1.1. Review by: Jacob David Tamarkin.
The American Mathematical Monthly 42 (4) (1935), 237-239.

In the whirlwind of tremendous development of the modern physics during the last thirty years, in their eagerness to obtain quantitative results with which to work, the physicists became used to disregarding the requirements of logical consistency, rigour and even clarity in their mathematical treatment of physical problems. The situation thus created is in many respects analogous to that in analysis at the beginning of the nineteenth century. It required a genius of the calibre of Abel or Cauchy to call the attention of analysts to the necessity of a critical attitude toward the methods of getting results. The present book is a welcome and successful step in the same direction with regard to quantum mechanics. It is devoted to a careful and critical study of logical and mathematical difficulties connected with the most fundamental notion of physics, namely that of measurement of physical quantities. It undertakes to give a sound axiomatic foundation to the colourful and diversified collections of rules of quantum mechanics, whose only justification in a great many cases was so far based simply on good luck. In undertaking this ambitious programme the author had at his disposal a powerful tool embodied in the theory of operations in Hilbert spaces, to which he himself had made so many fundamental contributions.

After a general introduction of Chapter I, the author gives in Chapter II a rapid but very clear exposition of the indispensable notions and facts of the theory of operations in Hilbert spaces. Chapter III is devoted to translating the notions and facts of the quantum-theoretical statistics into the language of operators in Hilbert spaces, thus preparing the ground for the subsequent axiomatic treatment. ...
...
Chapter IV establishes a small system of axioms from which the main body of the quantum theoretical statistics is derived. ...
...
Chapter V contains applications to thermodynamics and to questions of reversibility and equilibrium. Finally the last chapter VI is devoted to an axiomatic discussion of the general problem of measurements of physical quantities. This brief enumeration gives only an inadequate idea of the richness of material covered in the book, the appearance of which signifies an important step in the development of the most vital part of the structure of modern physics.

2.1. From the Preface.

This book contains an exposition and various applications of a mathematical theory of games. The theory has been developed by one of us since 1928 and is now published for the first time in its entirety. The applications are of two kinds: On the one hand to games in the proper sense, on the other hand to economic and sociological problems which, as we hope to show, are best approached from this direction. The applications which we shall make to games serve at least as much to corroborate the theory as to investigate these games. The nature of this reciprocal relationship will become clear as the investigation proceeds. Our major interest is, of course, in the economic and sociological direction. Here we can approach only the simplest questions. However, these questions are of a fundamental character. Furthermore, our aim is primarily to show that there is a rigorous approach to these subjects, involving, as they do, questions of parallel or opposite interest, perfect or imperfect information, free rational decision or chance influences.

2.2. Review by: Emil Julius Gumbel.
The Annals of the American Academy of Political and Social Science 239 (1945), 209-210.

The classical theory of games is based on the calculus of probabilities, and the mathematical approach to economics is the new science of econometrics. Both methods have in common the notion of a chance variable which is the fundament in the calculus of probabilities, and a supplement in econometrics. The mathematical tools in both sciences are essentially the same, differential and integral calculus. However, no theory of games used to be considered a part of economics. The authors, a brilliant mathematician and a well-known economist of the Austrian school, try a completely new start which interprets economics in terms of games of strategy. These games, which include games of pure chance, such as dice throwing, and rational games, such as chess, serve as models for economics as the usual physical models serve in physics. However, the tool is no longer calculus but mainly the theory of sets and groups. The authors justify their new method by an interesting historical analysis: the development of physics since Newton has been made possible only through the discovery of new mathematical tools. In the same way the completely new problem of n participants of a game, each trying to maximise his gain without being able to control the activity of the other participants, cannot be solved by the present mathematical methods. Therefore we need not wonder that the authors confine their study to the most elementary economic processes.
...

The whole approach is purely static, and there is a long way to travel to the point where this line of thought may furnish tangible results for the rational solution of economic problems. The reviewer doubts whether the method based essentially on a capitalist form of production covers all rational economics. The authors must be lauded for their courage and endurance in undertaking their highly theoretical endeavours. However, it seems questionable whether they will obtain adequate appreciation from the public. Page after page full of formulae discourage the reader. A non-mathematical summary of the main results would have meant an essential improvement without any sacrifice of rigidity.

2.3. Review by: Robert W Harrison.
Journal of Farm Economics 27 (3) (1945), 725-726.

Here is a work of utmost importance for all who earnestly seek to remove from the social sciences the atmosphere of vague description and generalisation which has grown up about them. Not only have the authors succeeded in demonstrating that the mathematical theory applicable to games of strategy may be a useful (and in certain cases ideal) tool m the study of human behaviour in economic and social situations, but, of even greater importance, they have made it unmistakably clear that progress toward a rigorous social science, as in the physical sciences, is apt to come only through a careful accumulation of exact knowledge first about familiar events and situations and later about the more complex and less obvious. They have reversed the method of attack used so frequently by economists and sociologists and instead of trying to build a mathematical system covering a complex segment of human affairs, or, as has sometimes been tried, one covering all man's activities, have been content to explore exactly a few simple and familiar problems. As a consequence of this modesty they have been able to develop a truly mathematical analysis of some simple problems and have pointed the way to exact knowledge of greater ones. Those who study their accomplishments will find them vastly more impressive than the former attempts at using mathematics in economics and sociology.

The development of the theory which this book presents has been in progress for some years. In 1928 von Neumann published the first phase. The present work brings into form the full theory and demonstrates its application. Seeing that the situations existing during games of strategy have many elements common to everyday economic and social situations, von Neumann applies the methods of point-set theory and topology. These methods enable him in Chapter II to set up a formal description of a game. Those familiar only with the usual attempts to use mathematics in economics and sociology which rest largely on the calculus of differential equations will immediately realise the uniqueness and great value of this new approach. The description of the game in exact terms, as outlined in Chapter II, demonstrates a method for analysing social behaviour in situations where each participant is dependent upon the behaviour of the other participants. This accomplished, it is possible to analyse such social factors as competition and cooperation, within, of course, the limits set by the method.

As the study progresses the rational strategy of the participants in various situations is studied and the pattern of behaviour analysed. Perhaps the most interesting conclusion is that in games with more than two persons, coalitions will generally appear. For economists dealing with various aspects of monopoly this is a conclusion of prime significance. The sociologist will also work this as the key to many of his problems. As the authors suggest, perhaps the first practical use of their work will be in the study of situations involving a few participants; for example, economic situations involving strong coalitions of trade unions, consumer cooperatives, cartels, etc. It is futile, however, to attempt prediction concerning the manner in which the theory of games will find its most fruitful use. This may come in the most unexpected manner. The explanation of economic affairs is still on a relatively primitive level compared with the older sciences, i.e., astronomy and physics. The authors foresee the difficulty of developing economics to a comparable level: "The importance of the social phenomena, the wealth and multiplicity of their manifestations, and the complexity of their structure, are at least equal to those in physics. It is therefore to be expected - or feared - that mathematical discoveries of a stature comparable to that of calculus will be needed in order to produce decisive success in this field. ... A fortiori it is unlikely that a mere repetition of the tricks which served us so well in physics will do for the social phenomena too."

Something should be said about the difficulty of understanding the theory of games. It is quite difficult to say just what level of mathematical knowledge is required to appreciate this work. Perhaps only an elementary level of training is required; but alone this will hardly suffice. A sympathy for mathematical reasoning will also be required. Whatever the case, an understanding of this work will repay the study. The first chapter, by the way, is an excellent essay on things economic and mathematical as they bear on each other and the progress of science, and can be enjoyed apart from the mathematical phases of the study.

2.4. Review by: E N.
The Journal of Philosophy 42 (20) (1945), 550-554.

This is a difficult technical book, the product of collaboration between au outstanding mathematician and a distinguished economist, and is addressed primarily to mathematicians and mathematical economists. It nevertheless merits some at tent ion from philosophers. For it initiates a radically new direction in constructing a mathematical analysis of human behaviour, and it also has important implications for the theory of inductive inference. The authors are convinced that the traditional approach in mathematical economies has in the main been unsuccessful because physics has been taken as the model from which mathematical techniques are to be borrowed. They believe, on the contrary, that quite different formal tools - in fact, those required for developing a general theory of games - show greater promise of being adequate to handling the theoretical problems of the economic market. Accordingly, central logical issues in the construction of a systematic theory of social behaviour arise in a new and arresting form; and philosophers who are prepared to struggle through a complicated mathematical discussion will find much in the book to reward their effort. No specific knowledge of advanced mathematics is presupposed; indeed, practically no use is made of the ideas of the differential calculus, and unfamiliar though advanced techniques (such as set theory, linear algebra, and the theory of groups) are fully explained as the need for them arises. Moreover, a careful study of the first four chapters (approximately 200 pages) will supply the reader with enough of the fundamental ideas to permit him to form a decent conception of the manner in which the theory of games is constructed.

