1.1. From the Preface.

In introducing the collection of problems forming the substance of this work, it is perhaps necessary to offer more explanations than is usually the case for a mathematical monograph. The problems listed are regarded as unsolved in the sense that the author does not know the answers. In this sense the structure of this small collection differs inherently from that of the well-known collection of problems by Pólya and Szego. The questions, drawn from several fields of mathematics, are by no means chosen to represent the central problems of these fields, but rather reflect the personal interests of the author. For the main part, the motif of the collection is a set-theoretical point of view and a combinatorial approach to problems in point set topology, some elementary parts of algebra, and the theory of functions of a real variable.

In spirit, the questions considered in the first part of this collection belong to a complex of problems represented in the Scottish Book. This was a list of problems compiled by mathematicians of Lwów in Poland before World War II, also containing problems written down by visiting mathematicians from other cities in Poland and from other countries. The author has recently translated this document into English and distributed it privately; the interest shown by some mathematicians in this collection encouraged him to prepare the present tract for publication. Many of the problems contained here were indeed first inscribed in the Scottish Book, but the greater part of the material is of later origin beginning with the years spent at Harvard (1936-1940) and a large proportion stems from recent years. appearing here for the first time. Many of the problems originated through conversations with others and were stimulated by the transitory interests of the moment in various mathematical centres. In addition, several problems were communicated by friends for inclusion in this collection. The last few chapters have a different character: the stress is on computations on calculating machines with examples of problems whose study through the use of this modem tool would have, in the author's opinion, great heuristic value.

Most of the problems were seriously considered and worked upon, but with different degrees of attention and time spent on attempts to solve them. Some have been studied by other mathematicians to whom they were communicated orally but others have not been thoroughly investigated and it would not surprise the author if a number admitted trivial solutions. Most of the problems are, so to say, of medium difficulty. A majority of them should definitely not fall into the category of mere exercises to be solved by routine applications of known lemmas and theorems. In fact one of the aims was a selection of "simple" questions in various domains of mathematics; simple, for example, in the sense that no elaborate definitions beyond those used in general courses on set theory, analysis, and algebra would be necessary for their understanding. The author believes that, on a purely heuristic level, a survey of this sort, if properly enlarged and deepened by others, could bring out the possible general and typical common "reasons" for the difficulties encountered in quite diverse branches of mathematics.

The present situation in mathematical research is perhaps different from that of previous epochs in its very great degree of specialisation. The connections between different fields are growing more tenuous, or else so general and purely formal, that they become illusory. It has been said that unsolved problems form the very life of mathematics; certainly they can illuminate and, in the best cases, crystallise and summarise the essence of the difficulties inherent in various fields. The very existence of mathematics can be considered as fruitful only because it produces simple and concise statements whose proofs are much more complicated in comparison. Moreover, Gödel's discovery of the existence of undecidable propositions in every consistent system of mathematics, including arithmetic, renders the "probably true" propositions all the more precious. The intriguing possibility which now exists a priori of undecidability lends an additional flavour, to some at least, of the unsolved mathematical problems.

The separation between mathematical research stimulated by pure mathematics alone and the ideas stemming from theoretical physics has been increasing in the development of these fields during the last few decades. This may seem at first sight surprising, since the ideas and models of reality employed nowadays in physics tend toward increasing abstractness. However, it appears that on the whole, applied mathematics, so-called, deals at the present time in the majority of cases, with questions of classical physics-or else, when it concerns itself with the new theories, its role is restricted to a purely technical intervention. On the conceptual level one does not have enough, it seems, of cross fertilisation of ideas! In the author's opinion it appears likely that in the near future the large class of concepts which have their origin in Cantor's set theory, which have influenced so many of the purely mathematical disciplines, will playa role in physical theory. The difficulties of the phenomena of divergence in present formulations of field theory may indicate the need for a type of mathematics capable of dealing with physical problems employing actual infinities ab initio. Several elementary problems are included here which are intended to indicate the nature of such possible formulations and the kind of mathematical schemes which may be of use in some future physical theories.

The set-theoretical motivation underlying the selection of questions in the various fields to which the problems refer influenced the choice of the more elementary problems and made the illustration of the more sophisticated ideas of recent years, in topology or algebra for example, impractical.

It is impossible to give detailed credit to all who have indirectly contributed to the set of ideas illustrated in the list of problems, but I would like to acknowledge in particular the pleasure of past collaboration with Banach, Borsuk, Kuratowski, Schreier, and Mazur in Poland, and John von Neumann, Garrett Birkoff, J C Oxtoby, P Erdős and C Everett in this country.

1.2. Review by: John C Oxtoby.
Bulletin of the American Mathematical Society 66 (5) (1960), 361-363.

This book is unique. There are other collections of problems, for instance, the well-known Aufgaben unà Lehrsâtze of Polya and Szegö, and Knopp's problem book, but these are collections of problems with solutions. The present book is a collection of unsolved problems, or at least ones for which the author does not know the answers. There is a closer parallel with Hubert's famous lecture on mathematical problems at the Paris Congress of 1900. In comparison, the present collection is less pretentious and more personal. The author does not pretend to forecast the lines of future development of mathematics. He does not even claim that the problems he proposes are central; merely that they reflect his personal interests. They consist of open questions that have arisen from his work in many different fields.

The problems considered are in the spirit of the so-called Scottish Book. During the 1930's a notably gregarious group of mathematicians in Lwów, Poland - including Banach, Steinhaus, Mazur, Orlicz, Schauder, Schreier, Ulam, and others - were accustomed to meet for long mathematical discussions in "The Scottish Coffee Shop." From time to time, problems which they posed to each other were written down in a notebook which was kept there for the purpose. (Sometimes the proposer would indicate his estimate of the difficulty of a problem by offering a prize for its solution; perhaps a bottle of wine, or two small beers!) Visiting mathematicians too were invited to add their problems to the collection. After the war this book was carried to Wroclaw, where the tradition was revived by some of the surviving members of the group. Many problems from the "New Scottish Book" have appeared in the problem section of Colloquium Mathematicum. A few years ago, Ulam circulated privately a translation made from a copy of the original Scottish Book. The interest aroused by this encouraged him to write the present book. The problems include many which he first inscribed in the Scottish Book, but a greater number stem from later years. In fact, the book constitutes a kind of mathematical autobiography. Each of the various fields in which the author has worked has contributed its share of problems. But despite their diversity there is an underlying unity. As the author puts it, "the motif of the collection is a set-theoretical point of view and a combinatorial approach to problems."

The problems are arranged in a logical order which is also roughly chronological. First come problems in set theory: product isomorphisms, projective algebras, logic, and abstract measure theory. Some interesting ideas concerning the existence of non-measurable projective sets are expressed here. The second chapter contains problems about groups and semi-groups. Then come problems about metric and topological spaces, problems concerning various kinds of invariance. There is a short chapter devoted to topological groups, especially the group $S_{∞}$ of all permutations of the integers, but actually notions arising from topological groups pervade much of the book. The chapter on analysis is largely concerned with functional equations, especially ones that are only approximately satisfied, and with questions of conjugacy of functions and transformations. The last two chapters, which constitute almost half of the book, are devoted to problems suggested by physics and by computing machines. Some very suggestive ideas concerning the possible role of actual infinities in physical theories, and the relevance of the notions of Cantorian set theory, are expressed. Questions concerning flows in phase space, and the topology of magnetic lines of force, are formulated. The final chapter "makes propaganda" (as the author would say) for the use of computing machines as a heuristic aid. Various conjectures are expressed concerning games, number theory, functional equations, and physical models, that can be tested by Monte Carlo methods. Partial answers obtained by the author and his collaborators are described. In some situations a combination of operator and machine can be more effective than a fully programmed machine. For example, by watching a display produced by the machine, the operator may be in position to steer it toward the location of a critical point more efficiently than would a search code. In such collaboration of machine and operator the author sees the most likely direction of progress in the immediate future.

The problems are neither numbered nor displayed. Some are completely specific, others merely outlined. The author considers that most are of "medium difficulty," but says that he would not be surprised if some should turn out to admit trivial solutions. The reader should perhaps be warned that a few problems are somewhat carelessly stated. For instance, the property of Baire is incorrectly defined on pages 11, 17, and 24; while on pages 19 and 23 the strong and weak properties are not distinguished from each other. On page 58 the discrete topology is classified as not locally compact. But in all these cases it is easy enough to see what was intended. In conclusion, this is not a book to read through at one sitting, nor is it one to plough through like a textbook. The book is valuable not only for the problems which is contains, but also for the glimpse it affords of a fertile mathematical imagination at work, and for the problem-centred approach to mathematics which it fosters.

1.3. Review by: Philip Rabinowitz.
Science, New Series 132 (3428) (1960), 665-666.

The unsolved problems in mathematics can be divided broadly into two classes. There are problems such as appear in number theory which are very easy to formulate and which require almost no mathematical background to understand. These "easy" problems usually turn out to be very difficult to solve. On the other hand, there are problems like those in this collection, which require the reader to have a good background in the subject matter in order to understand what they are about. While such problems are also difficult to solve, their difficulties are relatively easier to surmount. Thus several problems included here were solved in the interval between the writing and the publishing of the work.

The problems in this book come from various branches of mathematics - for example, set theory, algebra, topology, analysis, and mathematical physics. The presentation of the problems varies. In some cases there is a rather lengthy discussion of the problem and its motivation, with partial results and implications resulting from its solution. In other cases there is only a dry listing of problems and conjectures. And sometimes no particular problem is discussed; instead an entire area of research is suggested. Thus, the chapter on computing machines as a heuristic aid gives the author's ideas on the subject of man-machine cooperation in solving some outstanding problems in mathematics and mathematical physics.

Although the range of topics is wide there seem to be several unifying concepts, the principal one being that of transformation. There are also several scattered problems whose formulation is simple. However, the overwhelming majority of the problems are difficult both to formulate and to solve; they should provide Ph.D. advisers with sufficient material to offer to their aspiring young mathematicians.

1.4. Review by: C P Snow.
Scientific American 203 (3) (1960), 256.

Some years ago, before World War II, the author of this book, together with a number of leading Polish mathematicians then living in Lvóv, had a small mathematical club which met almost every day in one of the coffee houses near the university to discuss problems of common interest and to tell one another of their latest work and results. The suggestion was made that the more interesting problems be recorded, and so a large notebook was purchased and deposited with the headwaiter of the "Scottish Coffee House," who, upon demand, would bring it out of some secure hiding place, leave it at the table, and after the guests departed, return it to its secret location. The storm which swept over Europe spared neither the city of Lvóv, its mathematicians nor the "Scottish Book" (as it came to be called). First the Russians occupied the city, and certain of their mathematicians must have visited the very coffee house, for at the end of the Scottish Book they entered several problems (and even left prizes for their solution). The Germans were less kind. When they occupied the town, they killed a number of mathematicians, among tens of thousands of other Poles. The Scottish Book disappeared; in fact it was spirited away by one of the survivors and may have been buried, as had been agreed beforehand, near the goal post of a football field outside the city. At any rate, it survived. A typewritten copy was sent after the war by the mathematician Hugo Steinhaus to Ulam, and he had mimeographed copies made which were distributed to, and widely appreciated by, a number of U.S. mathematicians and physicists. Many of the problems in the Scottish Book appear in the present volume, but the greater part of the material is of later origin. There are questions, suggestions, matters not fully worked out. The problems deal with set theory, higher algebra, topology, group theory, theoretical physics, computing machines. A heady, difficult and challenging collection.

1.5. Review by: Richard Bellman.
Quarterly of Applied Mathematics 21 (4) (1964), 284.

The question as to which is of greater value to the development of mathematics, a good problem or a good solution, is very much like the chicken-and-egg paradox. The best of problems generate the best of solutions, and conversely. At the present time, dozens of universities of ever-increasing mathematical level are turning out hundreds of talented and highly trained practitioners capable of writing on and resolving clearly formulated mathematical problems. No comparable effort is being devoted to developing new areas for mathematical research, nor to new applications of known techniques in other parts of science.

Consequently, it is extremely refreshing and significant to see a book devoted purely to problems for which no solutions exist as yet. Even more important is the discussion of problem areas which still await precise formulation. The author frankly admits that the selection is subjective - as any such set must be. His stature as a mathematician guarantees its intrinsic worth.

To those interested in analysis and mathematical physics, which includes most of the readers of this journal, the book reads best from back to front. Probably the most fascinating chapter is the last, "Computing Machines as a Heuristic Aid." The description of this chapter as one devoted to the use of computers as experimental devices for mathematicians does not do justice to the many intriguing variations on this theme which Ulam presents. This chapter, at very least, should be required reading for any acolyte taking the vows of abstraction and obscurantism of the Bourbaki.

The seventh chapter, "Physical Systems," contains much of interest to those concerned with making mathematical models of physical processes, with particular reference to the mechanics of the continua and branching processes.

The sixth chapter, "Some Questions of Analysis," focuses on the basic problem of the stability of a mathematical model in a far deeper fashion than is customary. Although Hyers and Ulam have contributed some important results in this field, most of the basic questions are unanswered, and many have not even been asked.

The first five chapters are entitled in order, "Set Theory," "Algebraic Problems," "Metric Spaces," "Topological Spaces," and "Topological Groups." They contain a large number of explicit problems centring about point set topology, set theory and abstract spaces. With due respect for their historical importance, one has the feeling that there is little mathematical vitality left in these areas. Compared to the intensity that pervades the last three chapters, these chapters are rather the remembrance of things past.

Regardless of which parts of this book are particularly rewarding, everyone concerned with the present and future development of mathematics will find sources of inspiration in this scintillating volume.

1.6. Review by: Aryeh Dvoretzky.
Mathematical Reviews MR0120127 (22 #10884).

This fascinating little book is not exactly a collection of problems. It is of very uneven character ranging from very specific problems to minor discourses suggesting methods of attack on vast new topics. It is written in a kind of "off the cuff" manner and very relevant references are occasionally missing, e.g., to H Busemann [(1949), (1953)] in connection with the first problem on convex bodies on p. 38. This, however, does not detract from the value of the book, in which every mathematician would like to browse and which should certainly stimulate many younger mathematicians. The topics covered are those encountered by the author throughout his work and are dominated by the set-theoretic and combinatorial point of view. The book consists of eight chapters: Set theory (27 pp.); Algebraic problems (8); Metric spaces (15); Topological spaces (13); Topological groups (5); Some questions in analysis (19); Physical systems (31); Computing machines as a heuristic aid (30).

2.1. Review by: Editors.
Mathematical Review MR0280310 (43 #6031).

In this second (paperback) edition of a book that originally appeared under another title [A collection of mathematical problems, 1960], the author has added a three-page preface. In this he briefly discusses the work of P J Cohen on the continuum hypothesis, provides references for the solutions of some of the problems given in the original edition that have since been solved and formulates several new problems. He also promises a supplementary volume.

3.1. From the Publisher.

Fascinating study considers the origins and nature of mathematics, its development and role in the history of scientific thinking, impact of high-speed computers, 20th-century changes in the foundations of mathematics and mathematical logic, mathematisation of science and technology, much more. Compelling reading for anyone interested in the evolution of mathematical thought.

3.2. From the Introduction.

What is mathematics? How was it created and who were and are the people creating and practising it? Can one describe its development and its role in the history of scientific thinking and can one predict its future? This book is an attempt to provide a few glimpses into the nature of such questions and the scope and the depth of the subject.

Mathematics is a self-contained microcosm, but it also has the potentiality of mirroring and modelling all the processes of thought and perhaps all of science. It has always had, and continues to an ever increasing degree to have, great usefulness. One could even go so far as to say that mathematics was necessary for man's conquest of nature and for the development of the human race through the shaping of its modes of thinking.

For as far back as we can reach into the record of man's curiosity and quest of understanding, we find mathematics cultivated, cherished, and taught for transmittal to new generations. It has been considered as the most definitive expression of rational thought about the external world and also as a monument to man's desire to probe the workings of his own mind. We shall not undertake to define mathematics, because to do so would be to circumscribe its domain. As the reader will see, mathematics can generalise any scheme, change it, and enlarge it. And yet, every time this is done, the result still forms only a part of mathematics. In fact, it is perhaps characteristic of the discipline that it develops through a constant self-examination with an ever increasing degree of consciousness of its own structure. The structure, however, changes continually and sometimes radically and fundamentally. In view of this, an attempt to define mathematics with any hope of completeness and finality is, in our opinion, doomed to failure.

We shall try to describe some of its development historically and to survey briefly high points and trenchant influences. Here and there attention will focus on the question of how much progress in mathematics depends on "invention" and to what extent it has the nature of "discovery." Put differently, we shall discuss whether the external physical world, which we perceive with our senses and observe and measure with our instruments, dictates the choice of axioms, definitions, and problems. Or are these in essence free creations of the human mind, perhaps influenced, or even determined, by its physiological structure?

Like other sciences, mathematics has been subject to great changes during the past fifty years. Not only has its subject matter vastly increased, not only has the emphasis on what were considered the central problems changed but the tone and the aims of mathematics to some extent have been transmuted. There is no doubt that many great triumphs of physics, astronomy, and other "exact" sciences arose in significant measure from mathematics. Having freely borrowed the tools mathematics helped to develop, the sister disciplines reciprocated by providing it with new problems and giving it new sources of inspiration.

Technology, too, may have a profound effect on mathematics; having made possible the development of high-speed computers, it has increased immeasurably the scope of experimentation in mathematics itself.

The very foundations of mathematics and of mathematical logic have undergone revolutionary changes in modern times. In Chapter 2 we shall try to explain the nature of these changes.

Throughout mathematical history specific themes constantly recur; their interplay and variations will be illustrated in many examples.

The most characteristic theme of mathematics is that of infinity. We shall devote much space to attempting to show how it is introduced, defined, and dealt with in various contexts.

