In math and computer science, researchers have long understood that some questions are fundamentally unanswerable. Now physicists are explor
Only idealized systems that involve infinity — an infinitely long tape, an infinitely extensive grid of particles, an infinitely divisible space for placing pinballs and rubber ducks — can be truly undecidable. No one knows whether reality contains these sorts of infinities, but experiments definitely don’t. Every object on a lab bench has a finite number of molecules, and every measured location has a final decimal place. We can, in principle, completely understand these finite systems by systematically listing every possible configuration of their parts. So because humans can’t interact with the infinite, some researchers consider undecidability to be of limited practical significance. “There is no such thing as perfect knowledge, because you cannot touch it,” said Karl Svozil (opens a new tab), a retired physicist associated with the Vienna University of Technology in Austria. “These are very important results. They are very, very profound,” Wolpert said. “But they also ultimately have no implications for humans.” Other physicists, however, emphasize that infinite theories are a close — and essential — approximation of the real world. Climate scientists and meteorologists run computer simulations that treat the ocean as if it were a continuous fluid, because no one can analyze the ocean molecule by molecule. They need the infinite to help make sense of the finite. In that sense, some researchers consider infinity — and undecidability — to be an unavoidable aspect of our reality. “It’s sort of solipsistic to say: ‘There are no infinite problems because ultimately life is finite,’” Moore said. And so physicists must accept a new obstacle in their quest to acquire the foresight of Laplace’s demon. They could conceivably work out all the laws that describe the universe, just as they have worked out all the laws that describe pinball machines, quantum materials, and the trajectories of rubber ducks. But they’re learning that those laws aren’t guaranteed to provide shortcuts that allow theorists to fast-forward a system’s behavior and foresee all aspects of its fate. The universe knows what to do and will continue to evolve with time, but its behavior appears to be rich enough that certain aspects of its future may remain forever hidden to the theorists who ponder it. They will have to be satisfied with being able to discover where those impenetrable pockets lie. “You’re trying to discover something about the way the universe or mathematics works,” Cubitt said. “The fact that it’s unsolvable, and you can prove that, is an answer.”