According to the authors, the aim of economic science is to develop a theory of economic behaviour on the hypothesis that each participant in economic activity tries to obtain the maximum "utility" from his economic transactions. They adopt the simplifying assumption that utility can be expressed in monetary units; and they take their first task to be that of specifying a mathematically precise and psychologically reasonable definition of "rational behaviour" in the economic market. This definition requires, as might perhaps be expected, that each economic agent maximise his gains or minimise his losses. But the problem of finding a suitable plan of behaviour which will achieve this does not reduce to the ordinary type of maximum problem with which classical economics deals. For economic activity is regarded as a series of "moves" (carried on within a framework of fixed "rules of the game") in which the participants do not in general know the decisions which other participants may have taken and upon which the issue of a particular transaction may depend. The solution of the problem therefore consists in specifying a set of canons for each participant which will tell him how to behave so as to achieve a maximum of utility in every situation which may conceivably arise and independently of how the other participants decide to act.

Economic activity is thus conceived as a game, carried on be- tween a number of players; and most of the book is therefore devoted to constructing a general theory of games along the lines indicated in earlier publications by von Neumann.

2.5. Review by: Herbert A Simon.
American Journal of Sociology 50 (6) (1945), 558-560.

The Theory of Games is a rigorous mathematical development of a theory of games of strategy and an application of that theory to certain simple problems in economics. Although no explicit applications are made to sociology or political science, the schema is of such generality and breadth that it can undoubtedly make contributions of the most fundamental nature to those fields.

In the Foreword the authors, quite correctly, state: "The mathematical devices used are elementary in the sense that no advanced algebra, or calculus occurs ... . However, the reader .... will have to familiarise himself with the mathematical way of reasoning definitely beyond its routine, primitive phases." What is required of the reader, then, is not training in mathematics so much as "mathematical maturity." The reviewer found the Theory of Games at all times a model of clear and careful exposition.

Social scientists have for decades carried on a largely sterile debate as to the applicability of mathematical modes of thought to their discipline. Like all arguments about methodology, this one must in the long run be settled by results. To date, with the notable exception of mathematical economics, mathematical reasoning (as distinguished from the use of quantitative data, statistics) has not much to show in the way of results in the social sciences. To be sure, the single exception is a notable one - progress in economic theory is becoming more and more dependent upon the application of the calculus, and most of the gains of the last fifty years in the development of marginal analysis must be attributed to economists trained in mathematical modes of thought, even if they sometimes translated their thought into the more acceptable literary form for presentation.

In sociology and political science, attempts at a mathematical theory can be numbered on the fingers of one hand, and the results have heretofore certainly been negligible. ...
...

The Theory of Games is both more modest and infinitely more impressive in its results than any of these earlier attempts. It seeks merely to develop in systematic and rigorous manner a theory of rational human behaviour. Now the simplest setting in which human rationality is exercised is in the playing of games; hence the authors select as their starting-point a theory of games which von Neumann originated as early as 1928.

While most attempts at mathematisation in the social sciences have employed the tools of the calculus and differential equations, von Neumann moves in an entirely different direction and employs, instead, the mathematics of point-set theory and of topology. As a matter of fact, von Neumann insists - and his eminence among mathematicians lends great weight to his opinion on this point - that the lack of success of mathematics in the social sciences (which he certainly exaggerates, at least in relation to economics) has been due in large part to the use of tools which were developed in intimate connection with the growth of mathematical physics and which are not applicable to social theory. "It is therefore to be expected," he says, "that mathematical discoveries of a stature comparable to that of calculus will be needed in order to produce decisive success in this field .... it is unlikely that a mere repetition of the tricks which served us so well in physics will do for the social phenomena too."

2.6. Review by: Cedric Austen Bardell Smith.
The Mathematical Gazette 29 (285) (1945), 131-133.

This book is full of new and important ideas.

The mathematical theory of games in the past consisted chiefly of the application of mathematics to special games or kinds of games. This book, on the other hand, analyses the idea of a game in its most general form, and also the idea of a solution, and actually determines the solution in many important simple cases. Probably no general definition of a "game" has been given before, taking into account the possibilities of mixtures of chance moves and moves by players, varying amounts of information about the previous moves, and so on. So that is the first task the authors achieve. A specially important classification is into "zero-sum" games, in which money (or other measure of "utility ") simply changes hands at the end of the game, and "non-zero-sum" games, in which money may be created or destroyed.
...

The mathematics throughout the book is entirely modem in spirit, using matrices, theory of sets, axiomatic methods, etc., but the treatment is everywhere elegant and elementary, and does not assume a prior knowledge of these methods. The explanations are careful and detailed, and when attacking a difficult problem the authors have an admirable method of first giving a lucid account of the strategy they will adopt.

2.7. Review by: David Hawkins.
Philosophy of Science 12 (3) (1945), 221-227.

The literature of economic theory, like that of philosophy, abounds in prefaces and prolegomena. Methodology and analysis of concepts take an important place in a science which has not found the sure path of development. But there is no sure path for methodology either. The self-conscious methodology of social science has been largely a borrowing from that of physical science, where procedures have developed to a stage of considerable maturity. But the analogy falls down where guidance is most needed, at the points where social science is most likely to develop new concepts and new types of structure. Philosophers have not been lacking, indeed, to belittle the entire enterprise, and to deny the possibility of anything that could strictly be called social science.

The only remaining source for methodological principles in social science is social science itself. Methods are only relatively separable from their practice. What is so persuasive about the methodological principles of physics is not just their plausibility, but the fact that they are the principles which have been followed in, and in a sense produced by, the career of a successful science. No doubt there are important relations between the procedures of physical and social science; but it will be possible to see them clearly and exploit them properly only when the more recent science is able, in some degree, to stand on its own feet.

When therefore a work of social science appears which achieves positive and satisfactory results by a distinctly novel approach and procedure, it is bound to have considerable methodological value, quite apart from its scientific content. The impression that the Theory of Games has this kind of value is strengthened by the fact that it is centred around the analysis of an essentially conceptual difficulty; one which, moreover, is bound to appear if we reflect carefully upon the problems of social science. The work is, to be sure, only a theory of games but of games in their "strategic" aspect as rudimentary social phenomena. To lay the foundations of a mature deductive theory, even of such rudimentary phenomena, is in this reviewer's opinion, an important achievement.

2.8. Review by: W Edwards Deming.
Journal of the American Statistical Association 40 (230) (1945), 263-265.

Button, button; who's got the button? - a game, unless someone peeks. Poker - also a game. Tit-tat-toe, or naughts and crosses - not a game. Chess - trivial, and entertainment, yes, but not a game! Why not? Because, starting from a given configuration of the chessmen, it is obvious to anyone who thoroughly understands the game that A can win, or B can win, or A or B can force a draw. The entertainment in playing the game arises from the fact that the players do not fully understand what they are doing. Similarly in naughts and crosses, or in nimo, In them, as in chess, no probability enters. But in button, button, A may flip a coin or choose some system of probability to determine which hand he will put the button in. B may likewise adopt some system of probability for guessing which hand the button is in. The fact that it is B, not A; who is to guess which hand the button is in, affords no advantage to either player when each makes the most of his opportunities. Fair games are characterised by the fact that each player, when he plays his best, has a 50:50 expectation, but other expectations are possible and are treated.

In three-handed games it is usually advantageous for a pair of players to form a coalition, such as BC vs. A. It is then a two-handed game. C must not accept overpayment for his cooperation with B; if he does, A may underbid him. The problem of three or more players presents difficulties equal to the difficulties seen in three-body and multi-body problems in mechanics. The authors have formulated general necessary and sufficient conditions for solutions.