Contrary to a widespread opinion among non-scientists, mathematics is not a closed and perfect edifice. Mathematics is a science; it is also an art. The criteria of judgment in mathematics are always aesthetic, at least in part. The mere truth of a proposition is not sufficient to establish it as a part of mathematics. One looks for "usefulness," for "interest," and also for "beauty." Beauty is subjective, and it may seem surprising that there is usually considerable agreement among mathematicians concerning aesthetic values.

In one respect mathematics is set apart from other sciences: it knows no obsolescence. A theorem once proved never loses this quality though it may become a simple case of a more general truth. The body of mathematical material grows without revisions, and the increase of knowledge is constant.

In view of the enormous diversity of its problems and of its modes of application, can one discern an order in mathematics? What gives mathematics its unquestioned unity, and what makes it autonomous?

To begin with, one must distinguish between its objects and its method. The most primitive mathematical objects are positive integers 1, 2, 3, ... Perhaps equally primitive are points and simple configurations (e.g., straight lines, triangles). These are so deeply rooted in our most elementary experiences going back to childhood that for centuries they were taken for granted. Not until the end of the 19th century was an intricate logical examination of arithmetic (Peano, Frege, Russell) and of geometry (Hilbert) undertaken in earnest. But even while positive integers and points were accepted uncritically, the process (so characteristic of mathematics) of creating new objects and erecting new structures was going on.

From objects one goes on to sets of these objects, to functions, and to correspondences. (The idea of a correspondence or transformation comes from the still elementary tendency of people to identify similar arrangements and to abstract a common pattern from seemingly different situations.) And as the process of iteration continues, one goes on to classes of functions, to correspondences between functions (operators), then to classes of such correspondence, and so on at an ever accelerating pace, without end. In this way simple objects give rise to those of new and ever growing complexity.

The method consists mainly of the formalism of proof that hardly has changed since antiquity. The basic pattern still is to start with a small number of axioms (statements that are taken for granted) and then by strict logical rules to derive new statements. The properties of this process, its scope, and its limitations have been examined critically only in recent years. This study - metamathematics - is itself a part of mathematics. The object of this study may seem a rather special set of rules - namely, those of mathematical logic. But how all-embracing and powerful these turn out to be! To some extent then, mathematics feeds on itself. Yet there is no vicious circle, and as the triumphs of mathematical methods in physics, astronomy, and other natural sciences show, it is not sterile play. Perhaps this is so because the external world suggests large classes of objects of mathematical work, and the processes of generalisation and selection of new structures are not entirely arbitrary. The "unreasonable effectiveness of mathematics" remains perhaps a philosophical mystery, but this has in no way affected its spectacular successes.

Mathematics has been defined as the science of drawing necessary conclusions. But which conclusions? A mere chain of syllogisms is not mathematics. Somehow we select statements that concisely embrace a large class of special cases and consider some proofs to be elegant or beautiful. There is thus more to the method than the mere logic involved in deduction. There is also less to the objects than their intuitive or instinctive origins may suggest.

It is in fact a distinctive feature of mathematics that it can operate effectively and efficiently without defining its objects.

Points, straight lines, and planes are not defined. In fact, a mathematician of today rejects the attempts of his predecessors to define a point as something that bas "neither length nor width" and to provide equally meaningless pseudo-definitions of straight lines or planes.

The point of view as it evolved through centuries is that one need Dot know what things are as long as one knows what statements about them one is allowed to make. Hilbert's famous Grundlagen der Geometrie begins with the sentence: "Let there be three kinds of objects; the objects of the first kind shall be called 'points,' those of the second kind 'lines,' and those of the third 'planes.'" That is all, except that there follows a list of initial statements (axioms) that involve the words "point," "line," and "plane," and from which other statements involving these undefined words can now be deduced by logic alone. This permits geometry to be taught to a blind man and even to a computer! This characteristic kind of abstraction, which leads to a nearly total disregard of the physical nature of geometric objects, is not confined to the traditional boundaries of mathematics. Ernst Mach's critical discussion (which owes much to James Clerk Maxwell) of the notion of temperature is a case in point. To define temperature one needs the notions of thermal equilibrium and thermal contact, but to define these in logically acceptable terms is, at least, awkward and perhaps not even possible. An analysis shows that all one really needs is the transitivity of thermal equilibrium; i.e., the postulate, (sometimes called the zeroth law of thermodynamics) that if (A and B) and (A and C) are in thermal equilibrium, then so are (B and C). For completeness one also needs a kind of converse of the zeroth law, namely that if A, B, and C are in thermal equilibrium, then so are (A and B) and (A and C). Again, as in geometry, one need not know the (logically) precise meaning of terms, but only how to combine them into meaningful (i.e., allowable) statements.

But while we may operate reliably with undefined (and perhaps even undefinable) objects and concepts, these objects and concepts are rooted in apparent physical (or at least sensory) reality. Physical appearances suggest and even dictate the initial axioms; the same apparent reality guides us in formulating questions and problems.

To exist (in mathematics), said Henri Poincaré, is to be free from contradiction. But mere existence does not guarantee survival. To survive in mathematics requires a kind of vitality that cannot be described in purely logical terms.

In the following chapters we discuss a number of problems that not only have survived but have given birth to some of the most fruitful developments in mathematics. They range from the concrete to the abstract and from the very simple to the relatively complex. They were chosen to illustrate both the objects and the methods of mathematics, and should convince the reader that there is more to pure mathematics than is contained in Bertrand Russell's definition that "Pure mathematics is the class of all propositions of the form 'p implies q,' where p and q are propositions containing one or more variables, the same in the two propositions, and neither p nor q contains any constants except logical constants."

3.3. Review by: Kenneth O May.
Isis 60 (1) (1969), 112-113.

What is mathematics? How was it created and who were and are the people creating and practicing it? Can one describe its development and its role in history of scientific thinking and can one predict its future?" So begins the introduction to this Britannica Perspective prepared to commemorate the two-hundredth anniversary of the Encyclopaedia Britannica. From the title and these intriguing initial questions one might hope for a deep analysis, but Mark Kac and Stanislaw Ulam are not historians, philosophers, or mathematical critics. Rather they are distinguished practitioners, creative contributors, and expert expositors of the mathematical art. In their very next sentence they tell us to expect "a few glimpses into the nature of such questions and the scope and depth of the subject," and this is precisely what we get.

The first chapter, occupying two-thirds of the book, consists entirely of examples of mathematical problems and conceptions, some classic and others still of contemporary interest. These glimpses are not organised or related in any very obvious way, but they are seasoned with occasional comments and questions. Where one might expect to find an answer to the question or backing for a general statement, Kac and Ulam have inserted examples without explaining the connection. The same pattern is followed in the remaining chapters entitled "Themes, Trends, and Syntheses"; "Relations to Other Disciplines"; and "Summary and Outlook." Indeed, the last chapter consists of a page of general comment followed by eight pages describing some mathematical activities of interest today.

A surprising amount of nontrivial mathematics is compressed into this little book. Mathematics undergraduates will find it easy reading, and those with less training can do some judicious skipping. It is too bad that the authors omitted references to sources in which the reader, baffled or inspired, could find assistance or further challenge.

In their introduction and again at the beginning of the last chapter, the authors suggest that they have been guided by historical ideas, and they do indeed make a number of historical assertions in the course of their exposition. Historians will not be surprised that these reflect at best the mixture of fact and fancy that make up the historical tradition familiar to mathematicians. Their very first example is said to be Euclid's argument for the infinity of primes, but it is actually not the argument of Book IX, Proposition 20, nor is it even a modern version of that proof. Indeed, on this first page the authors have unhistorically mingled different ideas of infinity separated by time gaps of as much as 2,000 years and conceptual gaps equally large. Here and frequently throughout the book, the twentieth-century logical organisation of a concept is confused with its historical development. The authors repeat many ideas that are widely believed by mathematicians but that do not appear to correspond to the facts. For example, they state that the method by which mathematics has grown is characteristically deductive, although the fact is that mathematics as a whole has been organised deductively only in very recent times, and it has always grown and still grows primarily by extra-logical procedures, even though results once acquired must be fitted into a deductive system in order to be accepted.

However, it would be missing the point of the book to criticise it as if it were a piece of historical scholarship. On the contrary, it is grist for the historian's mill. By means of examples and impromptu comments, two mathematical insiders have offered a view of their science as they see it. The historian of science, and especially the historian of mathematics, can find in this both enlightenment and challenge.

For example, the authors look upon mathematics as a science and are well aware of extra-mathematical motivations, yet they, like most scientists, find the usefulness of mathematics mysterious. Certainly, one would not expect the mystery to be explained entirely in terms of the content of mathematics, but it should be well within the power of historians of science to show the inevitability of this usefulness by examining the way in which mathematics has developed as a part of culture.

3.4. Review by: David Booth.
The Journal of Symbolic Logic 36 (4) (1971), 677.

Logic is exhibited alive in this book, so that the reader may better learn to use it, rather than embalmed in that superfluous formalisation with which some writers beguile the inexperienced reader. The book consists of a polymathic collection of mathematical tales followed by some commentary; the examples are well chosen and presented without pedantry. Few of these examples, however, concern logic in their content; the book is more of an introduction to mathematical logic - though it also has many other uses - than a description of it.

In their commentary, the authors describe the directions and the distinguishing features of current research. They mention the most famous metamathematical results of arithmetic and set theory. They hope that there will eventually be more precise formalisations of the notion of an intuitively clear sentence of set theory. They suggest that set theory, unlike geometry, might be independent of any theory which describes our experience in "the real world"; no evidence is given for this presumption.

This book will be useful for students and gratifying to experienced mathematicians.

3.5. Review by: R H Thomason.
Science, New Series 163 (3867) (1969), 557-558.

This book is one of a series commemorating the bicentennial of the Encyclopaedia Britannica. Its title, which suggests a study of the interrelationships of mathematics and logic, is misleading. The book is actually a survey of mathematics, together with general methodological remarks. Due attention is given to logic as an area of mathematics, but it is not especially stressed.

The authors present their material by drawing on examples from all major areas of mathematics. Among the topics they discuss are proofs of impossibility (with special reference to geometrical constructions), elementary probability and measure theory, linear algebra, braid theory, Gödel's incompleteness theorem, and game and information theory. Although non-mathematicians may find the book difficult going in places, it should be understandable to most people with scientific training,

Kac and Ulam have produced an exciting and illuminating panorama of their subject, rather than a conglomeration. Their lively style and constant sense of interconnection make the treatment of each example a vivid lesson relating in some way to larger topics. Themes such as complex numbers and groups arc woven through the material in a way which illustrates nicely how underlying structures can appear in widely separated areas of mathematics. By often following their presentations of well-known results with unsolved problems arising from them, the authors manage to keep their material from appearing too settled and fixed, and succeed well in presenting a picture of mathematics as it appears to a working mathematician.

Although not useful as a source or reference volume, this book is a valuable contribution. It provides a perspective and distance which most modern scientists must struggle to obtain, and does this with grace and good sense. The philosophical points to be found in it will probably not strike the mathematician as remarkable, but they are refreshingly sound in comparison with the oversimplifications often made by philosophers when speaking of mathematics. And the authors' suggestions concerning what may be in store for mathematics, especially as regards the use of computers and of ideas from the life sciences, are thought-provoking and worthy of consideration.

3.6. Review by: J G Kemeny.
The American Mathematical Monthly 77 (3) (1970), 316.

A book attempting to explain the nature of mathematics, authored by two mathematicians as distinguished as Kac and Ulam, is an important event. Their approach is significantly different from that of many other popular books on mathematics. The strength of the book consists of a fascinating variety of excellent examples from the history of mathematics. On the other hand, little attempt is made to draw general conclusions about the nature of mathematics, and it is left as a challenge to the reader to draw his own conclusions as to what these varied examples have in common.

The book is not aimed at the general lay audience. The reviewer would strongly recommend it as supplementary reading for mathematics majors and for the college teacher who wants to broaden his view of the nature of mathematics.

Examples are drawn from at least a dozen major branches of mathematics. They are described succinctly, emphasising the essential and skipping irrelevant details. While this is one of the great strengths of the book, it will also mean that most readers will have difficulties with examples in some fields with which they are not familiar. Since most of the examples are independent of each other, there is little harm in skipping an individual example which proves too difficult. The reviewer found several occasions where a few additional comments might have made it easier to read the book. Also there are a few occasions where due to an attempt to simplify the subject matter as much as possible, it is possible for the reader to infer mathematically incorrect conclusions from the text.

The book is best summarised by the authors themselves: "To survive in mathematics requires a kind of vitality that cannot be described in purely logical terms. In the following chapters we discuss a number of problems that not only have survived but have given birth to some of the most fruitful developments in mathematics."

3.7. Review by: Reuben Louis Goodstein.
The Mathematical Gazette 54 (388) (1970), 174.

This book by two very eminent mathematicians is intended, not so much for the general reader, but for those who have had some training in mathematics. It constitutes a masterly survey of concepts and methods and ranges over a wide variety of mathematical topics, some of which are treated in some depth and others just touched upon; but each topic introduced is illuminated by the authors insight and expository skill.

The opening chapter is a series of examples designed to illustrate the diversity of mathematical problems and the ingenuity which has been displayed in their solution; these examples include some that are generally found in popular texts, like the irrationality of √2, but also many that are less familiar like the impossibility of decomposing a cube into a finite number of different cubes, Sperner's lemma in combinatorial topology with an application to Brouwer's fixed point theorem, concepts of probability theory and group theory, the Lorentz transformation in special relativity, the ergodic theorem and Markov chains.

The second chapter discusses the power of algebra in solving topological problems, with special reference to Artin's theory of braids, and passes on to discuss Gödel's proof of the incompleteness of axiomatic arithmetic, Turing computability and the use of modern computers, and finally Klein's model of non-Euclidean Geometry.

Chapter three considers the relationship of mathematics to the real world and to various branches of science, and includes a discussion of the resolution of the apparent conflict of the second law of thermodynamics with the molecular theory of matter, and the last, very short chapter looks at possible future developments.

I am confident that this stimulating book will be widely read, and I hope that when the time comes for a second edition, the authors will add some bibliographical notes.

3.8. Review by: William Spangler.
Reference Quarterly 8 (1) (1968), 63.

This is a book for the person with specialised knowledge. Prepared as one of the Britannica Perspectives in honour of Encyclopaedia Britannica's 200th anniversary, the book begins as a brief history of mathematics and surveys the high points of mathematical development. Later chapters illustrate the effects of technology on mathematics and explore the relations between mathematics and the empirical disciplines.

3.9. Review by: Haskell Brooks Curry.
Mathematical Reviews MR0232640 (38 #964).

This book is one of the "Perspectives" made to commemorate the 200th anniversary of the Encyclopaedia Britannica. It aims to give for intelligent nonspecialists some idea of the nature of mathematics and its role in human thought. It does this by surveying a sample of the more spectacular accomplishments of mathematics, some old and some quite new, and draws a general conclusion from discussion of the examples. It does not have as much to say about logic as its title would indicate. The reviewer found it stimulating reading. It also contains quite a bit of information; but unfortunately, since there is neither an index nor a detailed table of contents, this information is not available for reference.

4.1. From the Preface by W A Beyer.

Stanislaw Marcin Ulam was born in Lwów, Poland. He received his M.Sc. at the Polytechnic Institute there in 1932 and his Ph.D. in 1933. At the invitation of John von Neumann he came to the Institute for Advanced Study in Princeton in late 1935. He was a member of the Society of Fellows at Harvard from 1936 to 1940. After two years at the University of Wisconsin he joined the Los Alamos Scientific Laboratory in New Mexico where he remained until 1967, occasionally spending time at Harvard, MIT, University of Southern California, California at La Jolla, Colorado at Boulder, and IBM. In 1967 he became Professor of Mathematics at the University of Colorado, but continued to visit and consult at Los Alamos.

Ulam's career and interests have been unusually broad for a mathematician and it is not easy to survey his work. The commentaries by other mathematicians assembled at the end of this volume attest to Ulam's impact on the development of mathematics and other sciences since his first paper in 1929. In a brief introduction, one can do little more than mention some of the areas of pure and applied mathematics, technology, computation, physics, astronomy, and biology to which he has contributed. In addition to direct contributions to these fields, he has posed concise problems whose solutions or attempts at solution have advanced these fields. Some of these problems are discussed in his book A Collection of Mathematical Problems which is included in this volume.

Ulam's early mathematical work was in set theory, an area fundamental to modern mathematics. Perhaps his most influential contribution here was a paper in which he relates measure theory to general set theory and proves, among other things, that no countably additive measure function $m(A)$ exists, defined for all subsets of a set $E$ of cardinality $\aleph _{1}$, which vanishes for all subsets consisting of a single point, and for which $m(E) = 1$. With Schreier he made important contributions to group theory and with Borsuk he commenced the development of the notion of $\epsilon$-mappings; from this, he in later years (with D Hyers) laid the foundation for concepts of structural stability. Lomnicki and Ulam developed measure-theoretic foundations of probability theory, which were prior to and independent of Kolmogorov's book on the same subject.

A fundamental problem in science is to connect microscopic and macroscopic descriptions of matter. An important aspect of this is the ergodic hypothesis, which asserts roughly that the time averages of functions of material states almost always equal the corresponding phase averages. The time averages will usually, in contrast to the space averages, be inscrutable. G D Birkhoff showed in 1931 that the ergodic hypothesis can be replaced by the hypothesis that the only sets invariant under a given transformation are of measure 0 or 1. This contribution, however, fails to establish the relevance of the ergodic hypothesis to statistical mechanics. Oxtoby and Ulam in their long paper of 1941 succeeded in showing that the ergodic hypothesis holds in general except for an exceptional set of the first category. Whether the transformations of statistical mechanics are in the exceptional set is not known. Nevertheless the ergodic transformations are dense in the space of all transformations which they considered. Thus Oxtoby and Ulam came closer than anyone (at least until the last few years) in showing the relevance of the ergodic hypothesis to statistical mechanics. For a more complete discussion, see the commentaries by Oxtoby and Smale.

Ulam (with Hawkins and Everett) developed in the years 1944-48 the theory of multiplicative systems (now called branching processes). This work was motivated by applications for the atomic project at Los Alamos.

