Fermat's Library | Birds and Frogs annotated/explained version.

### TL;DR In this essay Freeman Dyson presents two types of math...

Freeman Dyson is an English-born American theoretical physicist and...

Science has evolved following both Cartesian and Baconian views sim...

The International Congress of Mathematicians (ICM) is the largest c...

The Bourbaki group was a group of young French mathematicians start...

#### Hilbert's 23 Mathematical Problems 1. Cantor's problem of t...

#### The Ten Commandments of Leo Szilard: 1. Recognize the conne...

Lie groups, named after Sophus Lie, lie at the intersection of alge...

Here there is a reference to Riemann hypothesis, one of Millenium m...

> ***"the history of mathematics is a history of horrendously diff...

In mathematics a Kakeya set, or Besicovitch set, is a set of points...

A hypocycloid is a curve generated by the trace of a fixed point on...

I'm not an expert on computational complexity — but I believe this ...

Here is a great resource if you want to learn more about Gödel's In...

John Von Neumann wrote ***First Draft of a Report on the EDVAC** in...

> ***Chaos: When the present determines the future, but the approxi...

You can read this paper here: [Period Three Implies Chaos - Li, T. ...

212 Notices of the AMs VoluMe 56, NuMber 2

Birds and Frogs

Freeman Dyson

ome mathematicians are birds, others

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.

212 Notices of the AMs Volume 56, Number 2

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.

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

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.

Francis Bacon

René Descartes

februAry 2009 Notices of the AMs 213

214 Notices of the AMs VoluMe 56, NuMber 2

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

februAry 2009 Notices of the AMs 215

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

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.

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.”

Abram Besicovitch

Hermann Weyl

Photograph of A. Besicovitch from the AMS archives. Photo of Hermann Weyl

Photograph of A. Besicovitch from AMS archives. Photo of H. Weyl from the archives of Peter Roquette, used with permisssion.

216 Notices of the AMs VoluMe 56, NuMber 2

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

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

februAry 2009 Notices of the AMs 217

classical world, [8]. All

that he needed to do, to

change a classical gauge

into a quantum gauge,

was to change real

numbers into complex

numbers. In quantum

mechanics, every quan-

tum of electric charge

carries with it a com-

plex wave function with

a phase, and the gauge

field is concerned with

the non-integrability of

measurements of phase.

The gauge field could

then be precisely identified with the electromag-

netic potential, and the law of conservation of

charge became a consequence of the local phase

invariance of the theory.

Weyl died four years after he returned from

Princeton to Zürich, and I wrote his obituary for the

journal Nature, [3]. “Among all the mathematicians

who began their working lives in the twentieth

century,” I wrote, “Hermann Weyl was the one who

made major contributions in the greatest number

of different fields. He alone could stand compari-

son with the last great universal mathematicians

of the nineteenth century, Hilbert and Poincaré.

So long as he was alive, he embodied a living con-

tact between the main lines of advance in pure

mathematics and in theoretical physics. Now he

is dead, the contact is broken, and our hopes of

comprehending the physical universe by a direct

use of creative mathematical imagination are for

the time being ended.” I mourned his passing, but

I had no desire to pursue his dream. I was happy

to see pure mathematics and physics marching

ahead in opposite directions.

The obituary ended with a sketch of Weyl as

a human being: “Characteristic of Weyl was an

aesthetic sense which dominated his thinking on

all subjects. He once said to me, half joking, ‘My

work always tried to unite the true with the beau-

tiful; but when I had to choose one or the other,

I usually chose the beautiful’. This remark sums

up his personality perfectly. It shows his profound

faith in an ultimate harmony of Nature, in which

the laws should inevitably express themselves in

a mathematically beautiful form. It shows also

his recognition of human frailty, and his humor,

which always stopped him short of being pomp-

ous. His friends in Princeton will remember him

as he was when I last saw him, at the Spring Dance

of the Institute for Advanced Study last April: a

big jovial man, enjoying himself splendidly, his

cheerful frame and his light step giving no hint of

his sixty-nine years.”

The fifty years after Weyl’s death were a golden

age of experimental physics and observational

astronomy, a golden

age for Baconian travel-

ers picking up facts, for

frogs exploring small

patches of the swamp

in which we live. Dur-

ing these fifty years, the

frogs accumulated a de-

tailed knowledge of a

large variety of cosmic

structures and a large

variety of particles and

interactions. As the

exploration of new ter-

ritories continued, the

universe became more

complicated. Instead of a grand design displaying

the simplicity and beauty of Weyl’s mathematics,

the explorers found weird objects such as quarks

and gamma-ray bursts, weird concepts such as su-

persymmetry and multiple universes. Meanwhile,

mathematics was also becoming more compli-

cated, as exploration continued into the phenom-

ena of chaos and many other new areas opened

by electronic computers. The mathematicians

discovered the central mystery of computability,

the conjecture represented by the statement P is

not equal to NP. The conjecture asserts that there

exist mathematical problems which can be quickly

solved in individual cases but cannot be solved

by a quick algorithm applicable to all cases. The

most famous example of such a problem is the

traveling salesman problem, which is to find the

shortest route for a salesman visiting a set of cit-

ies, knowing the distance between each pair. All

the experts believe that the conjecture is true, and

that the traveling salesman problem is an example

of a problem that is P but not NP. But nobody has

even a glimmer of an idea how to prove it. This is

a mystery that could not even have been formu-

lated within the nineteenth-century mathematical

universe of Hermann Weyl.