It has been said that the best mathematicians are also the most practical people in the world when they set their minds to application. When a mathematician like von Neumann turns his attention to practical problems, something is going to happen. Mathematicians now, as during the past few years, are leaving their imprints on techniques in psychology, medicine, marketing research, techniques of surveying in social and economic fields, and economics. The results may be expected to remake economic theory. However, the reader should be warned that the book under review is really different. The usual student of economics, without appreciation for mathematical processes and logic, will have difficulty until translations and interpretations have been made.

As the authors say there is no fundamental reason why mathematics should not be used in economics. The objections that have been made were also made centuries ago in fields where mathematics is now the chief instrument of analysis (physics, chemistry, biology). Precise measurements of energy and temperature in the natural sciences were the outcome and not the antecedents of mathematical theory in those fields. The lack of real success of mathematics and economics is largely due to a combination of unfavourable circumstances, some of which can gradually be removed. For one thing it is often not certain just what the problems are, and there is no point in using exact methods under such circumstances. But this state of affairs is disappearing. Concepts are improving, and data are fast accumulating for subjecting theories to test, thanks to the statistician who largely through training in the mathematical theories of sampling is making it possible to obtain data cheaper and quicker than ever before, not just as a mathematical exercise but to meet the ever-increasing demands of government and industry for better information in guiding policy and in the design of machines and marketing of products.

There has existed heretofore no satisfactory treatment of the question of rational behaviour. In fact, as the authors say it is sometimes claimed in the economic literature that discussions on the notions of utility and preference are altogether unnecessary. The authors in this book are attempting to obtain a real understanding of the problem of exchange by studying it from an altogether different angle - that is, from the perspective of a game of strategy. The authors make a definite contribution to economic theory by putting the theory of exchange on a systematic basis for two players - that is, producer and consumer. There exists in the literature a considerable amount of theoretical discussion purporting to show that the zones of in- determinateness (of rates of exchange) - which undoubtedly exist when the number of participants is small - narrow and disappear as the number increases. This would then provide a continuous transition into the ideal case of free competition-for a very great number of participants - where all solutions would, be sharply and uniquely determined. While it is to be hoped that this indeed turns out to be the case in sufficient generality, one cannot concede that anything like this contention has been established conclusively thus far. There is no getting away from it: the problem must be formulated, solved, and understood for small numbers of participants before anything can be proved about the changes of its character in any limiting case of large numbers, such as free competition.

2.9. Review by: Abraham Wald.
Mathematical Reviews MR0011937 (6,235k).

The book presents a new approach to the problem of rational behaviour. The view taken is that the problem of rational behaviour cannot be regarded as a simple maximum problem, but is of a type that arises in the theory of games of strategy. The main reasons for this attitude may be briefly summarised as follows. An essential characteristic of a social-economic organisation is that the participants enter into relations with each other and the result for each participant depends not only on his own actions but also on the actions of others. For example, in a social exchange economy where several persons exchange goods, the increase of utility a participant can achieve will depend on the behaviour of all the other participants. In general, the situation that arises may be characterised as follows. Each participant wishes to maximise a certain function of several variables. He controls, however, only a partial set of these variables, while the values of the remaining variables depend on the actions of the other participants, who may have different or even opposite interests. The problem of a rational choice of the variables for each participant cannot be regarded as a simple maximum problem. Instead, we have a peculiar mixture of several conflicting maximum problems. Problems of the same type arise in the theory of games of strategy.
...
The book will be of interest to sociologists and economists who are interested in a rigorous foundation of the theory of rational behaviour, in particular, the theory of exchange in a general market. [It may be of interest to mention that the theory of games has applications to statistics as well, since the general problem of statistical inference may be treated as a zero-sum two-person game.] The authors make an endeavour to present the material in a form which will make it understandable to readers without a special knowledge of any part of advanced mathematics. In spite of this, the book is not really elementary, because of the intricate nature of many of the mathematical deductions. Non-mathematical readers will find the excellent verbal expositions given parallel with every major mathematical deduction very helpful.

2.10. Review by: Tibor Barna.
Economica, New Series 13 (50) (1946), 136-138.

Professors Neumann and Morgenstern have written a book designed to become a fundamental textbook of economic theory. The essence of the book is not a refinement or summary of mathematical economics but an outright condemnation of the particular mathematical methods used in economics, and the substitution for them of an entirely different mathematical approach to the central problems of economic theory.

According to the authors, the unsuccessful use of mathematics in economics (in comparison with other sciences) was due not to inherent causes but to the fact that an incorrect mathematical technique has been used. For the solution of economic problems it is necessary to remove two preliminary obstacles: the inadequate clarity in the formulation of economic problems, and the insufficiency of the empirical background. While the removal of these obstacles is necessary, this book is on a more abstract level, being concerned with the mathematical treatment of common human behaviour of economic importance. The main possibility of progress is seen in the quantitative treatment of factors which were hitherto labelled "psychological" and considered to be outside the scope of economics. The particular type of mathematical technique (that of infinitesimal calculus) which has been applied to economics, is well suited to the "Robinson Crusoe" type of problem; here the problem is clearly that of maximisation. But when we deal with an exchange economy, with two or more participants, the nature of the problem changes because now each individual is attempting to maximise something which is also dependent on the action of others.

An exchange economy contains various interests, sometimes parallel, sometimes conflicting, and economic equilibrium is the result of the interplay of those interests. The description of such a system requires the application of a mathematical technique different from that successful in physics and other natural sciences; in the abstract sense an exchange economy resembles games of strategy. Hence the first step in working out an economic theory is the creation of a complete theory of games. The latter requires a new mathematical technique, scarcely applied in other sciences yet, on which Professor Neumann was working for over fifteen years. It is now for the first time that the theory of games is published in its completeness, and this forms the bulk of the book.

The mathematical technique is that of combinatorics and set theory. On the face of it this technique looks more difficult than the usual method of infinitesimal calculus, but probably this is only due to its unfamiliarity, not only to economists but to most scientists. On the other hand the mastering of the new technique does not require previous knowledge of higher mathematics; in fact the book explains all the mathematical concepts introduced from the beginning. There is therefore reason to hope that it may be taken up by economists; otherwise progress by this method of analysis remains unlikely.

The theory of games starts with two-person games, and slowly develops into tackling games with a great number of participants. In most games, of course, gains balance losses, while in economic society there is a positive difference corresponding to production. This difficulty is ingeniously overcome by introducing a "dummy" player, and by demonstrating that a game with a given number of participants where gains do not balance losses raises the same problems as a game with one more participant (the "dummy") where gains balance losses. As the theory of games proceeds from two-person to n-person games, the theory of economics advances from bilateral monopoly to the case of perfect competition, as a special case.

It is the main advantage of this approach that it gives to mathematics a more fundamental place in economics, from which it is possible to arrive at new truths, instead of merely translating literal economics into symbols. ...

2.11. Review by: Maurice George Kendall.
Journal of the Royal Statistical Society 107 (3-4) (1944), 293.

This unusual book is based on the thesis that the economic man attempts to maximise his share of the world's goods and services in the same way that a participant in a game involving many players attempts to maximise his winnings. The authors point out that the maximisation of individual wealth is not an ordinary problem in variational calculus, because the individual does not control, and may even be ignorant of, some of the variables. The general theory of social games, in their view, offers a simplified conceptual model of economic behaviour, and a study of that theory can do much to throw light on certain basic concepts of economics, notably that of utility.

The opening chapter of the book introduces this basic idea and seeks to establish the dualism between standards of economic behaviour and strategies in social games. Once this is done the book turns to the mathematics of games themselves for about 5oo pages, returning to economic interpretation at the end. Thus the bulk of the work is a complete account of the theory of Gesellschaftsspiele developed by von Neumann himself in the last seventeen years, and contains a good deal of material now published for the first time. The economic interpretation of the results is much shorter.

Although the mathematics are self-contained, in the sense that the theory is developed from primitive ideas, an extensive use is made of set-theory, functionals and, of course, mathematical notation. From the point of view of familiarity with the notation alone it is doubtful whether anyone but a fairly competent mathematician would follow the argument and appreciate what the authors are driving at. This was unavoidable, but seems a pity, as the book will remain closed to those economists who are unfamiliar with mathematical ideas and methods; whereas there are many stimulating suggestions in it for the theoretical economist. In the theory of games, for example, monopoly and duopoly appear as the simplest cases and free competition as a limiting case for a large number of players, which seems closer to current economic thought than the traditional method of treating the subjects in the contrary order. Again, the equations of classical economics made no allowance for the stochastic element, whereas in the formulation of a strategy due account can be taken of it on a rational basis.

