Birds and Frogs Freeman Dyson
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Birds and Frogs
Freeman Dyson
S
ome mathematicians are birds, others skill as a mathematician. In his later years he hired
are frogs. Birds fly high in the air and younger colleagues with the title of assistants to
survey broad vistas of mathematics out do mathematical calculations for him. His way of
to the far horizon. They delight in con- thinking was physical rather than mathematical.
cepts that unify our thinking and bring He was supreme among physicists as a bird who
together diverse problems from different parts of saw further than others. I will not talk about Ein-
the landscape. Frogs live in the mud below and see stein since I have nothing new to say.
only the flowers that grow nearby. They delight
in the details of particular objects, and they solve Francis Bacon and René Descartes
problems one at a time. I happen to be a frog, but At the beginning of the seventeenth century, two
many of my best friends are birds. The main theme great philosophers, Francis Bacon in England and
of my talk tonight is this. Mathematics needs both René Descartes in France, proclaimed the birth of
birds and frogs. Mathematics is rich and beautiful modern science. Descartes was a bird, and Bacon
because birds give it broad visions and frogs give it was a frog. Each of them described his vision of
intricate details. Mathematics is both great art and the future. Their visions were very different. Bacon
important science, because it combines generality said, “All depends on keeping the eye steadily fixed
of concepts with depth of structures. It is stupid on the facts of nature.” Descartes said, “I think,
to claim that birds are better than frogs because therefore I am.” According to Bacon, scientists
they see farther, or that frogs are better than birds should travel over the earth collecting facts, until
because they see deeper. The world of mathemat- the accumulated facts reveal how Nature works.
ics is both broad and deep, and we need birds and The scientists will then induce from the facts the
frogs working together to explore it. laws that Nature obeys. According to Descartes,
This talk is called the Einstein lecture, and I am scientists should stay at home and deduce the
grateful to the American Mathematical Society laws of Nature by pure thought. In order to deduce
for inviting me to do honor to Albert Einstein. the laws correctly, the scientists will need only
Einstein was not a mathematician, but a physicist the rules of logic and knowledge of the existence
who had mixed feelings about mathematics. On of God. For four hundred years since Bacon and
the one hand, he had enormous respect for the Descartes led the way, science has raced ahead
power of mathematics to describe the workings by following both paths simultaneously. Neither
of nature, and he had an instinct for mathematical Baconian empiricism nor Cartesian dogmatism
beauty which led him onto the right track to find has the power to elucidate Nature’s secrets by
nature’s laws. On the other hand, he had no inter- itself, but both together have been amazingly suc-
est in pure mathematics, and he had no technical cessful. For four hundred years English scientists
have tended to be Baconian and French scientists
Freeman Dyson is an emeritus professor in the School of Cartesian. Faraday and Darwin and Rutherford
Natural Sciences, Institute for Advanced Study, Princeton, were Baconians; Pascal and Laplace and Poincaré
NJ. His email address is dyson@ias.edu. were Cartesians. Science was greatly enriched by
This article is a written version of his AMS Einstein Lecture, the cross-fertilization of the two contrasting cul-
which was to have been given in October 2008 but which tures. Both cultures were always at work in both
unfortunately had to be canceled. countries. Newton was at heart a Cartesian, using
212 Notices
N otices of AMS
the AMS
of the Volume 56, Number 2
pure thought as Descartes intended, and using wave mechanics in 1926. Schrödinger was a bird
it to demolish the Cartesian dogma of vortices. who started from the idea of unifying mechanics
Marie Curie was at heart a Baconian, boiling tons with optics. A hundred years earlier, Hamilton had
of crude uranium ore to demolish the dogma of unified classical mechanics with ray optics, using
the indestructibility of atoms. the same mathematics to describe optical rays
In the history of twentieth century mathematics, and classical particle trajectories. Schrödinger’s
there were two decisive events, one belonging to idea was to extend this unification to wave optics
the Baconian tradition and the other to the Carte- and wave mechanics. Wave optics already existed,
sian tradition. The first was the International Con- but wave mechanics did not. Schrödinger had to
gress of Mathematicians in Paris in 1900, at which invent wave mechanics to complete the unification.
Hilbert gave the keynote address, Starting from wave optics as a model,
charting the course of mathematics he wrote down a differential equa-
for the coming century by propound- tion for a mechanical particle, but the
ing his famous list of twenty-three equation made no sense. The equation
outstanding unsolved problems. Hil- looked like the equation of conduction
bert himself was a bird, flying high of heat in a continuous medium. Heat
over the whole territory of mathemat- conduction has no visible relevance to
ics, but he addressed his problems to particle mechanics. Schrödinger’s idea
the frogs who would solve them one seemed to be going nowhere. But then
at a time. The second decisive event came the surprise. Schrödinger put
was the formation of the Bourbaki the square root of minus one into the
group of mathematical birds in France equation, and suddenly it made sense.
in the 1930s, dedicated to publish- Suddenly it became a wave equation
ing a series of textbooks that would instead of a heat conduction equation.
establish a unifying framework for Francis Bacon And Schrödinger found to his delight
all of mathematics. The Hilbert prob- that the equation has solutions cor-
lems were enormously successful in responding to the quantized orbits in
guiding mathematical research into the Bohr model of the atom.