Frank Yang and Yuri Manin

The last fifty years have been a hard time for

birds. Even in hard times, there is work for birds

to do, and birds have appeared with the courage to

tackle it. Soon after Weyl left Princeton, Frank Yang

arrived from Chicago and moved into Weyl’s old

house. Yang took Weyl’s place as the leading bird

among my generation of physicists. While Weyl

was still alive, Yang and his student Robert Mills

discovered the Yang-Mills theory of non-Abelian

gauge fields, a marvelously elegant extension of

Weyl’s idea of a gauge field, [11]. Weyl’s gauge field

was a classical quantity, satisfying the commuta-

tive law of multiplication. The Yang-Mills theory

had a triplet of gauge fields which did not com-

mute. They satisfied the commutation rules of the

three components of a quantum mechanical spin,

Chen Ning (Frank)

Yang

Yuri Manin

Photo of F. Yang courtesy of SUNY Stony Brook. Photo of Y. Manin courtesy of Northwestern University.

218 Notices of the AMs VoluMe 56, NuMber 2

the worlds of geometry and dynamics with his

concept of fluxions, nowadays called calculus. In

the nineteenth century Boole linked the worlds

of logic and algebra with his concept of symbolic

logic, and Riemann linked the worlds of geometry

and analysis with his concept of Riemann sur-

faces. Coordinates, fluxions, symbolic logic, and

Riemann surfaces are all metaphors, extending

the meanings of words from familiar to unfamiliar

contexts. Manin sees the future of mathematics

as an exploration of metaphors that are already

visible but not yet understood. The deepest such

metaphor is the similarity in structure between

number theory and physics. In both fields he sees

tantalizing glimpses of parallel concepts, symme-

tries linking the continuous with the discrete. He

looks forward to a unification which he calls the

quantization of mathematics.

“Manin disagrees with the Baconian story, that

Hilbert set the agenda for the mathematics of the

twentieth century when he presented his famous

list of twenty-three unsolved problems to the In-

ternational Congress of Mathematicians in Paris

in 1900. According to Manin, Hilbert’s problems

were a distraction from the central themes of

mathematics. Manin sees the important advances

in mathematics coming from programs, not from

problems. Problems are usually solved by apply-

ing old ideas in new ways. Programs of research

are the nurseries where new ideas are born. He

sees the Bourbaki program, rewriting the whole of

mathematics in a more abstract language, as the

source of many of the new ideas of the twentieth

century. He sees the Langlands program, unifying

number theory with geometry, as a promising

source of new ideas for the twenty-first. People

who solve famous unsolved problems may win big

prizes, but people who start new programs are the

real pioneers.”

The Russian version of Mathematics as Meta-

phor contains ten chapters that were omitted from

the English version. The American Mathematical

Society decided that these chapters would not be

of interest to English language readers. The omis-

sions are doubly unfortunate. First, readers of the

English version see only a truncated view of Manin,

who is perhaps unique among mathematicians in

his broad range of interests extending far beyond

mathematics. Second, we see a truncated view of

Russian culture, which is less compartmentalized

than English language culture, and brings math-

ematicians into closer contact with historians and

artists and poets.

John von Neumann

Another important figure in twentieth century

mathematics was John von Neumann. Von Neu-

mann was a frog, applying his prodigious tech-

nical skill to solve problems in many branches

of mathematics and physics. He began with the

which are generators of the simplest non-Abelian

Lie algebra

. The theory was later generalized so

that the gauge fields could be generators of any

finite-dimensional Lie algebra. With this general-

ization, the Yang-Mills gauge field theory provided

the framework for a model of all the known par-

ticles and interactions, a model that is now known

as the Standard Model of particle physics. Yang put

the finishing touch to it by showing that Einstein’s

theory of gravitation fits into the same framework,

with the Christoffel three-index symbol taking the

role of gauge field, [10].

In an appendix to his 1918 paper, added in 1955

for the volume of selected papers published to

celebrate his seventieth birthday, Weyl expressed

his final thoughts about gauge field theories (my

translation), [12]: “The strongest argument for my

theory seemed to be this, that gauge invariance

was related to conservation of electric charge in

the same way as coordinate invariance was related

to conservation of energy and momentum.” Thirty

years later Yang was in Zürich for the celebration

of Weyl’s hundredth birthday. In his speech, [12],

Yang quoted this remark as evidence of Weyl’s de-

votion to the idea of gauge invariance as a unifying

principle for physics. Yang then went on, “Sym-

metry, Lie groups, and gauge invariance are now

recognized, through theoretical and experimental

developments, to play essential roles in determin-

ing the basic forces of the physical universe. I have

called this the principle that symmetry dictates in-

teraction.” This idea, that symmetry dictates inter-

action, is Yang’s generalization of Weyl’s remark.

