数学世纪

The
Mathematical
Century
The
30 Greatest
Problems
of tre
Last 1 00 Years

 

At the beginning of the seventeenth century, two great philosophers,
Francis Bacon in England and Rene Descartes in France, proclaimed
the birth of modern science. Each of them described his vision of
the future. Their visions were very different.
Bacon said, "All depends on keeping the eye steadily fixed on the
facts of nature. " Descartes said, "I think, therefore I am."
According to Bacon, scientists should travel over the
earth collecting facts, until the accumulated facts reveal how
Nature works. The scientists will then induce from the facts
the laws that Nature obeys. According to Descartes, scientists
should stay at home and deduce the laws of Nature by pure
thought. In order to deduce the laws correctly, the scientists
will need only the rules oflogic and knowledge of the existence
of God. For four hundred years since Bacon and Descartes led
the way, science has raced ahead by following both paths simultaneously.
Neither Baconian empiricism nor Cartesian dogmatism has the power
to elucidate Nature's secrets by itself, but both together have been
amazingly successful. For four hundred years,
English scientists have tended to be Baconian and
French scientists Cartesian.

 

Faraday and Darwin and Rutherford were Baconians: Pascal
and Laplace and Poincare were Cartesians. Science was greatly
enriched by the cross-fertilization of the two contrasting national
cultures. Both cultures were always at work in both countries.
Newton was at heart a Cartesian, using pure thought as Descartes
intended, and using it to demolish the Cartesian dogma of vortices.
Marie Curie was at heart a Baconian, boiling tons of crude uranium
ore to demolish the dogma of the indestructibility of atoms.

Piergiorgio Odifreddi has done a superb job, telling the
story of twentieth-century mathematics in one short and readable volume.
My only complaint about this book is that Piergiorgio's account is a
little too Cartesian for my taste. It presents
the history of mathematics as more orderly and logical than I
imagine it. I happen to be a Baconian while Piergiorgio is a
Cartesian. We agree about the historical facts. We only disagree
about the emphasis. This book is Piergiorgio's version of the truth.
My version would be slightly different.

In the Cartesian version of twentieth-century mathematics,
there were two decisive events. The first was the International
Congress of Mathematicians in Paris in 1900, at which Hilbert
gave the keynote address, charting the course of mathematics
for the coming century by propounding his famous list of
twenty-three outstanding unsolved problems. The second decisive
event was the formation of the Bourbaki group of mathematicians
in France in the 1930s, dedicated to publishing aseries of textbooks
that would establish a unifYing framework for all of mathematics.
The Hilbert problems were enormously successful in guiding mathematical
research into fruitful directions. Some of them were solved and some
remain unsolved,but almost all of them stimulated the growth of new ideas
and new fields of mathematics. The Bourbaki project was equally
influential. It changed the style of mathematics for the next
fifty years, imposing a logical coherence that did not exist before,
and moving the emphasis from concrete examples to abstract generalities.
In the Bourbaki scheme of things, mathematics is the abstract structure
included in the Bourbaki textbooks. What is not in the textbooks is not
mathematics.Concrete examples, since they do not appear in the textbooks,
are not mathematics. The Bourbaki program was the extreme
expression of the Cartesian style of mathematics. It narrowed
the scope of mathematics by excluding all the beautiful flowers
that Baconian travelers might collect by the wayside. Fortunately,
Piergiorgio is not an extreme Cartesian. He allows many concrete examples
to appear in his book. He includes beautiful flowers such as sporadic
finite groups and packings of spheres in Euclidean spaces. He even
includes examples from applied as well as from pure mathematics.
He describes the Fields Medals, which are awarded at the quadrennial
International Congress of Mathematicians to people who solve concrete
problems as well as to people who create new abstract ideas.

