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Birds and Frogs
Freeman Dyson
are frogs. Birds fly high in the air and
survey broad vistas of mathematics out
to the far horizon. They delight in con-
cepts that unify our thinking and bring
together diverse problems from different parts of
the landscape. Frogs live in the mud below and see
only the flowers that grow nearby. They delight
in the details of particular objects, and they solve
problems one at a time. I happen to be a frog, but
many of my best friends are birds. The main theme
of my talk tonight is this. Mathematics needs both
birds and frogs. Mathematics is rich and beautiful
because birds give it broad visions and frogs give it
intricate details. Mathematics is both great art and
important science, because it combines generality
of concepts with depth of structures. It is stupid
to claim that birds are better than frogs because
they see farther, or that frogs are better than birds
because they see deeper. The world of mathemat-
ics is both broad and deep, and we need birds and
frogs working together to explore it.
This talk is called the Einstein lecture, and I am
grateful to the American Mathematical Society
for inviting me to do honor to Albert Einstein.
Einstein was not a mathematician, but a physicist
who had mixed feelings about mathematics. On
the one hand, he had enormous respect for the
power of mathematics to describe the workings
of nature, and he had an instinct for mathematical
beauty which led him onto the right track to find
nature’s laws. On the other hand, he had no inter-
est in pure mathematics, and he had no technical
skill as a mathematician. In his later years he hired
younger colleagues with the title of assistants to
do mathematical calculations for him. His way of
thinking was physical rather than mathematical.
He was supreme among physicists as a bird who
saw further than others. I will not talk about Ein-
stein since I have nothing new to say.
Francis Bacon and René Descartes
At the beginning of the seventeenth century, two
great philosophers, Francis Bacon in England and
René Descartes in France, proclaimed the birth of
modern science. Descartes was a bird, and Bacon
was a frog. 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 of logic 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 suc-
cessful. 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 Poincaré
were Cartesians. Science was greatly enriched by
the cross-fertilization of the two contrasting cul-
tures. Both cultures were always at work in both
countries. Newton was at heart a Cartesian, using
Freeman Dyson is an emeritus professor in the School of
Natural Sciences, Institute for Advanced Study, Princeton,
NJ. His email address is
dyson@ias.edu
.
This article is a written version of his AMS Einstein Lecture,
which was to have been given in October 2008 but which
unfortunately had to be canceled.
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S
ome mathematicians are birds, others
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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.
In the history of twentieth century mathematics,
there were two decisive events, one belonging to
the Baconian tradition and the other to the Carte-
sian tradition. The first was the International Con-
gress of Mathematicians in Paris in 1900, at which
Hilbert gave the keynote address,
charting the course of mathematics
for the coming century by propound-
ing his famous list of twenty-three
outstanding unsolved problems. Hil-
bert himself was a bird, flying high
over the whole territory of mathemat-
ics, but he addressed his problems to
the frogs who would solve them one
at a time. The second decisive event
was the formation of the Bourbaki
group of mathematical birds in France
in the 1930s, dedicated to publish-
ing a series of textbooks that would
establish a unifying framework for
all of mathematics. The Hilbert prob-
lems 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 Bour-
baki 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 math-
ematics. The Bourbaki program was the extreme
expression of the Cartesian style. It narrowed the
scope of mathematics by excluding the beautiful
flowers that Baconian travelers might collect by
the wayside.
Francis Bacon
wave mechanics in 1926. Schrödinger was a bird
who started from the idea of unifying mechanics
with optics. A hundred years earlier, Hamilton had
unified classical mechanics with ray optics, using
the same mathematics to describe optical rays
and classical particle trajectories. Schrödinger’s
idea was to extend this unification to wave optics
and wave mechanics. Wave optics already existed,
but wave mechanics did not. Schrödinger had to
invent wave mechanics to complete the unification.
Starting from wave optics as a model,
he wrote down a differential equa-
tion for a mechanical particle, but the
equation made no sense. The equation
looked like the equation of conduction
of heat in a continuous medium. Heat
conduction has no visible relevance to
particle mechanics. Schrödinger’s idea
seemed to be going nowhere. But then
came the surprise. Schrödinger put
the square root of minus one into the
equation, and suddenly it made sense.
Suddenly it became a wave equation
instead of a heat conduction equation.
And Schrödinger found to his delight
that the equation has solutions cor-
responding to the quantized orbits in
the Bohr model of the atom.