Weyl observed that gauge invariance is intimately

connected with physical conservation laws. Weyl

could not go further than this, because he knew

only the gauge invariance of commuting Abelian

fields. Yang made the connection much stronger

by introducing non-Abelian gauge fields. With

non-Abelian gauge fields generating nontrivial Lie

algebras, the possible forms of interaction between

fields become unique, so that symmetry dictates

interaction. This idea is Yang’s greatest contribu-

tion to physics. It is the contribution of a bird,

flying high over the rain forest of little problems

in which most of us spend our lives.

Another bird for whom I have a deep respect

is the Russian mathematician Yuri Manin, who

recently published a delightful book of essays with

the title Mathematics as Metaphor [5]. The book

was published in Moscow in Russian, and by the

American Mathematical Society in English. I wrote

a preface for the English version, and I give you

here a short quote from my preface. “Mathematics

as Metaphor is a good slogan for birds. It means

that the deepest concepts in mathematics are

those which link one world of ideas with another.

In the seventeenth century Descartes linked the

disparate worlds of algebra and geometry with

his concept of coordinates, and Newton linked

februAry 2009 Notices of the AMs 219

foundations of mathematics. He found the first

satisfactory set of axioms for set-theory, avoiding

the logical paradoxes that Cantor had encountered

in his attempts to deal with infinite sets and

infinite numbers. Von Neumann’s axioms were

used by his bird friend Kurt Gödel a few years later

to prove the existence of undecidable propositions

in mathematics. Gödel’s theorems gave birds a new

vision of mathematics. After Gödel, mathematics

was no longer a single structure tied

together with a unique concept of

truth, but an archipelago of structures

with diverse sets of axioms and di-

verse notions of truth. Gödel showed

that mathematics is inexhaustible. No

matter which set of axioms is chosen

as the foundation, birds can always

find questions that those axioms can-

not answer.

Von Neumann went on from the

foundations of mathematics to the

foundations of quantum mechanics.

To give quantum mechanics a firm

mathematical foundation, he created

a magnificent theory of rings of op-

erators. Every observable quantity is

represented by a linear operator, and

the peculiarities of quantum behav-

ior are faithfully represented by the

algebra of operators. Just as Newton

invented calculus to describe classi-

cal dynamics, von Neumann invented

rings of operators to describe quan-

tum dynamics.

Von Neumann made fundamental

contributions to several other fields,

especially to game theory and to the

design of digital computers. For the

last ten years of his life, he was deeply

involved with computers. He was so

strongly interested in computers that he decided

not only to study their design but to build one with

real hardware and software and use it for doing

science. I have vivid memories of the early days of

von Neumann’s computer project at the Institute

for Advanced Study in Princeton. At that time he

had two main scientific interests, hydrogen bombs

and meteorology. He used his computer during the

night for doing hydrogen bomb calculations and

during the day for meteorology. Most of the people

hanging around the computer building in daytime

were meteorologists. Their leader was Jule Char-

ney. Charney was a real meteorologist, properly

humble in dealing with the inscrutable mysteries

of the weather, and skeptical of the ability of the

computer to solve the mysteries. John von Neu-

mann was less humble and less skeptical. I heard

von Neumann give a lecture about the aims of his

project. He spoke, as he always did, with great con-

fidence. He said, “The computer will enable us to

divide the atmosphere at any moment into stable

regions and unstable regions. Stable regions we

can predict. Unstable regions we can control.” Von

Neumann believed that any unstable region could

be pushed by a judiciously applied small perturba-

tion so that it would move in any desired direction.

The small perturbation would be applied by a fleet

of airplanes carrying smoke generators, to absorb

sunlight and raise or lower temperatures at places

where the perturbation would be most

effective. In particular, we could stop

an incipient hurricane by identifying

the position of an instability early

enough, and then cooling that patch

of air before it started to rise and form

a vortex. Von Neumann, speaking in

1950, said it would take only ten years

to build computers powerful enough

to diagnose accurately the stable and

unstable regions of the atmosphere.

Then, once we had accurate diagno-

sis, it would take only a short time

for us to have control. He expected

that practical control of the weather

would be a routine operation within

the decade of the 1960s.

Von Neumann, of course, was

wrong. He was wrong because he

did not know about chaos. We now

know that when the motion of the

atmosphere is locally unstable, it is

very often chaotic. The word “chaotic”

means that motions that start close

together diverge exponentially from

each other as time goes on. When the

motion is chaotic, it is unpredictable,

and a small perturbation does not

move it into a stable motion that can

be predicted. A small perturbation

will usually move it into another cha-

otic motion that is equally unpredictable. So von

Neumann’s strategy for controlling the weather

fails. He was, after all, a great mathematician but

a mediocre meteorologist.

Edward Lorenz discovered in 1963 that the so-

lutions of the equations of meteorology are often

chaotic. That was six years after von Neumann

died. Lorenz was a meteorologist and is generally

regarded as the discoverer of chaos. He discovered

the phenomena of chaos in the meteorological con-

text and gave them their modern names. But in fact

I had heard the mathematician Mary Cartwright,

who died in 1998 at the age of 97, describe the

same phenomena in a lecture in Cambridge in 1943,

twenty years before Lorenz discovered them. She

called the phenomena by different names, but they

were the same phenomena. She discovered them in

the solutions of the van der Pol equation which de-

scribe the oscillations of a nonlinear amplifier, [2].