fruitful directions. Some of them were It turns out that the Schrödinger
solved and some remain unsolved, equation describes correctly every-
but almost all of them stimulated the thing we know about the behavior of
growth of new ideas and new fields atoms. It is the basis of all of chem-
of mathematics. The Bourbaki project istry and most of physics. And that
was equally influential. It changed the square root of minus one means that
style of mathematics for the next fifty nature works with complex numbers
years, imposing a logical coherence and not with real numbers. This dis-
that did not exist before, and moving covery came as a complete surprise,
the emphasis from concrete examples to Schrödinger as well as to every-
to abstract generalities. In the Bour- body else. According to Schrödinger,
René Descartes
baki scheme of things, mathematics is his fourteen-year-old girl friend Itha
the abstract structure included in the Junger said to him at the time, “Hey,
Bourbaki textbooks. What is not in the textbooks you never even thought when you began that so
is not mathematics. Concrete examples, since they much sensible stuff would come out of it.” All
do not appear in the textbooks, are not math- through the nineteenth century, mathematicians
ematics. The Bourbaki program was the extreme from Abel to Riemann and Weierstrass had been
expression of the Cartesian style. It narrowed the creating a magnificent theory of functions of
scope of mathematics by excluding the beautiful complex variables. They had discovered that the
flowers that Baconian travelers might collect by theory of functions became far deeper and more
the wayside. powerful when it was extended from real to com-
plex numbers. But they always thought of complex
Jokes of Nature numbers as an artificial construction, invented by
For me, as a Baconian, the main thing missing in human mathematicians as a useful and elegant
the Bourbaki program is the element of surprise. abstraction from real life. It never entered their
The Bourbaki program tried to make mathematics heads that this artificial number system that they
logical. When I look at the history of mathematics, had invented was in fact the ground on which
I see a succession of illogical jumps, improbable atoms move. They never imagined that nature had
coincidences, jokes of nature. One of the most got there first.
profound jokes of nature is the square root of Another joke of nature is the precise linearity
minus one that the physicist Erwin Schrödinger of quantum mechanics, the fact that the possible
put into his wave equation when he invented states of any physical object form a linear space.
February 2009 Notices of the AMS 213
Before quantum mechanics was invented, classical second commandment says: “Let your acts be di-
physics was always nonlinear, and linear models rected towards a worthy goal, but do not ask if they
were only approximately valid. After quantum can reach it: they are to be models and examples,
mechanics, nature itself suddenly became linear. not means to an end.” Szilard practiced what he
This had profound consequences for mathemat- preached. He was the first physicist to imagine
ics. During the nineteenth century Sophus Lie nuclear weapons and the first to campaign ac-
developed his elaborate theory of continuous tively against their use. His second commandment
groups, intended to clarify the behavior of classical certainly applies here. The proof of the Riemann
dynamical systems. Lie groups were then of little Hypothesis is a worthy goal, and it is not for us to
interest either to mathematicians or to physicists. ask whether we can reach it. I will give you some
The nonlinear theory of Lie groups was too compli- hints describing how it might be achieved. Here I
cated for the mathematicians and too obscure for will be giving voice to the mathematician that I was
the physicists. Lie died a disappointed man. And fifty years ago before I became a physicist. I will
then, fifty years later, it turned out that nature was talk first about the Riemann Hypothesis and then
precisely linear, and the theory of linear represen- about quasi-crystals.
tations of Lie algebras was the natural language of There were until recently two supreme unsolved
particle physics. Lie groups and Lie algebras were problems in the world of pure mathematics, the
reborn as one of the central themes of twentieth proof of Fermat’s Last Theorem and the proof of
century mathematics. the Riemann Hypothesis. Twelve years ago, my
A third joke of nature is the existence of quasi- Princeton colleague Andrew Wiles polished off
crystals. In the nineteenth century the study of Fermat’s Last Theorem, and only the Riemann Hy-
crystals led to a complete enumeration of possible pothesis remains. Wiles’ proof of the Fermat Theo-
discrete symmetry groups in Euclidean space. rem was not just a technical stunt. It required the
Theorems were proved, establishing the fact that discovery and exploration of a new field of math-
in three-dimensional space discrete symmetry ematical ideas, far wider and more consequential
groups could contain only rotations of order three, than the Fermat Theorem itself. It is likely that
four, or six. Then in 1984 quasi-crystals were dis- any proof of the Riemann Hypothesis will likewise
covered, real solid objects growing out of liquid lead to a deeper understanding of many diverse
metal alloys, showing the symmetry of the icosa- areas of mathematics and perhaps of physics too.
hedral group, which includes five-fold rotations. Riemann’s zeta-function, and other zeta-func-
Meanwhile, the mathematician Roger Penrose tions similar to it, appear ubiquitously in number
discovered the Penrose tilings of the plane. These theory, in the theory of dynamical systems, in
are arrangements of parallelograms that cover a geometry, in function theory, and in physics. The
plane with pentagonal long-range order. The alloy zeta-function stands at a junction where paths lead
quasi-crystals are three-dimensional analogs of in many directions. A proof of the hypothesis will
the two-dimensional Penrose tilings. After these illuminate all the connections. Like every serious
discoveries, mathematicians had to enlarge the student of pure mathematics, when I was young I
theory of crystallographic groups to include quasi- had dreams of proving the Riemann Hypothesis.
crystals. That is a major program of research which I had some vague ideas that I thought might lead
is still in progress. to a proof. In recent years, after the discovery of
A fourth joke of nature is a similarity in be- quasi-crystals, my ideas became a little less vague.
havior between quasi-crystals and the zeros of I offer them here for the consideration of any
the Riemann Zeta function. The zeros of the zeta- young mathematician who has ambitions to win
function are exciting to mathematicians because a Fields Medal.
they are found to lie on a straight line and nobody Quasi-crystals can exist in spaces of one, two,
understands why. The statement that with trivial or three dimensions. From the point of view of
exceptions they all lie on a straight line is the physics, the three-dimensional quasi-crystals are
famous Riemann Hypothesis. To prove the Rie- the most interesting, since they inhabit our three-
mann Hypothesis has been the dream of young dimensional world and can be studied experi-
mathematicians for more than a hundred years. mentally. From the point of view of a mathemati-
I am now making the outrageous suggestion that cian, one-dimensional quasi-crystals are much
we might use quasi-crystals to prove the Riemann more interesting than two-dimensional or three-
Hypothesis. Those of you who are mathematicians dimensional quasi-crystals because they exist in
may consider the suggestion frivolous. Those who far greater variety. The mathematical definition
are not mathematicians may consider it uninterest- of a quasi-crystal is as follows. A quasi-crystal
ing. Nevertheless I am putting it forward for your is a distribution of discrete point masses whose
serious consideration. When the physicist Leo Fourier transform is a distribution of discrete
Szilard was young, he became dissatisfied with the point frequencies. Or to say it more briefly, a
ten commandments of Moses and wrote a new set quasi-crystal is a pure point distribution that has
of ten commandments to replace them. Szilard’s a pure point spectrum. This definition includes
214 Notices of the AMS Volume 56, Number 2as a special case the ordinary crystals, objects is a quintessentially Baconian
Photograph of A. Besicovitch from AMS archives. Photo of H. Weyl from the archives of Peter Roquette, used with permisssion.