One would like to see the economic side of the subject developed further.

2.12. Review by: Louis O Kattsoff.
Social Forces 24 (2) (1945), 245-246.

This book is basically an analysis of games in more or less mathematical terms. There is nothing esoteric about the term "game." The authors mean by games such things as poker, chess, solitaire, etc. These are constantly used as illustrations of the developed theory. The authors consider economic behaviour to be an application of the theory of games of strategy. The contention is that economic behaviour can be considered as a specific type of game and can be described as such. The general theory of games would then be a general theory having economics as a special case. The discussion of economics takes up approximately 100 of the 625 pages of this difficult book. Those seeking a detailed analysis of economic problems and a discussion of the advantages of this abstract point of view over other approaches would be largely disappointed. The authors insist, however, that "the typical problems of economic behaviour become strictly identical with the mathematical notions of suitable games of strategy." In other words, economic theory is an interpretation of the symbols descriptive of certain games of strategy. And the chief aim of this book is to develop the mathematical theory of games necessary for this interpretation.
...

The book is a formidable one both intellectually and stylistically. The involved sentences, the numerous (and often unnecessary) footnotes, and the many misprints and typographical errors add to the cumbersome quality of the book

2.13. Review by: Louis Weisner.
Science & Society 9 (4) (1945), 366-369.

The theory of games of chance or entertainment has usually been treated by mathematicians from the standpoint of combinatorial analysis and probability. A different approach was made in 1928 by J von Neumann, who created a mathematical theory of games designed to deal with such questions as the strategies employed by the players, the maximum gains and minimum losses possible, the information available to each player at every stage, the forces which impel the formation of coalitions of players, as well as general theorems concerning games. In the volume under review this theory is expounded and developed systematically at great length by von Neumann and Morgenstern, and the results are applied to problems in economics.

The foundation of the theory is a set of axioms for games, the term "axiom" being used in its modern sense. Thus the axioms are not "self-evident truths," but primitive propositions from which it is proposed to derive all properties of games by deductive reasoning. They are not given a priori, but are the result of a brilliant analysis whose purpose it is to extract from particular games certain properties common to all games, which are sufficient to describe all games abstractly. The discovery of the axioms is no mean achievement; for it means that the theory of games may be treated as an abstract, deductive science, in the strict sense of the term, after the classical model of Euclidean geometry. ...
...

However optimistic von Neumann and Morgenstern may be, the discerning reader will not find it difficult to discover in their work inherent obstacles to the realisation of their dream. Their vision is bounded on all sides by the doctrine of marginal utility, and they dismiss rather lightly all other schools of economic thought. They explicitly exclude the economy of a communist society from their programme because their theory demands that combat and competition prevail in the distribution of the social product, and this condition is obviously not fulfilled in a communist society "since the interests of all the members of such a society are strictly identical". They limit their science of economics to that body of economic doctrine which is amenable to their theory of games. Hence they are silent on those problems of economics which fall within the sphere of production. This reticence is not peculiar to the authors but to the marginal utility school of economics to which they belong.

An adequate science of economics, comparable to the physical sciences, must not only provide rational explanations of the processes of production, consumption, exchange and distribution, but must also account for the instability of economic systems. Why did mercantilism yield to industrial capitalism? Why did industrial capitalism give way to imperialism? What are the characteristic features of the economy which will rise in the post-war era? The objection to the thesis of von Neumann and Morgenstern is not that their answers to these questions are unsatisfactory, but that their theory, being static and unhistorical, cannot provide any answers at all.

The most that can be expected of a "science of economics" which may result from the theory of games is a contribution to marginal utility economics. That doctrine, although widely accepted in academic circles and believed by many to have eternal validity, has been criticised for its over-emphasis on the market, its neglect of production, and its failure to give guidance in dealing with pressing problems such as crises and unemployment. These criticisms will probably be much sharper in the post-war epoch of capitalism, based upon full employment of labour and capital, the elimination of trade barriers and a consequent expansion of foreign trade, and the complete industrialisation of underdeveloped countries. The inability of marginal utility economics to make substantial contributions, even of a theoretical nature, to the requirements of the new epoch, may be expected to weaken the influence of that theory. For these reasons the reviewer does not share the authors' belief that their theory of games will result in an adequate science of economics.

2.14. Review by: L R Wilcox.
A Review of General Semantics 4 (2) (1947), 129-131.

Every so often one encounters the opinion that human behaviour is not amenable to scientific analysis and study. Undoubtedly semantic malfunctionings largely account for the existence and prevalence of such a view. For example, the terms 'soul,' 'free will,' 'too many variables,' and the like have been used so long that they have come to symbolise the uselessness of even attempting to apply scientific methods. Furthermore, the fact that science has not often been very successful in human affairs at first suggests, and later is thought to imply, that it can never be successful. Perhaps the most powerful weapon against the anti-science view would be the actual accomplishment of what is claimed to be impossible. While no one, and least of all von Neumann and Morgenstern, would claim 'success' in this direction, it must be said that their book represents an amazing and sizeable first step.
...

Although this book is not likely to be as widely read as it should be, because of its mathematical character, its potentially successful beginning of a scientific approach to theoretical economics should exert no mean influence on much of social and economic theory. One hopes that the new methods will pave the way toward lessening the gap between the physical and the social, with respect to both the nature of the intellectual machinery used and the extent of scientific accomplishment and understanding.

2.15. Review by: Arthur H Copeland.
Bulletin of the American Mathematical Society 51 (7) (1945), 498-504.

Posterity may regard this book as one of the major scientific achievements of the first half of the twentieth century. This will undoubtedly be the case if the authors have succeeded in establishing a new exact science - the science of economics. The foundation which they have laid is extremely promising. Since both mathematicians and economists will be needed for the further development of the theory it is in order to comment on the background necessary for reading the book. The mathematics required beyond algebra and analytic geometry is developed in the book. On the other hand the non-mathematically trained reader will be called upon to exercise a high degree of patience if he is to comprehend the theory. The mathematically trained reader will find the reasoning stimulating and challenging. As to economics, a limited background is sufficient.

The authors observe that the give-and-take of business has many of the aspects of a game and they make an extensive study of the strategy of games with this similarity in mind (hence the title of this book). In the game of life the stakes are not necessarily monetary; they may be merely utilities. In discussing utilities the authors find it advisable to replace the questionable marginal utility theory by a new theory which is more suitable to their analysis. They note that in the game of life as well as in social games the players are frequently called upon to choose between alternatives to which probabilities rather than certainties are attached. The authors show that if a player can always arrange such fortuitous alternatives in the order of his preferences, then it is possible to assign to each alternative a number or numerical utility expressing the degree of the player's preference f or that alternative. The assignment is not unique but two such assignments must be related by a linear transformation.
...
The book leaves much to be done but this fact only enhances its interest. It should be productive of many extensions along the lines of economic interpretation as well as of mathematical research. In fact the authors suggest a number of directions in which research might profitably be pursued.

3.1. From the Preface.

The second edition differs from the first in some minor respects only. We have carried out as complete an elimination of misprints as possible, and wish to thank several readers who have helped us in that respect. We have added an Appendix containing an axiomatic derivation of numerical utility, This subject was discussed in considerable detail, but in the main qualitatively, in Section 3. A publication of this proof in a periodical was promised in the first edition, but we found it more convenient to add it as an Appendix. Various Appendices on applications to the theory of location of industries and on questions of the four and five person games were also planned, but had to be abandoned because of the pressure of other work.

3.2. Review by: Leonid Hurwiez.
The Annals of Mathematical Statistics 19 (3) (1948), 436-437.

This review is devoted to the second edition of a book which from its first appearance was acknowledged to be a major contribution in the field of theory of rational behaviour. As is pointed out in the Preface, "the second edition differs from the first in some minor respects only". The main change is the addition of a proof (of "measurability" of utility) omitted in the first edition.

The book's objective is to solve the problem of rational behaviour in a very general type of situation.

It is, therefore, not surprising that its results are of relevance in many fields of knowledge, among them economics and statistical inference.