The van der Pol equation was important in World

John von Neumann

Mary Cartwright

Photograph of Mary Cartright courtesy of The Mistress and Fellows, Girton College, Cambridge.

220 Notices of the AMs VoluMe 56, NuMber 2

War II because nonlinear amplifiers fed power

to the transmitters in early radar systems. The

transmitters behaved erratically, and the Air Force

blamed the manufacturers for making defective

amplifiers. Mary Cartwright was asked to look into

the problem. She showed that the manufacturers

were not to blame. She showed that the van der Pol

equation was to blame. The solutions of the van der

Pol equation have precisely the chaotic behavior

that the Air Force was complaining about. I heard

all about chaos from Mary Cartwright seven years

before I heard von Neumann talk about weather

control, but I was not far-sighted enough to make

the connection. It never entered my head that the

erratic behavior of the van der Pol equation might

have something to do with meteorology. If I had

been a bird rather than a frog, I would probably

have seen the connection, and I might have saved

von Neumann a lot of trouble. If he had known

about chaos in 1950, he would probably have

thought about it deeply, and he would have had

something important to say about it in 1954.

Von Neumann got into trouble at the end of

his life because he was really a frog but everyone

expected him to fly like a bird. In 1954 there was

an International Congress of Mathematicians in

Amsterdam. These congresses happen only once

in four years and it is a great honor to be invited to

speak at the opening session. The organizers of the

Amsterdam congress invited von Neumann to give

the keynote speech, expecting him to repeat the act

that Hilbert had performed in Paris in 1900. Just as

Hilbert had provided a list of unsolved problems

to guide the development of mathematics for the

first half of the twentieth century, von Neumann

was invited to do the same for the second half of

the century. The title of von Neumann’s talk was

announced in the program of the congress. It was

“Unsolved Problems in Mathematics: Address by

Invitation of the Organizing Committee”. After the

congress was over, the complete proceedings were

published, with the texts of all the lectures except

this one. In the proceedings there is a blank page

with von Neumann’s name and the title of his talk.

Underneath, it says, “No manuscript of this lecture

was available.”

What happened? I know what happened, be-

cause I was there in the audience, at 3:00 p.m.

on Thursday, September 2, 1954, in the Concert-

gebouw concert hall. The hall was packed with

mathematicians, all expecting to hear a brilliant

lecture worthy of such a historic occasion. The

lecture was a huge disappointment. Von Neumann

had probably agreed several years earlier to give

a lecture about unsolved problems and had then

forgotten about it. Being busy with many other

things, he had neglected to prepare the lecture.

Then, at the last moment, when he remembered

that he had to travel to Amsterdam and say some-

thing about mathematics, he pulled an old lecture

from the 1930s out of a drawer and dusted it off.

The lecture was about rings of operators, a subject

that was new and fashionable in the 1930s. Noth-

ing about unsolved problems. Nothing about the

future. Nothing about computers, the subject that

we knew was dearest to von Neumann’s heart.

He might at least have had something new and

exciting to say about computers. The audience in

the concert hall became restless. Somebody said

in a voice loud enough to be heard all over the

hall, “Aufgewärmte Suppe”, which is German for

“warmed-up soup”. In 1954 the great majority of

mathematicians knew enough German to under-

stand the joke. Von Neumann, deeply embarrassed,

brought his lecture to a quick end and left the hall

without waiting for questions.

Weak Chaos

If von Neumann had known about chaos when he

spoke in Amsterdam, one of the unsolved prob-

lems that he might have talked about was weak

chaos. The problem of weak chaos is still unsolved

fifty years later. The problem is to understand

why chaotic motions often remain bounded and

do not cause any violent instability. A good ex-

ample of weak chaos is the orbital motions of the

planets and satellites in the solar system. It was

discovered only recently that these motions are

chaotic. This was a surprising discovery, upsetting

the traditional picture of the solar system as the

prime example of orderly stable motion. The math-

ematician Laplace two hundred years ago thought

he had proved that the solar system is stable. It

now turns out that Laplace was wrong. Accurate

numerical integrations of the orbits show clearly

that neighboring orbits diverge exponentially. It

seems that chaos is almost universal in the world

of classical dynamics.

Chaotic behavior was never suspected in the

solar system before accurate long-term integra-

tions were done, because the chaos is weak. Weak

chaos means that neighboring trajectories diverge

exponentially but never diverge far. The divergence

begins with exponential growth but afterwards

remains bounded. Because the chaos of the plan-

etary motions is weak, the solar system can survive

for four billion years. Although the motions are

chaotic, the planets never wander far from their

customary places, and the system as a whole does

not fly apart. In spite of the prevalence of chaos,

the Laplacian view of the solar system as a perfect

piece of clockwork is not far from the truth.

