快猫短视频

IT IS OFTEN said that there are only seven narrative lines for a novel, all
known to the ancient Greeks. There seem to be even fewer ways to write a
mathematical proof, and the ancient Greeks knew only one of these narrative
lines. That was the short, sweet, compellingly clever kind of argument that made
Euclid鈥檚 reputation. No one ever asks why such proofs are necessary. You set out
an interesting mathematical statement, prove that it鈥檚 right by some brilliant,
concise insight that occupies just a few lines of maths, and your reputation is
made. Everyone exclaims at the elegance of the maths, the beauty of the
mathematical world. Everyone now understands exactly why your statement is
right. Everyone is happy.

Paul Erd枚s, the eccentric but brilliant mathematician who collaborated
with more people than anyone else on the planet, thought the same way. He
reckoned that up in heaven God had a book that contained all the best proofs. If
Erd枚s was really impressed by a proof, he declared it to be 鈥渇rom The
Book鈥�. In his view, the mathematician鈥檚 job is to sneak a look over God鈥檚
shoulder and pass on the beauty of His creation to the rest of His creatures.
(See 鈥淪hort and sweet鈥�.)

But now it seems that this simple, elegant approach is just one of a number
of possible narrative lines for mathematical proofs. Take the proofs that have
been hitting the headlines in the past year or two. Rather than the short,
compelling story line known to the Greeks, these are blockbusters, running to
hundreds or even thousands of pages. What happened to the beauty of God鈥檚
creation? Are these mammoth proofs really necessary? And are they so vast only
because mathematicians are being too stupid to find the really short, really
clever versions written in The Book?

Well, one answer is that there鈥檚 nothing to say that every short, simple and
true statement should have a short, simple proof. In fact, there鈥檚 good reason
to believe the opposite. The Austrian-born mathematician Kurt G枚del proved
in principle that short statements can sometimes require long proofs. He just
didn鈥檛 know which statements these were鈥攁nd neither does anyone else.

Many of the most significant proofs of the past few years have certainly been
long and complicated. Take Fermat鈥檚 last theorem, famously cracked in 1996 by
British-born mathematician Andrew Wiles, working at Princeton University in New
Jersey. To solve the problem, Wiles had to use massive mathematical machinery,
battering the question into submission like a gnat beneath a steam hammer. But
far from being boring and unnecessary, the resulting proof is rich and
beautiful鈥攏ot a short story, like the proofs in The Book, but a War
and Peace.

The tale of how Fermat鈥檚 theorem came into being bears retelling. In 1637,
Pierre de Fermat, a French lawyer with more mathematical ability in his little
fingernail than most of us have in our entire heads, made a fateful annotation
in his personal copy of the Arithmetica of Diophantus. His note relates
to Pythagoras鈥檚 theorem that
a² + b² = c² for certain
whole numbers a, b and c. There are plenty of
different values of a, b and c for which this works
fine. Each combination, in fact, makes up the sides of right-angled triangles,
with c as the hypotenuse.

Fermat tried to do the same kind of thing with cubes or fourth powers, and
couldn鈥檛 find any examples. In other words, he couldn鈥檛 find an equation of the
form aⁿ + bⁿ = cⁿ,
where a, b and
c are any whole numbers at all, and n is a whole number bigger
than 2. Did that mean that no such equation could possibly exist? In the margin
of his book, Fermat wrote that he had found a marvellous proof that the
Pythagorean relationship only worked for powers of 2, but added that 鈥渢his
margin is too small to contain it鈥�.

Secret strategy

Such a proof, even though it couldn鈥檛 fit in a margin, would surely be
concise and elegant enough to earn a place in God鈥檚 aesthetic book wouldn鈥檛 it?
Yet for three and a half centuries mathematician after mathematician came a
cropper trying to find it. Then, in the late 1980s, British mathematician Andrew
Wiles at Princeton University in New Jersey began an extended attack on the
problem. He worked alone in the attic of his house, telling only a few select
colleagues who were sworn to secrecy.