which are periodic distributions with activity. It is an appropriate activity
periodic spectra. for mathematical frogs. We shall then
Excluding the ordinary crystals, find the well-known quasi-crystals
quasi-crystals in three dimensions associated with PV numbers, and
come in very limited variety, all of also a whole universe of other quasi-
them associated with the icosahedral crystals, known and unknown. Among
group. The two-dimensional quasi- the multitude of other quasi-crystals
crystals are more numerous, roughly we search for one corresponding to
one distinct type associated with each the Riemann zeta-function and one
regular polygon in a plane. The two- corresponding to each of the other
dimensional quasi-crystal with pentag- zeta-functions that resemble the Rie-
onal symmetry is the famous Penrose mann zeta-function. Suppose that
tiling of the plane. Finally, the one- Abram Besicovitch we find one of the quasi-crystals in
dimensional quasi-crystals have a far our enumeration with properties
richer structure since they are not tied that identify it with the zeros of the
to any rotational symmetries. So far as Riemann zeta-function. Then we have
I know, no complete enumeration of proved the Riemann Hypothesis and
one-dimensional quasi-crystals exists. we can wait for the telephone call
It is known that a unique quasi-crystal announcing the award of the Fields
exists corresponding to every Pisot- Medal.
Vijayaraghavan number or PV number. These are of course idle dreams.
A PV number is a real algebraic inte- The problem of classifying one-
ger, a root of a polynomial equation dimensional quasi-crystals is horren-
with integer coefficients, such that all dously difficult, probably at least as
the other roots have absolute value difficult as the problems that Andrew
less than one, [1]. The set of all PV Wiles took seven years to explore. But
numbers is infinite and has a remark- Hermann Weyl if we take a Baconian point of view,
able topological structure. The set the history of mathematics is a his-
of all one-dimensional quasi-crystals tory of horrendously difficult prob-
has a structure at least as rich as the lems being solved by young people too ignorant to
set of all PV numbers and probably much richer. know that they were impossible. The classification
We do not know for sure, but it is likely that a of quasi-crystals is a worthy goal, and might even
huge universe of one-dimensional quasi-crystals turn out to be achievable. Problems of that degree
not associated with PV numbers is waiting to be of difficulty will not be solved by old men like me.
discovered. I leave this problem as an exercise for the young
Here comes the connection of the one- frogs in the audience.
dimensional quasi-crystals with the Riemann
hypothesis. If the Riemann hypothesis is true, Abram Besicovitch and Hermann Weyl
then the zeros of the zeta-function form a one-
Let me now introduce you to some notable frogs
dimensional quasi-crystal according to the defini-
and birds that I knew personally. I came to Cam-
tion. They constitute a distribution of point masses
bridge University as a student in 1941 and had
on a straight line, and their Fourier transform is
the tremendous luck to be given the Russian
likewise a distribution of point masses, one at each
mathematician Abram Samoilovich Besicovitch
of the logarithms of ordinary prime numbers and
as my supervisor. Since this was in the middle
prime-power numbers. My friend Andrew Odlyzko
of World War Two, there were very few students
has published a beautiful computer calculation of
in Cambridge, and almost no graduate students.
the Fourier transform of the zeta-function zeros,
Although I was only seventeen years old and Besi-
[6]. The calculation shows precisely the expected
covitch was already a famous professor, he gave
structure of the Fourier transform, with a sharp
me a great deal of his time and attention, and we
discontinuity at every logarithm of a prime or
became life-long friends. He set the style in which
prime-power number and nowhere else.
I began to work and think about mathematics. He
My suggestion is the following. Let us pretend gave wonderful lectures on measure-theory and
that we do not know that the Riemann Hypothesis integration, smiling amiably when we laughed at
is true. Let us tackle the problem from the other his glorious abuse of the English language. I re-
end. Let us try to obtain a complete enumera- member only one occasion when he was annoyed
tion and classification of one-dimensional quasi- by our laughter. He remained silent for a while and
crystals. That is to say, we enumerate and classify then said, “Gentlemen. Fifty million English speak
all point distributions that have a discrete point English you speak. Hundred and fifty million Rus-
spectrum. Collecting and classifying new species of sians speak English I speak.”
February 2009 Notices of the AMS 215Besicovitch was a frog, and he became famous into a regular and an irregular component, that
when he was young by solving a problem in el- the regular component has a tangent almost
ementary plane geometry known as the Kakeya everywhere, and the irregular component has a
problem. The Kakeya problem was the following. projection of measure zero onto almost all direc-
A line segment of length one is allowed to move tions. Roughly speaking, the regular component
freely in a plane while rotating through an angle looks like a collection of continuous curves, while
of 360 degrees. What is the smallest area of the the irregular component looks nothing like a con-
plane that it can cover during its rotation? The tinuous curve. The existence and the properties of
problem was posed by the Japanese mathematician the irregular component are connected with the
Kakeya in 1917 and remained a famous unsolved Besicovitch solution of the Kakeya problem. One
problem for ten years. George Birkhoff, the lead- of the problems that he gave me to work on was
ing American mathematician at that time, publicly the division of measurable sets into regular and
proclaimed that the Kakeya problem and the four- irregular components in spaces of higher dimen-
color problem were the outstanding unsolved sions. I got nowhere with the problem, but became
problems of the day. It was widely believed that permanently imprinted with the Besicovitch style.
the minimum area was π /8, which is the area of a The Besicovitch style is architectural. He builds
three-cusped hypocycloid. The three-cusped hypo- out of simple elements a delicate and complicated
cycloid is a beautiful three-pointed curve. It is the architectural structure, usually with a hierarchical
curve traced out by a point on the circumference plan, and then, when the building is finished, the
of a circle with radius one-quarter, when the circle completed structure leads by simple arguments
rolls around the inside of a fixed circle with radius to an unexpected conclusion. Every Besicovitch
three-quarters. The line segment of length one can proof is a work of art, as carefully constructed as
turn while always remaining tangent to the hypo- a Bach fugue.