It turns out that the Schrödinger
equation describes correctly every-
thing we know about the behavior of
atoms. It is the basis of all of chem-
istry and most of physics. And that
square root of minus one means that
nature works with complex numbers
and not with real numbers. This dis-
covery came as a complete surprise,
to Schrödinger as well as to every-
body else. According to Schrödinger,
his fourteen-year-old girl friend 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 of functions of
complex variables. They had discovered that the
theory of functions became far deeper and more
powerful when it was extended from real to com-
plex 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.
René Descartes
Jokes of Nature
For me, as a Baconian, the main thing missing in
the Bourbaki program is the element of surprise.
The Bourbaki program tried to make mathematics
logical. 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
minus one that the physicist Erwin Schrödinger
put into his wave equation when he invented
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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 mathemat-
ics. 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 of little
interest either to mathematicians or to physicists.
The nonlinear theory of Lie groups was too compli-
cated 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 of linear represen-
tations 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.
A third joke of nature is the existence of quasi-
crystals. 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 dis-
covered, real solid objects growing out of liquid
metal alloys, showing the symmetry of the icosa-
hedral group, which includes five-fold rotations.
Meanwhile, the mathematician Roger Penrose
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 to include quasi-
crystals. That is a major program of research which
is still in progress.
A fourth joke of nature is a similarity in be-
havior between quasi-crystals and the zeros of
the Riemann Zeta function. The zeros of the zeta-
function are exciting to mathematicians because
they are found to lie on a straight line and nobody
understands why. The statement that with trivial
exceptions they all lie on a straight line is the
famous Riemann Hypothesis. To prove the Rie-
mann Hypothesis has been the dream of young
mathematicians for more than a hundred years.
I am now making the outrageous suggestion that
we might use quasi-crystals to prove the Riemann
Hypothesis. Those of you who are mathematicians
may consider the suggestion frivolous. Those who
are not mathematicians may consider it uninterest-
ing. Nevertheless I am putting it forward for your
serious consideration. When the physicist Leo
Szilard was young, he became dissatisfied with the
ten commandments of Moses and wrote a new set
of ten commandments to replace them. Szilard’s
second commandment says: “Let your acts be di-
rected towards a worthy goal, but do not ask if they
can reach it: they are to be models and examples,
not means to an end.” Szilard practiced what he
preached. He was the first physicist to imagine
nuclear weapons and the first to campaign ac-
tively against their use. His second commandment
certainly applies here. The proof of the Riemann
Hypothesis is a worthy goal, and it is not for us to
ask whether we can reach it. I will give you some
hints describing how it might be achieved. Here I
will be giving voice to the mathematician that I was
fifty years ago before I became a physicist. I will
talk first about the Riemann Hypothesis and then
about quasi-crystals.
There were until recently two supreme unsolved
problems in the world of pure mathematics, the
proof of Fermat’s Last Theorem and the proof of
the Riemann Hypothesis. Twelve years ago, my
Princeton colleague Andrew Wiles polished off
Fermat’s Last Theorem, and only the Riemann Hy-
pothesis remains. Wiles’ proof of the Fermat Theo-
rem was not just a technical stunt. It required the
discovery and exploration of a new field of math-
ematical ideas, far wider and more consequential
than the Fermat Theorem itself. It is likely that
any proof of the Riemann Hypothesis will likewise
lead to a deeper understanding of many diverse
areas of mathematics and perhaps of physics too.
Riemann’s zeta-function, and other zeta-func-
tions similar to it, appear ubiquitously in number
theory, in the theory of dynamical systems, in
geometry, in function theory, and in physics. The
zeta-function stands at a junction where paths lead
in many directions. A proof of the hypothesis will
illuminate all the connections. Like every serious
student of pure mathematics, when I was young I
had dreams of proving the Riemann Hypothesis.
I had some vague ideas that I thought might lead
to a proof. In recent years, after the discovery of
quasi-crystals, my ideas became a little less vague.
I offer them here for the consideration of any
young mathematician who has ambitions to win
a Fields Medal.
Quasi-crystals can exist in spaces of one, two,
or three dimensions. From the point of view of
physics, the three-dimensional quasi-crystals are
the most interesting, since they inhabit our three-
dimensional world and can be studied experi-
mentally. From the point of view of a mathemati-
cian, one-dimensional quasi-crystals are much
more interesting than two-dimensional or three-
dimensional quasi-crystals because they exist in
far greater variety. The mathematical definition
of a quasi-crystal is as follows. A quasi-crystal
is a distribution of discrete point masses whose
Fourier transform is a distribution of discrete
point frequencies. Or to say it more briefly, a
quasi-crystal is a pure point distribution that has
a pure point spectrum. This definition includes
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as a special case the ordinary crystals,
which are periodic distributions with
periodic spectra.