In both economics and statistics the problem of rational behaviour is a fundamental one. Thus one of the classical problems treated by the economic theory is that of profit maximisation by a firm. The firm is assumed to be maximising its net profit which is a function of prices of the product, materials used, etc., as well as the quantities used and produced. In the simplest case prices are taken as given; more generally they are assumed to be functions (known to the firm) of the quantities sold and purchased. But assuming this function to be known presupposes the knowledge of behaviour of other firms. This procedure has for a long time been regarded as highly unsatisfactory; it is analogous to elaborating the theory of rational behaviour of a poker player on the assumption that he knows the strategy of the other players!

It is the type of situation in which not only the behaviour of various individuals, but even their strategies, are interdependent, that is treated by von Neumann and Morgenstern. The essence of their solutions is to base the optimal strategy on the minimax principle. As applied to a game, the principle re- quires that one should choose a strategy which minimises the maximum loss that could be inflicted by the opponent.

3.3. Review by: Abraham Wald.
Mathematical Reviews MR0021298 (9,50f).

The second edition differs only slightly from the first. The changes consist essentially of eliminating the misprints discovered in the first edition and of adding an appendix which contains a derivation of the numerical character of utility from the axioms formulated in section 3. As the authors remark, the axioms in their present form do not allow for the possibility of a specific utility or disutility of gambling. To permit such a possibility, it would be necessary to modify some of the axioms. It seems likely that the really critical axiom that would have to be changed is the axiom (3:C:b) which expresses the combination rule for multiple chance alternatives. The mathematical difficulties in formulating a system which allows for a utility or disutility of gambling are likely to be considerable and, as is pointed out by the authors, the current method of using indifference curves would not diminish these difficulties.

4.1. From the Foreword.

The lectures on "Operator Theory", of which the present volume constitutes the first part, were given in the academic years 1933-34 and 1934-35, at the Institute for Advanced Study, The notes were prepared in these years by Dr Robert S Martin and Dr Charles C Torrance, respectively. They were multigraphed and distributed by the Institute for Advanced Study shortly thereafter, but the original edition has been completely exhausted for several years. The interest in these lecture notes appears to have been continuing, and therefore a new edition is now being brought out.

4.2. Review by: Paul R Halmos.
Mathematical Reviews MR0032011 (11,240f).

The author's lecture notes on functional operators, despite their limited circulation, were for a long time one of the major sources of measure-theoretic information in the United States. The present volume differs from the first part of the original edition only in that "typographical errors have been corrected and some notations and references have been elaborated." ... The first nine chapters contain an extremely thorough and detailed exposition of the classical theory of Lebesgue measure in finite-dimensional Euclidean space. Chapters X and XI, each of which is slightly longer than the first nine chapters together, cover the generalisation of the theory to arbitrary measure spaces. Among the useful material covered is the connection between set functions and point functions, an important and complicated subject which is frequently treated in the trivial one-dimensional case only. The usefulness of the book would be enhanced if it had an index; the somewhat repetitious lecture style of presentation makes it difficult to locate a particular topic.

4.3. Review by: Jacob Lionel Bakst Cooper.
The Mathematical Gazette 35 (312) (1951), 14-15.

These two volumes contain lectures given by the author at the Institute for Advanced Study in 1933-5. They have exerted considerable influence on students of Measure Theory and Operator Theory in the U.S.A., and are now reproduced, with some corrections, as Nos. 21 and 22 of "Annals of Mathematics Studies".

The two volumes can be read independently of one another: the first is not restricted by any particular needs of operator theory, but contains a complete account of the theory of integrals of summable functions. The second makes little use of matter in the first.

The first nine chapters of Volume I treat the theory of Lebesgue measure and integration in n-dimensional spaces by more or less classical methods, including the Vitali covering theorem and the theory of the differentiation of the integral. With the section on Fubini's theorem in the last chapter these chapters would cover pretty well all an analyst needs to know about Lebesgue integration. Chapters X and XI each occupy about one third of the volume, and deal with the generalisation of the theory to abstract spaces. Chapter X gives the theory of extension of measures for rings and other algebraic systems of sets, and also the theory of measures on product spaces: this theory is used in an ingenious fashion to give a definition of the ordinary Lebesgue integral and of Lebesgue-Stieltjes integrals free from the normal topological notions. Chapter XI deals with the general integral, discussing in great detail the derivative of one set function with respect to another. The discussion almost reverses the normal order, and though advantages are claimed for it the treatment seems rather heavy.

All definitions and proofs are given in great detail and the treatment is very thorough and accurate. It is likely that Honours mathematics would suffice for reading all save a few passages. However, the book has some of the defects of lecture notes: it is rather long, owing partly to repetitions of arguments, and lacks the remarks intended to orientate the reader which would be given verbally in a lecture. An index of definitions and of notations would also make it more readable; and references, if only by giving the names under which some of the classical theorems are known, would be very useful.

5.1. Review by: Paul R Halmos.
Mathematical Reviews MR0034514 (11,599e).

This volume contains chapter XII (Linear spaces), chapter XIII (Linear operators), and chapter XIV (Commutativity, reducibility). As the subtitle of the volume indicates, the principal subject is not the algebraic and analytic theory of operators, but rather their relation to the geometry of Hilbert space. Chapter XII treats the relations among various possible requirements on an inner product space (dimensionality, separability, completeness) and gives the standard measure-theoretic examples. Chapter XIII treats the natural elementary properties of and constructions based on operators (closure, extension, adjoint) in terms of their graphs. The main concern of chapter XIV is to provide the tools for a careful discussion of the matrices associated with operators.

5.2. Review by: Jacob Lionel Bakst Cooper.
The Mathematical Gazette 35 (312) (1951), 14-15.

These two volumes contain lectures given by the author at the Institute for Advanced Study in 1933-5. They have exerted considerable influence on students of Measure Theory and Operator Theory in the U.S.A., and are now reproduced, with some corrections, as Nos. 21 and 22 of "Annals of Mathematics Studies".

The two volumes can be read independently of one another: the first is not restricted by any particular needs of operator theory, but contains a complete account of the theory of integrals of summable functions. The second makes little use of matter in the first.

The second volume is devoted to the geometrical theory of orthogonal vector spaces, those in which a scalar product is defined. These spaces are defined axiomatically, and examples of spaces obeying various combinations of the axioms are given. This is followed by definitions of the main types of operators, and by what, if it is not stretching that abused adjective, we may call the elementary theory of these operators: their algebra and geometry: the analytic resolution theory is not touched. Most of the contents of Volume II are due to von Neumann; but although it contains much not in his published papers, it is far from exhausting or even giving the most important parts of his great contributions to the subject. There are, in fact, references to chapters yet to come.

All definitions and proofs are given in great detail and the treatment is very thorough and accurate. It is likely that Honours mathematics would suffice for reading all save a few passages. However, the book has some of the defects of lecture notes: it is rather long, owing partly to repetitions of arguments, and lacks the remarks intended to orientate the reader which would be given verbally in a lecture. An index of definitions and of notations would also make it more readable; and references, if only by giving the names under which some of the classical theorems are known, would be very useful.

6.1. From the Preface.

The object of this book is to present the new quantum mechanics in a unified representation which, so far as it is possible and useful, is mathematically rigorous. This new quantum mechanics has in recent years achieved in its essential parts what is presumably a definitive form: the so-called "transformation theory. Therefore the principal emphasis shall be placed. on the general and fundamental questions which have arisen in connection with this theory. In particular, the difficult problems of interpretation, many of which are even now not fully resolved, will be investigated in detail. In this context the relation of quantum mechanics to statistics and to the classical statistical mechanics is of special importance. However, we shall as a rule omit any discussion of the application of quantum mechanical methods to particular problems, as well as any discussion of special theories derived from the general theory - at least so far as this is possible without endangering the understanding of the general relationships. This seems the more advisable since several excellent treatments of these problems are either in print or in process of publication.

On the other hand, a presentation of the mathematical tools necessary for the purposes of this theory will be given, i.e., a theory of Hilbert space and the so-called Hermitian Operators. For this end, an accurate introduction to unbounded operators is also necessary, i.e., an extension of the theory beyond its classical limits (developed by Hilbert and E Hellinger, F Riesz, E Schmidt, O Toeplitz). The following may be said regarding the method employed in this mode of treatment: as a rule, calculations should be performed with the operators themselves (which represent physical quantities) and not with the matrices, which after the introduction of a (special and arbitrary) coordinate system in Hilbert space, result from them. This "coordinate free," i.e., invariant, method, with its strongly geometric language, possesses noticeable formal advantages.

