Dyson Freeman - On Birds and Frogs.pdf - Popularnonaukowe - antonif

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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215

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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