Wiles鈥檚 strategy, like that of many before him, was to assume that the
equation with a, b, c and n existed, and
then play with the numbers algebraically in the hope that they would lead him to
a contradiction. His starting point was an idea emanating from, among others,
Gerhard Frey of the University of Essen in Germany. Frey realised that you can
construct a type of cubic equation known as an elliptic curve from the three
roots a, b and c of Fermat鈥檚 鈥渋mpossible鈥� equation.
This was a brilliant idea, because mathematicians had been playing with elliptic
curves for more than a century, and had developed plenty of ways of manipulating
them. What鈥檚 more, mathematicians then realised that the elliptic curve made
from Fermat鈥檚 roots would have such strange properties that it would contradict
another conjecture鈥攌nown as the Taniyama-Shimura-Weil
conjecture鈥攚hich governs the behaviour of such curves.

The roots of Fermat鈥檚 equation would contravene the Taniyama-Shimura-Weil
conjecture, which means that proving the conjecture was right would show that
the roots could not exist. So for seven years, Wiles brought every big gun of
number theory to bear on the conjecture, until he came up with a strategy that
cracked it wide open. Although he worked alone, he didn鈥檛 invent the whole area
by himself. He kept in close touch with all new developments on elliptic curves,
and without a strong community of number theorists creating a steady stream of
new techniques he probably would not have succeeded. Even so, his own
contribution is massive, and it is propelling the subject into exciting new
territory.

Wiles鈥檚 proof has now been published in full. It is a little over 100 pages
long鈥攃ertainly too long to fit into a margin. Was it worth the effort?
Absolutely. The machinery that Wiles developed to crack Fermat鈥檚 last theorem is
extremely rich and beautiful. His ideas are opening up whole new areas of number
theory. Agreed, the story he had to tell was long, and only experts in the area
can understand it in any detail. But it makes no more sense to complain about
that than it would to complain that to read Tolstoy in the original you have to
be able to understand Russian.

There is a third narrative style for proofs鈥攐ne that has appeared only
in the past 30 years or so. This is the computer-assisted proof, and it is like
a fast-food outlet that serves billions of dull, repetitive burgers. It does the
job, but not prettily. There are often some clever ideas, but their job is to
reduce the problem to a massive, routine, calculation. This is then entrusted to
a computer, and if the computer says 鈥測es鈥� then the proof is complete.

Pack `em in

An example of this kind of proof turned up last year. In 1611, Johannes
Kepler was considering the ways that spheres could be packed together. He came
to the conclusion that the most efficient method鈥攖he one that packed as
many balls as possible into a given region鈥攚as the one that greengrocers
use to stack oranges. Make a flat layer in a honeycomb pattern, then stack
another such layer on top, sitting in the depressions of the first layer, and
continue like this forever. This pattern shows up in lots of crystals, and
physicists call it the face-centred cubic lattice.

It is often said that Kepler鈥檚 statement is 鈥渙bvious鈥�, but anyone who thinks
this way doesn鈥檛 appreciate the subtleties. For example, it is not even clear
that the most efficient arrangement includes a flat plane of spheres.
Greengrocers start their stacking from a flat surface, but you don鈥檛 have to
start like that. Even the two-dimensional version of the problem, which shows
that a honeycomb pattern is the most efficient way to pack equal circles in a
plane, wasn鈥檛 proved until 1947, by a Hungarian mathematician called Laszlo
Fejes T贸th. About ten years ago Wu-Yi Hsiang from the University
California at Berkeley announced a proof of the three-dimensional version, some
200 pages long, but gaps emerged in the proof and eventually other
mathematicians refused to accept it. Last year, however, Thomas Hales from the
University of Michigan in Ann Arbor announced a computer-assisted proof that
involved hundreds of pages of mathematics plus a vast quantity of supporting
computer calculations. It was initially published on his Web page, and is now
undergoing peer review for publication in a mathematical journal.