Dirac, in several papers, as well as in his recently published book, has given a representation of quantum mechanics which is scarcely to be surpassed in brevity and elegance, and which is at the same time of invariant character. It is therefore perhaps fitting to advance a few arguments on behalf of our method, which deviates considerably from that of Dirac.

The method of Dirac, mentioned above, (and this is overlooked today in a great part of quantum mechanical literature, because of the clarity and elegance of the theory) in no way satisfies the requirements of mathematical rigour - not even if these are reduced in a natural and proper fashion to the extent common elsewhere in theoretical physics. For example, the method adheres to the fiction that each self-adjoint operator can be put in diagonal form. In the case of those operators for which this is not actually the case, this requires the introduction of "improper" functions with self-contradictory properties. The insertion of such a mathematical "fiction" is frequently necessary in Dirac's approach, even though the problem at hand is merely one of calculating numerically the result of a clearly defined experiment. There would be no objection here if these concepts, which cannot be incorporated into the present day framework of analysis, were intrinsically necessary for the physical theory.

Thus, as Newtonian mechanics first brought about the development, of the infinitesimal calculus, which, in its original form, vas undoubtedly not self-consistent, so quantum mechanics might suggest a new structure for our "analysis of infinitely many variables" - i.e., the mathematical technique would have to be changed, and not the physical theory. But this is by no means the case. It should rather be pointed out that the quantum mechanical "Transformation theory" can be established in a manner which is just as clear and unified, but which is also without mathematical objections. It should be emphasised that the correct structure need not consist in a mathematical refinement and explanation of the Dirac method, but rather that it requires a procedure differing from the very beginning, namely, the reliance on the Hilbert theory of operators.

In the analysis of the fundamental questions, it will be shown how the statistical formulas of quantum mechanics can be derived from a few qualitative, basic assumptions. Furthermore, there will be a detailed discussion or the problem as to whether it is possible to trace the statistical character of quantum mechanics to an ambiguity (i.e., incompleteness) in our description of nature. Indeed, such an interpretation would be a natural concomitant of the general principle that each probability statement arises from the incompleteness of our knowledge. This explanation "by hidden parameters," as well as another, related to it, which ascribes the "hidden parameter" to the observer and not to the observed system, has been proposed more than once. However, it will appear that this can scarcely succeed in a satisfactory way, or more precisely, such an explanation is incompatible with certain qualitative fundamental postulates of quantum mechanics.

The relation of these statistics to thermodynamics is also considered. A closer investigation shows that the well known difficulties of classical mechanics, which are related to the "disorder" assumptions necessary for the foundation of thermodynamics, can be eliminated here.

6.2. Review by: A F Stevenson.
Quarterly of Applied Mathematics 14 (1) (1956), 34.

This is a translation of von Neumann's well-known book, Mathematische Grundlagen der Quantenmechanik which first appeared in 1932. The translator's preface is dated 1949, the long interval which has elapsed before publication not being explained. Since the original edition appeared, there have been important advances in Quantum Mechanics, notably in connection with the quantisation of wave fields. No attempt has been made to bring the book up to date in this regard, and the translation sticks very closely to the original German edition. Even the references to text-books have not been brought up to date.

The translation is somewhat too literal in many places, and the translator often employs phrases and words which, if not absolutely incorrect, have at least a strange and somewhat "foreign" flavour to them. Nevertheless, the meaning is always clear. One slight difference with the original which most people will probably consider an improvement, is that the footnotes have been placed in the body of the text instead of at the end of the book.

The translator and publisher have performed a service in making this classic available to a wider circle of English-speaking readers. It remains indispensable to those who desire a rigorous presentation of the foundations of the subject.

6.3. Review by: Paul Karl Feyerabend.
The British Journal for the Philosophy of Science 8 (32) (1958), 343-347.

Elementary quantum mechanics as presented in the early thirties was a highly successful theory, but from the point of view of rigour it suffered from various defects, mainly two: it used mathematical procedures that were not permissible, and even inconsistent; and its interpretation, viz. the Copenhagen-interpretation, although a useful guide for the theoretician, a good means of orientation for the experimentalist as well as an exciting subject for popularisation it was far from clear what exactly its presuppositions were.

The treatise at present under review, which appeared first in 1930, can be regarded as an attempt at a presentation of the theory (elementary quantum mechanics without spin) which is free from those two defects. The first defect is removed by developing an extension of the theory of the Hilbert-space and by a corresponding reformulation of the eigenvalue-problem. We shall not concern ourselves with this part of the treatise (chaps. I, II), which has found little appreciation among physicists as it 'involves a technique at once too delicate and too cumbersome for the ... average physicist' (E C Kemble). The second defect is partly removed by an attempt to base the whole theory upon the formalism together with 'a few general qualitative assumptions'. Although this attempt has not been completely successful, it is still of importance as an attempt (a) to discover just what exactly is implied by quantum mechanics and what must be left to speculation and (b) what are the assumptions which must be added to the formalism in order to turn it into a fully-fledged physical theory.

It has sometimes been thought that those assumptions are provided by the Copenhagen-interpretation. This view is incorrect as the Copenhagen-interpretation is neither necessary nor sufficient for connecting the formalism 'with reality', i.e. with the results of measurements. It is also fairly independent of the formalism, as it consists mainly in interpreting the quantum-postulate and in drawing conclusions from it which may be said to be elements of a new (and non-mechanistic) ontology. This at once raises the question to what extent this ontology is justified, not by speculation on the basis of the quantum of action, but by the theory as a whole; or, to be more precise, it raises the question which of those assertions are implied by the theory and which are based upon (ontological) speculation only.

Von Neumann's treatise attempts to provide an answer to this question.

7.1. From the Preface by Klara von Neumann.

To give the Silliman Lectures, one of the oldest and most outstanding academic lecture series in the United States, is considered a privilege and an honour among scholars all over the world. Traditionally the lecturer is asked to give a series of talks, over a period of about two weeks, and then to shape the manuscript of the lectures into a book to be published under the auspices of Yale University, the home and headquarters of the Silliman Lectures.

Early in 1955 my husband, John von Neumann, was invited by Yale University to give the Silliman Lectures during the spring term of 1956, some time in late March or early April. Johnny was deeply honoured and gratified by this invitation, despite the fact that he had to make his acceptance subject to one condition - namely, that the lectures be limited to one week only. The accompanying manuscript would, however, cover more fully his chosen topic - The Computer and the Brain - a theme in which he had been interested for a considerable time. The request to abbreviate the lecture period was made of necessity, as he had just been appointed by President Eisenhower as one of the members of the Atomic Energy Commission, a full-time job which does not permit even a scientist much time away from his desk in Washington. My husband knew, however, that he could find time to write the lectures, for he had always done his writing at home during the night or at dawn. His capacity for work was practically unlimited, particularly if he was interested, and the many unexplored possibilities of automata did interest him very much indeed; so he felt quite confident that he could prepare a full manuscript even though the lecture period would have to be somewhat cut. Yale University, helpful and understanding at this early period as well as later, when there was only sadness, sorrow, and need, accepted this arrangement, and Johnny started his new job at the Commission with the added incentive that he would continue his work on the theory of automata even if it was done a little en cache.

In the spring of 1955 we moved from Princeton to Washington, and Johnny went on leave of absence from the Institute for Advanced Study, where he had been Professor in the School of Mathematics since 1933.

