Episode 03 - The art of the good-enough answer

Tomlin Hawthorne

How many piano tuners are there in Chicago?

Tomlin Hawthorne

You don't know. I don't know. Nobody has ever known this off the top of their head, except possibly one very lonely official at the Piano Technicians Guild.

Tomlin Hawthorne

But I'd like you to say a number anyway. Out loud, if you're alone. Ten? A thousand? Fifty thousand?

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Keep hold of it. In about ten minutes you are going to work out the real answer, to within a factor of two or so, without looking anything up. And you'll be slightly annoyed at how easy it was.

Tomlin Hawthorne

Last episode I asked you to mark a strip of paper with rungs, one for each factor of ten, and place ten things on it. I predicted you'd be worst at the things nearest home. If you put the Moon about ten times too close, you are in the majority. It is thirty Earths away. In nearly every diagram you've seen, it's drawn about three Earths away, because otherwise it won't fit on the page.

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Today is a skills episode. There's one technique I want you to have before we go any further, because physicists use it constantly and almost never mention it. The question is: how do they know roughly what the answer is before they've calculated anything?

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The technique is named after Enrico Fermi, and to explain why, I need to take you to the New Mexico desert at half past five in the morning on the sixteenth of July, 1945.

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Fermi is about ten miles from a steel tower. On top of the tower is the first atomic bomb, which is about to be tested. Nobody knows how big the explosion will be. There was a betting pool among the scientists. The guesses ranged from a dud to something that would set the atmosphere alight, and I am told the second one was mostly a joke.

Tomlin Hawthorne

Fermi was one of the great physicists of the century. Italian, Nobel Prize, built the first nuclear reactor under a sports stand in Chicago. And he had a peculiar gift. He was world-class at both theory and experiment, which almost nobody is, and he had a horror of not knowing a number.

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So the bomb goes off. There's the flash, and then, because sound and pressure travel slower than light, a wait of about forty seconds.

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And during that wait, Fermi takes a sheet of paper and tears it into small pieces. When the blast wave arrives, he holds his hand up at about head height and lets the pieces fall.

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They drift sideways, and land about two and a half metres behind him. He paces it out.

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Then he does some arithmetic in his head and announces that the bomb was equivalent to about ten thousand tons of TNT.

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The official figure took weeks, using instruments that had cost a fortune. It came out around twenty thousand tons. He was within a factor of two, using litter.

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So let's do it. I'll go slowly, and you check each step against your own sense of things.

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Step one. You can't answer the big question, so you break it into small ones you can.

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How many people live in Chicago? You might not know exactly. It's a very big American city, not the biggest. Call it three million.

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How many people in a household? Two or three. Let's say three million people, a bit over a million households. Call it a million.

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What fraction of households has a piano? Think about the homes you've been in. Not one in two. Not one in a thousand. One in twenty feels about right. So fifty thousand pianos. Add some for schools, churches, bars. Still roughly fifty thousand.

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How often is a piano tuned? Should be once a year. Most are neglected. Let's say on average once a year anyway, since the concert halls do theirs constantly and make up for your aunt.

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So: fifty thousand tunings a year need doing.

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Now the other side. How many can one tuner do? A tuning takes a couple of hours, plus travel. Four a day is a full day. Five days a week, fifty weeks a year. That's a thousand tunings a year per tuner.

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Fifty thousand tunings, a thousand each. Fifty piano tuners.

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The real number, as far as anyone has checked, is somewhere in the range of fifty to a couple of hundred, depending on how you count part-timers.

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How did your guess at the top compare? If you said fifty thousand, you'd now see straight away that that's a tuner for every piano. They'd have to live with the instrument.

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I want to pick that apart, because something a bit magical happened and it's worth seeing why.

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Every one of those numbers was a guess. Chicago's population I got roughly right. The piano fraction I honestly made up. So how does a chain of made-up numbers arrive near the truth?

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Two reasons.

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The first is that the errors don't all point the same way. I probably guessed too high on some steps and too low on others. When you multiply them together, a too-high and a too-low partly cancel. It's the same reason a crowd guessing the weight of an ox does better than most of the individuals in it. For the whole thing to go badly wrong, I'd need to be wrong in the same direction every single time, and that takes real commitment.

Tomlin Hawthorne

The second reason is subtler. For each small question, I had a feel for the boundaries. Pianos: not half of homes, not one in a thousand. I couldn't tell you the answer, but I could fence it in. And you nearly always can. The trick, when you're stuck, is to stop asking "what is it?" and ask "what's obviously too small, and what's obviously too big?"

Tomlin Hawthorne

Then you take the middle. But here's the one technical point in today's episode: you take the middle on the ladder. Remember the rungs from last time? If the answer is somewhere between ten and a thousand, the fair guess is not five hundred. It's a hundred. One rung up from the bottom, one rung down from the top.

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That's the whole method. Break it up. Fence each piece in. Take the middle rung. Multiply.

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Now, you may be thinking this is familiar. If you have ever sat a consulting interview, or sized a market for a business plan, somebody asked you how many golf balls fit in a bus, or how many nappies are sold in Brazil. That's Fermi. The management consultancies took it from the physicists and, in the proper tradition, did not send a thank-you note.

Tomlin Hawthorne

And it's used the same way in both worlds. Not to get the answer. To find out whether an answer is even sane. Before a physicist spends six months on a calculation, they spend ten minutes on the back of an envelope, so that when the six-month answer comes out a million times too big, they know something has gone wrong and it isn't the universe.