cycloid with its two ends also on the hypocycloid. A few years after my apprenticeship with Be-
This picture of the line turning while touching the sicovitch, I came to Princeton and got to know
inside of the hypocycloid at three points was so Hermann Weyl. Weyl was a prototypical bird, just
elegant that most people believed it must give the as Besicovitch was a prototypical frog. I was lucky
minimum area. Then Besicovitch surprised every- to overlap with Weyl for one year at the Princeton
one by proving that the area covered by the line as Institute for Advanced Study before he retired
it turns can be less than for any positive . from the Institute and moved back to his old home
Besicovitch had actually solved the problem in in Zürich. He liked me because during that year I
1920 before it became famous, not even knowing published papers in the Annals of Mathematics
that Kakeya had proposed it. In 1920 he published about number theory and in the Physical Review
the solution in Russian in the Journal of the Perm about the quantum theory of radiation. He was one
Physics and Mathematics Society, a journal that of the few people alive who was at home in both
was not widely read. The university of Perm, a subjects. He welcomed me to the Institute, in the
city 1,100 kilometers east of Moscow, was briefly hope that I would be a bird like himself. He was dis-
a refuge for many distinguished mathematicians appointed. I remained obstinately a frog. Although
after the Russian revolution. They published two I poked around in a variety of mud-holes, I always
volumes of their journal before it died amid the looked at them one at a time and did not look for
chaos of revolution and civil war. Outside Russia connections between them. For me, number theory
the journal was not only unknown but unobtain- and quantum theory were separate worlds with
able. Besicovitch left Russia in 1925 and arrived at separate beauties. I did not look at them as Weyl
Copenhagen, where he learned about the famous did, hoping to find clues to a grand design.
Kakeya problem that he had solved five years ear- Weyl’s great contribution to the quantum theory
lier. He published the solution again, this time in of radiation was his invention of gauge fields. The
English in the Mathematische Zeitschrift. The Ka- idea of gauge fields had a curious history. Weyl
keya problem as Kakeya proposed it was a typical invented them in 1918 as classical fields in his
frog problem, a concrete problem without much unified theory of general relativity and electromag-
connection with the rest of mathematics. Besico- netism, [7]. He called them “gauge fields” because
vitch gave it an elegant and deep solution, which they were concerned with the non-integrability
revealed a connection with general theorems about of measurements of length. His unified theory
the structure of sets of points in a plane. was promptly and publicly rejected by Einstein.
The Besicovitch style is seen at its finest in After this thunderbolt from on high, Weyl did
his three classic papers with the title, “On the not abandon his theory but moved on to other
fundamental geometric properties of linearly things. The theory had no experimental conse-
measurable plane sets of points”, published in quences that could be tested. Then in 1929, after
Mathematische Annalen in the years 1928, 1938, quantum mechanics had been invented by others,
and 1939. In these papers he proved that every Weyl realized that his gauge fields fitted far bet-
linearly measurable set in the plane is divisible ter into the quantum world than they did into the
216 Notices of the AMS Volume 56, Number 2classical world, [8]. All astronomy, a golden
Photo of F. Yang courtesy of SUNY Stony Brook. Photo of Y. Manin courtesy of Northwestern University.
that he needed to do, to age for Baconian travel-
change a classical gauge ers picking up facts, for
into a quantum gauge, frogs exploring small
was to change real patches of the swamp
numbers into complex in which we live. Dur-
numbers. In quantum ing these fifty years, the
mechanics, every quan- frogs accumulated a de-
tum of electric charge tailed knowledge of a
carries with it a com- large variety of cosmic
plex wave function with structures and a large
a phase, and the gauge variety of particles and
field is concerned with interactions. As the
the non-integrability of Chen Ning (Frank) Yuri Manin exploration of new ter-
measurements of phase. Yang ritories continued, the
The gauge field could universe became more
then be precisely identified with the electromag- complicated. Instead of a grand design displaying
netic potential, and the law of conservation of the simplicity and beauty of Weyl’s mathematics,
charge became a consequence of the local phase the explorers found weird objects such as quarks
invariance of the theory. and gamma-ray bursts, weird concepts such as su-
Weyl died four years after he returned from persymmetry and multiple universes. Meanwhile,
Princeton to Zürich, and I wrote his obituary for the mathematics was also becoming more compli-
journal Nature, [3]. “Among all the mathematicians cated, as exploration continued into the phenom-
who began their working lives in the twentieth ena of chaos and many other new areas opened
century,” I wrote, “Hermann Weyl was the one who by electronic computers. The mathematicians
made major contributions in the greatest number discovered the central mystery of computability,
of different fields. He alone could stand compari- the conjecture represented by the statement P is
son with the last great universal mathematicians not equal to NP. The conjecture asserts that there
of the nineteenth century, Hilbert and Poincaré. exist mathematical problems which can be quickly
So long as he was alive, he embodied a living con- solved in individual cases but cannot be solved
tact between the main lines of advance in pure by a quick algorithm applicable to all cases. The
mathematics and in theoretical physics. Now he most famous example of such a problem is the
is dead, the contact is broken, and our hopes of traveling salesman problem, which is to find the
comprehending the physical universe by a direct shortest route for a salesman visiting a set of cit-
use of creative mathematical imagination are for ies, knowing the distance between each pair. All
the time being ended.” I mourned his passing, but the experts believe that the conjecture is true, and
I had no desire to pursue his dream. I was happy that the traveling salesman problem is an example
to see pure mathematics and physics marching of a problem that is P but not NP. But nobody has
ahead in opposite directions. even a glimmer of an idea how to prove it. This is
The obituary ended with a sketch of Weyl as a mystery that could not even have been formu-
a human being: “Characteristic of Weyl was an lated within the nineteenth-century mathematical
aesthetic sense which dominated his thinking on universe of Hermann Weyl.