Johnny was born in Budapest, Hungary, in 1903. Even in his early years he had shown a remarkable ability and interest in scientific matters, and as a child his almost photographic memory manifested itself in many unusual ways. Reaching college age, he studied first chemistry and then mathematics at the University of Berlin, the Technische Hohschule in Zurich, and the University of Budapest. In 1927 he was appointed Privatdozent at the University of Berlin, probably one of the youngest persons appointed to such a position in any of the German universities within the last few decades. Later Johnny taught at the University of Hamburg, and in 1930, for the first time, crossed the Atlantic, having accepted the invitation of Princeton University to become a guest lecturer for one year. In 1931 he became a member of the faculty of Princeton University, thus making his permanent home in the United States and becoming a citizen of the New World. During the 1920's and 30's Johnny's scientific interest was ranging widely, mostly in theoretical fields. His publications included works on quantum theory, mathematical logic, ergodic theory, continuous geometry, problems dealing with rings of operators, and many other areas of pure mathematics. Then, during the late thirties, he became interested in questions of theoretical hydrodynamics, particularly in the great difficulties encountered in obtaining solutions to partial differential equations by known analytical methods. This endeavour, carried forward when war clouds were darkening the horizon all over the world, brought him into scientific defence work and made him more and more interested in the applied fields of mathematics and physics. The interaction of shock waves, a very intricate hydrodynamic problem, became one of the important defence research interests, and the tremendous amount of calculations required to get some of the answers motivated Johnny to employ a high-speed computing machine for this purpose. The ENIAC, built in Philadelphia for the Ballistic Research Laboratories of Army Ordnance, was Johnny's first introduction to the vast possibilities of solving many yet unresolved questions with the aid of automation. He helped to modify some of the mathematical-logical design of the ENIAC, and from then until his last conscious hours, he remained interested in and intrigued by the still unexplored aspects and possibilities of the fast-growing use of automata.

In 1943, soon after the Manhattan Project was started, Johnny became one of the scientists who "disappeared into the West," commuting back and forth between Washington, Los Alamos, and many other places. This was the period during which he became completely convinced, and tried to convince others in many varied fields, that numerical calculations done on fast electronic computing devices would substantially facilitate the solution of many difficult, unsolved, scientific problems.

After the war, together with a small group of selected engineers and mathematicians, Johnny built, at the Institute for Advanced Study, an experimental electronic calculator, popularly known as the JONIAC, which eventually became the pilot model for similar machines all over the country. Some of the basic principles developed in the JONIAC are used even today in the fastest and most modern calculators. To design the machine, Johnny and his co-workers tried to imitate some of the known operations of the live brain. This is the aspect which led him to study neurology, to seek out men in the fields of neurology and psychiatry, to attend many meetings on these subjects, and, eventually, to give lectures to such groups on the possibilities of copying an extremely simplified model of the living brain for man-made machines. In the Silliman Lectures these thoughts were to be further developed and expanded.

During the post-war years Johnny divided his work among scientific problems in various fields. Particularly, he became interested in meteorology, where numerical calculations seemed to be helpful in opening entirely new vistas; part of his time was spent helping to make calculations in the ever-expanding problems of nuclear physics. He continued to work closely with the laboratories of the Atomic Energy Commission, and in 1952 he became a member of the General Advisory Commit- tee to the AEC.

On March 15, 1955, Johnny was sworn in as a member of the Atomic Energy Commission, and early in May we moved our household to Washington. Three months later, in August, the pattern of our active and exciting life, centred around my husband's indefatigable and astounding mind, came to an abrupt stop; Johnny had developed severe pains in his left shoulder, and after surgery, bone cancer was diagnosed. The ensuing months were of alternating hope and despair; sometimes we were confident that the lesion in the shoulder was a single manifestation of the dread disease, not to recur for a long time, but then indefinable aches and pains that he suffered from at times dashed our hopes for the future. Throughout this period Johnny worked feverishly - during the day in his office or making the many trips required by the job; at night on scientific papers, things which he had postponed until he would be through with his term at the Commission. He now started to work systematically on the manuscript for the Silliman Lectures; most of what is written in the following pages was produced in those days of uncertainty and waiting. In late November the next blow came: several lesions were found on his spine, and he developed serious difficulties in walking. From then on, everything went from bad to worse, though still there was some hope left that with treatment and care the fatal illness might be arrested, for a while at least.

By January 1956 Johnny was confined to a wheelchair, but still he attended meetings, was wheeled into his office, and continued working on the manuscript for the lecture. Clearly his strength was waning from day to day; all trips and speaking engagements had to be cancelled one by one, with this single exception - the Silliman Lectures. There was some hope that with X-ray treatments the spine might be, at least temporarily, sufficiently strengthened by late March to permit his traveling to New Haven and fulfilling this one obligation that meant so very much to him. Even so, the Silliman Lecture Committee had to be asked further to reduce the lectures to one or two at the most, for the strain of a whole week of lecturing would have been dangerous in his weakened condition. By March, however, all false hopes were gone, and there was no longer any question of Johnny being able to travel anywhere. Again Yale University, as helpful and understanding as ever, did not cancel the lectures, but suggested that if the manuscript could be delivered, someone else would read it for him. In spite of many efforts, Johnny could not finish writing his planned lectures in time; as a matter of tragic fate he could never finish writing them at all.

In early April Johnny was admitted to Walter Reed Hospital; he never left the hospital grounds again until his death on February 8, 1957. The unfinished manuscript of the Silliman Lectures went with him to the hospital, where he made a few more attempts to work on it; but by then the illness had definitely gained the upper hand, and even Johnny's exceptional mind could not overcome the weariness of the body.

I should like to be permitted to express my deep gratitude to the Silliman Lecture Committee, to Yale University, and to the Yale University Press, all of which have been so helpful and kind during the last, sad years of Johnny's life and now honour his memory by admitting his unfinished and fragmentary manuscript to the series of the Silliman Lectures Publications.

7.2. Review by: Stan Ulam.
Scientific American 198 (6) (1958), 127-130.

This book by John von Neumann is published more than a year after his death. It is hardly more than an introduction to a monograph on a subject which preoccupied him during the last years of his life. It has been prepared from an incomplete manuscript of the Silliman Lectures which he was to give at Yale University in 1956. As Mrs von Neumann explains in the introduction to the book, the lectures were written during his fatal illness. In places the presentation of the material lacks his characteristic style. Nevertheless the book, like everything von Neumann wrote, remains highly original and intensely stimulating.

Von Neumann became interested in the possibilities of electronic computing machines during the Second World War. In the beginning he was primarily concerned with the logic of the operation of such machines, but he was the first to devise a means by which a machine with fixed circuits could deal flexibly with a variety of mathematical problems. Before he had entered the field, the solution of each problem required a different set of connections.

It was the problems of mathematical physics that arose at the Los Alamos Scientific Laboratory which created the need to plan massive computations on the new machines. The solution of one of these problems may require billions of elementary steps: additions, multiplications, and so on. Beyond the mere bulk of repetitive computations the problem may involve combinations and logical operations of considerable complexity. The program of calculation must be prepared in advance, first in a general form. This is done by means of a flow diagram, which is then elaborated by a detailed set of instructions - a "code." This very scheme of a flow diagram and a code is due to von Neumann, along with many other concepts now commonplace in the art of computing.

It was von Neumann's belief that computing machines would not only help solve existing problems but also open new perspectives in mathematics and physics. Obviously the machines could he used to test tentative theories by computations too laborious and too lengthy for manual methods. But, more important, they could provide the imaginative investigator with new "experimental" material which could suggest new theories.

From the beginning von Neumann was intrigued by the similarities and differences between the operation of computing machines and the working of the human brain. He envisaged a general theory of automata, and even of organisms, but his ideas remained undeveloped. The present book is an approach to an understanding of the nervous system from the point of view of the mathematician. Von Neumann stated that, since he was not a nerve physiologist or a psychologist, he would restrict himself to the logical and statistical features of the elements of the brain. He felt that a mathematical study of the nervous system would not only lead to a better understanding of this system but also could affect the future development of mathematics itself. ... This little volume will leave the reader with a sharper awareness of the loss represented by von Neumann's premature death. It was not given to him to pursue these speculations and render them into mathematical form. But his ideas will have great value to further investigations.

We are only beginning to realise the potentialities of the large electronic computer. In the opinion of this reviewer the next steps in its evolution will involve not only greater speed and larger memory capacity but also a more intimate connection between the machine and its user. The usefulness of the machine would be greatly increased if the results of its computations could be quickly displayed and if its orders could be quickly changed. If the machine can be made to operate more like the human nervous system, it may stimulate the imagination of its user to a greater degree. Certainly it seems that future machines will perform many operations in parallel channels. Work is under way at several centres which will enable a machine to prove elementary theorems in certain simple mathematical domains. Such machines may be capable of a modicum of genuine learning. The material outlined in von Neumann's book awaits conversion into mathematics.

