Imagine, as a thought experiment, that you are blissfully reclined on a hammock next to the swimming pool of the Grand Hotel. A soft breeze stirs the air; a fountain gurgles nearby. You’re happily daydreaming, not a care in the world. But then you hear footsteps approaching and look up to see the hotel manager arguing with a man named Hilbert.
“Impossible! Impossible” the manager says forcefully.
“But I’m just asking for one room for the night,” Hilbert says.
“Impossible! No vacancies! You must try another hotel.”
“But you have an infinite number of rooms in the hotel,” Hilbert says.
“Yes, we do have an infinite number of rooms,” the manager says. “That’s why we call it the Grand Hotel. But all the rooms are full!”
“No problem,” Hilbert says. “Just ask each guest to move up a room. The guest in Room 1 moves up to Room 2, which is free because its occupant has moved to Room 3, whose occupant has moved to 4 – and so on and so on – to infinity. Leaving Room 1 gloriously empty for me. Could you have the porter take my suitcase there?”
“The porter would be happy to assist you – wait! Something’s wrong. How can Room 1 be free if we’re already full?”
“Take my word for it – I’m a mathematician,” Hilbert says. “In fact, I could tell you a simple way to accommodate any guests who arrives, even if an infinite number of them showed up all at once.”
“Impossible!” the manager says, but not so forcefully this time. He pauses, bewildered, and looks around for help. He and Hilbert both notice you lounging in the hammock.
“Let’s allow this well-rested guest to resolve the argument,” Hilbert says, turning to you with a smile. “With a little thought, I’m sure you could give the manager a simple rule – no more than a few words – that would enable him to make rooms available immediately for an infinite number of new arrivals.”
You sit up and adjust your sunglasses. A little thought doesn’t yield an answer.
“Give me half an hour,” you say. They retreat to the lobby, leaving you to ponder what rule could be made room for an infinite number of new guests. After ten minutes of fruitless rumination, you consider a new question: Should I keep trying to figure out this problem, or should I just go back to daydreaming?
The traditional answer, of course, is to keep trying. This particular problem, a paradox devised in the 1920s by the British mathematician David Hilbert, is the sort of mental exercise that’s supposed to be good for you, at least according to generations of disciplinarians. It builds character, the Victorians liked to say. Schoolchildren learned this virtue from the classic nineteenth-century textbook, McGuffey’s Reader:
Tis a lesson you should heed
Try, try again;
If at first you don’t succeed,
Try, try again;
Then your courage should appear,
For, if you will persevere,
You will conquer, never fear;
Try, try again.
But is that really the best way to solve problems and improve your mind? Not necessarily, according to social scientists who have been studying this question in recent years. While researchers have found new support for the old Victorian practices, they’ve also discovered evidence for the benefits of daydreaming. This activity, long dismissed as a waste of time (or worse — psychology textbooks warned it could lead to psychosis), has gained new respect from researchers like Jonathan Schooler, a psychologist at the University of California, Santa Barbara.
“If daydreaming is so problematic, why do we do it so much?” Dr. Schooler asks. “It’s an exemplification of what distinguishes us from other animals: the capacity to leave the here and now to and to engage in future thinking and distant planning about events that aren’t before us right now. It provides us with the opportunity for mental time travel, and it seems to be useful for creativity.”
During a typical day, about 30 percent of your time is given over to mind wandering, which is the type of daydreaming that researchers have been studying. It’s when your mind is taken up with “task-unrelated thoughts,” and it’s most likely to happen when you’re performing a repetitive but undemanding task, like taking a walk, jogging or knitting. It has repeatedly been shown to foster creativity, as in experiment reported last year by Dr. Schooler and several colleagues, including Benjamin Baird and Jonathan Smallwood at U.C. Santa Barbara.
The experiment involved a creativity exercise in which participants were given two minutes to suggest many unusual uses as possible for various objects – like, say, a brick, which might be used for a doorstop, a weapon or just about anything else that came to mind. After this mental workout, the participants took a short break. Some of them just rested, doing nothing. Some played a sort of video game that required them to concentrate intensely on numbers appearing on a screen, and others did an easier version of that game that didn’t require much concentration.
Afterwards, all the participants returned to the creativity exercise of imagining uses for the brick and for other objects. The ones who simply rested weren’t any more creative after the break; neither were the ones who spent the break intensely concentrating on the screen. But there was a sharp increase in creativity among the people who’d played the easy version of the video game, which encouraged mind wandering because it was a repetitive but undemanding task. As their minds wandered, they weren’t deliberately trying to come up with creative answers to the earlier exercise, but somehow it happened anyway. This period of incubation, as the researchers called it, enabled them to imagine new uses for a brick, like serving as desk paperweight, or a mock coffin for a Barbie funeral.
What goes on during this period of incubation? “One possibility is that mind wandering enhances creativity by increasing unconscious associative processing,” the researchers theorize. Another possibility, suggested by brain-imaging studies, is that mind wandering simultaneously taps into two normally separate networks in the brain. Whatever the cause, it can help you look for new approaches to thought experiments like the one at the Grand Hotel.
“Thought experiments and daydreaming actually would work very well together,” Dr. Schooler says. “Thought experiments, particularly difficult ones, will require incubation to solve, and mind wandering, seems to be an ideal way to encourage creative solutions following incubation. Some research still underway also suggests that mind-wandering may be particularly associated with overcoming impasses for creative individuals like writers and physicists.”
So if you find yourself at an impasse at the Grand Hotel, maybe you don’t have to try, try again – at least not right away. The best strategy may be to find a simple task that’s not too demanding mentally, like a leisurely swim in the hotel pool. You might find those extra rooms more quickly than if you kept trying to concentrate on the thought experiment.
But you still need to get back to those thought experiments at some point, because they serve a purpose, too. The Victorians were right about mental exercises building character and strengthening willpower. Recent experiments have shown willpower is more than just a metaphor: there really is a source of physical energy in the brain that we draw on when we resist temptations, force ourselves to concentrate, make decisions and exert any form of self-control. The psychologist Roy Baumeister, of Florida State University, compares willpower to a muscle that gets fatigued temporarily when it’s used, but that can also be strengthened over time by repeated exercise.
So the more often you force yourself to try thought experiments, the stronger you get, and the longer you can think about these problems before you get fatigued and hit an impasse. If you persevere long enough to find rooms for an infinite number of new arrivals at the Grand Hotel, you can take on other thought experiments – and inevitably you’ll end up daydreaming, too. You can’t have one without the other, as Dr. Schooler observes in his answer to this Big Question.
“I can imagine a world without thought experiments, but I can’t imagine one without daydreaming,” Dr. Schooler says. “So it probably is more useful to society to promote the former, as people will do the latter no matter what.”
In that spirit, here are some challenges to strengthen your mental muscle.
Questions for Discussion:
1) What is the simple rule that will enable the manager of the Grand Hotel to make rooms instantly available for an infinite number of new guests?
2) How, if Achilles starts a race one step behind a tortoise, can Achilles ever catch up? As noted in this ancient Greek paradox of Zeno, by the time Achilles takes the first step, the tortoise will have moved a slight distance to a new position. By the time Achilles reaches this new position, the tortoise will have moved yet again, to a second position. And by the time Achilles reaches this second position, the tortoise will have moved yet again – and so on, and so on, for an infinite number of moves.
3) How about some of the other famous thought experiments, as presented in these charming versions by the Open University?
4) Has daydreaming helped you work though any of these thought experiments? Or solved any other problems that once had you stumped?
5) What are your favorite thought experiments?
6) Is it time for a dip in the Grand Hotel pool?