all subjects. He once said to me, half joking, ‘My
work always tried to unite the true with the beau- Frank Yang and Yuri Manin
tiful; but when I had to choose one or the other, The last fifty years have been a hard time for
I usually chose the beautiful’. This remark sums birds. Even in hard times, there is work for birds
up his personality perfectly. It shows his profound to do, and birds have appeared with the courage to
faith in an ultimate harmony of Nature, in which tackle it. Soon after Weyl left Princeton, Frank Yang
the laws should inevitably express themselves in arrived from Chicago and moved into Weyl’s old
a mathematically beautiful form. It shows also house. Yang took Weyl’s place as the leading bird
his recognition of human frailty, and his humor, among my generation of physicists. While Weyl
which always stopped him short of being pomp- was still alive, Yang and his student Robert Mills
ous. His friends in Princeton will remember him discovered the Yang-Mills theory of non-Abelian
as he was when I last saw him, at the Spring Dance gauge fields, a marvelously elegant extension of
of the Institute for Advanced Study last April: a Weyl’s idea of a gauge field, [11]. Weyl’s gauge field
big jovial man, enjoying himself splendidly, his was a classical quantity, satisfying the commuta-
cheerful frame and his light step giving no hint of tive law of multiplication. The Yang-Mills theory
his sixty-nine years.” had a triplet of gauge fields which did not com-
The fifty years after Weyl’s death were a golden mute. They satisfied the commutation rules of the
age of experimental physics and observational three components of a quantum mechanical spin,
February 2009 Notices of the AMS 217which are generators of the simplest non-Abelian the worlds of geometry and dynamics with his
Lie algebra A2
. The theory was later generalized so concept of fluxions, nowadays called calculus. In
that the gauge fields could be generators of any the nineteenth century Boole linked the worlds
finite-dimensional Lie algebra. With this general- of logic and algebra with his concept of symbolic
ization, the Yang-Mills gauge field theory provided logic, and Riemann linked the worlds of geometry
the framework for a model of all the known par- and analysis with his concept of Riemann sur-
ticles and interactions, a model that is now known faces. Coordinates, fluxions, symbolic logic, and
as the Standard Model of particle physics. Yang put Riemann surfaces are all metaphors, extending
the finishing touch to it by showing that Einstein’s the meanings of words from familiar to unfamiliar
theory of gravitation fits into the same framework, contexts. Manin sees the future of mathematics
with the Christoffel three-index symbol taking the as an exploration of metaphors that are already
role of gauge field, [10]. visible but not yet understood. The deepest such
In an appendix to his 1918 paper, added in 1955 metaphor is the similarity in structure between
for the volume of selected papers published to number theory and physics. In both fields he sees
celebrate his seventieth birthday, Weyl expressed tantalizing glimpses of parallel concepts, symme-
his final thoughts about gauge field theories (my tries linking the continuous with the discrete. He
translation), [12]: “The strongest argument for my looks forward to a unification which he calls the
theory seemed to be this, that gauge invariance quantization of mathematics.
was related to conservation of electric charge in “Manin disagrees with the Baconian story, that
the same way as coordinate invariance was related Hilbert set the agenda for the mathematics of the
to conservation of energy and momentum.” Thirty twentieth century when he presented his famous
years later Yang was in Zürich for the celebration list of twenty-three unsolved problems to the In-
of Weyl’s hundredth birthday. In his speech, [12], ternational Congress of Mathematicians in Paris
Yang quoted this remark as evidence of Weyl’s de- in 1900. According to Manin, Hilbert’s problems
votion to the idea of gauge invariance as a unifying were a distraction from the central themes of
principle for physics. Yang then went on, “Sym- mathematics. Manin sees the important advances
metry, Lie groups, and gauge invariance are now in mathematics coming from programs, not from
recognized, through theoretical and experimental problems. Problems are usually solved by apply-
developments, to play essential roles in determin- ing old ideas in new ways. Programs of research
ing the basic forces of the physical universe. I have are the nurseries where new ideas are born. He
called this the principle that symmetry dictates in- sees the Bourbaki program, rewriting the whole of
teraction.” This idea, that symmetry dictates inter- mathematics in a more abstract language, as the
action, is Yang’s generalization of Weyl’s remark. source of many of the new ideas of the twentieth
Weyl observed that gauge invariance is intimately century. He sees the Langlands program, unifying
connected with physical conservation laws. Weyl number theory with geometry, as a promising
could not go further than this, because he knew source of new ideas for the twenty-first. People
only the gauge invariance of commuting Abelian who solve famous unsolved problems may win big
fields. Yang made the connection much stronger prizes, but people who start new programs are the
by introducing non-Abelian gauge fields. With real pioneers.”
non-Abelian gauge fields generating nontrivial Lie The Russian version of Mathematics as Meta-
algebras, the possible forms of interaction between phor contains ten chapters that were omitted from
fields become unique, so that symmetry dictates the English version. The American Mathematical
interaction. This idea is Yang’s greatest contribu- Society decided that these chapters would not be
tion to physics. It is the contribution of a bird, of interest to English language readers. The omis-
flying high over the rain forest of little problems sions are doubly unfortunate. First, readers of the
in which most of us spend our lives. English version see only a truncated view of Manin,
Another bird for whom I have a deep respect who is perhaps unique among mathematicians in
is the Russian mathematician Yuri Manin, who his broad range of interests extending far beyond
recently published a delightful book of essays with mathematics. Second, we see a truncated view of
the title Mathematics as Metaphor [5]. The book Russian culture, which is less compartmentalized
was published in Moscow in Russian, and by the than English language culture, and brings math-
American Mathematical Society in English. I wrote ematicians into closer contact with historians and
a preface for the English version, and I give you artists and poets.