We see the same phenomena of weak chaos in

the domain of meteorology. Although the weather

in New Jersey is painfully chaotic, the chaos has

firm limits. Summers and winters are unpredict-

ably mild or severe, but we can reliably predict

that the temperature will never rise to 45 degrees

Celsius or fall to minus 30, extremes that are

often exceeded in India or in Minnesota. There

februAry 2009 Notices of the AMs 221

is no conservation law of physics that forbids

temperatures from rising as high in New Jersey

as in India, or from falling as low in New Jersey

as in Minnesota. The weakness of chaos has been

essential to the long-term survival of life on this

planet. Weak chaos gives us a challenging variety

of weather while protecting us from fluctuations

so severe as to endanger our existence. Chaos

remains mercifully weak for reasons that we do

not understand. That is another unsolved problem

for young frogs in the audience to take home. I

challenge you to understand the reasons why the

chaos observed in a great diversity of dynamical

systems is generally weak.

The subject of chaos is characterized by an

abundance of quantitative data, an unending sup-

ply of beautiful pictures, and a shortage of rigor-

ous theorems. Rigorous theorems are the best way

to give a subject intellectual depth and precision.

Until you can prove rigorous theorems, you do not

fully understand the meaning of your concepts.

In the field of chaos I know only one rigorous

theorem, proved by Tien-Yien Li and Jim Yorke in

1975 and published in a short paper with the title,

“Period Three Implies Chaos”, [4]. The Li-Yorke

paper is one of the immortal gems in the literature

of mathematics. Their theorem concerns nonlinear

maps of an interval onto itself. The successive posi-

tions of a point when the mapping is repeated can

be considered as the orbit of a classical particle.

An orbit has period

N if the point returns to its

original position after

N mappings. An orbit is

defined to be chaotic, in this context, if it diverges

from all periodic orbits. The theorem says that if a

single orbit with period three exists, then chaotic

orbits also exist. The proof is simple and short. To

my mind, this theorem and its proof throw more

light than a thousand beautiful pictures on the

basic nature of chaos. The theorem explains why

chaos is prevalent in the world. It does not explain

why chaos is so often weak. That remains a task

for the future. I believe that weak chaos will not

be understood in a fundamental way until we can

prove rigorous theorems about it.

String Theorists

I would like to say a few words about string theory.

Few words, because I know very little about string

theory. I never took the trouble to learn the subject

or to work on it myself. But when I am at home at the

Institute for Advanced Study in Princeton, I am sur-

rounded by string theorists, and I sometimes listen

to their conversations. Occasionally I understand a

little of what they are saying. Three things are clear.

First, what they are doing is first-rate mathemat-

ics. The leading pure mathematicians, people like

Michael Atiyah and Isadore Singer, love it. It has

opened up a whole new branch of mathematics,

with new ideas and new problems. Most remark-

ably, it gave the mathematicians new methods to

solve old problems that were previously unsolvable.

Second, the string theorists think of themselves

as physicists rather than mathematicians. They

believe that their theory describes something real

in the physical world. And third, there is not yet

any proof that the theory is relevant to physics.

The theory is not yet testable by experiment. The

theory remains in a world of its own, detached

from the rest of physics. String theorists make

strenuous efforts to deduce consequences of the

theory that might be testable in the real world, so

far without success.

My colleagues Ed Witten and Juan Maldacena

and others who created string theory are birds,

flying high and seeing grand visions of distant

ranges of mountains. The thousands of hum-

bler practitioners of string theory in universities

around the world are frogs, exploring fine details

of the mathematical structures that birds first

saw on the horizon. My anxieties about string

theory are sociological rather than scientific. It is

a glorious thing to be one of the first thousand

string theorists, discovering new connections and

pioneering new methods. It is not so glorious to

be one of the second thousand or one of the tenth

thousand. There are now about ten thousand

string theorists scattered around the world. This

is a dangerous situation for the tenth thousand

and perhaps also for the second thousand. It may

happen unpredictably that the fashion changes

and string theory becomes unfashionable. Then it

could happen that nine thousand string theorists

lose their jobs. They have been trained in a narrow

specialty, and they may be unemployable in other

fields of science.

Why are so many young people attracted to

string theory? The attraction is partly intellectual.

String theory is daring and mathematically elegant.

But the attraction is also sociological. String theory

is attractive because it offers jobs. And why are

so many jobs offered in string theory? Because

string theory is cheap. If you are the chairperson

of a physics department in a remote place without

much money, you cannot afford to build a modern

laboratory to do experimental physics, but you can

afford to hire a couple of string theorists. So you

offer a couple of jobs in string theory, and you

have a modern physics department. The tempta-

tions are strong for the chairperson to offer such

jobs and for the young people to accept them.

This is a hazardous situation for the young people

and also for the future of science. I am not say-

ing that we should discourage young people from

working in string theory if they find it exciting. I

am saying that we should offer them alternatives,

so that they are not pushed into string theory by

economic necessity.

Finally, I give you my own guess for the future

of string theory. My guess is probably wrong. I

have no illusion that I can predict the future. I tell

222 Notices of the AMs VoluMe 56, NuMber 2

you my guess, just to give you something to think

about. I consider it unlikely that string theory will

turn out to be either totally successful or totally

useless. By totally successful I mean that it is a

complete theory of physics, explaining all the de-

tails of particles and their interactions. By totally

useless I mean that it remains a beautiful piece of

pure mathematics. My guess is that string theory

will end somewhere between complete success

and failure. I guess that it will be like the theory

of Lie groups, which Sophus Lie created in the

nineteenth century as a mathematical framework

for classical physics. So long as physics remained

classical, Lie groups remained a failure. They were

a solution looking for a problem. But then, fifty

years later, the quantum revolution transformed

physics, and Lie algebras found their proper place.

