Free Letters to a Young Mathematician by Ian Stewart Page B

hoping to be let off the hook. A student who knows how to construct proofs never asks what they’re for. In fact, a student who knowshow to do long multiplication in his head while doing a handstand also never asks why that’s worth doing.
People who enjoy performing an activity hardly ever feel the need to question its worth; the enjoyment alone is enough. So the student who asks why we need proofs is probably having trouble understanding them, or constructing his own. He is hoping you will answer, “There’s no need to worry about proofs. They’re totally useless. In fact, I’ve taken them off the syllabus and they won’t come up in the test.”
    Ah, in your dreams.
    It’s still a good question, and if I leave it at what I’ve just said, I’m ducking the issue just as blatantly as any proofophobic student.
    Mathematicians need proofs to keep them honest. All technical areas of human activity need reality checks.
It is not enough to believe that something works, that it is a good way to proceed, or even that it is true. We need to know why it’s true. Otherwise, we don’t know anything at all.
    Engineers test their ideas by building them and seeing whether they hold up or fall apart. Increasingly they do this in simulation rather than by building a bridge and hoping it won’t fall down, and in so doing they refer their problems back to physics and mathematics, which are the sources of the rules employed in their calculations and the algorithms that implement those rules. Even so, unexpected problems can turn up. The Millennium Bridge, a footbridge across the Thames in London, looked fine in the computer models. When it opened and people started to use it, it suddenly started to sway alarmingly from side to side. It was still safe, it wasn’t going to fall down, but crossing it wasn’t an enjoyable experience. At that point it became clear that the simulations had modeled people as smoothly moving masses; they had ignored the vibrations induced by feet hitting the deck.
    The military learned long ago that when soldiers cross a bridge, they should fall out of step. The synchronized impact of several hundred right feet can set up vibrations and do serious damage. I suspect this fact was known to the Romans. No one expected a similar kind of synchronization to arise when individuals walked across the bridge at random. But people on a bridge respond to the movement of the bridge, and they do so in a similar way and at roughly the same time. So when the bridge moved slightly—perhaps in response to a gust of wind— the people on it started to move in synchrony. The more closely the footsteps of the people became synchronized, the more the bridge moved, which in turn increased the synchronization of the footsteps. Soon the whole bridge was swaying from side to side.
    Physicists use mathematics to study what they amusingly call the real world. It is real, in a sense, but much of physics addresses rather artificial aspects of reality, such as a lone electron or a solar system with only one planet. Physicists are often scathing about proofs, partly out of fear, but also because experiment gives them a very effective way to check their assumptions and calculations.
If an intuitively plausible idea leads to results that agree with experiment, there’s not much point in holding the entire subject up for ten, fifty, or three hundred years until someone finds a rigorous proof. I agree entirely. For example, there are calculations in quantum field theory that have no rigorous logical justification, but agree with experiment to an accuracy of nine decimal places. It would be silly to suggest that this agreement is an accident, and that no physical principle is involved.
    The mathematicians’ take on this, though, goes further. Given such impressive agreement, it is equally silly not to try to find out the deep logic that justifies the calculation. Such understanding should advance physics directly, but if not, it will