Ulam has long been concerned with the possibility that other mathematical structures might better describe our space-time on the submicroscopic level than does Euclidean geometry. In developing suitable alternative structures, it is necessary to take account of the Lorentz group. In their paper Everett and Ulam develop a theory of vector spaces over a $p$-adic field and an associated Lorentz group. Beltrametti discusses the relation of their paper to modern physics in more detail in the commentary.

In the area of technology Ulam holds, with Everett, a patent for propelling very large space vehicles by a series of small external nuclear explosions. This idea has developed into what is now called the Orion project. He seems to have been the first to propose extracting gravitational energy from planets to propel space vehicles. This idea is now being used in the "flyby" missions to the outer planets and will provide part of the energy for the first spacecraft to travel beyond the planets. Ulam holds, with Teller, the patent disclosure for the first thermonuclear weapon.

From the earliest days of electronic computers, Ulam has been active in their application to mathematical and physical problems. He proposed (with Fermi and von Neumann) and developed Monte Carlo techniques as a means of computing solutions to probabilistic problems, and extended the applications to non-probabilistic problems. With Fermi and Pasta he studied by computer the time evolution of a model of a nonlinear vibrating string. To the surprise of the investigators the system did not evolve as one might have predicted; i.e., the energy did not equipartition itself among the modes. Instead, the system tended to return to its initial state. Ulam participated in writing the first computer program to play a game of chess. He made use of computers to study heuristically problems in number theory.

The discovery that biological organisms are encoded by discrete finite sequences over an alphabet of four letters was of great interest to Ulam, as it would be to any mathematician. Biological organisms, usually seemingly very complex, would seem to require very long codes. With Schrandt, Ulam studied the possibilities of simple recursively defined codes giving rise to complex objects. Thus it might be possible that short codes could define objects that might appear to be complex. With Beyer, Smith, and Stein, Ulam studied applications of the concept of distance between finite sequences as a means of reconstructing the evolutionary history of biological organisms.

The present volume contains reprints of Ulam's major papers in pure mathematics and his studies of the applications of computers to nonlinear computation in game theory and physics. The editors hope to prepare a future volume covering his contributions to biology, fluid mechanics, nuclear physics, astronomy, and pattern recognition.

4.2. Review by: Editors.
Mathematical Review MR0441664 (56 #67).

This book contains 52 articles by the author dealing with a variety of mathematical topics in pure mathematics and the application of computers to game theory and physics, together with commentaries presented by a number of distinguished specialists in these areas. The articles are reprinted from the original journals.

Also included is a bibliography (listing 108 items) and a complete reprinting of Ulam's book [A collection of mathematical problems, 1960].

5.1. From the Publisher.

Member of the famous Polish mathematical school. One of the first to use and advocate computers for scientific research. Co-author of the paper that was the basis for the construction of the hydrogen bomb. Originator of ideas for nuclear propulsion of space vehicles. Unique for his fundamental contributions to the most abstract and abstruse fields of mathematics.

5.2. Review by: Martin Gardner.
The New York Times (9 May 1976).

Biographical history, as taught in our public schools, is still largely a history of boneheads: ridiculous kings and queens, paranoid political leaders, compulsive voyagers, ignorant generals - the flotsam and jetsam of historical currents. The men who radically altered history, the great creative scientists and mathematicians, are seldom mentioned if at all.

Imagine Aristotle revivified and visiting Manhattan. Nothing in our social, political, economic, artistic, sexual or religious life would mystify him, but he would be staggered by our technology. Its products - skyscrapers, cars, airplanes, television, pocket calculators - would have been impossible without calculus. Who invented calculus?

Let's jump to a more recent discovery, one that has now given humanity a means for obliterating its own history. Who invented the H‐bomb? Fermi? Oppenheimer? Teller? General Groves? No, it was a Polish mathematician named Stanislaw Ulam.

To this day we don't know what came into Ulam's head because it's still top secret. The most he says about it in his autobiography is that it was an "iterative scheme" which modified a previous and unworkable plan of Teller's. Well - the "adventures" (even though they take place mostly under his hat) of a man who invented the H‐bomb are surely no less worth reading about than the adventures of the man who ordered the first atom bomb dropped on a city.

Ulam was born in 1909 at Lwów, Poland. One of his earliest memories is playing on an Oriental rug at the age of 4 while his father, a lawyer, watched and smiled. "He smiles," Ulam remembers thinking, "because he thinks I am childish, but I know these are curious patterns. I know something my father does not know."

A few years later Ulam was "drunk," as he puts it, with the curious patterns of mathematics. His student years at the Polytechnic Institute in Lwów threw him into contact with the eminent Polish mathematicians. A year at Cambridge put him in touch with the British mathematicians. "Which is more important, ideas or things?" he asked Alfred North Whitehead, who lived above him. "Ideas about things" was the instant reply. Another symbolic memory. Ulam always shuttled back and forth easily between pure and applied mathematics.

It was the great Hungarian mathematician John von Neumann, "Johnny" as Ulam called him, who invited Ulam to the Institute for Advanced Study at Princeton. This was followed by teaching at several universities. At the University of Wisconsin he became a citizen and married Françoise, a beautiful French girl he had met at Cambridge. Today he heads the mathematics department at the University of Colorado.

It was Johnny who also brought Ulam to Los Alamos to work on the atom bomb. One of Ulam's most seminal ideas at Los Alamos was the Monte Carlo method, a way of simulating physical processes such as chain reactions by using a computer to generate random numbers. Ulam's mind was always making such leaps, leaps that sometimes generated entirely new branches of mathematics. You will find his major papers on pure mathematics in a 710‐page tome published two years ago by M.I.T. Press. A volume of equal bulk is planned for his contributions to various sciences.

Mathematicians enjoy humorous word play (it has a combinatorial flavour), and Ulam is no exception. Some of his jokes are on himself, such as the time when a telephone operator asked him to "hold the wire." Ulam was in a New York phone booth, newly arrived from Poland. "Which wire should hold?" he asked in all seriousness. A programmer at Los Alamos, pretty and well‐endowed, had a habit of unfolding computer printouts in front of Ulam and Enrico Fermi, and holding them below her low‐cut blouse. "How do they look?" she would ask. "Marvellous!" Ulam would exclaim.

Dozens of eminent mathematicians and physicists walk through Ulam's pages, and his book is rich in stories about them. His most moving accounts are of the deaths of von Neumann and Fermi. When Fermi died in Chicago, Ulam could only weep and think of Plato's account of the death of Socrates. Niels Bohr impressed Ulam as a man of "great wisdom" but without the genius of Newton or Einstein. Oppenheimer was a brilliant but "sad" man who lacked the "ultimate creative spark of originality." Teller seems to Ulam "a comedian whose ambition is to be a great tragedian or vice‐versa."

Although Ulam's political views are liberal, he does not consider in this book the deep moral dilemmas over how the explosive results of science should be controlled by society. Of the great controversy that raged over whether the H‐bomb should be built Ulam has little to say. But he does speculate at length about the surprising fact that mathematics, a creation of the mind, so accurately fits the outside world.

For Ulam this correspondence is a mystery related somehow to the fantastic amount of uniformity in the universe. The formulas of physics are compressed descriptions of nature's weird repetitions. The accuracy of those formulas, coupled with nature's tireless ability to keep on doing everything the same way, gives them their incredible power. Ulam has never ceased to be amazed by "how a few scribbles on a blackboard … could change the course of human affairs." That this kind of symbol manipulation, in the hands of absent‐minded intellects, can shape history for both good and evil is the apocalyptic centre of Ulam's story.

Behind the scenes, invisible to all but a few, are the discoverers of these curious patterns in the cosmic carpet. They scribble their hieroglyphics on the back of a menu and men go to the moon, harness the atom, crack the genetic code, transform the planet's face. It is no tribute to our culture that on those rare occasions when creative mathematician tells the story of his life, men and women who fancy themselves educated would rather read about the amours of narcissistic actresses and political humbugs.

5.3. Review by: Albert S J Tarka.
The Polish Review 21 (3) (1976), 255-257.

This review betrays the writer's own unexpected enthusiasm and a frank impatience to attract as many readers as possible to this most unusual book. Ulam's autobiography will probably be reviewed many times, and certainly by mathematicians and scientists like himself, as it should be. But its impact is strong enough and broad enough to affect any reader who risks opening the book. This is precisely why the least likely kind of person - a humanist - is hereby attempting to induce Polish Review readers to do the same: to read one of the most exciting books of our age!

Permit me to confess that the title of the book did not move me. Throughout my academic experience I had expended a great deal of energy and imagination in trying to avoid as many courses in mathematics and related subjects (statistics, logic, and the like) as possible without jeopardising eligibility for graduation. To this day, I feel that "maths" is probably the most unattractive word in the English language! Then why even consider a book like this? Because a patient and persistent friend was persuasive enough to overcome my carefully nurtured reluctance. I am grateful to him. A thumbnail sketch would tell us that S M Ulam was born in Lwów in the year 1909. He was early in recognising his interest in the curious patterns of mathematics, and he focused on this field for study during his student years. His interest and productivity eventually carried him to the prestigious universities of England and the United States. World War II took him abruptly from the world of mathematicians to that of physicists when he joined the Manhattan Project to participate in the beginnings of the Nuclear Age. He was the inventor of the H-bomb.

Stanislaw Marcin Ulam, the man who has written hundreds of mathematical treatises, writes the story of his life in a direct and simple way that reflects the concerns, interests, conflicts, loyalties and great warmth of a remarkable human being. The impact of his story, however, moves beyond biography to become an exciting affirmation of historical importance. First, it tells us that warm, concerned and sensitive human beings ushered this world into the atomic age. Second, it reminds us again that a fettered nation once sent her sons to these shores to battle for the freedom her own people could not enjoy; that at another turning point in human history, when the flower and culture of that same nation was being systematically destroyed by a new generation of oppressors, the sons of that nation again contributed to the battle of the United States and to the wealth and progress of the world by making possible the technological revolution we are experiencing today and which tremendously affected the capacity and potential of man in this universe.

The book carries the reader to his own discoveries. A real excitement develops early in the reading, when Ulam writes about his young teacher at the Polytechnic Institute in Lwów who had just come from Warsaw and who was a student of Sierpinski, Mazurkiewicz and Janiszewski,... and one realises that these men are members of the now legendary Polish School of Mathematics!

One is struck by the great number of Poles who appear on these pages. Ulam recalls a discussion with his friend "Johnny" von Neuman, the great Hungarian mathematician, with the remark that it will be left for historians of science to discover and explain what catalysed the emergence of so many brilliant individuals from the region of the Carpathian Mountains which was a part of Hungary, Czechoslovakia and Poland. Later, in her book Illustrious Immigrants, Laura Fermi was to express surprised admiration at the large percentage of Polish mathematicians in the United States who contributed so much to the flourishing of this field.

With his gentle humour, Ulam can taunt an American host about his waspish prejudices when he asks why it is that the Pilgrims "landed" in America while the present European immigrants and scientific refugees merely "arrived". But the prejudice that was felt by many of the intellectual immigration wave of World War II was very real in its effects on lives and careers. Even after the war, when the achievements of the Manhattan Project were already a matter of record, Ulam and his wife - now American citizens - considered returning to his former university post in Madison, Wisconsin. Concerned about his chances for promotion (to meet growing family responsibilities) and tenure, a friend responded frankly, "No reason to beat around the bush. Were you not a foreigner, it would be much easier and your career would develop faster." So, Ulam looked elsewhere.

The tenderness and humanity of this giant of science becomes touchingly clear in the simple accounts of the illness and death of two close associates - Fermi and von Neuman. And his heroism emerges equally clearly during a series of events which he describes as the most shattering experiences in his life. Sudden unexplained headaches resulted in emergency surgery to relieve a severe and dangerous pressure on his brain. The illness, never really to be identified, was tentatively diagnosed as a kind of virus encephalitis. For several long days his wife and friends were concerned about his survival. Ulam's concern was even deeper and extended well into his convalescence: Was his thinking process impaired! Because - at least for mathematicians and physicists - a good memory forms a large part of their talent. Whatever questions he had about his recovery should have been dispelled by a telegram inviting him back to Los Alamos, where he returned to continue the work which he had advanced in the Manhattan Project. The dimensions and facets of Stan Ulam's life and work are truly remarkable, and perhaps much of this is due to his attitude and approach to both. "... I also believe that changing fields of work ... is rejuvenation. If one stays too much with the same subfields or the same narrow class of problems, a sort of self-poisoning prevents acquisition of new points of view and one becomes stale."

The inevitable question of moral responsibility in an atomic age comes to the mind of the reader, and Ulam responds to it directly. "Contrary to those people who were violently against the bomb on political, moral or sociological grounds, I never had any questions about doing purely theoretical work. I did not feel it is immoral to try to calculate physical phenomena. Whether it was worthwhile strategically was an entirely different aspect of the problem - in fact the crux of a historical, political or sociological question of the gravest kind - and had little to do with the physical or technological problem itself. Even the simplest calculation in purest mathematics can have terrible consequences. Without the invention of infinitesmal calculus most of our technology would be impossible. Should we say therefore that calculus is bad?

It may be that many PIAS members, remembering Ulam's Jurzykowski Award in 1966, will see more than a biography in Adventures of a Mathematician. Indeed, Stan M Ulam may be more than a distinguished scientist. In a very real sense Ulam becomes a symbol, albeit an illustrious one, of the group of scholars and intellectuals whose number in any age or nation never attains statistical significance in the sociological sense, but on whose in vestment and contribution the very growth of culture depends. In keeping with this special Bicentennial Issue of the Polish Review, Ulam reminds us of that significant intellectual minority that must be recognised in the assessment of the role of Poland and Polonia in the history of our country, even of the world itself.

This once reluctant reviewer fully recognises his subjective reaction to a volume he once considered unattractive. But if the Reader requires a more rational inducement, it may suffice to quote two statements from the excellent review by Martin Gardner (regular contributor to the Scientific American) which appeared in the May 6th edition of The New York Times. "Biographical history ... is still largely a history of boneheads: ridiculous kings and queens, paranoid political leaders, compulsive voyagers, ignorant generals - the flotsam and jetsam of historical currents. The men who radically altered history, the great creative scientists and mathematicians, are seldom mentioned, if at all ... It is no tribute to our culture that on those rare occasions when a creative mathematician tells the story of his life, men and women who fancy themselves educated would rather read about the amours of narcissistic actresses and political humbug." We hope Gardner is too stern in his judgement.

5.4. Review by: Bernard Norling
Polish American Studies 34 (2) (1977), 71-75.

This book is the autobiography of Stanislaw Ulam, a prominent member of the famous school of Polish mathematicians centred at the University of Lwów between the World Wars. Many of its members perished in World War II. Interestingly, Ulam, a European with a sense of history, does not say that they were "victims of genocide" or that they died "in the Nazi holocaust" but simply that they "were murdered by the Germans." He might have added that the monstrous totalitarian regimes of our century level in more ways than one. Hating nothing so much as independent thought, they concentrate on exterminating or neutralising dissident intellectuals. But Ulam does not say this; he notes merely that many European scientists of the 1930s were impractical men who did not comprehend what the events of that decade portended in world affairs. Ulam himself did comprehend. He left Poland in the mid-1930s, came to Princeton ("that way station for displaced European scientists,"), gravitated to Harvard, then to Wisconsin, and finally to Los Alamos, New Mexico in 1943, where he spent much of the next two decades working intermittently on the Atomic and Hydrogen Bombs. He travelled extensively, occasionally returned briefly to academic life, and since 1967, has been Chairman of the Department of Mathematics at the University of Colorado.

Ulam pushed hard to build the H-Bomb and was for a time one of a triumvirate who supervised that project. Then or afterwards he had no qualms about it. He notes matter-of-factly that the human mind always seeks knowledge, that nobody can ever foresee to what use even the simplest mathematical calculations will be put in unknowable circumstances in the future, that the Russians and others would eventually have built H-Bombs whether Americans did or not, and that in the late 1940s the difference between the prospective H-Bomb and the huge A-Bombs that we already knew how to build did not seem great anyway.

Ulam knew most of the leading scientists and mathematicians of his generation: von Neumann, Fermi, Banach, Bethe, Bohr, Feynman, Teller, Oppenheimer, Frisch, Weisskopf, Segre and innumerable lesser lights. He was involved in most of the scientific and political controversies that surrounded the building of the H-Bomb, most notably the Oppenheimer affair. He is remarkably generous in his reminiscences about these events and the men who figured in them. He offers a few observations about Einstein's well developed ego, the obstinacy and ambition of Edward Teller, the animosity between Oppenheimer and AEC Director Lewis Strauss, and the inevitable increase in the average hat size of scientists in and after World War II when for the first time in their lives they were suddenly flooded with public attention and lucrative government contracts. But that is all. He condemns no one in the H-Bomb and Oppenheimer controversies, pays tribute to the mind and skills of Teller even though he (Ulam) had much trouble with Teller, lavishes praise on his close friend von Neumann ("Johnny" throughout the book), and on Fermi, Bethe, and Banach, all of whom he admired immensely. He describes his own political position as fundamentally liberal but about halfway between "hawkish" scientists like von Neumann and Teller who were obviously apprehensive about Russian intentions and the "dovish" scientists whose main interest was the possibility of internationalising nuclear weapons. One of his reasons is especially interesting, coming from a Pole whose people have been caught for a thousand years between Russia and Germany. Ulam says that considering all the trouble the Russians have had merely maintaining a communist regime in Poland, he could never believe that they wanted to invade western Europe. Suppose they should succeed in communising West Germany! What would be the result? A united East and West Communist Germany that would immediately try to wrest leadership of world communism from Russian hands!