here a short quote from my preface. “Mathematics
as Metaphor is a good slogan for birds. It means John von Neumann
that the deepest concepts in mathematics are Another important figure in twentieth century
those which link one world of ideas with another. mathematics was John von Neumann. Von Neu-
In the seventeenth century Descartes linked the mann was a frog, applying his prodigious tech-
disparate worlds of algebra and geometry with nical skill to solve problems in many branches
his concept of coordinates, and Newton linked of mathematics and physics. He began with the
218 Notices of the AMS Volume 56, Number 2foundations of mathematics. He found the first divide the atmosphere at any moment into stable
satisfactory set of axioms for set-theory, avoiding regions and unstable regions. Stable regions we
the logical paradoxes that Cantor had encountered can predict. Unstable regions we can control.” Von
in his attempts to deal with infinite sets and Neumann believed that any unstable region could
infinite numbers. Von Neumann’s axioms were be pushed by a judiciously applied small perturba-
used by his bird friend Kurt Gödel a few years later tion so that it would move in any desired direction.
to prove the existence of undecidable propositions The small perturbation would be applied by a fleet
in mathematics. Gödel’s theorems gave birds a new of airplanes carrying smoke generators, to absorb
vision of mathematics. After Gödel, mathematics sunlight and raise or lower temperatures at places
was no longer a single structure tied where the perturbation would be most
together with a unique concept of effective. In particular, we could stop
truth, but an archipelago of structures an incipient hurricane by identifying
with diverse sets of axioms and di- the position of an instability early
verse notions of truth. Gödel showed enough, and then cooling that patch
that mathematics is inexhaustible. No of air before it started to rise and form
Photograph of Mary Cartright courtesy of The Mistress and Fellows, Girton College, Cambridge.
matter which set of axioms is chosen a vortex. Von Neumann, speaking in
as the foundation, birds can always 1950, said it would take only ten years
find questions that those axioms can- to build computers powerful enough
not answer. to diagnose accurately the stable and
Von Neumann went on from the unstable regions of the atmosphere.
foundations of mathematics to the Then, once we had accurate diagno-
foundations of quantum mechanics. sis, it would take only a short time
To give quantum mechanics a firm for us to have control. He expected
mathematical foundation, he created John von Neumann that practical control of the weather
a magnificent theory of rings of op- would be a routine operation within
erators. Every observable quantity is the decade of the 1960s.
represented by a linear operator, and Von Neumann, of course, was
the peculiarities of quantum behav- wrong. He was wrong because he
ior are faithfully represented by the did not know about chaos. We now
algebra of operators. Just as Newton know that when the motion of the
invented calculus to describe classi- atmosphere is locally unstable, it is
cal dynamics, von Neumann invented very often chaotic. The word “chaotic”
rings of operators to describe quan- means that motions that start close
tum dynamics. together diverge exponentially from
Von Neumann made fundamental each other as time goes on. When the
contributions to several other fields, motion is chaotic, it is unpredictable,
especially to game theory and to the and a small perturbation does not
design of digital computers. For the Mary Cartwright move it into a stable motion that can
last ten years of his life, he was deeply be predicted. A small perturbation
involved with computers. He was so will usually move it into another cha-
strongly interested in computers that he decided otic motion that is equally unpredictable. So von
not only to study their design but to build one with Neumann’s strategy for controlling the weather
real hardware and software and use it for doing fails. He was, after all, a great mathematician but
science. I have vivid memories of the early days of a mediocre meteorologist.
von Neumann’s computer project at the Institute Edward Lorenz discovered in 1963 that the so-
for Advanced Study in Princeton. At that time he lutions of the equations of meteorology are often
had two main scientific interests, hydrogen bombs chaotic. That was six years after von Neumann
and meteorology. He used his computer during the died. Lorenz was a meteorologist and is generally
night for doing hydrogen bomb calculations and regarded as the discoverer of chaos. He discovered
during the day for meteorology. Most of the people the phenomena of chaos in the meteorological con-
hanging around the computer building in daytime text and gave them their modern names. But in fact
were meteorologists. Their leader was Jule Char- I had heard the mathematician Mary Cartwright,
ney. Charney was a real meteorologist, properly who died in 1998 at the age of 97, describe the
humble in dealing with the inscrutable mysteries same phenomena in a lecture in Cambridge in 1943,
of the weather, and skeptical of the ability of the twenty years before Lorenz discovered them. She
computer to solve the mysteries. John von Neu- called the phenomena by different names, but they
mann was less humble and less skeptical. I heard were the same phenomena. She discovered them in
von Neumann give a lecture about the aims of his the solutions of the van der Pol equation which de-
project. He spoke, as he always did, with great con- scribe the oscillations of a nonlinear amplifier, [2].
fidence. He said, “The computer will enable us to The van der Pol equation was important in World
February 2009 Notices of the AMS 219War II because nonlinear amplifiers fed power from the 1930s out of a drawer and dusted it off.
to the transmitters in early radar systems. The The lecture was about rings of operators, a subject
transmitters behaved erratically, and the Air Force that was new and fashionable in the 1930s. Noth-
blamed the manufacturers for making defective ing about unsolved problems. Nothing about the
amplifiers. Mary Cartwright was asked to look into future. Nothing about computers, the subject that
the problem. She showed that the manufacturers we knew was dearest to von Neumann’s heart.
were not to blame. She showed that the van der Pol He might at least have had something new and
equation was to blame. The solutions of the van der exciting to say about computers. The audience in
Pol equation have precisely the chaotic behavior the concert hall became restless. Somebody said
that the Air Force was complaining about. I heard in a voice loud enough to be heard all over the
all about chaos from Mary Cartwright seven years hall, “Aufgewärmte Suppe”, which is German for
before I heard von Neumann talk about weather “warmed-up soup”. In 1954 the great majority of
control, but I was not far-sighted enough to make mathematicians knew enough German to under-
the connection. It never entered my head that the stand the joke. Von Neumann, deeply embarrassed,
erratic behavior of the van der Pol equation might brought his lecture to a quick end and left the hall
have something to do with meteorology. If I had without waiting for questions.
