THERE can鈥檛 be many people that would turn down a Nobel prize. So after the International Mathematical Union announced this week that it was awarding what some consider the mathematical equivalent, the prestigious Fields medal, to the Russian mathematician Grigori Perelman, it may seem surprising that Perelman has decided to refuse it. He was supposed to have received the award this week from King Juan Carlos of Spain at the International Congress of Mathematicians in Madrid, but as 快猫短视频 went to press, he looked set to turn it down.
Perelman was to have been awarded the Fields medal for work made public four years ago that proves a century-old idea about the nature of four-dimensional geometry, the Poincar茅 conjecture. There is more at stake in Perelman鈥檚 snub than wounded pride. In a confusing twist, two Chinese mathematicians, Huai-Dong Cao of Lehigh University in Pennsylvania and Xi-Ping Zhu of Harvard University, published 鈥渁 first written account of a complete proof鈥� of the Poincar茅 conjecture in June, according to The Asian Journal of Mathematics, where it appeared. The underlying issue that emerges when you add together Perelman鈥檚 work and attitude, the Chinese claims, and the problems of attributing proper credit, is that mathematicians are finding it increasingly difficult to decide whether or not something has been proved.
Proof is supposed to be what sets mathematics apart from the other sciences. Traditionally, the subject has not been an evolutionary one in which the fittest theory survives. New insights don鈥檛 suddenly overturn the theorems of the previous generation. The subject is like a huge pyramid, with each generation building on the secure foundations of the past. The nature of proof means that mathematicians, to use Newton鈥檚 words, really do stand on the shoulders of giants.
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In the past, those shoulders have been extremely steady. After all, in no other science are the discoveries of the Ancient Greeks still as valid today as they were at the time. Euclid鈥檚 2300-year-old proof that there are infinitely many primes is perhaps the first great example of a watertight proof.
It works like this. Suppose a mathematician comes with a finite list of primes and claims there are no more. Euclid showed that there must be a prime missing from the list. Multiply all the primes on the list together and then add one to this number. This new number is not divisible by any of the primes on the list because you always get remainder one. So Euclid鈥檚 new number is either another prime itself or divisible by a prime that is missing from the list. If you add this new prime to the list, repeating Euclid鈥檚 trick will always show that any finite list is missing a prime.
Euclid鈥檚 proof is rapier-like in its uncompromising destruction of anyone who thinks there are only a finite number of primes. It is also surprisingly simple, which means you can check it.
鈥淣othing is able to garner Perelman鈥檚 cooperation, not even the million-dollar prize set aside for solving the conjecture鈥�
Since Euclid鈥檚 time, proofs have become ever more sophisticated, with some now extending over thousands of pages. They still have to be checked, of course, and that has become a daunting task. The proof of the classification of finite simple groups, a kind of periodic table of mathematical symmetry, for example, was announced in 1982. Stretching to over 10,000 pages, it was authored by hundreds of mathematicians 鈥� and it turned out to be incomplete. In the early 1990s mathematicians trying to master the argument in its entirety discovered that a portion of the proof was missing. After battling for some years the gap was finally plugged in 2004, but it took a paper whose proof was more than 1200 pages long.
Such logical holes are one thing, but mathematicians have also had to come to terms with the possibility of a new kind of error: computer programming mistakes. In 1977, the four-colour theorem, a suggestion that any political map can be shaded using just four colours without any borders sharing the same two colours, became the first major theorem to be proved with the help of a computer. The proof has s urvived nearly 30 years without someone redrawing the boundaries of Europe and finding they need five colours, but the possibility still remains that a glitch is hiding somewhere in the mass of computer code that could kill the proof.
A further twist on the computer proof issue came in 1998, when Thomas Hales of the University of Pittsburgh in Pennsylvania announced a computer-assisted proof of the Kepler conjecture. This confirmed mathematically what every grocer intuitively knows: that a hexagonal pyramid is the most efficient way to stack oranges. Hales proved that no other configuration can fit more oranges into a given space than this hexagonal lattice.
鈥淗e doesn鈥檛 want to be used as a symbol to attract younger mathematicians to a profession he has become disillusioned with鈥�
In the proof Hales showed how the problem can be reduced to a large but finite number of calculations that would confirm that the grocer鈥檚 symmetrical stack of oranges is the most efficient. 鈥淟arge number鈥� hardly covers it, however: the calculations are so numerous that Hales needed the help of a computer to complete the proof. In the end, it consisted of 250 pages of conventional mathematical argument and over 3 gigabytes of computer code and data.
Trial by jury
So how do we know Hales鈥檚 proof is correct? If we鈥檙e being picky, we don鈥檛. The Annals of Mathematics, the premier mathematical journal, appointed a committee of 12 referees to check the proof. Most papers get one referee. The referees reported back that, due to the difficulty of checking all the computer data, they were only 99 per cent certain that the proof was correct. Mathematicians are used to 100 per cent certainty. This is why the involvement of computers has unsettled some in the community. Even deciding how to publish such a proof took much debate. Eventually the journal accepted the 250 pages of theoretical argument (Annals of Mathematics, vol 162, p 1063) but banished the computer data to another journal for publication.
