Further Reading J. S. Bell. 1964. On the Einstein-Podolsky-Rosen paradox. Physics 1:195. A simple discussion of Bell's inequality and its experimental verification can be found in "The Quantum Theory and Reality" by Bernard d'Espagnat. Scientific American, November 1979, p. 128. Alain Aspect, Philippe Grangier, and Gerard Roger. 1982. Experimental realization of Einstein-PodolskyRosen-Bohm Gedankenexperiment: A new violation of Bell's inequalities. Physical Review Letters 49:91. L. Boltzmann. 1868. Studien über des Gleichgewicht der lebendigen Kraft zwichen bewegten materiellen Punkten. Wiener Berichte 58:517. L. Boltzmann. 1872. Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen. Wiener Berichte 66: 275. J. C. Maxwell. 1860. Illustrations of the dynamical theory of gases. Part I: On the motions and collisions of perfectly elastic spheres. Philosophical Magazine and Journal of Science, Fourth Series 19: 19. J. C. Maxwell. 1860. Illustrations of the dynamical theory of gases. Part II. On the process of diffusion of two or more kinds of moving particles among one another. Philosophical Magazine and Journal of Science, Fourth Series 20:21. J. C. Maxwell. 1860. Illustrations of the dynamical theory of gases. Part III. On the collision of perfectly elastic bodies of any form. Philosophical Magazine and Journal of Science, Fourth Series 20: 33. J. Clerk Maxwell. 1867. On the dynamical theory of gases. Philosophical Transactions of the Royal Society of London 157:49. George D. Birkhoff. 1966. Dynamical Systems. American Mathematical Society Colloquium Publications, Volume 9. Providence, Rhode Island: American Mathematical Society. See also Lectures on Ergodic Theory, by Paul R. Halmos. Tokyo: Mathematical Society of Japan, 1956. J. C. Oxtoby and S. M. Ulam. 1941. Measure-preserving homeomorphisms and metrical transitivity. Annals of Mathematics 42:874. V. I. Arnold and A. Avez. 1968. Ergodic Problems of Classical Mechanics. New York: W. A. Benjamin, Inc. Ja. G. Sinai. 1963. On the foundations of the ergodic hypothesis for a dynamical system of statistical mechanics. Soviet Mathematics-Doklady 4: 1818. Ja. G. Sinai. 1967. Ergodicity of Boltzmann's gas model. In Statistical Mechanics: Foundations and Applications (Proceedings of the 1. U. P. A. P. Meeting, Copenhagen, 1966), edited by Thor A. Bak. New York: W. A. Benjamin, Inc. J. Bellissard and M. Vittot. 1985. Invariant tori for an infinite lattice or coupled classical rotators. Universite de Aix-Marseille preprint CPT-85 P1796. Douglas M. Eardley and Vincent Moncrief. 1982. The global existence of Yang-Mills-Higgs fields in 4-dimensional Minkowski space. Communications in Mathematical Physics 82: 193. E. Fermi, J. Pasta, and S. Ulam. 1955. Studies of nonlinear problems. Los Alamos Scientific Laboratory report LA-1940. Also in Stanislaw Ulam: Sets, Numbers, and Universes, edited by W. A. Beyer, J. Mycielski, and G.-C. Rota. Cambridge, Massachusetts: The MIT Press, 1974. A. Patrascioiu, E. Seiler, and I. O. Stamatescu. 1985. Non-ergodicity in classical electrodynamics. Physical Review A 31: 1906. Adrian Patrascioiu. 1983. Beyond the mystery of quantum mechanics. In Asymptotic Realms of Physics: Essays in Honor of Francis E. Low, edited by Alan H. Guth, Kerson Huang, and Robert L. Jaffe. Cambridge, Massachusetts: The MIT Press. Adrian Patrascioiu. 1984. Are there any ergodic local relativistic field theories? Physics Letters 104A: 87. A. Patrascioiu. 1981. On the nature of quantum mechanics. Institute for Advanced Study report. BIOLOGY Ulan lam's genius for addressing basic questions in biology through simple mathematical models as aforady resen encountered. In the matematics section mathematical Mycielski introduced Ulam's notion of "genealogical" distance, a measure of shared ancestry. This was one among several extensions of the theory of branching processes invented by Ulam to answer questions in population dynamics and evolution. Another area that has had even more impact is Ulam's early work on cellular automata in which he and Schrandt demonstrated how complex patterns can evolve from simple initial conditions by applying a few simple recursion rules over and over again. The idea behind these computer studies hinges on a basic question in developmental biology: How does a human being develop from a single cell, a single fertilized egg? Certainly, with 1012 cells and only 106 genes, there are not nearly enough genes for each cell to have its own gene. Stan proposed that genes encode not just simple rules, but rules of a