been a bird rather than a frog, I would probably
have seen the connection, and I might have saved Weak Chaos
von Neumann a lot of trouble. If he had known If von Neumann had known about chaos when he
about chaos in 1950, he would probably have spoke in Amsterdam, one of the unsolved prob-
thought about it deeply, and he would have had lems that he might have talked about was weak
something important to say about it in 1954. chaos. The problem of weak chaos is still unsolved
Von Neumann got into trouble at the end of fifty years later. The problem is to understand
his life because he was really a frog but everyone why chaotic motions often remain bounded and
expected him to fly like a bird. In 1954 there was do not cause any violent instability. A good ex-
an International Congress of Mathematicians in ample of weak chaos is the orbital motions of the
Amsterdam. These congresses happen only once planets and satellites in the solar system. It was
in four years and it is a great honor to be invited to discovered only recently that these motions are
speak at the opening session. The organizers of the chaotic. This was a surprising discovery, upsetting
Amsterdam congress invited von Neumann to give the traditional picture of the solar system as the
the keynote speech, expecting him to repeat the act prime example of orderly stable motion. The math-
that Hilbert had performed in Paris in 1900. Just as ematician Laplace two hundred years ago thought
Hilbert had provided a list of unsolved problems he had proved that the solar system is stable. It
to guide the development of mathematics for the now turns out that Laplace was wrong. Accurate
first half of the twentieth century, von Neumann numerical integrations of the orbits show clearly
was invited to do the same for the second half of that neighboring orbits diverge exponentially. It
the century. The title of von Neumann’s talk was seems that chaos is almost universal in the world
announced in the program of the congress. It was of classical dynamics.
“Unsolved Problems in Mathematics: Address by Chaotic behavior was never suspected in the
Invitation of the Organizing Committee”. After the solar system before accurate long-term integra-
congress was over, the complete proceedings were tions were done, because the chaos is weak. Weak
published, with the texts of all the lectures except chaos means that neighboring trajectories diverge
this one. In the proceedings there is a blank page exponentially but never diverge far. The divergence
with von Neumann’s name and the title of his talk. begins with exponential growth but afterwards
Underneath, it says, “No manuscript of this lecture remains bounded. Because the chaos of the plan-
was available.” etary motions is weak, the solar system can survive
What happened? I know what happened, be- for four billion years. Although the motions are
cause I was there in the audience, at 3:00 p.m. chaotic, the planets never wander far from their
on Thursday, September 2, 1954, in the Concert- customary places, and the system as a whole does
gebouw concert hall. The hall was packed with not fly apart. In spite of the prevalence of chaos,
mathematicians, all expecting to hear a brilliant the Laplacian view of the solar system as a perfect
lecture worthy of such a historic occasion. The piece of clockwork is not far from the truth.
lecture was a huge disappointment. Von Neumann We see the same phenomena of weak chaos in
had probably agreed several years earlier to give the domain of meteorology. Although the weather
a lecture about unsolved problems and had then in New Jersey is painfully chaotic, the chaos has
forgotten about it. Being busy with many other firm limits. Summers and winters are unpredict-
things, he had neglected to prepare the lecture. ably mild or severe, but we can reliably predict
Then, at the last moment, when he remembered that the temperature will never rise to 45 degrees
that he had to travel to Amsterdam and say some- Celsius or fall to minus 30, extremes that are
thing about mathematics, he pulled an old lecture often exceeded in India or in Minnesota. There
220 Notices of the AMS Volume 56, Number 2is no conservation law of physics that forbids solve old problems that were previously unsolvable.
temperatures from rising as high in New Jersey Second, the string theorists think of themselves
as in India, or from falling as low in New Jersey as physicists rather than mathematicians. They
as in Minnesota. The weakness of chaos has been believe that their theory describes something real
essential to the long-term survival of life on this in the physical world. And third, there is not yet
planet. Weak chaos gives us a challenging variety any proof that the theory is relevant to physics.
of weather while protecting us from fluctuations The theory is not yet testable by experiment. The
so severe as to endanger our existence. Chaos theory remains in a world of its own, detached
remains mercifully weak for reasons that we do from the rest of physics. String theorists make
not understand. That is another unsolved problem strenuous efforts to deduce consequences of the
for young frogs in the audience to take home. I theory that might be testable in the real world, so
challenge you to understand the reasons why the far without success.
chaos observed in a great diversity of dynamical My colleagues Ed Witten and Juan Maldacena
systems is generally weak. and others who created string theory are birds,
The subject of chaos is characterized by an flying high and seeing grand visions of distant
abundance of quantitative data, an unending sup- ranges of mountains. The thousands of hum-
ply of beautiful pictures, and a shortage of rigor- bler practitioners of string theory in universities
ous theorems. Rigorous theorems are the best way around the world are frogs, exploring fine details
to give a subject intellectual depth and precision. of the mathematical structures that birds first
Until you can prove rigorous theorems, you do not saw on the horizon. My anxieties about string
fully understand the meaning of your concepts. theory are sociological rather than scientific. It is
In the field of chaos I know only one rigorous a glorious thing to be one of the first thousand
theorem, proved by Tien-Yien Li and Jim Yorke in string theorists, discovering new connections and
1975 and published in a short paper with the title, pioneering new methods. It is not so glorious to
“Period Three Implies Chaos”, [4]. The Li-Yorke be one of the second thousand or one of the tenth
paper is one of the immortal gems in the literature thousand. There are now about ten thousand
of mathematics. Their theorem concerns nonlinear string theorists scattered around the world. This
maps of an interval onto itself. The successive posi- is a dangerous situation for the tenth thousand
tions of a point when the mapping is repeated can and perhaps also for the second thousand. It may
be considered as the orbit of a classical particle. happen unpredictably that the fashion changes
An orbit has period N if the point returns to its and string theory becomes unfashionable. Then it
original position after N mappings. An orbit is could happen that nine thousand string theorists
defined to be chaotic, in this context, if it diverges lose their jobs. They have been trained in a narrow
from all periodic orbits. The theorem says that if a specialty, and they may be unemployable in other
single orbit with period three exists, then chaotic fields of science.
orbits also exist. The proof is simple and short. To Why are so many young people attracted to
my mind, this theorem and its proof throw more string theory? The attraction is partly intellectual.
light than a thousand beautiful pictures on the String theory is daring and mathematically elegant.
basic nature of chaos. The theorem explains why But the attraction is also sociological. String theory
chaos is prevalent in the world. It does not explain is attractive because it offers jobs. And why are
why chaos is so often weak. That remains a task so many jobs offered in string theory? Because
for the future. I believe that weak chaos will not string theory is cheap. If you are the chairperson
be understood in a fundamental way until we can of a physics department in a remote place without
prove rigorous theorems about it. much money, you cannot afford to build a modern
laboratory to do experimental physics, but you can
String Theorists afford to hire a couple of string theorists. So you
I would like to say a few words about string theory. offer a couple of jobs in string theory, and you
Few words, because I know very little about string have a modern physics department. The tempta-
theory. I never took the trouble to learn the subject tions are strong for the chairperson to offer such
or to work on it myself. But when I am at home at the jobs and for the young people to accept them.
