There is a moment in a dream when you realize that things don’t add up: You know that geese don’t speak English, and yet there you are chatting away with one about the price of gasoline. You’re sure that you never went to flight school, so why are you piloting this Cessna over Dubuque?
Then you wake up.
But sometimes, you don’t. Sometimes, as you confront two seemingly unassailable, clashing truths, you realize that you’re not dreaming at all—you’ve just encountered a paradox, and it’s a wake-up of an entirely different kind.
For centuries, paradoxes have helped physicists and philosophers challenge and deepen their understanding of how our world works. Paradoxes reveal assumptions and prejudices we never knew we had and open hidden hatches into new physics.
“There’s no getting around it: The universe is really strange, and paradoxes hit you with that,” says Anthony Aguirre, a physicist at the University of California, Santa Cruz.
For Aguirre, that’s a good thing: “That feeling of mystery is really what’s exciting in physics. You know there is something fun, interesting, and potentially important to be gained by going down that road.”
“Sometimes I consider that my knowledge is broken up into tectonic plates of understanding on the Earth of my total knowledge–a small part of the total universe of possible understanding,” says physicist Robert Nemiroff, who gives a special lecture on paradoxes to his students at Michigan Technological University. “Sometimes, I learn something that demands that two plates collide–both plates cannot be used to understand this new thought. This new thought can frequently be coined as a paradox. If resolved, these plates can lock into a larger plate of greater understanding, if I am lucky.”
“Paradoxes heighten what’s at stake conceptually,” says MIT science historian David Kaiser, who adds that physicists like Niels Bohr, John Wheeler, and Albert Einstein all deployed paradoxes strategically to underline mathematical contradictions that others deemed inconsequential. “Paradoxes are one way of grappling with what the equations really say.”
Here are the stories of three paradoxes from far-flung times and places in the history of physics and math. Though they have all been resolved, they remind us just how weird the universe really is. It is a dream from which we will never wake up. But who would want to?
How is it that anyone or anything ever moves from point A to point B? This simple question is the crux of a paradox first posed by the ancient Greek philosopher Zeno, and it has made generations of math students question the nature of reality every time they walk across a room.
Here’s one rendition of the paradox: Say you want to walk down the hallway from your bedroom to your bathroom. First, you have to cover half the distance between the rooms. Next, you’ll need to cross half the remaining distance. As you continue down the hall, you will always have half the previous distance left to cross. Though you will move ever closer to the threshold of the bathroom door, you will never actually reach it.
Obviously, we don’t spend our entire lives stranded in hallways. Why not? The answer is at the heart of calculus: It turns out that infinitely long sequences of numbers can actually have finite sums. This means that even though we must cross an infinite number of progressively smaller “chunks” of space on our way to the bathroom, the time it takes to do so is finite. That’s why we eventually get there.
Yet Zeno’s paradox also reflects one of the biggest questions in physics today: Is space—or spacetime—continuous, or is it broken up into discrete chunks? In Zeno’s world, space was continuous: It could be subdivided into smaller pieces on and on into infinity. Yet we know this isn’t how matter works. If you split a cookie in half over and over again (as many guilty sweet-tooths have no doubt tried at home) you will eventually be left with the indivisible components (electrons and quarks) of one atom. Eat them or don’t, but you cannot divide them in half.
This is also the moral of the story of quantum mechanics: The energy contained in all the particles that make up the universe is quantized. Why should spacetime be any different? In fact, some of the leading theories of quantum gravity predict that, on the tiniest scales, spacetime should break down into discrete chunks. Like a pointillist painting, spacetime may look perfectly smooth from afar, but up close it dissolves into pixels. There are currently experiments in the works to test this prediction.
Why is the sky dark at night? Before you say that it’s because the sun has set, consider that it took the greatest minds in science more than three hundred years to resolve this paradox. Everyone from Einstein to Edgar Allan Poe was swept up by this apparently simple puzzle.
The history of Olbers’ paradox goes back at least as far as Johannes Kepler, who posed it in 1610. If the universe is infinite, argued Kepler, containing an infinite number of stars distributed evenly across the sky, then every point on the night sky should be illuminated by starlight. The brightness of any individual star, as seen from Earth, fades in proportion to the square of its distance from Earth, but the number of stars at a given distance from Earth increases in proportion to the square of the distance from Earth, so it is a wash. The night sky, therefore, should be just as bright as the daylight sky. To Kepler, this meant that the universe must not be infinite after all.