They became the key to understanding the central

role of symmetries in the quantum world. I expect

that fifty or a hundred years from now another

revolution in physics will happen, introducing new

concepts of which we now have no inkling, and the

new concepts will give string theory a new mean-

ing. After that, string theory will suddenly find

its proper place in the universe, making testable

statements about the real world. I warn you that

this guess about the future is probably wrong. It

has the virtue of being falsifiable, which accord-

ing to Karl Popper is the hallmark of a scientific

statement. It may be demolished tomorrow by

some discovery coming out of the Large Hadron

Collider in Geneva.

Manin Again

To end this talk, I come back to Yuri Manin and

his book Mathematics as Metaphor. The book

is mainly about mathematics. It may come as a

surprise to Western readers that he writes with

equal eloquence about other subjects such as the

collective unconscious, the origin of human lan-

guage, the psychology of autism, and the role of

the trickster in the mythology of many cultures.

To his compatriots in Russia, such many-sided

interests and expertise would come as no surprise.

Russian intellectuals maintain the proud tradition

of the old Russian intelligentsia, with scientists

and poets and artists and musicians belonging to

a single community. They are still today, as we see

them in the plays of Chekhov, a group of idealists

bound together by their alienation from a super-

stitious society and a capricious government. In

Russia, mathematicians and composers and film-

producers talk to one another, walk together in the

snow on winter nights, sit together over a bottle of

wine, and share each others’ thoughts.

Manin is a bird whose vision extends far be-

yond the territory of mathematics into the wider

landscape of human culture. One of his hobbies

is the theory of archetypes invented by the Swiss

psychologist Carl Jung. An archetype, according to

Jung, is a mental image rooted in a collective un-

conscious that we all share. The intense emotions

that archetypes carry with them are relics of lost

memories of collective joy and suffering. Manin is

saying that we do not need to accept Jung’s theory

as true in order to find it illuminating.

More than thirty years ago, the singer Monique

Morelli made a recording of songs with words by

Pierre MacOrlan. One of the songs is La Ville Morte,

the dead city, with a haunting melody tuned to

Morelli’s deep contralto, with an accordion singing

counterpoint to the voice, and with verbal images

of extraordinary intensity. Printed on the page, the

words are nothing special:

“En pénétrant dans la ville morte,

Je tenait Margot par le main…

Nous marchions de la nécropole,

Les pieds brisés et sans parole,

Devant ces portes sans cadole,

Devant ces trous indéfinis,

Devant ces portes sans parole

Et ces poubelles pleines de cris”.

“As we entered the dead city, I held Margot by

the hand…We walked from the graveyard on our

bruised feet, without a word, passing by these

doors without locks, these vaguely glimpsed holes,

these doors without a word, these garbage cans

full of screams.”

I can never listen to that song without a dispro-

portionate intensity of feeling. I often ask myself

why the simple words of the song seem to resonate

with some deep level of unconscious memory, as

if the souls of the departed are speaking through

Morelli’s music. And now unexpectedly in Manin’s

book I find an answer to my question. In his chap-

ter, “The Empty City Archetype”, Manin describes

how the archetype of the dead city appears again

and again in the creations of architecture, litera-

ture, art and film, from ancient to modern times,

ever since human beings began to congregate in

cities, ever since other human beings began to

congregate in armies to ravage and destroy them.

The character who speaks to us in MacOrlan’s song

is an old soldier who has long ago been part of an

army of occupation. After he has walked with his

wife through the dust and ashes of the dead city,

he hears once more:

“Chansons de charme d’un clairon

Qui fleurissait une heure lointaine

Dans un rêve de garnison”.

“The magic calls of a bugle that came to life for

an hour in an old soldier’s dream”.

The words of MacOrlan and the voice of Mo-

relli seem to be bringing to life a dream from our

collective unconscious, a dream of an old soldier

wandering through a dead city. The concept of the

collective unconscious may be as mythical as the

concept of the dead city. Manin’s chapter describes

the subtle light that these two possibly mythical

februAry 2009 Notices of the AMs 223

concepts throw upon each other. He describes the

collective unconscious as an irrational force that

powerfully pulls us toward death and destruction.

The archetype of the dead city is a distillation of

the agonies of hundreds of real cities that have

been destroyed since cities and marauding armies

were invented. Our only way of escape from the

insanity of the collective unconscious is a collec-

tive consciousness of sanity, based upon hope

and reason. The great task that faces our contem-

porary civilization is to create such a collective

consciousness.

References

[1] M. J. Bertin et al., Pisot and Salem Numbers,

Birkhäuser Verlag, Basel, 1992.

[2]

M. L. Cartwright and J. E. Littlewood, On non-

linear differential equations of the second order, I,

Jour. London Math. Soc. 20 (1945), 180–189.

[3]

Freeman Dyson, Prof. Hermann Weyl, For.Mem.R.S.,

Nature 177 (1956), 457–458.

[4]

Tien-Yien Li and James A. Yorke, Period three implies

chaos, Amer. Math. Monthly 82 (1975), 985–992.