When Poland fell in 1939 Ulam says he knew a curtain had fallen over his past and henceforth he would be an American. His observations about his adopted land are thoughtful; sometimes sharp. From the first he liked the free atmosphere in America and appreciated the willingness of so many Americans to coach and instruct both each other and foreigners. He was particularly impressed by the readiness of brilliant young American scientists at Los Alamos to be "team players," to subordinate their personal interests to the good of the overall project. Europeans would have been less unselfish, more prone to personal empire building. In the 1940s, he marvelled at the energy and enterprise Americans showed attacking any problem, qualities which seem to him to have declined by the 1970s. Like many European intellectuals whose first experience of America is in Ivy League universities, he fully expected the University of Wisconsin to be an intellectual wilderness. He was pleasantly surprised, doubly so when the professors there did not put on airs as some of their compatriots had done at Harvard. When he assesses American education in general though, Ulam is a typical European. Teaching undergraduates is usually a bore, teaching itself "is sometimes done in a partial trancelike state," too many of the students are inept and uninterested, serious academic work begins too late, and the rate of academic improvement at such places as USC, where Ulam spent 1945-46, is too slow.

Ulam's own mathematical ability must have been extraordinary, particularly his capacity to ask the right kinds of questions of those engaged in nuclear research. What comes through equally clearly in the book is the vast range of his intellectual interests. He is fascinated by human mental processes and adverts to the subject repeatedly. He speculates on the qualities of mind necessary for success in mathematics, regarding persistence, concentration, and memory as crucial. Like so many others, he worries that the proliferation of human knowledge is proceeding at such a rate that most educated people no longer understand what those in other areas of intellectual endeavour are doing. In particular, he fears that the whole world of mathematics soon will be split into enclaves where the inhabitants of one will hardly know what their colleagues are doing in others and nobody will understand mathematics as a whole. He is much impressed by the past generation's accomplishments in biology and astronomy and marvels at how strange the universe now seems compared with the way we have grown accustomed to thinking of it in the context of past observations and hypotheses. He thinks the presence of life elsewhere in the universe likely and ruminates about how discovery of such life would revolutionise all human thought about religion.

Yet, for all his speculation about the nature and destiny of the cosmos and his work on the most fearsome weapons systems in all history, and though most of his family disappeared in World War II, Ulam emerges from these pages as a light hearted, perpetually optimistic man who must have charmed many of those who knew him well. He admits that he has always been extremely self-confident, even egotistical, and that others have so regarded him, but he draws the sting with numerous self deprecatory jokes and stories. Even when engaged in the most intense intellectual labour on a project that might shape human destiny for millennia he still liked to play poker and chess, to eat and drink, to formulate multi-lingual (and usually risqué) jokes, to invent games and private languages with his friends (especially von Neumann), and to talk endlessly. He mentions, only a couple of sentences apart, some crucial work he did on the H-Bomb and the fact that one of his poker playing pals had won the unprecedented sum (for their game) of $80.00. He interrupts a discussion of detailed mathematical calculations involved in certain phases of nuclear research to remark on the delightful contours of one of the female programmers at Los Alamos. Another time he interrupted a galaxy of political and intellectual dignitaries who were discussing the possibility of re-using booster rocket engines by observing that this sounded like using the same condom twice.

In summary, Adventures of a Mathematician reflects the paradoxical character and puckish personality of its author. Portions of it are incomprehensible save to mathematicians, yet it is an exceptionally pleasant book to read, one this reviewer had difficulty putting down. The momentous issues in which Ulam was involved are not slighted; his judgments about his many famous colleagues are often acute but never savage or vengeful; his reflections on both his adopted country and the human condition are sometimes profound, sometimes subtle, but always keen; and his high spirits shine forth on many pages. One wishes for more precise information about the details of what went on at Los Alamos in the 1940s and 1950s and what Ulam contributed to it, but much such information is still classified. Altogether, Ulam's autobiography is a sprightly reminiscence by an extraordinarily talented man who played a major role in one of the great enterprises of our century.

5.5. Review by: Philip Morrison.
Scientific American 236 (6) (1977), 136.

In 1949 radioactivity from the first Russian fission-bomb test contaminated the filters of an Air Force reconnaissance plane. Soon the theorists were set hard at work at Los Alamos and at Princeton: Could a fusion bomb be made? At Los Alamos "we started to work each day for four to six hours with slide rule, pencil, and paper." At Princeton the brand-new computer MANIAC was trying to catch up with the Los Alamos mathematician pair, the witty Stan Ulam and his taciturn partner of pre-war days, Cornelius Everett. (Everett "used to say, 'I never make mistakes' and this was true. ..." Once when Edward Teller maintained that Everett had made an error of a factor of $10^{4}$, Everett became annoyed; eventually "Edward had to admit that it was he who was at fault.") Nothing worked; the device on paper would not ignite. Enrico Fermi and Ulam next showed that the paper explosion would not spread, and John von Neumann soon reported MANIAC'S concurrence: "Icicles are forming." After a year's elaboration of the basic theory of thermonuclear explosion around a design that could not work Ulam had an extraordinary idea. He proposed a new scheme (which is still classified today), "a repetition of certain arrangements " that became the turning point for H-bomb work, and perhaps for all of us. A similar iterative scheme has presumably since been hit on in secrecy by ingenious people in four other countries.

Three distinguished mathematicians of our time have written well-known autobiographies in English: G H Hardy, Norbert Wiener and now Stanisław Ulam. Ulam, the youngest, knew both of the others. Hardy's donnish disdain for the applications of mathematics and his enthusiasm for both cricket and militant atheism, like Wiener's touching expression of the life of an ex-prodigy who was childishly in need of repeated reassurance of his ability, display almost total eccentricity of style and thought. Ulam is not an eccentric but an urbane original; some of his mathematical inventions have led to profound consequences, one might say to the choice between life and death.

What Ulam tells us of his mind and his times is generally fascinating. He makes little effort to draw us into the mathematical content of his deep and varied work. He is, however, transparently honest, and he is effective in portraying his impatient, ironic and quizzical style, his ambitions, his estimates of others, his interests and his opinions with "a frankness and truthfulness which are sometimes a little strong but never really shocking."

His wordplay and comment draw now on the Latin of his excellent classical studies, now on the logical Jewish jokes of Central European cafes. When it was mentioned that Fermi would soon come to wartime Los Alamos, "immediately I intoned: 'Annuncio vobis gaudi urn maximum, papam habemus,''' as they say in St Peter's when the white smoke heralds the election of a pope. Was not Fermi the infallible pope of the physicists? Later, when some mathematicians seeking Government contracts asserted the clear utility of their beloved work for national security, Ulam was reminded of the Jew who wanted to pray on Yom Kippur but tried to sneak in without paying for his seat on that crowded occasion by explaining he wanted only to deliver an urgent message. "But the guard refused, telling him: 'Ganev, Sie wollen beten' ['You thief! You really want to pray']. This, we like to think, was a nice abstract illustration of the point." A score of such tales enliven Ulam's book, as they embellished the speech of Ulam and von Neumann throughout the years. Stefan Banach, Enrico Fermi, George Gamow and above all von Neumann were friends and colleagues of Ulam's. Banach was the master of his youth as a gifted student in Lwów, one of the centres of the great blossoming of Polish mathematics between the wars. "I recall a session with Mazur and Banach at the Scottish Café which lasted seventeen hours without interruption except for meals." Ideas and proofs flowed from those conversations. Then the decision came to leave war-threatened Europe. (Of more than one of his Lwów group Ulam writes: "Murdered by the Germans.") Ulam was first a Harvard research fellow and then a faculty member at Wisconsin, a rising mathematical worker.

It is von Neumann who came closest to Ulam. They had met first in Warsaw well before the war, and they were parted only by von Neumann's death. The two men were congenial, complementary, intimate. Von Neumann displayed a not uncommon "admiration for people who had power "; indeed, Ulam thinks that in von Neumann there lay "a hidden admiration for people or organisations that could be tough and ruthless." During von Neumann's last days Ulam would read to him, in Greek, Thucydides' gripping tale of the expedition against Melos, "a story he liked especially." Von Neumann was "remarkably universal" and yet avoided "tangents from the main edifice of mathematics." He died too young of cancer, a strict Catholic near death although he had been an agnostic in life, with an enormous reputation and every honour of the mathematical world, yet "not entirely ... a mathematician's mathematician."

A mathematician such as Ulam works without external aids: no props, no equipment. Without chalk or pencil he may be at work even while walking, eating or talking. He seeks analogies between analogies. Such a person lives by this inner search and by the aid and the appraisal of a small set of peers. No wonder there is a certain detachment, a self-centred world view, an echo of fatalism. A few generous friendships of the kind that are candidly shared here with us are given to the lucky ones, but a certain coolness informs Ulam's estimate of men he found unclear, including Niels Bohr and Robert Oppenheimer. Most touching is the moment when Ulam, grievously ill with a virus inflammation of the brain, slowly recovers consciousness and speech. "One morning the surgeon asked me what 13 plus 8 were. The fact that he asked such a question embarrassed me so much that I just shook my head. Then he asked me what the square root of twenty was, and I replied: about 4.4. He kept silent, then I asked, 'Isn't it?' I remember Dr Rainey laughing, visibly relieved, and saying 'I don't know.' ''

Readers owe Ulam a debt for a book of reminiscent perceptions that have rarely been matched. A plausible conjecture suggests that we owe its coherence of form largely to an acute and sensitive Parisienne, Françoise Ulam.

5.6. Review by: Loretta M Taylor.
The Mathematics Teacher 70 (8) (1977), 699.

S M Ulam, a great mathematician and scientist, has been one of the giants of the nuclear and space age, contributing significantly to the development of nuclear and computer research. His interest in learning, memory, and the function of the brain is evidenced by his preoccupation with these phenomena, and it may be that his greatest contribution to society is even yet to come, perhaps in this area of research.

In writing this autobiography, Ulam has related his personal and professional life, focusing on his association with many great scientists. Names familiar to the general public, such as von Neumann, Fermi, Teller, and Oppenheimer, and some not so familiar to the general public, such as Banach, Birkoff, and Hilbert, come to life in the pages of the book. Details pertaining to their lives, personalities, capabilities, and contributions range from sketchy to significant, depending on the degree of his association with them. So many names appear in the book that it gives the appearance of a "Scientific Registry" or "Who's Who." It seems that virtually everyone who was anyone in science during this period of time is mentioned some where.

Following eight years of schooling at the gymnasium in Lwów, Poland, where he was born in 1909, Ulam studied engineering at the Lwów Polytechnic Institute. He received his master's degree there in 1932 and wrote his doctor's examination there in 1933. His was the first doctorate ever awarded at the institute. Lwów was a remarkable centre for mathematicians, and he spent much time in coffee houses and inns discussing mathematics with such giants as Mazur and Banach. One such discussion he described as lasting for seventeen hours. His personal and professional association with his colleagues and associates seems to have been a continuing and most important facet of his life.

Ulam first came to the United States in 1935 and became a citizen in 1941. He spent parts of his professional career at Princeton, Harvard, the University of Wisconsin, the University of Southern California, and the University of Colorado. He also spent much time in what seems to be his favourite place in the world, the pinon pine country of New Mexico at Los Alamos, where he was involved in the Manhattan Project.

The audience for which this book would have appeal is sizable and varied; however, there are those who might not appreciate the degree of self-confidence to which Ulam so unashamedly admits and who might label it as extreme egotism, as did his very close friend and associate, von Neumann. The description of people and events and occasional lapses into philosophy as presented by Ulam should be fully appreciated by those individuals knowledgeable in mathematics and science and in the historical development of these fields during this century. Almost any intelligent lay person with an interest in history and the capacity to appreciate the contributions that brilliant minds have made to society should also enjoy this autobiography.

Lastly, this book should serve well as a source of inspiration to the youngster with an interest in mathematics or science. Ulam himself was so intrigued and inspired at a young age by the writings of Jules Verne and H G Wells that he must have written with the desire to influence bright young minds himself.

5.7. Review by: N Metropolis.
Science, New Series 193 (4253) (1976), 568-569.

It was inevitable that S M Ulam would tell his story. A native of Lwów, Poland, he was a member of its distinguished school of mathematics; later studies and global events led to a series of residences spanning two continents, and his scientific interests spread into physics and then into mathematical biology. Such activity implied, of course, a host of new colleagues. A conversationalist of first magnitude with ample scientific curiosity, Ulam would have much to write. Apart from the time it would take away from scientific meditation, the only real obstacle to the realisation of the Adventures would be his general impatience. The storyteller, fortunately, has prevailed.

The present volume belongs to a relatively new genre that attempts to get behind the scenes to describe some of the idiosyncrasies, the weaknesses, and the strengths of segments of the scientific community. Laura Fermi's delightful book Atoms in the Family is exemplary; she has provided one perspective on researchers' lives outside the laboratory, thereby partially satisfying the public curiosity rendered acute by the relatively recent glitter of science. A companion volume to this is Emilio Segre's Enrico Fermi, Physicist, which is along more traditional lines but has the sparkle derived from the writer's "being there" as a collaborator over an extended period, starting from earliest times. A somewhat different, in parts more specialised, approach is taken in James Watson's famous dramatisation of the events leading to one of the great discoveries in biology, The Double Helix. Ulam reaches for all three objectives: to describe some of the nontechnical background in several scientific fields; to sketch a few biographies; and to summarise some developments in mathematics and thermonuclear physics. The reader is made privy to the ambitions, frustrations, successes, and bits of professional gossip associated with an interesting collection of superior minds.

The Adventures form a natural sequence of four parts: the early years as a student, then the postdoctoral phase and academia in America, followed by the extended Los Alamos period, and, finally, back to the university in 1967.

Most scientists are aware of the brilliance emanating from Budapest in the 1920's and early 1930's created by Szilard, Teller, von Neumann, and Wigner. But except among their confreres, the Polish school of mathematics, developed at Lwów and Warsaw, has until now remained relatively obscure. Ulam is changing all that, and such names as Banach, Kuratowski, Mazur, Sierpinski, and Steinhaus, along with Tarski and Kac, who immigrated here, will begin to have a more familiar ring. It was probably Stefan Banach who, in that formative period, played the central role for the budding mathematician Ulam, whose modus vivendi, but for imminent events that were to upset the great globe itself, might well have been otium cum dignitate.

The American scene for Ulam is identified with John von Neumann. It was von Neumann who arranged an invitation for Ulam to Princeton's Institute for Advanced Study, thus continuing a series of meetings, started in Europe, that was to lead to a deep friendship. Eight years later, it was von Neumann who suggested to Ulam the possibility of participation in the Los Alamos venture. The wartime period was, strangely enough, a relatively quiet time for the mathematician slowly turning physicist without abandoning his love for abstract mathematics. The war over, Ulam had a brief interlude at the University of Southern California (and an episode of near tragedy) before returning to Los Alamos to renew acquaintance with Enrico Fermi and, of more interest, Edward Teller. Some commentary is given on the Teller-Ulam collaboration in thermonuclear studies. Because the work is classified the reporting is necessarily of a peripheral nature and does not provide much illumination concerning the contributions of each. Later historians may obtain a clearer picture.

Ulam has a feeling for words, and they reveal an intense compassion on exceptional occasions. Perhaps the most moving descriptions are the accounts of last visits with Fermi and with von Neumann, which remain vivid more than two decades later. The last chapter, "Random reflections," will be of particular interest to the professionals. Here is a short summary of new directions and aspirations at a technical level that gives the lie to the oft-expressed supposition that the fertile imagination goes first, usually by age 30.

The writing has an engaging quality, conversational, if not chatty, rather than studied. One gets the impression that the manuscript was, at least in part, created in recording sessions. Less satisfying is the fact that sometimes a sequence of episodes is not smoothly joined, so that on occasion the author's unabashedness makes the treatment seem more discrete than discreet. Maxwell Evarts Perkins would no doubt have enjoyed coping with the manuscript. A collection of 25 photographs lends a personal touch in another dimension.

The book captures the spirit and tenor of scientific interaction in a succinct and piquant manner. It will not, however, meet the historiographer's criteria for a well-documented publication; the author remarks that he has never kept notes or a diary of any kind. But no one will deny that here is a fascinating kaleidoscope of an exciting time.

5.8. Review by: John C Oxtoby.
Mathematical Review MR0485098 (58 #4954).

The mathematical scene in pre-World War II Poland and the United States as it appeared to a young Polish mathematician, and the intellectual climate of Los Alamos during the war and post-World War II years, furnish the background and settings for this engrossing autobiography. The book provides a valuable supplement to the author's selected works [Sets, numbers, and universes: selected works, 1974]. In the course of relating his varied and often humorous adventures the author describes in greater or less detail the personalities, appearance and working habits of dozens of well-known mathematicians and physicists, often through anecdotes or remembered conversations. He also expresses his views about mathematics, its relations with physics and biology, and the role of computers, and speculates about the working of the human brain. The author's close association with von Neumann over a period of twenty years forms a central theme of the book and provided the initial incentive for writing it. Banach and Fermi are among the other leading characters. The portrait of the author that emerges will be fully recognisable to those who know him and will no doubt delight other readers as well.

5.9. Review by: H E Robbins.
Bulletin of the American Mathematical Society 84 (1) (1978), 107-110.

I. Ulam is a magic name in modern mathematics. One thinks of Leonardo's letter to the Duke of Milan:

Most Illustrious Lord;

... Item: In case of need I will make big guns, mortars, and light ordnance of fine and useful forms, out of the common type.

Item: I can carry out sculpture in marble, bronze, or clay, and also I can do in painting whatever may be done, as well as any other, be he who he may ....

And so he could.

In Ulam's writing, as in Leonardo's, scarcely a mention of mother and father. At eleven Ulam began to be known as a bright child who understood the special theory of relativity. He was an A student but did not study much, active in sports, played bridge, poker, and chess. At 15 he absorbed the calculus, number theory, and set theory. At 18, when he matriculated from gymnasium, the choice of profession presented difficulties. His father wanted him to join his successful law practice, while Ulam longed for a university career. But university positions in Poland were almost impossible to obtain if one's family, however wealthy and culturally assimilated, had a Jewish background. As a compromise, Ulam entered Lwów, Polytechnic Institute to study engineering.