Given the complexity of many modern proofs, perhaps it isn鈥檛 surprising that even now, four years on from the initial excitement over Perelman鈥檚 announcement, mathematicians are still cagey about whether he has really proved the Poincar茅 conjecture. In fact the award citation speaks of Perelman鈥檚 contributions to geometry and other revolutionary insights, and not of the Poincar茅 conjecture directly. The award committee is non-committal about whether the Poincar茅 proof is complete. It states that 鈥渢he mathematical community is still in the process of checking his work to ensure that it is entirely correct and that the conjectures have been proved鈥�. Nevertheless, the decision to award Perelman the Fields medals will be widely regarded as some validation of his proof.
Widely, but not exclusively. The Chinese paper, which stretches to 318 pages, claims to have completed what Perelman only started. Perelman鈥檚 preprints provided 鈥済uidelines鈥�, according to the Chinese newspaper the People鈥檚 Daily. 鈥淕uidelines are totally different to complete proof of theories,鈥� Yang Le, a member of the Chinese Academy of Sciences, told the newspaper.
It seems that the one person who doesn鈥檛 really care about what鈥檚 going on is Perelman. In 1996 he snubbed the European Mathematical Society when he turned down a prize; it looks like he鈥檚 done it again. John Ball, president of the International Mathematical Union, spent two days with him in St Petersburg in Russia, trying to persuade him to accept. Although Perelman鈥檚 reasons for turning down the prize are complicated, they centre on his feelings of isolation from the mathematical community and his desire not to be used as a symbol to attract younger mathematicians to a profession he has personally become disillusioned with.
Nothing so far seems to be able to garner Perelman鈥檚 cooperation 鈥� not even the million-dollar prize set aside for the person or people that prove the Poincar茅 conjecture. To collect the prize, offered by the Clay Mathematics Institute, the paper must be refereed by a reputable mathematical journal and survive the scrutiny of the wider mathematical community for two years after it is published. Perelman has not submitted his proof to a journal. Instead it remains in preprints that he posted on the web. It seems he has proved the conjecture to his own satisfaction and made the proof freely available to others, and for him that is enough.
If the Clay prize is ever claimed, the million dollars will be divided according to how much of the puzzle individual people have completed, and when they do this Clay鈥檚 legal and mathematical team are going to find themselves walking a tricky tightrope. There is an even more difficult task facing mathematics, however. Mathematicians are beginning to engage with the increasingly complex issue of what exactly constitutes a proof. Perhaps the subject is moving into a more Darwinian age of survival of the fittest. Hopefully what will outlive all the controversy over the Poincar茅 conjecture is the new dawn in our understanding of the geometry of space that Perelman has brought. Despite all the controversy, his work is both astounding and profound, and fully deserving of the highest recognition.
Beauty and the brute
In the mathematical universe, computers and brute force can only get you so far. Consider one of the great unsolved problems about prime numbers: can you generate a sequence of numbers in which the difference between all the numbers is the same 鈥� an arithmetic progression 鈥� of any length you want in which all the numbers are prime numbers?
For example, 3, 5, 7 is an arithmetic progression of primes of length three. For a progression of length four, you could take 5, 11, 17, 23, four primes that differ by 6. The largest known sequence has 23 primes in a row. If you start at the prime 56,211,383,760,397 and count on 44,546,738,095,860 you get another prime. Count on another 44,546,738,095,860 and you get a third prime. If you keep doing this you get 23 primes in an arithmetic progression. This was discovered (using a computer) in 2004 by Markus Frind, Paul Jobling and Paul Underwood.
However, none of this constitutes a proof that you can get any length sequence you want. The mathematical universe that is observable by experiment represents an infinitesimal and often unrepresentative fraction of the infinite expanse of numbers. Mathematicians have been deceived in the past by seemingly convincing but ultimately incomplete data.
In 2004, Terence Tao of the University of California, Los Angeles (who, as it happens, also won a Fields medal this week), in collaboration with Ben Green of the University of Bristol, laid the issue to rest beautifully, proving that it is theoretically possible to find a suitable sequence of any length you want somewhere in the universe of numbers. Compared with many proofs, Tao and Green鈥檚 proof is far from arduous in its length and complexity, but they needed about 50 pages and relied on the proofs of many other authors. The only frustrating thing is that it is a non-constructive proof: it tells you the sequences exist but it doesn鈥檛 tell you how to find them.
The Poincar茅 conjecture
Take a two-dimensional piece of malleable rubber sheeting. You can wrap it up to make the surface of a ball or roll it up and bend the resulting tube round to make a bagel shape with a hole in the middle, called a torus. These are two fundamentally different things, for the following reasons. If I tie a lasso around the ball, then I can shrink the loop in the lasso until it vanishes by sliding the lasso towards one of the 鈥減oles鈥�; every such loop can be shrunk to a point. But on the bagel, the loop in a lasso that is tied through the hole in the middle and back round cannot be shrunk to nothing.
Henri Poincar茅 knew 100 years ago that the ball was the only way to wrap up the two-dimensional sheet to make a shape around which the loop of a lasso could always be shrunk to nothing. The Poincar茅 conjecture concerns the same question one dimension up. Imagine wrapping up three-dimensional space in four dimensions to create various shapes, then tying lassos onto them. Are there 4D shapes around which lasso loops can always be shrunk to nothing? Grigori Perelman has shown that there is only one: the 4D equivalent of the sphere, known as the hypersphere.