[5]

Yuri I. Manin, Mathematics as Metaphor: Selected

Essays, American Mathematical Society, Providence,

Rhode Island, 2007. [The Russian version is: Manin,

Yu. I., Matematika kak Metafora, Moskva, Izdatyelstvo

MTsNMO, 2008.]

[6]

Andrew M. Odlyzko, Primes, quantum chaos and

computers, in Number Theory, Proceedings of a Sym-

posium, National Research Council, Washington DC,

1990, pp. 35–46.

[7]

Hermann Weyl, Gravitation und elektrizität, Sitz.

König. Preuss. Akad. Wiss. 26 (1918), 465–480.

[8]

———

, Elektron und gravitation, Zeits. Phys. 56

(1929), 350–352.

[9]

———

, Selecta, Birkhäuser Verlag, Basel, 1956,

p. 192.

[

10] Chen Ning Yang, Integral formalism for gauge

fields, Phys. Rev. Letters 33 (1974), 445–447.

[

11] Chen Ning Yang and Robert L. Mills, Conservation

of isotopic spin and isotopic gauge invariance, Phys.

Rev. 96 (1954), 191–195.

[12]

———

, Hermann Weyl’s contribution to physics, in

Hermann Weyl, 1885–1985, (K. Chandrasekharan, ed.),

Springer-Verlag, Berlin, 1986, p. 19.

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The IMA is an NSF funded institute

From June 15-26, 2009 the IMA will host an intensive

short course designed to efficiently provide researchers

in the mathematical sciences and related disciplines the

basic knowledge prerequisite to undertake research in

applied algebraic topology. The course will be taught by

Gunnar Carlsson (Department of Mathematics, Stanford

University) and Robert Ghrist (Department of Electrical

and Systems Engineering, Department of Mathematics,

University of Pennsylvania). The primary audience for the

course is mathematics faculty. No prior background in

applied algebraic topology is expected. Participants will

receive full travel and lodging support during the

workshop.

For more information and to apply:

Application deadline: April 1, 2009

Instructors:

New Directions Short Course

Applied Algebraic Topology

June 15-26, 2009

Gunnar Carlsson (Stanford University)

Robert Ghrist (University of Pennsylvania)