Hales鈥檚 approach was to write down a list of all the possible ways to arrange
suitable small clusters of spheres, then prove that whenever the cluster is not
what you find in the face-centred cubic lattice it can be compressed by
rearranging the spheres slightly. Conclusion: the only incompressible
arrangement鈥攖he one that fills space most efficiently鈥攊s the
conjectured one. This is how T贸th handled the two-dimensional case, and
he needed to list about 50 possibilities. Hales had to deal with thousands, and
the computer had to verify an enormous list of inequalities that took up 3
gigabytes of computer memory.

One of the earliest proofs to use this brute-force computer method was the
proof of the four-colour theorem. Almost a century and a half ago, the British
mathematician Francis Guthrie asked whether every possible two-dimensional map
containing any arrangement of countries can be coloured using only four colours,
with neighbouring countries always getting different colours. It sounds simple,
but the proof turned out to be elusive. Eventually, in 1976, American
mathematicians Kenneth Appel and Wolfgang Haken found it. By trial and error and
hand calculations they first came up with a list of nearly 2000 configurations
of countries. Then they enlisted the computer to prove that the list is
鈥渦navoidable鈥�, meaning that every possible map must contain countries arranged
in the same way as at least one configuration in the list.

The next step was to show that each of these configurations is 鈥渞educible鈥�.
That is, a part of each configuration can be shrunk down until it disappears,
leaving a simpler map. Crucially, the shrinking must ensure that if the simpler
map left behind can be coloured with four colours, the original one can be as
well.

Now imagine the simplest possible map that would require five or more
colours鈥攖he so-called 鈥渕inimal criminal鈥�. Like all maps, this must contain
at least one of the 2000 reducible configurations. Shrink this configuration and
you obtain a map that is simpler than the minimal criminal, and must therefore
be law-abiding, needing only four colours. But that means the minimal criminal
would only need four colours too. The only way out of this contradiction is if
no criminals exist.

Actually, the process involves more general techniques than just shrinking
regions, but you get the idea. Matching every configuration with a way to shrink
it involved a huge computer calculation, which took about 2000 hours on the
fastest computer then available, but nowadays takes maybe an hour. But at the
end, Appel and Haken had their answer.

Computer-assisted proofs raise a number of problems: issues of taste, of
creativity, of technique, and of philosophy. Some philosophers feel that the
brute-force methods of computer proofs mean they are not actually proofs at all,
in the traditional sense. Others point out that this kind of massive but routine
exercise is what computers do very well, and human beings very badly. If a
computer and a human being both carry out a huge calculation and get different
answers, the smart money is on the computer.

Any one calculation by the computer is usually trivial and dull. It鈥檚 only
when you string them together that they are worth anything at all. If Wiles鈥檚
proof of Fermat鈥檚 last theorem is rich in ideas and form鈥攍ike War and
Peace鈥攖he computer proofs are more like telephone directories. And
who would ever want to read those? In fact, for the Appel-Haken and Hales proofs
life is, quite literally, too short for anyone to read, let alone check
them.

But the proofs are not devoid of elegance and insight. After all, you have to
be clever about how you set up the problem for the computer to tackle. What鈥檚
more, once you know the conjecture is right, you can try to find a more elegant
way to prove it. This might sound strange, but it is much easier to prove
something you already know is right. In mathematical common rooms you will
occasionally overhear conversations in which someone suggests鈥攐nly partly
as a joke鈥攖hat it might be a good idea to spread rumours that some
important problem has been solved, in the hope that this might make a proof
easier for someone else to find. Does this mean that eventually mathematicians
may find God鈥檚 proofs for Kepler, Fermat and the rest? It would be wonderful if
they did. But maybe they won鈥檛. Perhaps there are no proofs of those theorems in
The Book. There is no reason why every theorem that is simple to state must have
a simple proof. We all know that many other tremendously difficult problems are
deceptively easy to state: 鈥渓and on the Moon鈥�, 鈥渃ure cancer鈥�. Why should maths
be different?