Excluding the ordinary crystals,
quasi-crystals in three dimensions
come in very limited variety, all of
them associated with the icosahedral
group. The two-dimensional quasi-
crystals are more numerous, roughly
one distinct type associated with each
regular polygon in a plane. The two-
dimensional quasi-crystal with pentag-
onal symmetry is the famous Penrose
tiling of the plane. Finally, the one-
dimensional quasi-crystals have a far
richer structure since they are not tied
to any rotational symmetries. So far as
I know, no complete enumeration of
one-dimensional quasi-crystals exists.
It is known that a unique quasi-crystal
exists corresponding to every Pisot-
Vijayaraghavan number or PV number.
A PV number is a real algebraic inte-
ger, a root of a polynomial equation
with integer coefficients, such that all
the other roots have absolute value
less than one, [1]. The set of all PV
numbers is infinite and has a remark-
able topological structure. The set
of all one-dimensional quasi-crystals
has a structure at least as rich as the
set of all PV numbers and probably much richer.
We do not know for sure, but it is likely that a
huge universe of one-dimensional quasi-crystals
not associated with PV numbers is waiting to be
discovered.
Here comes the connection of the one-
dimensional quasi-crystals with the Riemann
hypothesis. If the Riemann hypothesis is true,
then the zeros of the zeta-function form a one-
dimensional quasi-crystal according to the defini-
tion. They constitute a distribution of point masses
on a straight line, and their Fourier transform is
likewise a distribution of point masses, one at each
of the logarithms of ordinary prime numbers and
prime-power numbers. My friend Andrew Odlyzko
has published a beautiful computer calculation of
the Fourier transform of the zeta-function zeros,
[6]. The calculation shows precisely the expected
structure of the Fourier transform, with a sharp
discontinuity at every logarithm of a prime or
prime-power number and nowhere else.
My suggestion is the following. Let us pretend
that we do not know that the Riemann Hypothesis
is true. Let us tackle the problem from the other
end. Let us try to obtain a complete enumera-
tion and classification of one-dimensional quasi-
crystals. That is to say, we enumerate and classify
all point distributions that have a discrete point
spectrum. Collecting and classifying new species of
Abram Besicovitch
objects is a quintessentially Baconian
activity. It is an appropriate activity
for mathematical frogs. We shall then
find the well-known quasi-crystals
associated with PV numbers, and
also a whole universe of other quasi-
crystals, known and unknown. Among
the multitude of other quasi-crystals
we search for one corresponding to
the Riemann zeta-function and one
corresponding to each of the other
zeta-functions that resemble the Rie-
mann zeta-function. Suppose that
we find one of the quasi-crystals in
our enumeration with properties
that identify it with the zeros of the
Riemann zeta-function. Then we have
proved the Riemann Hypothesis and
we can wait for the telephone call
announcing the award of the Fields
Medal.
These are of course idle dreams.
The problem of classifying one-
dimensional quasi-crystals is horren-
dously difficult, probably at least as
difficult as the problems that Andrew
Wiles took seven years to explore. But
if we take a Baconian point of view,
the history of mathematics is a his-
tory of horrendously difficult prob-
lems being solved by young people too ignorant to
know that they were impossible. The classification
of quasi-crystals is a worthy goal, and might even
turn out to be achievable. Problems of that degree
of difficulty will not be solved by old men like me.
I leave this problem as an exercise for the young
frogs in the audience.
Hermann Weyl
Abram Besicovitch and Hermann Weyl
Let me now introduce you to some notable frogs
and birds that I knew personally. I came to Cam-
bridge University as a student in 1941 and had
the tremendous luck to be given the Russian
mathematician Abram Samoilovich Besicovitch
as my supervisor. Since this was in the middle
of World War Two, there were very few students
in Cambridge, and almost no graduate students.
Although I was only seventeen years old and Besi-
covitch was already a famous professor, he gave
me a great deal of his time and attention, and we
became life-long friends. He set the style in which
I began to work and think about mathematics. He
gave wonderful lectures on measure-theory and
integration, smiling amiably when we laughed at
his glorious abuse of the English language. I re-
member only one occasion when he was annoyed
by our laughter. He remained silent for a while and
then said, “Gentlemen. Fifty million English speak
English you speak. Hundred and fifty million Rus-
sians speak English I speak.”
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Besicovitch was a frog, and he became famous
when he was young by solving a problem in el-
ementary plane geometry known as the Kakeya
problem. The Kakeya problem was the following.