From the first, mathematics took complete possession of him. Kuratowski quickly recognised the young student's gifts and took special pains with him. The names of Mazur, Lomnicki, Borsuk, Kacmarz, Nikliborc, Tarski, Schauder, Averbach, Schreier, Steinhaus, and above all Banach dominated a euphoric period of feverish activity. At 23 Ulam was sufficiently well known to be an invited speaker at the Zurich congress. Meeting foreign mathematicians for the first time, he found them nervous and given to facial twitches, or short and old, like Hilbert; certainly less impressive than his fellow Poles. Returning to Lwów, Ulam wrote a master's thesis which among other things outlined what is now category theory, and at 24 won his doctorate with a thesis in measure theory. But still there were no prospects of a university position for him in Poland.

Financed by his parents he visited Menger in Vienna, Hopf in Zurich, Cartan in Paris, and Hardy in Cambridge. Returning to Poland, he began a correspondence with von Neumann who invited him to visit the Institute at Princeton. In December 1935, Ulam sailed on the Aquitania for New York.

It was von Neumann whom Ulam came to admire above all others as a mathematician and kindred spirit. (The book was originally intended as a biography of von Neumann.) Things really began to happen when Ulam met G D Birkhoff at von Neumann's house and was in due course invited to Harvard as a Junior Fellow for three years. But soon after Ulam's return in 1939 from his customary three month visit to Poland, Hurewicz telephoned to say in sombre tones "Warsaw has been bombed, the war has begun."

Next spring, when things looked darkest, it was Birkhoff who came to the rescue again by securing for Ulam an instructorship at Madison. This was no easy matter, for there were many emigres by then and even modest positions were hard to find. At Madison he was promoted quickly to assistant professor, a position which held good hope for the future. He became an American citizen, married, and in 1943 rejoined von Neumann at Los Alamos, ignorant until he arrived of just what was going on there.

For Ulam, the transition from pure mathematics to applied physics was remarkably easy. (Not so for von Neumann, who had little physical intuition.) The physicist Otto Frisch in his first visit to Los Alamos from embattled Britain wrote "I also met Stan Ulam early on, a brilliant Polish topologist with a charming French wife. At once he told me that he was a pure mathematician who had sunk so low that his latest paper actually contained numbers with decimal points!"

II. Although Ulam's three intellectual heroes were Banach, von Neumann, and Fermi, none of them is portrayed so vividly in the book as Birkhoff. The Ulam-Birkhoff relationship seems to have been somewhat ambiguous on both sides.

"He liked the way I got almost furious when - in order to draw me out - he attacked his son Garrett's research on generalised algebras and more formal abstract studies of structures. I defended it violently. His smile told me that he was pleased that the worth and originality of his son's work was appreciated.

In discussing the general job situation, he would often make sceptical remarks about foreigners. I think he was afraid that his position as the unquestioned leader of American mathematics would be weakened by the presence of such luminaries as Hermann Weyl, Jacques Hadamard, and others. He was also afraid that the explosion of refugees from Europe would fill the important academic positions, at least on the Eastern seaboard. He was quoted as having said, 'If American mathematicians don't watch out, they may become hewers of wood and carriers of water.'

Even after Birkhoffs death the American suspicion of foreigners - even those who as Ulam describes himself were "not unpresentable" - continued to cause trouble. When the war ended in 1945 and Ulam wanted to return to Madison, chairman R E Langer answered when Ulam inquired about his chances for promotion and tenure: "No reason to beat around the bush, were you not a foreigner, it would be much easier and your career would develop faster."

At the time Ulam was 36, by any standards an outstandingly creative mathematician, pleasant and courteous in manner, and well supplied by now with friends in high places. How is it that all this did not suffice to overcome the Wisconsin xenophobia, nor to secure for him then or later a position commensurate with his talents from some leading American university? Surely there is a mystery here.

Before 1945 mathematicians were about as numerous in the academic world as professors of French literature, and their importance in the military-industrial-intellectual complex about as great. During the next twenty years American mathematics was a growth industry, since mathematicians had contributed essentially to making the weapons on which our safety now depended and would be needed in the future to keep ahead of possible rivals. Contrary to Birkhoff's fear, the refugees had created several jobs for American mathematicians for every one they occupied. Only the German rocket engineers imported after the war had a comparable effect.

III. Turned down by Wisconsin, Ulam spent an unhappy year at U. S. C, interrupted by a mysterious illness which brought him close to death, and in 1946 returned to Los Alamos. There he proposed the Monte Carlo method in a conversation with von Neumann. "Little did we know in 1946 that computing would become a fifty-billion-dollar industry annually by 1970." Teller and von Neumann were emotionally committed to constructing an H bomb at all costs. Ulam was not so obsessed, but it was he who thought of a way to make it work. "Contrary to those people who were violently against the bomb on political, moral, or sociological grounds, I never had any questions about doing purely theoretical work. ... I sincerely felt it was safer to keep these matters in the hands of scientists and people who are accustomed to objective judgments rather than in those of demagogues or jingoists, or even well-meaning but technically uninformed politicians."

In 1967 Ulam returned to university life at Boulder and became an elder statesman of government science.

IV. Some readers will be put off by the frequent examples of mathematical humour characteristic of Ulam and his friends. Thus of Erdős: "Once he stopped to caress a sweet little child and said in his special language: 'Look, Stan. What a nice epsilon.' A very beautiful young woman, obviously the child's mother, sat nearby, so I replied 'But look at the capital epsilon.' This made him blush with embarrassment.' " In fact, these episodes provide almost the only evidence of the humanity of the characters portrayed in this book. Erdős apart, they are preoccupied with seeking recognition of their precise rightful place in the official pecking order. It is a pity that this aspect of the world of mathematicians is so much emphasised in a book for the general reader; the more pity if indeed the emphasis is justified. The appearance of being thinking machines on the make, without discernible relation to parents, spouses, or children, and oblivious to the human concerns of our times, may be due in part to foreign systems of higher education that were devised to turn out idiot savants in the sciences as being more likely to be useful to the state. But if mathematical intelligence is strongly associated with emotional deprivation and social alienation, then even we earthy, super-honest, solid, and simple native Americans - the qualities that Ulam admires in us - are in for trouble.

6.1. Review by: Editors.
Mathematical Reviews MR0599363 (82a:00007).

The 1968 edition has been reviewed [see above]. As indicated in a note on the abridged edition, "In order to bring this important book before a wider public, this abridged edition has been prepared by removing the more difficult mathematics, so that it needs little more than a knowledge, or even a reminiscence, of school mathematics to follow the arguments."

7.1. From the Preface.

In writing a preface to another edition of this book I cannot resist the temptation to compare the present with the guesses and timid predictions I made about the future of science as it looked to me ten years ago. If anything, the present looks even more exciting than I had hoped. It is wonderful to observe how many unforeseen or unforeseeable facts and ideas have emerged. While I shall mention just a few of the many developments in recent science, it is important to realise that the rate at which we comprehend the universe is as vital as what we finally understand.

Progress in science and technology has proceeded at an ever-increasing pace, making the short period since I wrote this book as significant as any in the history of science. To see this one has only to think of the landings on the moon, the now commonplace launching of satellites, and the enormous discoveries made in astronomy and in the study of the earth itself.

Most notable has been the exponential growth in the technology of electronic computers, whose use pervades many aspects of daily life. Now elements of a "metatheory" of computing are being out lined and problems of computability in the general sense, especially with respect to its limits, are being studied successfully.

I wonder what John von Neumann's reaction would have been had he lived to see it all. He prophesied the growing importance of the computer's role, but even he would probably have been amazed at the scope of the computer age and the rapidity of its appearance.

One could say that after the atomic age there came the computer age, which, in turn, made the space age possible. All space vehicles - rockets, satellites, projectiles, shuttles, and so on - depend on the feasibility of very fast calculations that must be instantly transmitted to them in outer space to correct their orbits. Before the advent of the fastest electronic computers this kind of remote control was not possible.

Recently a great wealth of observations in physics and astronomy has increased the perplexity of the description of the universe. The enigma of quasars is still unresolved. These quasi-stellar objects seem to be billions of light-years away with an intrinsic luminosity hundreds of times greater than that of the galaxies in their foreground. In the few years since I wrote this book, vast "empty regions" hundreds of millions of light-years wide have been found. These areas make us question the sameness and isotropy of the universe suggested by the apparent uniformity of the cosmic radiation remaining from the Big Bang. It is now widely believed that black holes do exist. They may explain the behaviour of several observed astronomical objects. In addition, growing evidence supports the theory that violent processes cause gigantic explosions in star-like objects and galaxies.

To a mathematician like myself, the question, "Is the universe in space finite and bounded, or does it extend indefinitely?" remains the number-one problem of cosmogony and cosmology.

In physics, the number of new, fundamental, or primary particles is constantly increasing. Quarks seem more and more to represent real, not merely mathematical, constituents of matter, but their number and nature remain unverifiable, and scientists are considering the existence of subparticles, such as gluons.

Since the first publication of this book it has become more likely, it seems to me, that there might be an infinite chain of descending structures. To paraphrase a well-known statement about fleas, large quarks have bigger quarks on their backs to bite them, big ones have bigger ones, and so on ad infinitum.

There is also much speculation about the identity of or similarity between the different forces of nature. Certainly there is a strong analogy between electromagnetic forces and so-called weak interactions. There may even be a mathematical analogy between these forces, nuclear forces, and gravitational forces.

Mathematics remains the tool for investigating problems such as these. Electronic computers have helped immensely in solving complex calculations, and a great many new results have appeared in pure mathematical disciplines such as number theory, algebra, and geometry. The broadening range of "constructive" mathematical methods, such as the Monte Carlo method, indicates that a theory of complexity may soon affect many branches of mathematics and stimulate new points of view. Some physical problems such as the study and interpretation of particle collision on the new, miles-long accelerators call for gigantic Monte Carlo modelling.

Presently in vogue is the study of nonlinear transformations and operations. These began in the Los Alamos Laboratory, which now has a special centre devoted to nonlinear phenomena. This centre recently held an International conference on chaos and order. For the most part this work concerns the behaviour of iterations - repetitions of a given function or flow. These problems require guidance from what are essentially mathematical experiments.

Trials on a computer can give a mathematician a feeling or intuition of the qualitative behaviour of transformations. Some of this work continues a study mentioned in Chapter 12, and some follows work Paul Stein, I, and others have done in the intervening years.

While much of physics can be studied using linear equations in an infinite number of variables (as in quantum theory), many problems-hydrodynamics included are not linear. It is becoming more and more likely that there may be nonlinear principles in the foundations of physics. As Enrico Fermi once said, "It does not say in the Bible that all laws of nature are expressible linearly!"

To an amateur physicist such as I am the increasing mathematical sophistication of theoretical physics appears to bring about a decrease in the real understanding of both the small- and the large-scale universe. The increasing fragmentation may be due in part to neglect in the teaching of the history of science and certainly to the growth of specialisation and overspecialisation in various branches of science, in mathematics in particular. Although I am supposed to be a fairly well-read mathematician, there are now hundreds of new books whose very titles I do not understand.

I would like to devote a few words to what is manifestly the age of biology. I believe these past sixteen years have seen more significant advances in biology than in other sciences. Each new discovery brings with it a different set of surprises. Genes that were supposed to be fixed and immutable now appear to move. The portion of the code defining a gene may "jump," changing its location on the chromosome.

We now know that some segments of the genetic code do not express formulae for the manufacture of proteins. These sometimes longish sequences, called introns, lie between chromosome segments that do carry instructions. What purpose introns serve is still unclear.

The success of gene splicing - the insertion or removal of specific genes from a chromosome - has opened a new world of experimentation. The application of gene manipulation to sciences such as agriculture, for example, may have almost limitless benefits. In medicine we can already produce human-type insulin from genetically altered bacteria. Scientists have agreed to take precautions against accidentally creating dangerous new substances in gene-splicing experiments. This seems to satisfy the professional biologists. Still, there is a great debate over whether to allow unregulated genetic engineering, with all its possible consequences.

My article "Some Ideas and Prospects in Biomathematics" is an example of some of my own theoretical work in this area. It concerns ways of comparing DNA codes for various specific proteins by considering distances between them. This leads to some interesting mathematics that, inter alia, may be used to outline possible shapes of the evolutionary tree of organisms. The idea of using the different codes for a cytochrome C was suggested and first investigated by the biologist Emanuel Margoliash.

At Los Alamos, a group led by George Bell, Walter Goad, and other biologists is using computers to study the vast number of DNA codes now experimentally available. The group was recently awarded a contract by the National Institute of Health to establish a library of such codes and their interrelations.

It is well known that gradual changes, no matter how extensive, are barely noticeable while they occur. Only after a certain amount of time does one become aware of any transformation. One morning in Los Alamos during the war, I was thinking about the imperceptible changes in my own life in the past years that had led to my coming to this strange place. I was looking at the blue New Mexico sky where a few white clouds were moving slowly, seemingly retaining their shape. When I looked away for a minute and back up again, I noticed that they now had completely different shapes. A couple of hours later I was discussing the changes in physical theories with Richard Feynman. Suddenly he said, "It is really like the shape of clouds; as one watches them they don't seem to change, but if you look back a minute later, it is all very different." It was a curious coincidence of thoughts.

Changes are still taking place in my personal life. In 1976 I retired from the University of Colorado to become professor emeritus, a sobering title. At the same time, I accepted a position as research professor at the University of Florida in Gainesville, where I still spend a few months every year, mostly during the winter when it is not too hot.

My wife, Françoise, and I sold our Boulder house and bought another one in Santa Fe, which has become our base. From Santa Fe I commute three or four times a week to the Los Alamos Laboratory. Its superb scientific library and computing facilities allow me to continue working in some of the areas of science mentioned above. Françoise acts as my "Home Secretary," as I call her, alluding to the title of the British Interior Secretary. We still travel quite extensively and I continue to lecture in various places.

We are fortunate that our daughter Claire also lives in Santa Fe with her husband, Steven Weiner, an orthopaedic surgeon. Their daughter, now five, gives me occasion to wonder at how remarkable the learning processes of small children are, how a child learns to speak and use phrases analogous to and yet different from the ones it has heard. Observing Rebecca speak provides me with additional impulses and examples for describing a mathematical schema for analogy in general.

My collaborator, Dan Mauldin, a professor at North Texas State University, has recently edited an English version of The Scottish Book mentioned in Chapter 2. We are now collaborating on a collection of new unsolved problems. This book will have a different emphasis from that of my Collection of Mathematical Problems, published in 1960. The new collection will deal more with mathematical ideas connected to theoretical physics and biological schemata.

Many of the people mentioned in this book have since died, or left, as my friend Paul Erdős prefers to say: Kazfmir Kuratowski, my former professor; Karol Borsuk and Stanislaw Mazur, my Polish colleagues; my cousins Julek Ulam in Paris and Marysia Harcourt-Smith; in Boulder, Jane Richtmyer, who helped with the first writing of this book; George Gamow and his wife Barbara; my collaborators John Pasta and Ed Cashwell of the Monte Carlo experiments; and here in Los Alamos (within a few months of each other) the British physicist Jim Tuck and his wife Elsie. As Horace said, "Omnes eadem idimur omnium versatur urna ... sors exltura. ..."

A few weeks ago I was invited to give a Sunday talk at the Los Alamos Unitarian Church on the subject of "Pure Science in Los Alamos." The discussion that followed centred on problems that are of growing concern nowadays: the relation of science to morality; the good and the bad in scientific discoveries. Around 1910, Henri Poincaré, the famous French mathematician, had considered such dilemmas in his Dernières Pensées. The questions were less disturbing then. Now, the release of nuclear energy and the possibility of gene manipulation have complicated the problems enormously.

I was asked what would have happened had the Los Alamos studies proved that it was impossible to build an atomic bomb. The world, of course, would be a less dangerous place in which to live, without the risk of suicidal war and total annihilation. Unfortunately, proofs of impossibility are almost non-existent in physics. In mathematics, on the contrary, they provide some of the most beautiful examples of pure logic. (Think of the Greeks' proof that the square root of two cannot be a rational number, the quotient of two integers!) Humanity, it seems, is not emotionally or mentally ready to deal with these enormous increases in knowledge, whether they involve the mastery of energy sources or the inanimate and primitive life processes.

Someone in the audience wondered if some of the current research on the human brain might not ultimately lead to a wiser and better world. I would like to think so, but this possibility lies too far in the future to even guess at.

In the short span of my life great changes have taken place in the sciences. Seventy years amounts to some 2 percent of the total recorded history of mankind. I mentioned this once to Robert Oppenheimer at Princeton. He replied, "Ah! but one-fiftieth is really a large number, except to mathematicians!"

Sometimes I feel that a more rational explanation for all that has happened during my lifetime is that I am still only thirteen years old, reading Jules Verne or H G Wells, and have fallen asleep.

8.1. Preface by Martin Gardner.

Stanislaw Marcin Ulam, or Stan as his friends called him, was one of those great creative mathematicians whose interests ranged not only over all fields of mathematics, but over the physical and biological sciences as well. Like his good friend "Johnny" von Neumann, and unlike so many of his peers, Ulam is unclassifiable as a pure or applied mathematician. He never ceased to find as much beauty and excitement in the applications of mathematics as in working in those rarefied regions where there is a total unconcern with practical problems.

In his Adventures of a Mathematician Ulam recalls playing on an oriental carpet when he was four. The curious patterns fascinated him. When his father smiled, Ulam remembers thinking: "He smiles because he thinks I am childish, but I know these are curious patterns. I know something my father does not know."

The incident goes to the heart of Ulam's genius. He could see quickly, in flashes of brilliant insight, curious patterns that other mathematicians could not see. "I am the type that likes to start new things rather than improve or elaborate," he wrote. "I cannot claim that I know much of the technical material of mathematics. What I may have is a feeling for the gist, or maybe only the gist of the gist."