higher logical type that change the simple instructions, in other words "rules for the change of rules" that become operative in response to events outside the cell but in contact with it. Stan began investigating this idea by making what he called kindergarten rules and applying them to a small number of cells. In one set of rules, cells multiply along a straight line until they meet another cell at which point their line of propagation rotates by 45 degrees. Rules of this sort were run on the computer to produce patterns resembling those found in nature, such as the vein distribution in a leaf or the pattern of a capillary bed. (Several figures from these studies with Schrandt are shown on the next page.) Recent work by Gerald Edelman of Rockefeller University on morphogenesis lends credence to Ulam's basic idea of "rules for the change of rules.” Edelman's work suggests that form arises through an interaction involving adhesion molecules on the cell surface that alter the primary processes of cell development. "In this case, the modifications of the rules correspond to the developmental process of induction. For during induction, as a result of associations between adjacent groups of cells, particular cells undergo alterations in their properties through the process of cell differentiation, and these alterations, in turn, modify their subsequent interactions." (This quote is from an article by Leif Finkel and Edelman in volume 10, numbers 2 and 3 of Letters in Mathematical Physics, a special volume in memory of Stan Ulam.) Ulam's cellular automaton models of growth patterns were just a start. Now cellular automata of various kinds are being used to model the complex networks associated with food webs and kin selection, and even neurons in our brain. Stan was deeply interested in the organization and function of the human brain and the structure of memory. In the Gamow Memorial Lecture of October 5, 1982 (which is published here for the first time), he outlined some speculations about the mechanism A Gamow Memorial Lecture delivered at the University of Colorado, Boulder, on October 5, 1982 M y choice of subject for this talk may seem strange, since I am not a psychologist, a physiologist, or a neurologist, merely a mathematician and an amateur, a dilettante, in the workings of the brain. However, it is fitting that I give such a talk in memory of the late George Gamow, a friend of mine. Though by training a physicist, he was able to make famous contributions in other sciences, such as astronomy and biology, that interested him toward the end of his life. He was, like me, an amateur, a dilettante, in biology. Nevertheless one of the most important discoveries of recent times in that field is due to him. It was Gamow who first pointed out that ordered arrangements of four chemical units-four "letters"-along the DNA double helix, or chain, as he called it, might be codes for many biological pro cesses, and that the codes for the manufacture of proteins might consist of threeor four-letter "words." What I want to do today is talk about several of my own speculations, with some mathematical symbolism, concerning the operation of the brain. I believe that discoveries and breakthroughs within the next twenty years will lead to a better understanding of the mechanisms of the brain, of the processes of thought. It will not be a complete understandingthat would be too much to hope forbut it will give us some ideas of how the nervous system operates in lower animals and in humans. Mathematicians may help in reaching this understanding, although for the time being I think that 99 percent of the progress will come from physiological and anatomical experiments. However, mathe by S. M. Ulam matics can be useful, for it is clear that the similarities between electronic computers and the nervous system are of great importance. Another friend of mine, the late John von Neumann, was one of the pioneers in the planning and building of electronic computers. His book The Computer and the Brain, which was published posthumously in 1957, is still one of the most elegant and understandable general introductions to the subject. I remember the discussions we had on how the advent of computers would enlarge the scope of experimentation in mathematical and physical sciences and about his specific interest in the partial analogies between computers, as they were planned in the early forties, and the processes of deductive thinking. We saw each other frequently at the time, either in Los Alamos or in |