A line segment of length one is allowed to move
freely in a plane while rotating through an angle
of 360 degrees. What is the smallest area of the
plane that it can cover during its rotation? The
problem was posed by the Japanese mathematician
Kakeya in 1917 and remained a famous unsolved
problem for ten years. George Birkhoff, the lead-
ing American mathematician at that time, publicly
proclaimed that the Kakeya problem and the four-
color problem were the outstanding unsolved
problems of the day. It was widely believed that
the minimum area was
π/
8, which is the area of a
three-cusped hypocycloid. The three-cusped hypo-
cycloid is a beautiful three-pointed curve. It is the
curve traced out by a point on the circumference
of a circle with radius one-quarter, when the circle
rolls around the inside of a fixed circle with radius
three-quarters. The line segment of length one can
turn while always remaining tangent to the hypo-
cycloid with its two ends also on the hypocycloid.
This picture of the line turning while touching the
inside of the hypocycloid at three points was so
elegant that most people believed it must give the
minimum area. Then Besicovitch surprised every-
one by proving that the area covered by the line as
it turns can be less than
for any positive
.
Besicovitch had actually solved the problem in
1920 before it became famous, not even knowing
that Kakeya had proposed it. In 1920 he published
the solution in Russian in the
Journal of the Perm
Physics and Mathematics Society
, a journal that
was not widely read. The university of Perm, a
city 1,100 kilometers east of Moscow, was briefly
a refuge for many distinguished mathematicians
after the Russian revolution. They published two
volumes of their journal before it died amid the
chaos of revolution and civil war. Outside Russia
the journal was not only unknown but unobtain-
able. Besicovitch left Russia in 1925 and arrived at
Copenhagen, where he learned about the famous
Kakeya problem that he had solved five years ear-
lier. He published the solution again, this time in
English in the
Mathematische Zeitschrift
. The Ka-
keya problem as Kakeya proposed it was a typical
frog problem, a concrete problem without much
connection with the rest of mathematics. Besico-
vitch gave it an elegant and deep solution, which
revealed a connection with general theorems about
the structure of sets of points in a plane.
The Besicovitch style is seen at its finest in
his three classic papers with the title, “On the
fundamental geometric properties of linearly
measurable plane sets of points”, published in
Mathematische Annalen
in the years 1928, 1938,
and 1939. In these papers he proved that every
linearly measurable set in the plane is divisible
into a regular and an irregular component, that
the regular component has a tangent almost
everywhere, and the irregular component has a
projection of measure zero onto almost all direc-
tions. Roughly speaking, the regular component
looks like a collection of continuous curves, while
the irregular component looks nothing like a con-
tinuous curve. The existence and the properties of
the irregular component are connected with the
Besicovitch solution of the Kakeya problem. One
of the problems that he gave me to work on was
the division of measurable sets into regular and
irregular components in spaces of higher dimen-
sions. I got nowhere with the problem, but became
permanently imprinted with the Besicovitch style.
The Besicovitch style is architectural. He builds
out of simple elements a delicate and complicated
architectural structure, usually with a hierarchical
plan, and then, when the building is finished, the
completed structure leads by simple arguments
to an unexpected conclusion. Every Besicovitch
proof is a work of art, as carefully constructed as
a Bach fugue.
A few years after my apprenticeship with Be-
sicovitch, I came to Princeton and got to know
Hermann Weyl. Weyl was a prototypical bird, just
as Besicovitch was a prototypical frog. I was lucky
to overlap with Weyl for one year at the Princeton
Institute for Advanced Study before he retired
from the Institute and moved back to his old home
in Zürich. He liked me because during that year I
published papers in the
Annals of Mathematics
about number theory and in the
Physical Review
about the quantum theory of radiation. He was one
of the few people alive who was at home in both
subjects. He welcomed me to the Institute, in the
hope that I would be a bird like himself. He was dis-
appointed. I remained obstinately a frog. Although
I poked around in a variety of mud-holes, I always
looked at them one at a time and did not look for
connections between them. For me, number theory
and quantum theory were separate worlds with
separate beauties. I did not look at them as Weyl
did, hoping to find clues to a grand design.
Weyl’s great contribution to the quantum theory
of radiation was his invention of gauge fields. The
idea of gauge fields had a curious history. Weyl
invented them in 1918 as classical fields in his
unified theory of general relativity and electromag-
netism, [7]. He called them “gauge fields” because
they were concerned with the non-integrability
of measurements of length. His unified theory
was promptly and publicly rejected by Einstein.
After this thunderbolt from on high, Weyl did
not abandon his theory but moved on to other
things. The theory had no experimental conse-
quences that could be tested. Then in 1929, after
quantum mechanics had been invented by others,
Weyl realized that his gauge fields fitted far bet-
ter into the quantum world than they did into the
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