For me, as a Baconian, the main thing missing in this book
is the element of surprise. When I look at the history of mathematics,
I see a succession of illogical jumps, improbable coincidences,
jokes of nature. One of the most profound jokes of
nature is the square root of -1 that the physicist Erwin Schr6dinger
put into his wave equation in 1926. This is barely mentioned in
Piergiorgio's discussion of quantum mechanics in his
chapter 3, section 4. The Schr6dinger equation describes correctly
everything we know about the behavior of atoms. It is
the basis of all of chemistry and most of physics. And that
square root of -1 means that nature works with complex numbers
and not with real numbers. This discovery came as a complete surprise,
to Schr6dinger as well as to everybody else. According to Schrbdinger,
his fourteen-year-old girlfriend Itha Junger said to him at the time,
"Hey, you never even thought when you began that so much sensible stuff
would come out of it." All through the nineteenth century, mathematicians from
Abel to RIemann and Weierstrass had been creating a magnificent theory
offunctions of complex variables. They had discovered that the theory of
functions became far deeper and more powerful when it was extended from real
to complex numbers. But they always thought of complex numbers as an
artificial construction, invented by human mathematicians as a useful
and elegant abstraction from real life. It never entered their
heads that this artificial number system that they had invented
was in fact the ground on which atoms move. They never imagined that nature
had got there first.

Another joke of nature is the precise linearity of quantum
mechanics, the fact that the possible states of any physical object
form a linear space. Before quantum-mechanics was invented,
classical physics was always nonlinear, and linear models were
only approximately valid. After quantum mechanics,
nature itself suddenly became linear. This had profound consequences
for mathematics. During the nineteenth century, Sophus Lie developed
his elaborate theory of continuous groups,intended to clarifY the
behavior of classical dynamical systems. Lie groups were then oflittle
interest either to mathematicians or to physicists. The nonlinear theory
was too complicated for the mathematicians and too obscure for the physicists.
Lie died a disappointed man. And then, fifty years later, it turned out
that nature was precisely linear, and the theory oflinear representations
of Lie algebras was the natural language of particle physics. Lie groups
and Lie algebras were reborn as one of the central themes of twentieth-century
mathematics, as discussed in Piergiorgio's chapter 2, section 12.

A third joke of nature is the existence of quasi-crystals,
briefly discussed by Piergiorgio at the beginning of his chapter
3. In the nineteenth century, the study of crystals led to a complete
enumeration of possible discrete symmetry-groups in Euclidean space.
Theorems were proved, establishing the fact that in three-dimensional
space discrete symmetry groups could contain only rotations of order
three, four, or six. Then in 1984 quasi-crystals were discovered, real
solid objects growing out ofliquid metal alloys, showing the symmetry of
the icosahedral group which includes fivefold rotations. Meanwhile,
the mathematician Roger Penrose had discovered the Penrose tilings of
the plane. These are arrangements of parallelograms that cover
a plane with pentagonal long-range order. The alloy quasi-crystals are
three-dimensional analogs of the two-dimensional Penrose tilings.
After these discoveries, mathematicians had to enlarge the theory of
crystallographic groups so as to include quasi-crystals. That is a major
program of research which is still in progress.

I mention in conclusion one of my favorite Baconian dreams,
the possible connection between the theory of one-dimensional
quasi-crystals and the theory of the Riemann zeta-function.
A one-dimensional quasi-crystal need not have any symmetry.
It is defined simply as a non periodic arrangement of mass-points
on a line whose Fourier transform is also an arrangement of mass-points
on a line. Because of the lack of any requirement of symmetry, quasi-crystals
have much greater freedom to exist in one dimension than in two or three.
Almost nothing is known about the possible abundance of one-dimensional
quasi-crystals. Likewise, not much is known about the
zeros of the Riemann zeta-function, described by Piergiorgio
in his chapter 5, section 2. The RIemann Hypothesis, the statement that
all the zeros of the zeta-function with trivial exceptions lie on a certain
straight line in the complex plane, was conjectured by Riemann in 1859.
To prove it is the most famous unsolved problem in the whole of mathematics.
One fact that we know is that, if the Riemann Hypothesis is true, then
the zeta-function zeros on the critical line are a quasi-crystal
according to the definition. If the RIemann Hypothesis is true,
the zeta-function zeros have a Fourier transform consisting of
mass-points at logarithms of all powers of prime integers and
nowhere else. This suggests a possible approach to the proof
of the Riemann Hypothesis. First you make a complete classification of
all one-dimensional quasi-crystals, and write down a list of them.
Collecting and classifYing new species of objects is a quintessentially
Baconian activity. Then you look through the list and see whether the
zeta-function zeros are there. If the zeta-function zeros are there,
then you have proved the Riemann Hypothesis, and you need only wait for
the next International Congress of Mathematicians to collect your Fields
Medal. Of course the difficult part of this approach is collect-
ing and classifYing the quasi-crystals. I leave that as an exercise
for the reader.

Freeman Dyson
Institute for Advanced Study
Princeton, New Jersey, USA