In 1823, Heinrich Olbers drew up a different solution to the paradox that now bears his name. Olbers argued that as the light from each star makes its way toward Earth, it runs into interstellar dust and gas that absorb some its energy. Stars that are sufficiently far enough away from Earth would therefore be “cut off” from us.
The problem with Olbers’ logic is that dust and gas must spit back out the energy that they have absorbed, leaving us with the same problem we started with.
This is where Poe enters the picture. In his prose poem “Eureka,” published in 1848, he inserted time into the equation: What if light from distant stars just hasn’t had enough time to reach us yet? Poe wrote:
…yet so far removed from us are some of the “nebulae” that even light, speeding with this velocity, could not and does not reach us, from those mysterious regions, in less than 3 millions of years.
He may have had the details wrong, but the idea was right: Because stars have not been shining forever, and because it takes time for light to travel from here to there, there is a certain horizon from beyond which light has not yet reached us. Today, we know that the universe had a beginning (the Big Bang) and that the universe is expanding, causing light from distant stars to get stretched out (“redshifted”) beyond the visible part of the spectrum and into the infrared and radio bands, compounding the dark-sky effect.
“Olbers’ paradox is based on such a mind-numbingly simple observation,” says Aguirre. “What impresses me is the sheer amount of time that went by as people came up with one complicated and wrong solution after the next.”
The Twin Paradox
When Einstein proposed that time and space were elastic, it was weird enough. But the twin paradox challenges our understanding of Einstein’s rules even further. Imagine two twins, one a space traveler and the other an avowed homebody. The traveler sets out on a mission to a distant planet in a newfangled rocket that zips along at close to the speed of light. It’s a round-trip journey, so when she gets back, she is eager for a reunion with her twin. She wants to share the amazing stories of her voyage, of course, but she’s also looking forward to gloating about one favorable side effect of life in the cosmic fast lane: Because time passes more slowly for objects traveling close to the speed of light, the traveling twin anticipates that though she has hardly aged a bit, her stay-at-home sister will be sporting many years’ worth of new wrinkles. (Both twins are a bit vain.)
The stay-at-home sister is just as excited to see her twin. She knows her special relativity, too, and reasons that from her point of view, she was traveling close to light speed aboard spaceship Earth, while her sister sat complacently aboard her stationary vessel. Therefore, she will be the young-looking one, and will have fun counting up all her sister’s gray hairs.
And here is the paradox: Einstein tells us that no observer is “more correct” than any other, but the sisters can’t both be younger than each other. Still, Einstein wasn’t wrong. The solution to the paradox is that the sisters’ journeys were not actually identical. The traveling twin did not keep up a constant velocity throughout her entire trip. She accelerated to get up to speed, and then she changed directions—another kind of acceleration—before decelerating to get back into orbit around Earth. So the traveler, not her stay-at-home sister, gets the anti-aging benefits of time dilation.
Paradox solved? In one sense, yes. But in another sense, says Aguirre, the twin paradox reveals a deeper conundrum within the laws of relativity. The crux of the resolution, he says, is that even while velocity is relative, acceleration is absolute. “Where does this absolute non-accelerated reference frame come from? Einstein tried to do away with it in general relativity, but even a century later the question is largely open.”
Will today’s paradoxes be tomorrow’s truisms? Paradoxes arise when equations clash with our intuition about reality, says Kaiser. But intuition can change. “Newtonian mechanics did not look or sound ‘intuitive’ in the 17th century,” Kaiser points out. Today, we take it for granted that Newton’s conceptions of speed, gravity, and mass are in harmony with human intuition, but perhaps intuition itself has been reshaped by Newton’s view of the world. Will the apparent paradoxes of quantum theory and relativity one day feel just as natural as Newton’s laws? Is it possible to “grow out” of a paradox?
“Even once you know the golden thread that unravels the seeming contradiction, paradoxes are still appealing,” says Kaiser, who thinks of them as a kind of mental bodybuilding. “Paradoxes are like a much more satisfying version of Sudoku or a crossword puzzle.” The best part: “In the process we might really learn something about how the universe works.”
Editor’s picks for further reading
arXiv: Paradox in Physics, the Consistency of Inconsistency
In this article, Dragoljub A. Cucic classifies paradoxes in physics and reviews their utility.
Edge: The Paradox
In this essay, Anthony Aguirre argues that a better understanding of paradoxes would improve our cognitive toolkit.
FQXi: Black Holes: Paradox Regained
Stephen Hawking conceded his bet on the black hole information paradox, but the debate continues.