Of course Ulam was being too modest. He knew a great deal about the technical side of math. But it was seeing the gist, the inner core of a problem, that enabled him to open so many new roads-roads that often led to new branches of mathematics. To mention only three: Cellular automata theory, which he proposed to von Neumann; the Monte Carlo method of solving intractable problems, not only in probability theory but in areas such as number theory where the method would not have been thought applicable; and his work on nonlinear processes that anticipated today's interest in solitons and "chaos." We still do not know - it remains a government secret-what jumped into Ulam's head that made it possible for Edward Teller to build an H-bomb.

To me the most fascinating passages in this collection of what Ulam liked to call his "little notes" are those that touch on deep philosophical mysteries. To what extent are physical laws, the patterns of nature's carpet, "out there," independent of you and me? To what extent are they the free creations of human minds? Why is there such an incredible "fit" between elegant theories such as relativity and quantum mechanics and the way the universe behaves? In chapter 2 Ulam recalls telling his friend Enrico Fermi how astonished he was when, just out of high school, he learned how fantastically accurate is the Schrödinger equation of quantum mechanics in giving spectral lines. "You know Stan," Fermi replied, "it has no business being that good."

But the equations of physics are that good. Even more surprising, they are often equations of a very low order. Is this because the patterns of nature are, as Einstein believed, basically simple (whatever simplicity means here), or will the laws of the future be increasingly complicated? "It does not say in the Bible," Fermi once remarked (chapter 2), "that the fundamental equations of physics must be linear."

Some physicists believe they are on the verge of creating a grand unified theory in which a single superforce will explain all known forces and particles. There will then be nothing new for physics to discover, and science will turn to life and the human mind as the only phenomena not yet understood. Ulam did not share this view. Repeatedly he cites new developments in physics, logic, and mathematics - especially the work of Kurt Gödel - to suggest that no finite set of laws will ever embrace all there is.

In Ulam's vision the universe is infinitely mysterious and always full of yet-to-be-discovered wonders. Going down into the microworld there may be endless structural levels, wheels within wheels, and the same may be true in the other direction. Space and time, on a level smaller than a particle, may be discontinuous - a kind of "foam" (as John Wheeler calls it) of holes within holes, and subject to strange non-Euclidean topologies of which we now have no inkling. Similarly, the universe in the large may be embedded in vaster regions where space and time are utterly unlike the spacetime of classical relativity.

It may be true that any day now physicists will find a way to unify the four forces of nature, perhaps even explain how the explosion that created our universe was an inevitable consequence of fluctuations in a primordial quantum vacuum. But the vacuum of quantum mechanics is a far cry from metaphysical "nothing." Its fluctuations obey rigid quantum laws which must be "there" to sustain the eternal dance of virtual particles that bubble out of the energy sea and quickly die. Will the human mind ever learn all there is to know about nature's laws?

Ulam saw the search as endless, both in science and mathematics. 'just as animals play when they are young," he writes in chapter 13, "in preparation for situations arising later in their lives, it may be that mathematics is to a large extent a collection of games. In this light, it has the same role and may be the only way to change the individual or collective human mind to prepare it for a future that nobody can now imagine." This poetic sense of an open future with mysteries forever unresolved, with new surprises forever turning up, pervades all of Ulam's nontechnical writings.

Gian-Carlo Rota, who coedited this volume, has spoken of a writing style that he calls "Ulamian" - a mix of crystal clear prose, subtle humour, and graceful phrasing. The style was the product of a collaboration. Ulam's French-born wife, Françoise, was what she calls in her introduction her husband's "live word processor." It was this happy collaboration that resulted in Ulam's marvellous autobiography. As Ulam phrased it in his introduction, it was Françoise who "managed to decrease substantially the entropy" of his memoirs. The information, speculation and philosophical insights in the book you now hold are all Ulam's, but for the pleasure you experience in reading its essays you have a remarkable duo to thank.

8.2. Review by: William Aspray.
Isis 79 (4) (1988), 702-703.

Stanislaw Ulam is among the more interesting figures of twentieth-century mathematics because of his contributions to logic and probability theory, his fundamental role in the development of the technologies of computing and atomic energy, and his close personal associations with several great figures of modern mathematics. This collection of essays supplements the autobiography Ulam published in 1976 by making readily available many of his nontechnical writings, which were widely scattered across the literatures of mathematics, computer science, biology, physics, and history of science. Ulam referred to these essays as his "little notes," his attempts to sketch fundamental patterns of nature, often through attention to small and neglected details.

A survey of the topics encountered here indicates the breadth of Ulam's interests: the relations of mathematics to physics, the philosophy of science, the mathematics of computation, parallel processing and cellular automata, computer chess, mathematical biology, atomic weapons and nuclear propulsion, and profiles of mathematical colleagues. There is too much material to allow a detailed description, but two general themes can be identified. One is the influence of the Polish School on Ulam's mathematical development. This is seen most clearly in his portraits of Marian Smoluchowski, Kazimierz Kuratowski, and Stefan Banach but is also evident in his essay "Physics for Mathematicians" and elsewhere. Perhaps more important is the influence of his mathematical collaborator and friend, John von Neumann. There are perceptive essays on von Neumann's scientific career, his theory of automata, and his contributions to computing. Other essays on cellular automata, Monte Carlo, and parallel processing address aspects of their collaborative research.

Ulam is reported to have said that "whatever is worth saying, can be stated in fifty words or less." He has heeded his own advice, and the style that results is well suited to the essay format: aphoristic, but clear and punctuated by adroitly chosen references to the history and philosophy of science. This is a book to read not only if you are interested in the life and work of Ulam.

8.3. Review by: Mary L Crowley.
The Mathematics Teacher 80 (8) (1987), 689-691.

In this collection of twenty-three short essays, Stanislaw Ulam enlightens the reader about the origins and applicability of twentieth century mathematics and about how computers can serve as tools for mathematicians and physicists, with glimpses into the lives of several famous mathematician physicists of the twentieth century. Subdividing these essays into categories is somewhat artificial - the writings on mathematics involve physics, the writings about computers involve mathematics, the writings on individuals reflect their scientific contributions. I shall discuss the book as if it had two major components: the sciences and the biographies.

With a writing style that is lively and never condescending, Ulam navigates the lay reader through numerous scientific discussions. Number theory, the Monte Carlo method, nonlinearity, computer simulations, physics, and philosophy are among the ports of call. I warmed immediately to the writer when, in chapter 1, "The Applicability of Mathematics," he presented a rationale for mathematical research into highly abstract and specialised areas. He suggested that these studies should be thought of as "'patrols' sent into the unknown." Some of these explorations turn out to have immediate applications; others do not appear to have current application but might be useful in the future. At the very least, these ventures increase the general body of mathematical knowledge. Couldn't this serve as an answer to my students when they ask, "What good is mathematics to me?"

Ulam's discussion of the question Is chess an art or a science? held my interest, despite the fact that I am not a chess player. That discussion, those on patterns of growth, and those on biomathematics are excellent examples of mathematical modelling and problem solving. His discussion on the computer's role in assisting with proof was also of particular interest to me. This discussion was part of an excellent and all-too short essay on the computer in mathematics.

The one area of Ulam's writing in which I was slightly disappointed was his discussion of the problem of morality in scientific work. I wanted him to explore further the issue of the inherent good and evil in inventions, especially as he was one of the key figures in the nuclear projects at Los Alamos.

As one interested in mathematical biography, I found the reflections on the mathematicians and scientists to be of great interest. With a keen sense of detail and human interest, Ulam presents profiles of von Neumann, Gamow, Smoluchowski, Kuratowski, and Banach. The most attention (three essays) centres on Ulam's close personal friend, John von Neumann. In these, as in all the biographical essays, a nice balance is achieved between the description of the personal persona and the professional persona of the individual.

After reading this rich collection of observations, I also felt that I had been introduced to Ulam. Like his book, he is hard to categorise. Is he a pure or an applied mathematician? Is he a mathematician or a scientist? Is he a philosopher? A biographer? He is, it seems, all of these.

This book will have particular appeal to lay mathematical physicists and those interested in the history of mathematics and the computer. Ulam's writing style is vigorous, his use of language is entertaining, and his topics are timely. Thank goodness he did not entirely heed the advice we are told he gave others: "whatever is worth saying, can be stated in fifty words or less."

8.4. Review by: Andrew D Booth.
Mathematical Reviews MR0874755 (88d:01038).

Stan Ulam was well known to all of those who worked with the von Neumann group at Princeton during the 1940s and 1950s. Although his work at Los Alamos was not generally available, he nevertheless enjoyed a wide reputation as the originator of the "Monte Carlo" method for simulating complex systems. This book contains reprints of a large number of his more popular essays on topics which range from biography through speculations on the nature of natural selection and from automata theory to space flight.

The book opens with a "Memorial Service" for Ulam by Gian-Carlo Rota and continues with six chapters in which Ulam explains his philosophy of the importance of mathematics and especially of its relevance to physics, engineering and biology. It becomes clear that Ulam was convinced of the value of digital computers as an aid to mathematical thought and invention.

Chess enthusiasts will appreciate the chapter on early chess playing programs and the analysis of the needs of a Master-class program. The essay was written in 1957 and shows great perceptiveness on the part of its author.

Three chapters discuss parallel computation and "patterns of growth". The latter would today be called cellular automata and, considering that they date from 1957 to 1970, again reveal remarkable insight.

Biomathematics and the computer in genetics form the subjects of the next chapters. Again there is material which was ahead of its time, for example, the analysis of the possibility of natural selection having generated man in the time since the "big bang" is entirely topical in the light of recent work by Hoyle and his collaborators.

A chapter on the origins of the H-bomb is of considerable interest as an historical document in the history of electronic computation; it leads naturally to "The Orion project": a thermonuclear drive for space travel.

Three chapters deal with the life and scientific contributions of John von Neumann. They are of great interest in that it was John who brought Ulam to the USA.

The book ends with biographical sketches of Gamow, Smoluchowski, Kuratowski (who was Ulam's Doctoral supervisor) and Stephan Banach. Ulam was a stimulating writer, the essays are all readable even for non-specialists and the book is both a fine testimonial to a great mathematician and a useful historical document.

9.1. From the Foreword by A R Bednarek and Françoise Ulam.

"Good mathematicians see analogies between theorems or theories, the very best ones see analogies between analogies." Stefan Banach.

Stanisław Ulam's affiliation with Los Alamos National Laboratory spanned over two-thirds of his professional life. There was no aspect of its mathematical activity during this period in which he was not involved, either centrally or tangentially.

His catholic view of the role of mathematics vis-à-vis other sciences extended far beyond into the mathematical and scientific community at large, as did his genius for problem formulation and for applying the most abstract ideas from the foundations of mathematics to computing, physics, and biology. In addition he possessed the ability to excite others many of them not trained as mathematicians - and involve them in his researches. The impact of his work is still felt both at the Laboratory and in those larger communities, for he liked to disseminate his ideas orally in an ever widening round of lectures and seminars from where they took on a life of their own. The Monte Carlo method which he originated with von Neumann in order to study neutron scattering and other nuclear problems at Los Alamos is one such example. Its offshoots are now so universal that they are even applied to regulate traffic lights!

His influence, along with that of John von Neumann, the brilliant Hungarian mathematician, contributed to the establishment at the Laboratory of an atmosphere and a tradition that fostered and supported an exceptional - if not unique - interaction between mathematics and science. Extensive testimony and documentation concerning the integral role that Stan Ulam played in this interaction can be found in "From Cardinals to Chaos. Reflexions on the Life and Legacy of Stanislaw Ulam" published by Cambridge University Press in 1989.

From 1944 until his death in 1984. while connected with Los Alamos in a variety of ways, from staff member, to group leader, to research advisor, to 'no-fee consultant' - one of his favourite expressions - he wrote Laboratory reports (many with the help of trusted and talented collaborators) that show a breadth of scientific interests unusual for a mathematician. They cover pioneering work, in the horse-and-buggy days of computing, on mathematical modelling of physical processes, nuclear rocketry, space travel, and biomathematics. (Another eleven, weapons related reports, with Evans, Everett, Fermi, Metropolis, von Neumann, Richtmyer, Teller, Tuck, and others are still classified and unavailable for publication.)

Mathematically speaking, three motifs run through Ulam's theoretical and applied work: the iteration or composition of functions, or relations; the use of evolving computer capability in the exploration of analytically intractable problems: and the introduction of probabilistic approaches - while knowing that most practical applications are made in the presence of uncertainty. The fusion of these themes is characteristic of the central contributions of this collection.

As to the quotation from which the title of this book is derived, one must remember that Ulam held Banach, along with von Neumann and Fermi, "as one of the three great men whose intellects impressed me the most." Stefan Banach was an outstandingly original Polish mathematician and one of the founders of the now famous Lwów school of mathematics. Banach was Ulam's friend and mentor in Poland before World War II. His influence on Ulam was profound and Ulam liked to quote his comment on the ability of some mathematicians to see "analogies between analogies." There is no question that in the practice of his craft and his art, Ulam was guided by this principle, and that he, in turn, epitomised its application. In addition, to Ulam the idea of analogy was itself amenable to mathematical discussion.

In 1983. when D Sharp and M Simmons, editors of this Los Alamos Science series, asked Ulam to gather his unclassified - and declassified - reports for publication in one volume, they intended to omit a few that had appeared elsewhere. After his death in 1984, it was decided to publish them all as many represent preliminary studies of subjects that were subsequently expanded elsewhere, leading, in several instances, to the development of new and extensive theories.

Ulam had dictated brief introductory notes and a sketch for a preface which he intended to develop. Rather than put words in his mouth, it was thought more appropriate to reproduce his notes in their short and unpolished form. It was also decided to leave the style and substance of the reports untouched, as evidence that scientific advances do not usually arise in their final, definite form. More often than not they are the product of sequences of tentative, sometimes repetitive, and even at times inaccurate steps. Two appendices complete the volume: a list of Ulam's publications and a brief biographical chronology. (More detailed biographical material can be found in his autobiography. "Adventures of a Mathematician," as well as in "From Cardinals to Chaos.")

This collection represents an important complement to the selection of papers and problems, mostly in pure mathematics, published by MIT Press in 1974 as "Sets, Numbers, Universes," and to a volume of essays, "Science, Computers, and People," published by Birkhäuser in 1985. And, whereas these two books are composed of papers readily available albeit scattered in the scientific literature, the Los Alamos Reports have been for the most part difficult of access and little known. Their historic value is therefore very real.

As mentioned earlier, many of these reports and much of Ulam's work was done in collaboration. He liked to stress the importance of working with collaborators, with whom, he said, the nature of their "shared ideas and techniques" depended "on the personality and experience of the individuals." We add a few words about these colleagues as well as about the work they and Ulam engaged in.

As early as 1944, when he joined the Manhattan Project, Ulam and David Hawkins (Chapter 1) were "playing" with the notion of branching processes, or multiplicative systems, as they called them, motivated by their application to atomic physics.

David Hawkins, a philosopher of science by profession and mathematician by inclination, was, in Ulam's words "the best amateur mathematician I know," and they became fast friends. Hawkins is presently professor emeritus at the University of Colorado.

The work in the Ulam-Hawkins report was subsequently developed with Everett in the three extensive reports grouped in Chapter 3, which are reproduced here for the first time. While clearly motivated by the need to understand neutron multiplication in fission processes, the reports lay the foundations for - in their own words - a "formalism general enough to include as special cases the multiplication of bacteria, radioactive decay, cosmic ray showers, diffusion theory and the theory of trajectories in mechanical systems."

C J Everett, who died in 1987, was a mathematician with whom Ulam worked on a conceptual as well as technical level in Wisconsin and at Los Alamos, where he became a member of Ulam's group. An eccentric, shy, and witty man, he was quite probably the only person who ever opted for bus transportation to come to Los Alamos for a hiring interview, and he was known for having turned in a monthly progress report - in which staff members were supposed to describe their research - which said tersely "progress was made on last month's progress report."

The first written proposals for the Monte Carlo method put together in a 1946 "report" called "Statistical Methods in Neutron Diffusion" appear in Chapter 2. This method of approaching precise but intractable problems through the introduction of random processes and probabilistic experimentation, has found wide application not only in areas close to those motivating its origin but others more removed, such as operations research, and combinatorics. In fact, the "report" - of which only eight copies were made - consists of two letters and handwritten calculations photographed and stapled together. Its cover specifies that the "work" was "done" by Ulam and von Neumann and "written" by von Neumann and Robert Richtmyer - then head of the Laboratory's Theoretical Division. Its informality attests to the casual manner in which information was disseminated through the Laboratory at the time.

The long term professional and personal rapports between von Neumann and Ulam need not be recounted here - references to them can be found in the books already mentioned. Suffice it to say that though there exist few papers and abstracts under their joint names, von Neumann's extensive correspondence with Ulam attests to their interacting interests in pure mathematics, in pioneering computer technology and techniques, and in cellular automata and the brain. (The correspondence is now stored in the archives of the Philosophical Society in Philadelphia.)

Ulam's collaboration with Enrico Fermi initiated the computer simulations of nonlinear dynamical systems that lead to the evolution of a major field of research popularly labelled "chaos theory." Fermi called this work, which was developed with the programming assistance of John Pasta, "a minor discovery," a modest understatement given the seminal character of this investigation (Chapter 5.) Chapters 10 and 11, with P R Stein, address the subject of nonlinear transformations in greater detail.

Fermi, with whom Ulam became acquainted in Los Alamos during the war, was a man of simple tastes and life style. The Ulams had an opportunity to sample this while motoring together across France one summer. Feeling ill at ease during a lunch in a recommended temple of gastronomy, Fermi decreed he would select the night's lodgings. Meandering through a picturesque valley he chose a modest inn by a babbling brook where, after dinner, sitting under the stars they discussed physics and new mathematical problems to experiment with after the vibrating string calculations. However the night's encounter with fleas, bedbugs, and mosquitoes made him admit the next morning that the higher-class hostelry next door that Ulam had eyed, might perhaps have provided a more restful night.