www.ima.umn.edu/2008-2009/ND6.15-26.09

Discussion

I'm not an expert on computational complexity — but I believe this is backwards. Assuming $P \neq NP,$ shouldn't the TSP be in $NP$ but not $P$? > ***"the history of mathematics is a history of horrendously difficult problems being solved by young people too ignorant to know that they were impossible"*** Here is a great resource if you want to learn more about Gödel's Incompleteness Theorems: [Stanford Encyclopedia of Philosophy - Gödel's Incompleteness Theorems](https://plato.stanford.edu/entries/goedel-incompleteness/) You can read this paper here: [Period Three Implies Chaos - Li, T. Y. and Yorke, J. A. ](http://www2.hkedcity.net/sch_files/a/sfx/sfx-nwc/public_html/period_three_implies_chaos.pdf) A hypocycloid is a curve generated by the trace of a fixed point on a small circle that rolls within a larger circle. Below you can see how a three cusped hypocycloid, also known as a deltoid, is generated. ![deltoid](https://i.imgur.com/9EUlq8L.gif) Learn more here: - [Wolfram - Hypocycloid](http://mathworld.wolfram.com/Hypocycloid.html) - [Wikipedia - Deltoid](https://en.wikipedia.org/wiki/Deltoid_curve) John Von Neumann wrote ***First Draft of a Report on the EDVAC** in 1945 for the ENIAC project. It consists of the first published description of the logical design of a computer using the stored-program concept. More interesting reads: - [First Draft of a Report on the EDVAC](https://www.wiley.com/legacy/wileychi/wang_archi/supp/appendix_a.pdf) - [Fermat's Library - Von Neumann's First Computer Program](http://fermatslibrary.com/s/von-neumanns-first-computer-program) - [Fermat's Library - The Beginning of the Monte Carlo Method](http://fermatslibrary.com/s/the-beginning-of-the-monte-carlo-method) Here there is a reference to Riemann hypothesis, one of Millenium mathematics problems: https://www.youtube.com/watch?v=d6c6uIyieoo Freeman Dyson is an English-born American theoretical physicist and mathematician. He is professor emeritus at the Institute for Advanced Study and worked on several areas including quantum electrodynamics, solid-state physics, astronomy and nuclear engineering. ![dyson](https://i.imgur.com/qbQx3OU.jpg) ### TL;DR In this essay Freeman Dyson presents two types of mathematicians: birds and frogs. Birds are Caretesian, they fly high and are interested in the unification of our thinking. Frogs are Baconian, they live in the mud and delight in the details of particular objects. Dyson gives examples of famous birds and frogs and their different way of thinking about mathematics. Mathematics only advances thanks to a combination of birds and frogs. The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics and is held every 4 years. During the congress prizes like the Fields Medals, the Nevanlinna Prize, the Gauss Prize, and the Chern Medal. The second International Congress of Mathematicians was held in Paris in 1900. Here is a list of all past ICMs: [International Mathematical Union - Previous ICMs](https://www.mathunion.org/icm/past-icms) Science has evolved following both Cartesian and Baconian views simultaneously. ### Birds - Cartesian view - deduce laws from pure thought (Descartes "Cogito Ergo Sum") - they delight in the concepts that unify our thinking and bring together diverse problems from different topics. ### Frogs - Baconian view - collect facts until they have enough information to understand the workings of Nature - they delight in the details of particular objects - they solve problems one at a time > ***Chaos: When the present determines the future, but the approximate present does not approximately determine the future.*** Edward Lorenz Chaos theory is a branch of mathematics focusing on the behavior of dynamical systems that are highly sensitive to initial conditions. In 1963, Lorenz developed a simple math model for atmospheric convection. \begin{eqnarray*} x' &=& σ(y-x)\\ y' &=& x(ρ-z)-y\\ z' &=& xy-βz \end{eqnarray*} The equations relate the properties of a 2-D fluid layer uniformly warmed from below and cooled from above. Here is a set of ***chaotic solutions called The Lorenz attractor***: ![Lorenz attractor](https://video.twimg.com/tweet_video/DXxTVG7WkAEWFsW.mp4) In mathematics a Kakeya set, or Besicovitch set, is a set of points in a plane that contains a unit line segment in every direction. The Kakeya problem **asks whether there is a minimum area that plane such that a needle of unit length can be turned in all directions** (through 360°). Here is an interesting video by Numberphile about Kakeya's Needle problem: [![kakeya video](https://i.imgur.com/lhK56xD.jpg)](https://www.youtube.com/watch?v=j-dce6QmVAQ) Lie groups, named after Sophus Lie, lie at the intersection of algebra and geometry. They are smooth manifolds obeying the following properties: - the properties of a group - the group operations are differentiable A Lie group is a group of continuous symmetries of mathematical objects and structures. Lie groups are widely used in mathematics and modern theoretical physics. Learn more here: - [Wolfram - Lie Groups](http://mathworld.wolfram.com/LieGroup.html) - [Wikipedia - Lie Groups](https://en.wikipedia.org/wiki/Lie_group) The Bourbaki group was a group of young French mathematicians started in the 1930s which led to the publication of a influential collection of books in the field of mathematics. The founding members of the group included Henri Cartan, Claude Chevalley, Jean Coulomb, Jean Delsarte, Jean Dieudonné, Charles Ehresmann, Szolem Mandelbrojt, René de Possel, and André Weil. Bourbaki's collection of books is more than seven thousand pages long. The first chapter appeared in 1935, and new ones continued to appear until the early 1980s. In its final form it comprised the following: - I. Theory of Sets - II. Algebra - III. General Topology - IV. Functions of a Real Variable - V. Topological Vector Spaces - VI. Integration - Lie Groups and Lie Algebras - Commutative Algebra - Spectral Theories - Differential and Analytic Manifolds #### The Ten Commandments of Leo Szilard: 1. Recognize the connections of things and laws of conduct of men, so that you may know what you are doing. 2. Let your acts be directed towards a worthy goal, but do not ask if they will reach it; they are to be models and examples, not means to an end. 3. Speak to all men as you do to yourself, with no concern for the effect you make, so that you do not shut them out from your world; lest in isolation the meaning of life slips out of sight and you lose the belief in the perfection of creation. 4. Do not destroy what you cannot create. 5. Touch no dish, except that you are hungry. 6. Do not covet what you cannot have. 7. Do not lie without need. 8. Honor children. Listen reverently to their words and speak to them with infinite love. 9. Do your work for six years; but in the seventh, go into solitude or among strangers, so that the memory of your friends does not hinder you from being what you have become. 10. Lead your life with a gentle hand and be ready to leave whenever you are called. Read on here: [Letter by Mrs Szilard on the 10 commandments (July 1964)](https://library.ucsd.edu/dc/object/bb64521048/_1.pdf) #### Hilbert's 23 Mathematical Problems 1. Cantor's problem of the cardinal number of the continuum. 2. The compatibility of the arithmetic axioms. 3. The equality of two volumes of two tetrahedra of equal bases and equal altitudes. 4. Problem of the straight line as the shortest distance between two points. 5. Lie's concept of a continuous group of transformations without the assumption of the differentiability of the functions defining the group. (i.e., are continuous groups automatically differential groups?) 6. Mathematical treatment of the axioms of physics. 7. Irrationality and transcendence of certain numbers. 8. Problems (with the distribution) of prime numbers. 9. Proof of the most general law of reciprocity in any number field. 10. Determination of the solvability of a diophantine equation. 11. Quadratic forms with any algebraic numerical coefficients. 12. Extension of Kronecker's theorem on abelian fields. 13. Impossibility of the solution of the general equation of the 7th degree. 14. Proof of the finiteness of certain complete systems of functions. 15. Rigorous foundation of Schubert's calculus. 16. Problem of the topology of algebraic curves and surfaces. 17. Expression of definite forms by squares. 18. Building space from congruent polyhedra. 19. Are the solutions of regular problems in the calculus of variations always necessarily analytic? 20. The general problem of boundary curves. 21. Proof of the existence of linear differential equations having a prescribed monodromic group. 22. Uniformization of analytic relations by means of automorphic functions. 23. Further development of the methods of the calculus of variations.

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