know.

Another came later that century, when physicists realized how much “chaotic” systems amplified any uncertainties. A demon might be able to predict the weather in 50 years, but only with an infinite knowledge of the present all the way down to every beat of every butterfly’s wing.

In recent years, a third limitation has been percolating through physics—in some ways the most dramatic yet. Physicists have found it in collections of quantum particles, along with classical systems like swirling ocean currents.

Known as undecidability, it goes beyond chaos. Even a demon with perfect knowledge of a system’s state would be unable to fully grasp its future.

“I give you God’s view,” said Toby Cubitt, a physicist turned computer scientist at University College London and part of the vanguard of the current charge into the unknowable, and “you still can’t predict what it’s going to do.”

Eva Miranda, a mathematician at the Polytechnic University of Catalonia (UPC) in Spain, calls undecidability a “next-level chaotic thing.”

Undecidability means that certain questions simply cannot be answered. It’s an unfamiliar message for physicists, but it’s one that mathematicians and computer scientists know well. More than a century ago, they rigorously established that there are mathematical questions that can never be answered, true statements that can never be proved. Now physicists are connecting those unknowable mathematical systems with an increasing number of physical ones and thereby beginning to map out the hard boundary of knowability in their field as well.

These examples “place major limitations on what we humans can come up with,” said David Wolpert, a researcher at the Santa Fe Institute who studies the limits of knowledge but was not involved in the recent work. “And they are inviolable.”

The Blackest of Boxes:

A striking example of unknowability came to physics in 1990 when Cris Moore, then a graduate student at Cornell University, designed an undecidable machine with a single moving part.

His setup—which was purely theoretical—resembled a highly customizable pinball machine. Imagine a box, open at the bottom. A player would fill the box with bumpers, move the launcher to any position along the bottom of the box, and fire a pinball into the interior. The contraption was relatively simple. But as the ball ricocheted around, it was secretly performing a computation.

“I give you God’s view, and you still can’t predict what it’s going to do.”

Toby Cubitt, University College London

Moore had become fascinated with computation after reading Gödel, Escher, Bach, a Pulitzer Prize–winning book about systems that reference themselves. The system that most captured his imagination was an imaginary device that had launched the field of computer science, the Turing machine.

Defined by the mathematician Alan Turing in a landmark 1936 paper, the Turing machine consisted of a head that could move up and down an infinitely long tape, reading and writing 0s and 1s in a series of steps according to a handful of simple rules telling it what to do.

One Turing machine, following one set of rules, might read two numbers and print their product.

Another, following a different set of rules, might read one number and print its square root. In this way, a Turing machine could be designed to execute any sequence of mathematical and logical operations.

Today we would say that a Turing machine executes an “algorithm,” and many (but not all) physicists consider Turing machines to define the limits of calculation itself, whether performed by computer, human or demon.

Moore recognized the seeds of Turing machine behavior in the subject of his graduate studies: chaos. In a chaotic system, no detail is small enough to ignore. Adjusting the position of a butterfly in Brazil by a millimeter, in one infamous metaphor, could mean the difference between a typhoon striking Tokyo and a tornado tearing through Tennessee.

Uncertainty that starts off as a rounding error eventually grows so large that it engulfs the entire calculation. In chaotic systems, this growth can be represented as movement across a written-out number: Ignorance in the one-tenths place spreads left, eventually moving across the decimal point to become ignorance in the tens place.

Moore designed his pinball machine to complete the analogy to the Turing machine. The starting position of the pinball represents the data on the tape being fed into the Turing machine.

Crucially (and unrealistically), the player must be able to adjust the ball’s starting location with infinite precision, meaning that specifying the ball’s location requires a number with an endless procession of numerals after the decimal point.

Only in such a number could Moore encode the data of an infinitely long Turing tape.

Then the arrangement of bumpers steers the ball to new positions in a way that corresponds to reading and writing on some Turing machine’s tape.

Certain curved bumpers shift the tape one way, making the data stored in distant decimal places more significant in a way reminiscent of chaotic systems, while oppositely curved bumpers do the reverse.

The ball’s exit from the bottom of the box marks the end of the computation, with the final location as the result.

Moore equipped his pinball machine setup with the flexibility of a computer—one arrangement of bumpers might calculate the first thousand digits of pi, and another might compute the best next move in a game of chess, but in doing so, he also infused it with an attribute that we might not typically associate with computers: unpredictability.

Some algorithms stop, outputting a result. But others run forever. (Consider a program tasked with printing the final digit of pi.) Is there a procedure, Turing asked, that can examine any program and determine whether it will stop? This question became known as the halting problem.

Turing showed that no such procedure exists by considering what it would mean if it did. If one machine could predict the behavior of another, you could easily modify the first machine—the one that predicts behavior—to run forever when the other machine halts.

And vice versa: It halts when the other machine runs forever. Then—and here’s the mind-bending part—Turing imagined feeding a description of this tweaked prediction machine into itself. If the machine stops, it also runs forever. And if it runs forever, it also stops. Since neither option could be, Turing concluded, the prediction machine itself must not exist.

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Photograph: Kristina Armitage/Quanta Magazine