In the area of space technology, Ulam investigated schemes for nuclear rocketry with Everett and with Conrad Longmire, a physicist from the mountains of Tennessee who played a mean banjo. Chapter 7 describes a way to propel very large space vehicles by a series of small external nuclear explosions which later developed into Project Orion. Chapter 9 deals with the propulsion of space vehicles by extraction of gravitational energy from planets. Schemes based on a similar idea are now used in "flyby" missions to the outer planets and to provide part of the energy for spacecrafts going beyond the planets.

A study of patterns of growth, with Robert Schrandt (Chapter 12) investigates how simple recursively defined codes can give rise to complex objects. Such studies have become a growth industry of their own in the improved computer graphics world of today.

Several other reports are devoted to biomathematical questions. Their findings have opened new fields of biomathematical research. Abstract schemata of mathematics are applied to pattern recognition with the help of computers investigating, for example, the way visual pictures are recognised. Using metrics in molecular biology shows how a new mathematical concept of distance between finite sequences or objects can be applied to reconstruct the evolutionary history of biological organisms.

Closely involved with Ulam in this work was Paul Stein, a physicist turned mathematician under Ulam's influence who became an invaluable collaborator able to implement and develop the gist of Ulam's directions. William Beyer, a gifted fellow mathematician, also collaborated on a conceptual and technical level in the biological investigations.

John Pasta, Mary Tsingou-Menzel, Robert Schrandt, and Myron Stein lent their talents to creative programming, at a time when the art was in its infancy and pre-microchip-era machines with names like "Eniac," "Maniac," "Johnniac," presented storage and timing constraints. Pasta, who died in 1980, was a self-made man of Italian descent. He had furthered his education and became a physicist while working on the New York city police force.

The mathematician Al Bednarek, one of the editors of this volume and co-author of this foreword, also collaborated with Ulam on problems of parallel computation (Chapter 18). He is a former chairman of the mathematics department at the University of Florida.

Shortly after his arrival at Los Alamos in 1944, Ulam was asked by a colleague what it was that he was doing. Since at the time he was a very pure mathematician and had not yet familiarised himself with the nature of the work, his Socratic answer was "I supply the necessary don't know how!" Stan Ulam's "necessary don't know how" as well as his modestly unenunciated "know how" are sorely missed by all who were privileged to have known him or worked with him.

9.2. Sketch of a Preface by Ulam.

The collection of these reports, which appeared over the considerable span of years that I spent at Los Alamos, concerns a great variety of topics. Its very heterogeneous nature illustrates the diversity of the programmes and of the areas of research that interested the laboratory.

Before World War II it was almost exclusively in the universities, in the graduate schools of the larger institutions that scientific research used to take place. The Bureau of Standards and a very few large industrial companies such as Bell Telephone, General Electric, and some pharmaceutical firms were the exception to the rule.

This little collection may bear witness, in a very modest way, to the wide-ranging changes, which are still going on in the organisation and practice of research in this country and abroad. Because of the novel problems which confronted its scientists during the wartime establishment of Los Alamos, the need arose for research and ideas in domains contiguous to its central purpose. This trend continues unabated to the present.

Problems of a complexity surpassing anything that had ever existed in technology rendered imperative the development of electronic computing machines and the invention of new theoretical computing methods. There, consultants like von Neumann played an important role in helping enlarge the horizon of the innovations, which required the most abstract ideas derived from the foundations of mathematics as well as from theoretical physics. They were and still are invested in new, fruitful ways.

An enormous number of technological and theoretical innovations were initiated at this laboratory during these forty years. To mention but a few, besides the advances in computing, one can name research on nuclear propulsion of rockets and space vehicles, in molecular biology, and on the technology of separating cells.

The growing importance of research laboratories such as this one became a not exclusively American phenomenon. For instance the aspect of academic research has changed almost beyond recognition in France. What used to be, before World War II, almost exclusively the province of universities, has now shifted to the French National Center of Research (Centre National de Recherche.)

This collection of Los Alamos Reports ranges over almost four decades and may illustrate, I hope, how a mathematical turn of mind, a mathematical habit of thinking, a way of looking at problems in different subfields of physics, astronomy, or biology can suggest general insights and not just offer the mere use of techniques. Ideas derived from even very pure mathematical fields can provide more than mere "service work," they may help provide true conceptual contributions from the very beginning.

The period in question has seen the origin and development of the art of computing on a scale which vastly surpasses the breadth and depth of the numerical work of the past. In at least two different and separate ways the availability of computing machines has enlarged the scope of mathematical research. It has enabled us to attempt to gather, through heuristic experiments, impressions of the morphological nature of various mathematical concepts such as the behaviour of solutions of certain nonlinear transformations, the properties of some combinatorial systems, and some topological curiosities of seemingly general behaviours. It has also enabled us to throw light on the behaviour of solutions of many problems concerning complicated systems, by allowing numerical computations of very elaborate special physical problems, using both Monte Carlo type experiments and extensive but "intelligently chosen" brute force approaches, in hydrodynamics for example.

A number of such experiments have revealed, surprisingly, a nonclassical ergodic behaviour of several dynamical systems. They have showed unexpected regularities in certain flows of dynamical systems, in the mechanics of many-body problems, and in continuum mechanics. Recently they have been applied to the study of elementary particle physics set-ups and interactions.

And now there appear some most exciting vistas in the applications of mathematics to biology that deal with both the construction and the evolution of living systems, including problems of the codes, which seem to define the basic properties of organisms and ultimately may provide us with a partial understanding of the working and evolution of the nervous system and some of the powers of the brain itself.

In addition these reports show the varying involvements of my collaborators and myself. I particularly want to stress the importance of the role of collaborators. An ever increasing number of publications of mathematical research is proof of the advantages derived when two or more authors share ideas and techniques. The nature of this exchange varies from case to case, depending on the personality and experience of the individuals.

These few very sketchy remarks are merely intended to emphasise how the necessity of defence work at the frontiers of science has continued to this day to stimulate research in a multitude of directions.

S M Ulam, Santa Fe, February 1984.

9.3. Review by: Dieter H Mayer.
Mathematical Reviews MR1123485 (92i:01033).

This is a collection of reports known up to now only to very few people, on the work done by Ulam and collaborators at the Los Alamos National Laboratory in the years 1944 to 1982. Some of these reports remain classified to the present day, others have been declassified since their appearance and are now accessible to the general public. In a sense they complement other selections of the work of Ulam like the ones in his book Sets, numbers, and universes: selected works [1974], mostly in pure mathematics, and of more general thoughts and reflections of this remarkable scientist as in From cardinals to chaos [1989] or in another book by Ulam, Science, computers and people (1986).

The papers in the present collection mainly treat problems arising from the applied sciences such as physics, biology, etc. The main themes of the present papers recurring again and again are: the iteration of functions and their limit behaviour, the need for computers to understand their complex structure, and the development of probabilistic methods in the treatment of systems of physics or biology which are somehow perturbed by external noise that is hard to describe in detail. Surprisingly enough, these are exactly the themes one can find in present-day journals of physics or other fields engaged in the discussion of nonlinear phenomena. This shows how far-reaching Ulam's ideas and feelings for scientifically important developments really have been. Starting with three papers on the foundations of discrete multiplicative random processes in one and several dimensions originating in the problem, in nuclear physics, of understanding the proliferation of neutrons in nuclear reactions from one to the next generation, the iteration of functions makes its first appearance. There it turns out that the generating function for the distribution of the number of neutrons in the kth generation is just the kth iterate of the generating function in the first generation. The natural problem then is the behaviour of these iterations for large k. In modern terms, this is just the asymptotic behaviour of a finite-dimensional dynamical system with discrete time steps. Since there are almost no general methods available to treat this problem analytically, it was clear to Ulam that only computers could provide us with the necessary hints to ask the right questions. And indeed several reports deal just with the numerical treatment of rather simple polynomial, low-dimensional transformations. What Ulam and his co-workers found in this connection can be considered the origin of what today is called chaos theory: rather complex structures which by no means can be predicted from the simple form of the transformations. In Report 12, another object of great interest of our days is discussed; namely, cellular automata and the wonderful space-time structures they can generate from simple deterministic rules. Other papers in the volume discuss such diverse problems as the possibility of accelerating rockets by nuclear explosions or of extracting energy from the gravitational field for space vehicles, an idea which indeed has been used recently for satellites travelling to outer space, and the problem of defining the notion of complexity for numbers or other algorithms. The remaining reports are mainly on problems in the field of biology, such as the evolution of populations, and possible mechanisms for recognition and discrimination. In Report 20, "On the notion of analogy and complexity in some constructive mathematical schemata", Ulam quotes another great Polish mathematician, S. Banach, who often remarked that "good mathematicians see analogies between theorems and theories, while the very best ones see analogies between analogies". The remarkable scientist Ulam certainly belongs to this latter category, as the papers in the volume under review prove in such a convincing manner. The book should not be missing on the desk of anyone interested in the application of mathematics to the sciences.

10.1. From the Publisher.

The autobiography of mathematician Stanislaw Ulam, one of the great scientific minds of the twentieth century, tells a story rich with amazingly prophetic speculations and peppered with lively anecdotes. As a member of the Los Alamos National Laboratory from 1944 on, Ulam helped to precipitate some of the most dramatic changes of the post-war world. He was among the first to use and advocate computers for scientific research, originated ideas for the nuclear propulsion of space vehicles, and made fundamental contributions to many of today's most challenging mathematical projects.

With his wide-ranging interests, Ulam never emphasised the importance of his contributions to the research that resulted in the hydrogen bomb. Now Daniel Hirsch and William Mathews reveal the true story of Ulam's pivotal role in the making of the "Super," in their historical introduction to this behind-the-scenes look at the minds and ideas that ushered in the nuclear age. An epilogue by Françoise Ulam and Jan Mycielski sheds new light on Ulam's character and mathematical originality.

10.2. Preface by William G Mathews and Daniel O Hirsch.

It is still an unending source of surprise for me to see how a few scribbles on a blackboard or on a sheet of paper could change the course of human affairs.

This remark of Stanislaw Ulam's is particularly appropriate to his own career. Our world is very different today because of Ulam's contributions in mathematics, physics, computer science, and the design of nuclear weapons.

While still a schoolboy in Lwów, then a city in Poland, he signed his notebook "S Ulam, astronomer, physicist and mathematician." Of these early interests perhaps it was natural that the talented young Ulam would eventually be attracted to mathematics; it is in this science that Poland has made its most distinguished intellectual contributions in this century. Ulam was fortunate to have been born into a wealthy Jewish family of lawyers, businessmen, and bankers who provided the necessary resources for him to follow his intellectual instincts and his early talent for mathematics. Eventually Ulam graduated with a doctorate in pure mathematics from the Polytechnic Institute at Lwów in 1933. As Ulam notes, the aesthetic appeal of pure mathematics lies not merely in the rigorous logic of the proofs and theorems, but also in the poetic elegance and economy in articulating each step in a mathematical presentation. This very fundamental and aristocratic form of mathematics was the concern of the school of Polish mathematicians in Lwów during Ulam's early years.

The pure mathematicians at the Polytechnic Institute were not solitary academic recluses; they discussed and defended their theorems practically every day in the coffeehouses and tearooms of Lwów. This deeply committed community of mathematicians, in pursuing their work through collective discussion in public, allowed talented young students like Ulam to observe the intellectual excitement and creativity of pure mathematics. Eventually young Ulam could participate on an equal footing with some of the most distinguished mathematicians of his day. The long sessions at the cafes with Stefan Banach, Kazimir Kuratowski, Stanislaw Mazur, Hugo Steinhaus, and others set the tone of Ulam's highly verbal and collaborative style early on. Ulam's early mathematical work from this period was in set theory, topology, group theory, and measure. His experience with the lively school of mathematics in Lwów established Ulam's lifelong, highly creative quest for new mathematical and scientific problems.

As conditions in pre-war Poland deteriorated, Ulam welcomed opportunities to visit Princeton and Harvard, eventually accepting a faculty position at the University of Wisconsin. As United States involvement in World War II deepened, Ulam's students and professional colleagues began to disappear into secret government laboratories. Following a failed attempt to contribute to the Allied war effort by enlisting in the U.S. military, Ulam was invited to Los Alamos by his friend John von Neumann, one of the most influential mathematicians of the twentieth century. It was at Los Alamos that Ulam's scientific interests underwent a metamorphosis and where he made some of his most far-reaching contributions.

On his very first day at Los Alamos he was asked to work with Edward Teller's group on the "Super" bomb project, an early attempt to design a thermonuclear or hydrogen bomb. Except for Teller's small group, the scientists at Los Alamos were working on the design and construction of an atomic bomb based on the energy released by the fission or breakup of uranium or plutonium nuclei. Although there was a general consensus at Los Alamos that the fission bomb would have to precede the Super for which it would serve as an ignition device, Teller was already preoccupied with the Super and refused to work on the fission bomb calculations. As a means of retaining Teller at Los Alamos, Robert Oppenheimer as lab director allowed Teller to work on the Super bomb with several scientists and assistants. Teller's assignment for Ulam on his arrival at Los Alamos was to study the exchange of energy between free electrons and radiation in the extremely hot gas anticipated in thermonuclear bombs. Ironically, this first-day problem for Ulam in 1943 would later become a critical part of Ulam's work with Cornelius Everett in 1950 in which he demonstrated that Teller's design for the Super bomb was impractical.

This first problem in theoretical physics was the beginning of a major scientific transition for Ulam from the esoteric, abstract world of pure mathematics to a quite different kind of applied mathematics necessary to visualise and solve problems in physics. The mathematics relevant to the physical problems at Los Alamos involved differential and integral equations that describe the motion of gas, radiation, and particles. The transition from pure mathematics to physics is seldom attempted and very rarely accomplished at Ulam's level. The creative process and the initial guesswork that lead to significant new ideas in physics involve an added dimension of taste and judgment extending beyond the rigorous logic of mathematics alone. Physical intuition which "very few mathematicians seem to possess to any great degree" is constrained by knowledge of natural phenomena determined from experiment. Ulam claims not to have experienced this "gap between the mode of thinking in pure mathematics and the thinking in physics." Indeed, in these memoirs Ulam discusses his transition from pure mathematics to mathematical physics and hopes that his analysis "of thinking in science is one of the possible interests of this book."

Ulam could hardly have been in better company to learn physics. During the war years the scientists assembled at Los Alamos represented a Who's Who of modem physical science. The large number of eminent physicists - Hans Bethe, Niels Bohr, Enrico Fermi, Richard Feynman, Ernest Lawrence, J Robert Oppenheimer, and so on - formed an intellectual powerhouse of physics that has not been surpassed before or since.

During the war years Ulam contributed to the development of the fission bomb with statistical studies on the branching and multiplication of neutrons responsible for initiating and sustaining the chain reaction and energy release in uranium or plutonium. A critical problem on which Ulam worked with von Neumann was the detailed calculation of the implosion or compression of a sphere of uranium effected by an extern al chemical detonation. When uranium is compressed the small number of naturally occurring neutrons created by random fissions of uranium nuclei collide more easily with other uranium nuclei. Some of these collisions result in further fissions, multiplying further the number of neutrons until a rapid chain reaction ensues, ultimately releasing an extraordinary amount of energy in a powerful explosion. In order to predict the amount of energy released, Los Alamos scientists needed to estimate the detailed behaviour of the uranium as it was being compressed. Although this problem was conceptually straightforward, accurate solutions were not possible using standard mathematical analyses. This problem was quite literally at the secret core of atomic bomb research at Los Alamos - even the word "implosion" was classified during the war.

But Ulam's most remarkable achievement at Los Alamos was his contribution to the post-war development of the thermonuclear or hydrogen bomb in which nuclear energy is released when two hydrogen or deuterium nuclei fuse together. Ulam was a participant at a Los Alamos meeting in April, 1946, at which the wartime efforts on the Super bomb were discussed and evaluated. The conceptual idea of the "Classical Super" was to heat and ignite some part of a quantity of liquid deuterium by using an atomic bomb. The thermal energy deposited in this part would initiate deuterium reactions which would in turn heat adjacent regions. inducing further thermonuclear reactions, until the detonation would propagate through the entire amount of deuterium fuel. Deuterium, a heavier isotope of hydrogen having an extra neutron in its nucleus, was preferred since it reacts at significantly lower temperatures than ordinary hydrogen. Tritium, a third and even heavier form of hydrogen with two neutrons, reacts at even lower temperatures but, unlike deuterium, is virtually non-existent in nature and was extremely expensive to make in nuclear reactors.

The evaluation of Teller's Super project at the 1946 meeting was guardedly optimistic, but the participants were aware of major technical uncertainties and potential difficulties with the Super design. In discussing the conclusions reached at the meeting J Carson Mark has written, "The estimates available of the behaviour of the various steps and links in the sort of device considered were rather qualitative and open to question in detail. The main question of whether there was a specific design of that type which would work well was not answered." Studies prior to 1946 had established that the net balance of energy gains over losses in the Super bomb was marginal; there was no large margin of design flexibility for which a successful detonation could be guaranteed. According to Mark,

As it was, the studies of this question had merely sufficed to show that the problem was very difficult indeed; that the mechanisms by which energy would be created in the system and uselessly lost from it were comparable; and that because of the great complexity and variety of processes which were important, it would require one of the most difficult and extensive mathematical analyses which had ever been contemplated to resolve the question - with no certainty that even such an attempt could succeed in being conclusive.

The uncertainties regarding the ignition and sustenance of fusion reactions in the Super bomb design as developed by Teller's group during the war years were still present in late 1949 and early 1950. Nevertheless, this was the hydrogen bomb design that Teller lobbied for in Washington and that formed the basis of President Truman's decision early in 1950 to accelerate work on the fusion bomb.