Institute for Advanced Study in Princeton, I am sur- This is a hazardous situation for the young people
rounded by string theorists, and I sometimes listen and also for the future of science. I am not say-
to their conversations. Occasionally I understand a ing that we should discourage young people from
little of what they are saying. Three things are clear. working in string theory if they find it exciting. I
First, what they are doing is first-rate mathemat- am saying that we should offer them alternatives,
ics. The leading pure mathematicians, people like so that they are not pushed into string theory by
Michael Atiyah and Isadore Singer, love it. It has economic necessity.
opened up a whole new branch of mathematics, Finally, I give you my own guess for the future
with new ideas and new problems. Most remark- of string theory. My guess is probably wrong. I
ably, it gave the mathematicians new methods to have no illusion that I can predict the future. I tell
February 2009 Notices of the AMS 221you my guess, just to give you something to think Jung, is a mental image rooted in a collective un-
about. I consider it unlikely that string theory will conscious that we all share. The intense emotions
turn out to be either totally successful or totally that archetypes carry with them are relics of lost
useless. By totally successful I mean that it is a memories of collective joy and suffering. Manin is
complete theory of physics, explaining all the de- saying that we do not need to accept Jung’s theory
tails of particles and their interactions. By totally as true in order to find it illuminating.
useless I mean that it remains a beautiful piece of More than thirty years ago, the singer Monique
pure mathematics. My guess is that string theory Morelli made a recording of songs with words by
will end somewhere between complete success Pierre MacOrlan. One of the songs is La Ville Morte,
and failure. I guess that it will be like the theory the dead city, with a haunting melody tuned to
of Lie groups, which Sophus Lie created in the Morelli’s deep contralto, with an accordion singing
nineteenth century as a mathematical framework counterpoint to the voice, and with verbal images
for classical physics. So long as physics remained of extraordinary intensity. Printed on the page, the
classical, Lie groups remained a failure. They were words are nothing special:
a solution looking for a problem. But then, fifty
“En pénétrant dans la ville morte,
years later, the quantum revolution transformed
Je tenait Margot par le main…
physics, and Lie algebras found their proper place.
Nous marchions de la nécropole,
They became the key to understanding the central
Les pieds brisés et sans parole,
role of symmetries in the quantum world. I expect
Devant ces portes sans cadole,
that fifty or a hundred years from now another
Devant ces trous indéfinis,
revolution in physics will happen, introducing new
Devant ces portes sans parole
concepts of which we now have no inkling, and the
Et ces poubelles pleines de cris”.
new concepts will give string theory a new mean-
“As we entered the dead city, I held Margot by
ing. After that, string theory will suddenly find
the hand…We walked from the graveyard on our
its proper place in the universe, making testable
bruised feet, without a word, passing by these
statements about the real world. I warn you that
doors without locks, these vaguely glimpsed holes,
this guess about the future is probably wrong. It
these doors without a word, these garbage cans
has the virtue of being falsifiable, which accord-
full of screams.”
ing to Karl Popper is the hallmark of a scientific
I can never listen to that song without a dispro-
statement. It may be demolished tomorrow by
portionate intensity of feeling. I often ask myself
some discovery coming out of the Large Hadron
why the simple words of the song seem to resonate
Collider in Geneva.
with some deep level of unconscious memory, as
Manin Again if the souls of the departed are speaking through
Morelli’s music. And now unexpectedly in Manin’s
To end this talk, I come back to Yuri Manin and
book I find an answer to my question. In his chap-
his book Mathematics as Metaphor. The book
ter, “The Empty City Archetype”, Manin describes
is mainly about mathematics. It may come as a
how the archetype of the dead city appears again
surprise to Western readers that he writes with
and again in the creations of architecture, litera-
equal eloquence about other subjects such as the
ture, art and film, from ancient to modern times,
collective unconscious, the origin of human lan-
ever since human beings began to congregate in
guage, the psychology of autism, and the role of
cities, ever since other human beings began to
the trickster in the mythology of many cultures.
congregate in armies to ravage and destroy them.
To his compatriots in Russia, such many-sided
The character who speaks to us in MacOrlan’s song
interests and expertise would come as no surprise.
is an old soldier who has long ago been part of an
Russian intellectuals maintain the proud tradition
army of occupation. After he has walked with his
of the old Russian intelligentsia, with scientists
wife through the dust and ashes of the dead city,
and poets and artists and musicians belonging to
he hears once more:
a single community. They are still today, as we see
them in the plays of Chekhov, a group of idealists “Chansons de charme d’un clairon
bound together by their alienation from a super- Qui fleurissait une heure lointaine
stitious society and a capricious government. In Dans un rêve de garnison”.
Russia, mathematicians and composers and film- “The magic calls of a bugle that came to life for
producers talk to one another, walk together in the an hour in an old soldier’s dream”.
snow on winter nights, sit together over a bottle of The words of MacOrlan and the voice of Mo-
wine, and share each others’ thoughts. relli seem to be bringing to life a dream from our
Manin is a bird whose vision extends far be- collective unconscious, a dream of an old soldier
yond the territory of mathematics into the wider wandering through a dead city. The concept of the
landscape of human culture. One of his hobbies collective unconscious may be as mythical as the
is the theory of archetypes invented by the Swiss concept of the dead city. Manin’s chapter describes
psychologist Carl Jung. An archetype, according to the subtle light that these two possibly mythical
222 Notices of the AMS Volume 56, Number 2You can also read