Experts often get rather strong feelings about possible proofs: either that
the best-known one can鈥檛 be simplified, or that alternative methods that someone
is proposing can鈥檛 possibly work. Often they are right, but sometimes their
judgment can be affected by knowing too much. Think of a mountain. Zigzag paths
up its slopes are the natural way to climb it, but if it鈥檚 a high mountain, with
glaciers and crevasses and the like, the 鈥渙bvious鈥� path may be exceedingly long
and complicated. It鈥檚 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. Just occasionally, someone invents such a
piece of machinery and proves all the experts wrong.

Shrink to fit

On the other hand, remember G枚del and his discovery that some proofs
simply have to be long. Perhaps the four-colour theorem and Fermat鈥檚 last
theorem are examples of these. For the four-colour theorem, it is possible to do
some back-of-the-envelope calculations which show that if you want to use the
current approach鈥攆inding a list of unavoidable configurations and then
eliminating them one at a time by some 鈥渟hrinking鈥� process鈥攖hen nothing
radically shorter is possible. But that, in effect, is just counting the likely
crevasses. It doesn鈥檛 rule out a helicopter.

Which brings us back to Fermat鈥檚 scribbled note. If these massive tomes are
the best we can do, why did he write what he did? Surely he can鈥檛 have stumbled
across a 200-page proof, and jotted down that it didn鈥檛 quite fit into the
margin.

I have an alternative theory. Godfrey Hardy, a brilliant Cambridge
mathematician, was definitely no atheist, but he was not conventionally
religious either. Hardy was convinced God had it in for him. So whenever he
travelled by boat鈥攚hich he hated鈥攈e would send a telegram: 鈥淗ave
just proved Riemann hypothesis. No room to give details here.鈥� The Riemann
hypothesis, which relates prime numbers to complex analysis, was, and still is,
the most important unsolved problem in mathematics. Hardy was convinced God
would not let the boat sink, because if that happened Hardy might be given
posthumous credit for possibly having found the proof.

Perhaps Fermat had a similar idea. Or maybe he just wanted to be famous. If
so, it certainly worked.

MANY proofs do have all the characteristics they would need to appear in
God鈥檚 book鈥攁 brilliant, incisive idea creating a short, snappy proof that
you can digest at a single sitting like an exquisitely written short story. John
Milnor from the State University of New York at Stony Brook is a virtuoso at
such proofs. For example, he found an answer to a question first raised in the
1960s: can you hear the shape of a drum鈥攖hat is, can you reconstruct the
shape of an object from the pattern of its vibrations? In 1992, Milnor proved
that the answer, in high dimensions at least, is no. He found two distinct
16-dimensional tori which give rise to the same vibrational pattern, and his
entire proof occupies one short journal page.

Another example is the solution to the Seifert conjecture arrived at by
Krystyna Kuperberg of Auburn University in Alabama. This concerns a mathematical
object known as a 3-sphere, which sounds bizarre but is very familiar (and
entirely unremarkable) to mathematicians. It is like the surface of an ordinary
sphere, except that everything is beefed up by one dimension. In other words, it
is the three-dimensional 鈥渉ypersurface鈥� of a hypersphere in four-dimensional
space. Now imagine a fluid flowing through the 3-sphere in such a way that the
paths followed by the fluid particles are smooth curves with no sharp corners.
Several decades ago, Seifert asked whether every possible smooth flow like this
would necessarily include either a fixed point where the speed of flow is zero,
or a closed loop. His conjecture was that the answer is yes, and most
mathematicians were convinced that he was right. The main evidence was negative:
a lot of mathematicians had tried to find a smooth flow without fixed points and
closed loops, but nobody had succeeded.

Kuperberg proved them wrong. The answer to Seifert鈥檚 question is no. Her
solution overcame what everyone considered to be the biggest
obstacle鈥攄ealing with smoothness鈥攚ith a brilliant trick in which she
made the flow 鈥渆at its own tail鈥�, as a result of which smoothness ceased to be
an issue. Armed with this key idea, any professional could reconstruct the
entire proof from the half-page description of it published in
快猫短视频(鈥淗airy balls in higher dimensions鈥�,
13 November 1993, p 18).