The two main questions about the Super design were (1) whether it would be possible to ignite some of the deuterium to get the thermonuclear reactions started, and (2) whether a thermonuclear reaction in the liquid deuterium, once started, would be self-sustaining or, alternatively, would slow down and fizzle away if the rate that energy is lost from the reacting regions exceeded that produced by the reactions. The ignition of the Super would require a gun-type atomic bomb trigger in which two subcritical masses of fissionable uranium would be rapidly united to form a supercritical explosive mass, as in the Hiroshima bomb. The ignition problem was difficult. The unusually high temperatures required for ignition would require a trigger A-bomb that would need to have a yield, reach temperatures, and use a quantity of fissionable material substantially in excess of the bombs in the arsenal in 1950. Even under the most favourable circumstances, the deuterium could not be ignited directly. It was thought that a small amount of tritium could be used to help initiate deuterium-burning in the region initially heated by the fission bomb.

The first a major difficulty for the Super, the problem of ignition, was attacked by Ulam on his own initiative but in collaboration with Cornelius Everett, a mathematical colleague of Ulam's at the University of Wisconsin who had come to Los Alamos after the war at Ulam's invitation. These calculations followed in detail the initial evolution of the nuclear reactions in tritium and deuterium and included an estimate of the heating of the unburnt nuclear fuel by the hot reacting regions with allowances made for the energy lost due to expansion and radiation. The Ulam-Everett calculation was tedious and exacting. While each step of the computation was understood, the complex interplay among the many components involved made the whole calculation extremely difficult. The exchange of energy between electrons and radiation. Ulam's first problem at Los Alamos, was just one part of this monumental calculation. For several months Ulam and Everett worked in concentrated effort from four to six hours a day. Since each step in the calculation depended on the previous work, it was necessary to complete each stage virtually without error; fortunately, freedom from error was one of Everett's specialties. It is hard to imagine today that these calculations were performed with slide rules and old-fashioned manually operated mechanical desk calculators. Ulam and Everett had to make many approximations and educated guesses in order for a solution to be possible at all. By this time Ulam had clearly mastered the physical intuition and judgment needed to make sensible estimates. When the calculation was finished, however, their conclusion was negative. The deuterium could not be ignited without spectacular amounts of tritium, amounts sufficient to make the entire Super project impractical and uneconomical. Within a few months the conclusions of the Ulam-Everett calculation were confirmed by von Neumann using an early electronic computer at Princeton.

The second uncertainty in the design of the Super was the question of the propagation of the deuterium-burning region throughout the entire amount of liquid deuterium. Would the fusion reaction be self-sustaining assuming that the ignition difficulty could somehow be overcome? This fundamental problem was solved by Ulam in collaboration with the brilliant physicist Enrico Fermi. Again using slide rules and desk calculators - and great care in making the appropriate physical approximations - they reached another negative conclusion; the heat lost form the deuterium burning region was too great to sustain the reaction. In discussing the conclusions of the Ulam-Fermi calculations, Fermi noted cautiously that "if the cross-sections for the nuclear reactions could somehow be two or three times larger than what was measured and assumed, the reaction could behave more successfully." In fact the cross-sections (which characterise the rate that the reactions can occur) used by both Teller's group and by Ulam and Fermi in 1950 were larger and therefore more optimistic than the more accurate cross-sections obtained experimentally by James Tuck in the following year. In recent years the Ulam-Everett calculation has been redone in a much more refined manner using modem computers that have confirmed the marginal character of the self-sustaining propagation.

Within months of president Truman's directive to expedite the development of a thermonuclear bomb, the two basic assumptions of Teller's Super model were shown by Ulam and his colleagues to be incorrect. A crash program had begun on a project that was fundamentally flawed and which had never been seriously tested prior to Ulam's work. According to Hans Bethe, Teller "was blamed at Los Alamos for leading the Laboratory, and indeed the whole country, into an adventurous program on the basis of calculations which he himself must have known to have been very incomplete." The energy re leased by the deuterium reaction would be lost before adjacent regions could be ignited since, in Ulam's explanation, "the hydrodynamical disassembly proceeded faster than the build-up and maintenance of the reaction." Teller, who had worked on the Super during the war years and who later became a one-man political action committee urging a crash program for its construction, was distraught and practically undone by the Ulam-Everett-Fermi conclusions. Teller has written that "Ulam's work indicated that we were on the wrong track, that the hydrogen bomb design we thought would work best would not work at all."

The crisis of disappointment following these developments was quite stunningly resolved by Ulam in February 1951 when he suggested a means of compressing the deuterium sufficiently to allow both ignition and self-sustaining propagation. According to Hans Bethe, director of the theoretical division at Los Alamos during the war, Ulam's idea was to use "the propagation of [a] mechanical shock" (compression) wave from a fission explosion to induce a strong compression in the thermonuclear fuel, which would subsequently explode with great violence. The advantages of compression in helping to make thermonuclear reactions more efficient had been discussed even as early as the April 1946 meeting, but were never taken seriously since the compression required was far greater than could be achieved with chemical explosions. When Ulam told Teller of his idea of using a fission bomb to compress the deuterium just prior to its ignition, Teller immediately perceived the value of the idea. However, Teller suggested that the implosion could be achieved more conveniently by the action of radiation, with a so-called "radiation implosion," rather than with the mechanical shock proposed by Ulam (which would also have worked). The new idea for the hydrogen bomb, known euphemistically as the "Teller-Ulam device," was rapidly accepted by Los Alamos scientists and government officials. Since first proposed by Ulam, the coupling of a primary fission explosion with a secondary fusion explosion by means of implosion has been a standard feature of thermonuclear bombs.

All of these details concerning the origins of the hydrogen bomb, to the extent that we can put them together from declassified information, underscore Ulam as far more influential than has previously been known. Not only was he the first to dismantle the earlier Super concept which had been so inflexibly proposed for many years, he provided the key idea that resolved the difficulties of both ignition and propagation. In this instance, more than any other in Ulam's scientific career, he demonstrated "how a few scribbles on a blackboard or on a sheet of paper" have quite radically and irreversibly changed "the course of human affairs."

In view of the impact that the arsenal of nuclear weapons has had on world affairs, it is intriguing that Ulam returns in his autobiography several times to discuss the mindset and social role of weapons scientists who sequester themselves in top secret laboratories to invent and construct instruments of potential mass destruction. Most of the scientists who worked at Los Alamos during World War II were shocked by the annihilation of Japanese cities and elected to return to academic life after the war. It is likely that many of those who stayed on at Los Alamos or returned later were inherently apolitical and, like Ulam, were "mainly interested in the scientific aspects of the work," having "no qualms about returning to the laboratory to contribute to further studies of the development of atomic bombs." Although Ulam later felt that the stockpile of nuclear weapons had grown larger than necessary in his view there was nothing intrinsically "bad" about the mathematics or the laws of nature used in creating new weapons. Knowledge itself is without moral content. In particular, Ulam "never had any questions about doing purely theoretical work" on nuclear weapons, leaving to others their construction and application to political and military ends.

Ulam makes a curious distinction between the acquisition of knowledge concerning new instruments of mass destruction by scientists and its wider dissemination: "I sincerely felt it was safer to keep these matters in the hands of scientists and people who are accustomed to objective judgments rather than in those of demagogues or jingoists, or even well-meaning but technically uninformed politicians." However, in a government-funded laboratory such as Los Alamos, the symbiosis that exists between weapons technology and political decisions is inescapable. While Ulam insists that "one should not initiate projects leading to possibly horrible ends," it would nevertheless be "unwise for the scientists to tum away from problems of technology" since "this could leave it in the hands of dangerous and fanatical reactionaries." In spite of these apparent contradictions, Ulam's justifications of his role in weapons development provide us with one of the few insights into the personal attitudes of a Los Alamos scientist toward the end products of his work.

By virtue of his defence work at the Los Alamos Laboratory, Ulam enjoyed many advantages not available to academic scientists. Chief among these was his early access to the most powerful and fastest computers in existence. For several decades after the war, the computing facilities at the national weapons labs far exceeded those available to university scientists working on non-classified research. This was an advantage that Ulam exploited in a variety of remarkable ways.

The growth of powerful computers was initially driven by the war effort. At the beginning of World War II there were no electronic computers in the modem sense, only a few electromechanical relay machines. During the war, scientists at the University of Pennsylvania and at the Aberdeen Proving Ground in Maryland developed the ENIAC. the Electronic Numerical Integrator and Computer, which had circuitry specifically designed for computing artillery firing tables for the Army. By modem standards, this early computer was extremely slow and elephantine: the ENlAC operating at the University of Pennsylvania in 1945 weighed thirty tons and contained about eighteen thousand vacuum tubes with 500,000 soldered connections. While on a visit to the University of Pennsylvania in 1944, John von Neumann was inspired to design an electronic computer that could be programmed in the modem sense, one which could be instructed to perform any calculation and would not be restricted to computing artillery tables. The new computer would have circuits that could perform sequences of fundamental arithmetic operations such as addition and multiplication. Von Neumann desired a more flexible computer to solve the mathematically difficult A-bomb implosion problem being discussed at Los Alamos. The first electronic computer at Los Alamos, however, known as the MANIAC (Mathematical Analyzer, Numerical Integrator and Computer), was not available until 1952.

One of Ulam's early insights was to use the fast computers at Los Alamos to solve a wide variety of problems in a statistical manner using random numbers, a method which has become appropriately known as the Monte Carlo method. It occurred to Ulam during a game of solitaire that the probability of various outcomes of the card game could be determined by programming a computer to simulate a large number of games. Newly selected cards could be chosen from the remaining deck at random, but weighted by the probability that such a card would be the next selected. The computer would use random numbers whenever an unbiased choice was necessary. When the computer had played thousands of games, the probability of winning could be accurately determined. In principle the probability of solitaire success could be rigorously calculated using probability theory rather than computers. However, this approach is impossible in practice since it would involve too many mathematical steps and exceedingly large numbers. The advantage of the Monte Carlo method is that the computer can be efficiently programmed to execute each step in a particular game according to known probabilities and the final outcome can be determined to any desired precision depending on the number of sample games computed. The game of solitaire is an example of how the Monte Carlo method can be used to solve otherwise intractable problems with brute computational power.

An early application of the Monte Carlo method using high speed computers was to study the propagation of neutrons in fission bombs. This was accomplished by randomly picking the position of a radioactive nucleus that would release a neutron, then randomly selecting the neutron's energy, its direction of motion, and the distance the neutron would travel before either escaping or colliding with the nucleus of another atom. In the latter event, the neutron would either be scattered, absorbed, or could induce nuclear fission according to probabilities again selected with random numbers. In this manner, after many neutron life experiences had been calculated, it was possible to determine the number of neutrons at any energy moving in a particular direction at any position in the apparatus. The Monte Carlo method is also well-suited to computing the equilibrium properties of materials, in estimating the efficiency of radiation or particle detectors having complicated geometries, and in simulating experimental data for a wide variety of physical problems.

Another early use of computer technology in which Ulam made contributions is the problem of determining the motion of compressible material. Indeed, it was the calculation of imploding compression waves in the fissionable core of atomic bombs that initially attracted Los Alamos scientists to the advantages of fast computers. One of Ulam's contributions was his idea to represent the compressible material with an ensemble of representative points whose motion could be determined by the computer. Along similar lines, Ulam performed the first studies of the subtly complex collective motion of stars in a cluster, each mutually attracted to all the others by gravitational forces. The applications of computers to both compressible material and stellar systems along lines first explored by Ulam are major areas of research interest today.

Of particular interest is Ulam's farsighted computer experiment in the mid-fifties with John Pasta and Enrico Fermi on the oscillations of a chain of small masses connected with slightly nonlinear springs. A nonlinear spring is one that does not quite stretch in exact proportion to the amount of force applied. When the group of masses simulated by the computer was started out in a particular rather simple motion, Ulam and his colleagues discovered to their amazement that the masses eventually returned nearly to the original motion but only after having gone through a bizarre and totally unanticipated intermediate evolution. Today computer studies of such non-linear systems have become a major area of interdisciplinary scientific investigations. Many strange properties of dynamical systems have been discovered which have led to a deeper understanding of the long-term properties of nonlinear systems obeying deceptively simple physical laws.

A related computer experiment inspired by Ulam was the study of iterative nonlinear mappings. The computer is provided with a (nonlinear) rule for transforming one point in a mathematically defined region of space into another, then the same rule is applied to the new point and the process is continued for many iterations. When examined after only a few iterations. the pattern is generally uninteresting, but when a computer is used to generate thousands of iterations, Ulam and his colleague Paul Stein observed that a variety of strange patterns can result. In some cases after many iterations the points converge to a single point or are ordered along a curve within the given region of space. In other cases the successive images of iterated points appear to have disordered. chaotic properties. The final pattern of iterated images can be sensitive to the initial point chosen in generating them as well as the rules for (nonlinear) iteration. In recent years this early work of Ulam and Stein has been greatly extended at Los Alamos, now a major centre of nonlinear studies.

Ulam also had an interest in the application of mathematics to biology. One example that may have biological relevance is the subfield of cellular automata founded by von Neumann and Ulam. As an example of this class of problems imagine dividing a plane into many small squares like a checkerboard with several objects placed in nearby squares. Then specify rules for the appearance of new objects (or the disappearance of old objects) in each square depending on whether adjacent squares are occupied or not. With each application of the rules to all the squares, the pattern of occupied squares evolves with time. Depending on the initial configuration and the rules of growth, some computer generated cellular automata evolve into patterns resembling crystals or snowflakes, others seem to have an ever-changing motion as if they were alive. In some cases colonies of self-replicating patterns expand to fill the available space like the growth of coral or bacteria in a petri dish.

Stanislaw Ulam was a man of many ideas and a fertile imagination. His creative and visionary talent planted intellectual seeds from Lwów to Los Alamos which have nourished into new disciplines of study throughout the world. Ulam's scientific work was characterised by a singularly verbal style of inquiry begun during his early experiences in the coffeehouses of Lwów. The use of written material was also less essential for Ulam due to his formidable memory - he was able to recite many decades later the names of his classmates and to quote Greek and Latin poetry learned as a schoolboy. Ulam's verbal and socially interactive approach was in fact well suited to the research environment at Los Alamos. Talented colleagues there were available to collaborate with Ulam, to provide the missing details of the ideas he sketched out, and to prepare the scientific papers and reports which changed the course of human affairs.

William G Mathews
Daniel O Hirsch

11.1. From the Publisher.

What is mathematics? How was it created and who were and are the people creating and practicing it? Can one describe its development and role in the history of scientific thinking and can one predict the future? This book is a thought-provoking attempt to answer such questions and to suggest the scope and depth of the subject.

The volume begins with a discussion of problems involving integers in which ideas of infinity appear and proceeds through the evolution of more abstract ideas about numbers and geometrical objects. The authors show how mathematicians came to consider groups of general transformations and then, looking upon the sets of such subjects as spaces, how they attempted to build theories of structures in general. Also considered here are the relations between mathematics and the empirical disciplines, the profound effect of high-speed computers on the scope of mathematical experimentation, and the question of how much mathematical progress depends on "invention" and how much on "discovery." For mathematicians, physicists, or any student of the evolution of mathematical thought, this highly regarded study offers a stimulating investigation of the essential nature of mathematics.

11.2 Review by: Editors.
Mathematical Reviews MR1217082 (94b:00003).

This is an unabridged, unaltered republication of the authors' 1968 booklet written in the "Perspective" series to help commemorate the 200th anniversary of the Encyclopaedia Britannica.

12.1. From the Publisher.

Ulam, famous for his solution to the difficulties of initiating fusion in the hydrogen bomb, devised the well-known Monte-Carlo method. Here he presents challenges in the areas of set theory, algebra, metric and topological spaces, and topological groups. Issues in analysis, physical systems, and the use of computers as a heuristic aid are also addressed.

12.2. From the Preface.

In the small space allotted for a preface to this book in its paperback edition, I have had to restrict myself to just the briefest comments.

In my opinion by far the most important mathematical event of the last four years is the discovery by Paul Cohen. The continuum hypothesis is independent of the axioms of set theory as usually presented. This means that one could, if one wanted to, assume that there are intermediate powers between $\aleph _{1}$ and $c$. Cohen's result opens a way to a class of large and "free," so to say, non-Cantorian, set theories.

It may be that other unsolved problems of set theory have a similar status. Here perhaps is a new question in pure set theory, formulated in terms of infinite games. Given is an abstract noncountable set $E$. Two mathematicians - A and B - play as follows: A can divide $E$ in any way he pleases into two disjoint sets. B has to choose one of these sets and divide it into two disjoint sets in any manner he likes. Then it is A's turn. He also has to choose a set, divide it again, present to B for choice, and so on. Let us say that A wins if the intersection of all chosen sets is vacuous. Is there a strategy for one or the other player to win? Possibly this problem and others of its sort may also be undecidable in the system of axioms, usually employed for set theory.

13.1. From the Publisher.

During his forty-year association with the Los Alamos National Laboratory, mathematician Stanislaw Ulam wrote many Laboratory Reports, usually in collaboration with colleagues. Some of them remain classified to this day. The rest are gathered in this volume and for the first time are easily accessible to mathematicians, physical scientists, and historians. The timeliness of these papers is remarkable. They contain seminal ideas in such fields as nonlinear stochastic processes, parallel computation, cellular automata, and mathematical biology. The collection is of historical interest as well, During and after World War II, the complexity of problems at the frontiers of science surpassed any technology that had ever existed. Electronic computing machines had to be developed and new computing methods had to be invented based on the most abstract ideas from the foundations of mathematics and theoretical physics. To these problems and others in physics, astronomy, and biology, Ulam was able to bring both general insights and specific conceptual contributions. His fertile ideas were far ahead of their time, and ranged over many branches of science. In fact, his mathematical versatility fulfilled the statement of his friend and mentor, the great Polish mathematician Stefan Banach, who claimed that the very best mathematicians see "analogies between analogies." Introduced by A R Bednarek and Françoise Ulam, these Los Alamos reports represent a unique view of one of the twentieth century's intellectual masters and scientific pioneers. This title is part of UC Press's Voices Revived program, which commemorates University of California Press's mission to seek out and cultivate the brightest minds and give them voice, reach, and impact. Drawing on a backlist dating to 1893, Voices Revived makes high-quality, peer-reviewed scholarship accessible once again using print-on-demand technology. This title was originally published in 1990.