Tomlin Hawthorne

Let me show you the version of this that I think is close to art.

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Two years after that test in the desert, the American government released a set of photographs of the fireball. A sequence, frame after frame, each stamped with the time in thousandths of a second and with a scale bar. The size of the explosion was still top secret. The pictures, they reckoned, gave nothing away.

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In England, a physicist named Geoffrey Taylor looked at the photographs in a magazine. And he reasoned like this. What could the size of that fireball possibly depend on? The energy released, obviously. The time since the explosion. And the density of the air it has to push out of the way. And really nothing else.

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Now, each of those is measured in some unit. Energy, seconds, kilograms per cubic metre. And it turns out there is only one way to combine those three things so that the units come out as a length. One way. The arithmetic is forced.

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So he read the radius off the scale bar, read the time off the stamp, looked up the density of air, and solved for the energy. He published the answer: about seventeen thousand tons of TNT.

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It was correct. The Americans were, by all accounts, not delighted. He had worked out their most closely guarded number from a magazine.

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I think that's one of the most elegant things anyone has ever done with a photograph. And the idea in it is one we'll use again: quantities in physics carry units, and the units are not decoration. They constrain what the answer is allowed to be. You can't add a distance to a time any more than you can add a Tuesday to a colour.

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Let's do one together that has some wonder in it.

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You've probably heard the claim that every breath you take contains a molecule from Julius Caesar's dying breath. It sounds like the kind of thing that's printed on a tea towel. Let's find out whether it's true.

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First: how many molecules in one breath? A breath is about a litre of air. And a litre of air, here's a number you'd have to be told, holds about twenty-five thousand billion billion molecules. On the ladder, that's twenty-two rungs. A one with twenty-two zeros.

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So Caesar, on the Ides of March, breathes out ten-to-the-twenty-two molecules. And in two thousand years, the wind has stirred them evenly through the whole atmosphere.

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Second: how many breaths' worth of air is the whole atmosphere? This you can estimate, but I'll just give it to you. It's about ten-to-the-twenty-two as well. A little less.

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And look at what's just happened. Those two numbers are nearly the same.

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The number of molecules in one breath is about equal to the number of breaths in the sky.

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So Caesar's molecules are spread through the atmosphere at a concentration of a few per breath. Each time you inhale, you take in, on average, a handful of them. Probably not the same ones twice.

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It's true. The tea towel is right.

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And I'd like you to sit with that, because the fact about Caesar is charming but it isn't the point. It works for anyone. Cleopatra. Whoever painted the caves at Lascaux. The person you miss most.

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The point is what it tells you about atoms. A breath seems like nothing. And it contains so many molecules that when you dilute them into the entire sky of a planet, there's still one in every lungful. That's how small they are, and how many. We'll come back to atoms properly in episode eleven, but I wanted you to have felt the number first, in your own chest.

Tomlin Hawthorne

The myth of the week is going to be especially for those of you who trained in the humanities or in law. I say this with love.

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The myth is that it is irresponsible to give a number you can't defend.

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You were taught, rightly, not to assert things without evidence. So when someone asks how many piano tuners, the careful, educated response is: I couldn't possibly say.

Tomlin Hawthorne

In physics that is the worst available answer. It sounds modest and tells nobody anything. A physicist would much rather hear "about a hundred, and I could be off by five times either way." That's a real claim. It can be checked, improved, argued with. It's Anaximander again, from episode one: put something on the table that can be wrong.

Tomlin Hawthorne

Being off by a factor of two is perfectly respectable. Fermi was off by two with the bomb. The thing to be embarrassed about is not knowing whether the answer is a thousand or a billion, because that means you don't yet understand the question.

Tomlin Hawthorne

So for the rest of this series, when I ask you to guess, guess. Commit. Write it down. A wrong number you committed to will teach you ten times more than a right one you were handed.

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A question to take with you.

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There are places where a rough number beats a careful argument, and I'd say they're more common than we admit. How many people does this policy actually affect: thousands, or millions? Is this risk one in a hundred, or one in a hundred million? A lot of public shouting would quieten down if both sides did ten minutes on an envelope first.

Tomlin Hawthorne

But there are also places where a rough number is dangerous. Where it gives a false air of rigour to something that can't be measured, or where the one-in-a-million case is a person. You'll have examples from your own field, and I'd guess you've seen both kinds of mistake.

Tomlin Hawthorne

So where is the line? When should we insist on a number, and when should we be suspicious of one?

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And your challenge. Two estimates, ten minutes each, no looking things up until you've finished.

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One. How many times will your heart beat in your lifetime? Break it up. Beats a minute, minutes in a day, days in a year, years you're hoping for.

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Two. What does a cloud weigh? An ordinary white fair-weather cloud, the fluffy sort a child would draw. I'll give you one fact, because nobody could guess it: a cloud holds about half a gram of water in each cubic metre. About a raindrop's worth per fridge. You'll need to decide how big a cloud is.

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Write your answer down before you check. I'll tell you now that the second one surprises almost everybody, and that afterwards you will look at the sky a little differently, and perhaps not stand directly underneath.

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Next time, we meet the man who started modern physics by writing a play, and who worked out what a moving object does when you leave it alone, which is not what you think, and not what anybody thought for two thousand years.