f ebruAry 2009 N otices of the AMs 221

is no co ns erv ati on la w o f p hys ic s t hat fo rb ids

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 sev ere as to en dan ger our ex ist enc e. Cha os

remains mercifully weak for reasons that we do

not understa nd. 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 su bje ct o f c haos is ch arac ter ize d 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 f ield of chaos I k now o nly one rigor ous

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

1975 and published in a short paper with the title,

“Peri od Thre e Implie s Chaos ”, [4]. The Li- Yorke

paper is one of the immortal gems in the literature

of math emat ics. Their theo rem con cerns nonlin ear

map s of an in ter val onto its elf . Th e suc ces sive 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

origi nal pos ition af ter

N  m appings. An orb it 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 be autiful picture s 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 word s about stri ng theor y.

Few word s, bec ause I kno w ver y lit tle about st ring

the ory . I never too k the troubl e to lea rn the subje ct

or to wo rk on it my sel f. Bu t wh en I am at ho me at th e

Ins tit ute for Adv ance d Study in Princ eton, I am sur-

rou nde d by strin g theo ris ts, and I someti mes liste n

to their conver sation s. Occasi onally I under stan d a

lit tle of wha t the y are sayin g. Thr ee thing s are clear .

Fir st, what they are doing is firs t-rate mat hem at-

ics . The lea din g pure ma them aticia ns, peopl e like

Mic hae l Atiya h and Isad ore Singer , love it . It has

open ed up a whole new bra nch of ma them atic s,

wit h new idea s and new probl ems. Most remar k-

abl y, it gave the mathe mat icia ns new method s to

so lv e old pro bl em s tha t we re pr ev ious ly unso lva bl e.

Second, the str ing theo rists think of thems elves

as physici sts rather tha n mathemati cia ns. The y

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 t he rest o f physics . String theorist s 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,

flyi ng high and s eeing grand v isions of di stant

ra n ge s o f m ou nt a in s . Th e t ho u sa n ds o f h um -

bler practitioners of string theory in universities

around the world are frogs, exploring fine details

of the ma them ati cal str uct ures th at bird s f irst

sa w o n the h ori zon . My an xie tie s abo ut st rin g

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

t ho us a nd . T h er e a re no w a bo u t t en th ou s an d

string theorists scattered around the world. This

is a dangerous situation for the tenth thousand

and perhaps also for the second thousand. It may

happe n unpre dictably that t he fash ion chan ges

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 you ng p eopl e at trac ted to

string theory? The attraction is partly intellectual.

Stri ng theor y is dar ing and mathema tically eleg ant .

But the attractio n is also soci ologi cal. String theory

is attractive because it offers jobs. And why are

so many jo bs 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

labor atory to do experim ental 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 fo r the young people to acc ept th em.

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 gue ss is probably wrong. I

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