configuration that can be shrunk again and so on. The upshot is that if you can find a way to shrink every possible configuration, you have your proof. Matching every configuration with a way to “shrink” it like this involved a huge but routine computer calculation, which in those days took about two thousand hours on the fastest available computer. (Nowadays it takes maybe an hour.) But in the end, Appel and Haken had their answer.
Computer-assisted proofs raise issues of taste, creativity, technique, and philosophy. Some philosophers feel that with their brute-force methods, they are not proofs in the traditional sense. Yet this kind of massive but routine exercise is what computers were invented to do. It’s what they’re good at, and it’s what humans are very poor at. If a computer and a human being both carry out a huge calculation and get different answers,the smart money is on the computer. But it must be said that any one bit of the proof, any one calculation by the computer, is usually trivial and extremely dull. It’s only when you string them together that they’re worth anything. If Wiles’s proof of Fermat’s last theorem is rich in ideas and form—like War and Peace —the computer proofs are more like telephone directories. In fact, for the Appel–Haken proof, and even more so Hales’s proof, life is—literally—too short to read the whole thing in full detail, let alone check it.
Still, these proofs are not devoid of elegance and insight. You have to be pretty clever about how you set up the problem for the computer to tackle. What’s more, once you know the conjecture is right, you can set about trying to find a more elegant way to prove it. This might sound strange, but it’s well known among mathematicians that it’s much easier to prove something you already know is true. In mathematical common rooms worldwide, you will occasionally hear someone suggest— only partly as a joke—that it might be a good idea to spread rumors that some important problem has been solved, in the hope that this might speed up its actual solution. It’s a bit like crossing the Atlantic. For Christopher Columbus this was desperately hard, but it was easier for John Cabot, sailing just five years later, because he knew what Columbus had found.
Does this mean that eventually mathematicians may find God’s proofs for Kepler and the four color theorem?Maybe, but maybe not. It’s a bit naive to imagine that every theorem that is simple to state must therefore have a simple proof. We all know that many tremendously difficult problems are deceptively easy to state: “land on the moon,” “cure cancer.” Why should math be different?
Experts often get rather passionate about proofs, either that the best-known one can’t be simplified, or that alternative methods that someone is proposing can’t possibly work. Often they’re right, but sometimes their judgment can be affected by knowing too much. If you’re an experienced mountain climber proposing to scale a high peak, with glaciers and crevasses and the like, the “obvious” path may be exceedingly long and complicated.
It’s natural, too, to assume that the sheer cliff face, which seems to be the only alternative route, is simply unclimbable. But it may be possible to invent a helicopter that can swing you quickly and easily up to the top.
The experts can see the crevasses and the cliff, but they may miss a good idea for the design of a helicopter. Occasionally someone invents such a piece of machinery out of the blue, and proves all the experts wrong.
On the other hand, think of Gödel. We know that some proofs simply have to be long, and perhaps the four color theorem and Fermat’s last theorem are examples.
For the four color theorem, it is possible to do some back-of-the-envelope calculations that show that if you want to use the current approach—finding a list of “unavoidable”configurations and then eliminating them one at a time by
Computer-assisted proofs raise issues of taste, creativity, technique, and philosophy. Some philosophers feel that with their brute-force methods, they are not proofs in the traditional sense. Yet this kind of massive but routine exercise is what computers were invented to do. It’s what they’re good at, and it’s what humans are very poor at. If a computer and a human being both carry out a huge calculation and get different answers,the smart money is on the computer. But it must be said that any one bit of the proof, any one calculation by the computer, is usually trivial and extremely dull. It’s only when you string them together that they’re worth anything. If Wiles’s proof of Fermat’s last theorem is rich in ideas and form—like War and Peace —the computer proofs are more like telephone directories. In fact, for the Appel–Haken proof, and even more so Hales’s proof, life is—literally—too short to read the whole thing in full detail, let alone check it.
Still, these proofs are not devoid of elegance and insight. You have to be pretty clever about how you set up the problem for the computer to tackle. What’s more, once you know the conjecture is right, you can set about trying to find a more elegant way to prove it. This might sound strange, but it’s well known among mathematicians that it’s much easier to prove something you already know is true. In mathematical common rooms worldwide, you will occasionally hear someone suggest— only partly as a joke—that it might be a good idea to spread rumors that some important problem has been solved, in the hope that this might speed up its actual solution. It’s a bit like crossing the Atlantic. For Christopher Columbus this was desperately hard, but it was easier for John Cabot, sailing just five years later, because he knew what Columbus had found.
Does this mean that eventually mathematicians may find God’s proofs for Kepler and the four color theorem?Maybe, but maybe not. It’s a bit naive to imagine that every theorem that is simple to state must therefore have a simple proof. We all know that many tremendously difficult problems are deceptively easy to state: “land on the moon,” “cure cancer.” Why should math be different?
Experts often get rather passionate about proofs, either that the best-known one can’t be simplified, or that alternative methods that someone is proposing can’t possibly work. Often they’re right, but sometimes their judgment can be affected by knowing too much. If you’re an experienced mountain climber proposing to scale a high peak, with glaciers and crevasses and the like, the “obvious” path may be exceedingly long and complicated.
It’s natural, too, to assume that the sheer cliff face, which seems to be the only alternative route, is simply unclimbable. But it may be possible to invent a helicopter that can swing you quickly and easily up to the top.
The experts can see the crevasses and the cliff, but they may miss a good idea for the design of a helicopter. Occasionally someone invents such a piece of machinery out of the blue, and proves all the experts wrong.
On the other hand, think of Gödel. We know that some proofs simply have to be long, and perhaps the four color theorem and Fermat’s last theorem are examples.
For the four color theorem, it is possible to do some back-of-the-envelope calculations that show that if you want to use the current approach—finding a list of “unavoidable”configurations and then eliminating them one at a time by
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