**Weak Chaos**

If von Neumann had known about chaos when he spoke in Amsterdam, one of the unsolved problems 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 example 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 mathematician 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 integrations 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 planetary 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 unpredictably 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 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 supply of beautiful pictures, and a shortage of rigorous 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 positions 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 surrounded 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 mathematics. 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 remarkably, 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 humbler 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 temptations 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 saying 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 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 details 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 meaning. 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 according 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 language, 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 superstitious society and a capricious government. In Russia, mathematicians and composers and filmproducers 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 beyond 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 unconscious 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 disproportionate 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 chapter, “The Empty City Archetype”, Manin describes how the archetype of the dead city appears again and again in the creations of architecture, literature, 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 Morelli 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 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 collective consciousness of sanity, based upon hope and reason. The great task that faces our contemporary civilization is to create such a collective consciousness.

**References
**

- M. J. Bertin et al., Pisot and Salem Numbers, Birkhäuser Verlag, Basel, 1992.
- M. L. Cartwright and J. E. Littlewood, On nonlinear differential equations of the second order, I, Jour. London Math. Soc. 20 (1945), 180–189.
- Freeman Dyson, Prof. Hermann Weyl, For.Mem.R.S., Nature 177 (1956), 457–458.
- Tien-Yien Li and James A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975), 985–992.
- 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.]
- Andrew M. Odlyzko, Primes, quantum chaos and computers, in Number Theory, Proceedings of a Symposium, National Research Council, Washington DC, 1990, pp. 35–46.
- Hermann Weyl, Gravitation und elektrizität, Sitz. König. Preuss. Akad. Wiss. 26 (1918), 465–480.
- ——— , Elektron und gravitation, Zeits. Phys. 56 (1929), 350–352.
- ——— , Selecta, Birkhäuser Verlag, Basel, 1956, p. 192.
- Chen Ning Yang, Integral formalism for gauge fields, Phys. Rev. Letters 33 (1974), 445–447.
- Chen Ning Yang and Robert L. Mills, Conservation of isotopic spin and isotopic gauge invariance, Phys. Rev. 96 (1954), 191–195.
- ——— , Hermann Weyl’s contribution to physics, in Hermann Weyl, 1885–1985, (K. Chandrasekharan, ed.), Springer-Verlag, Berlin, 1986, p. 19.

**Adapted from NOTICES OF AMS 2/2009**