7.3. Review by: A Chapanis.
The Quarterly Review of Biology 35 (3) (1960), 254-255.

Early in 1955 John von Neumann was invited to give the Silliman Lectures at Yale University during the spring of 1956. An unfortunate illness never allowed him to deliver those lectures or even to finish the manuscript which was to be the basis for his lectures. This small book is based on his unfinished and fragmentary manuscript.

The book is divided into two main parts. The first discusses the two principal types of modern computing machines, analogue and digital, and their characteristics. The second part of the book discusses the brain, its functioning, and characteristics in machine terms.

It is difficult to say how much good it does to make detailed comparisons between the brain and computing machines. Although there are some superficial analogies between the two kinds of mechanism, the differences turn out to be much more impressive than the similarities. If you are interested in such speculations, however, you will find this an interesting and well-written document.

7.4. Review by: H P.
Mathematical Tables and Other Aids to Computation 13 (67) (1959), 226-228.

This small volume constitutes the last contribution of one of the great scientists of our time, John von Neumann. The material was prepared for delivery at Yale University during the spring of 1956 in the Silliman Lectures series. However, the lectures were never given, owing to the increasing severity of the illness which fin ally took von Neumann's life on February 8, 1957. The subject is one which attracted his interest for a number of years. It is a subject in which he was eminently qualified, by virtue of his great genius and his contributions in the fields of highspeed calculators and the logic of automata. In spite of the preliminary nature of this work, it is destined to become the nucleus of a new field of research which will challenge the minds of men for many years to come - the comparative study of the human brain and man-made automata.

The book is divided in two parts. In the first part von Neumann discusses the basic principles underlying the design of modern computing machines, both analogue and digital ...
...
In the second part von Neumann compares the functioning of the human brain with the operation of a modern computer, bringing out the areas of "similarity and dissimilarity between these two kinds of automata." ...
...
Looking deeper into the more intrinsic areas of comparison, von Neumann is led to the conclusion that the basic internal language used by the brain is undoubtedly quite different from the mathematical language with which we are acquainted. He arrives at this conclusion primarily on the basis of the argument that the information stored in the brain lacks sufficient accuracy to enable it to carry out mathematical and logical processes in such "depth" as would be required if the language used were based on conventional mathematical symbols. Von Neumann conjectures that the language used by the brain is probably statistical in nature in which correlation processes play an important role. ... He concludes the book with the remark: "Thus logics and mathematics in the central nervous system, when received as languages, must structurally be essentially different from those languages to which our common experience refers. It also ought to be noted that the language here involved may well correspond to a short code in the sense described earlier, rather than to a complete code: when we talk mathematics, we may be discussing a secondary language, built on the primary language truly used by the central nervous system. Thus the outward forms of our mathematics are not absolutely relevant from the point of view of evaluating what the mathematical or logical language truly used by the central nervous system is. However, the above remarks about reliability and logical and arithmetical depth prove that whatever the system is, it cannot fail to differ considerably from what we consciously and explicitly consider as mathematics."

7.5. Review by: Reuben Louis Goodstein.
The Mathematical Gazette 44 (348) (1960), 159.

When one of the greatest mathematicians of our times died in February 1957 he left unfinished the manuscript of the Silliman Memorial Lecture which he had been invited to give in the Spring Term of 1956, and in honour of his memory the Silliman Lecture Committee have published these preparatory notes in the series of Silliman Lectures Publications. These notes consist of a non-technical account of digital and analogue computers, their use and nature, and a comparison of computers with the brain in relation to speed and size. A moving preface by von Neumann's widow briefly traces his career from his birth in Budapest in 1903 to the onset of bone cancer in 1955.

7.6. Review by: Eduardo R Caianiello.
Mathematical Reviews MR0130066 (23 #B3099).

Although unrevised and partly incomplete because of the author's untimely death, these Silliman Lectures are an outstanding example of J von Neumann's insight, brilliance and clarity. In a language that any intelligent layman can understand, he speaks in the first part, concisely and precisely, of analogue and digital computers; in the second, of neuro-anatomical and physiological data concerning nervous systems. Similarities and dissimilarities are brought forth vividly on the basis of quantitative estimates of speed, complexity and energy consumption in both; particularly stimulating is the analysis of the different logical and arithmetical depth of operations in the machine and in the brain, as related to their different structures. The book ends with a remark which, in this reviewer's opinion, will become the more significant as years will pass: that the very language and body of concepts of extant logics and mathematics are not absolute necessities, but as accidental and history-bound as Greek or Sanskrit; thus, we have to look for the logics of the brain itself, which is determined by its anatomy and physiology.

7.7. Review by: A H Taub.
Isis 51 (1) (1960), 94-96.

This little book contains the last scientific work of the incomparable John von Neumann, as well as a moving preface by his wife, Klari, in which she relates the tragic circumstances under which much of this book was written. As in all of von Neumann's works, in this one there may be found penetrating analyses of various problems and deep insights into possible methods of their solution.

This book is concerned with "an approach toward the understanding of the nervous system from the mathematician's point of view." The last phrase refers to the logical and statistical techniques of mathematical investigations as they are used in the discipline called "information theory." It was one of von Neumann's purposes in writing this book to point out "that a deeper mathematical study of the nervous system ... will affect our understanding of the aspects of mathematics itself that are involved. In fact it may alter the way at which we look on mathematics and logics proper."

The book consists of two parts. The first one contains a discussion of computing machines, analogue, digital and mixed ones. Their nature and the principles underlying their organisation are presented in clear precise terms that an interested layman can follow. Included in the description of properties of automatic computers is an extremely cogent discussion of the necessity of extreme precision in the numbers manipulated by such devices.

It is pointed out that not only is this need determined by the extremely large number of arithmetic operations which these devices are capable of performing and actually do carry out in solving problems but that the operations performed in the course of a calculation may amplify errors that are introduced in earlier operations. Thus unless great care is used in controlling the amplification of errors, even the high precision of existing computers is not sufficient for many problems.

In part two, von Neumann discusses the points of similarity and dissimilarity between the human nervous system and modern automatic computers. Many of the similarities between these two kinds of automata have been pointed out previously by von Neumann and by other authors, as have some of the obvious dissimilarities in size and speed. This material is reviewed and summarised in this book. However, the author's primary aim is to develop some of the deep-lying dissimilarities which involve the principles of functioning and control and of over-all organisation.
...

In the section entitled, "The Logical Structure of the Nervous System," the author returns to point (3) above and points out that the nervous system must have an arithmetical as well as a logical part and that the needs of arithmetics in it are just as important as those of logics. He states that "the nervous system is a computing machine which man- ages to do its exceedingly complicated work on a rather low level of precision . . .but also to a rather high level of reliability." The following important question is then raised: "What essential inferences about the arithmetical and logical structure of the computing ma- chine that the nervous system represents can be drawn from these apparently somewhat conflicting observations?"

Von Neumann drew two inferences: (1) That natural automata have a radically different system of notation from the ones we are familiar with in ordinary arithmetics and mathematics which enable them to carry out with little arithmetic and logical depth tasks which can be presently posed to artificial automata only by using processes which involve great arithmetic and logical depth; and (2) the logics and mathematics in the central nervous system, when viewed as language, must structurally be essentially different from those languages to which our common experience refers.

The discovery and exploitation of these notations and languages are endeavours calling for great intellectual effort. The successful completion of these efforts will have great impact on biology, physiology and mathematics. It is extremely unfortunate that von Neumann himself will not be able to contribute further to the elucidation of these most important questions whose presentation alone is a noteworthy achievement.

8.1. From the Foreword by Israel Halperin.

This book reproduces the notes of lectures on Continuous Geometry given by John von Neumann at Princeton. Part I was given during the academic year 1935- 36, and Parts II and III were given during the academic year 1936-37. The notes were prepared, while the lectures were in progress, by L Roy Wilcox, and multigraphed copies were distributed by the Institute for Advanced Study. The supply was soon exhausted, and the notes have not been reproduced until now.

In the present edition many slips in typing have been corrected, in Part I with the help of Wallace Givens. I have inserted a few editorial remarks and I have made a small number of changes in the text. These changes, together with some comments, are listed at the back of the book. Of these changes, only one is essential and it was authorised explicitly by von Neumann.

Continuous geometry was invented by von Neumann in the fall of 1935. His previous work on rings of operators in Hilbert space, partly in collaboration with F J Murray, had led to the discovery of a new mathematical structure which possessed a dimension function. The new structure had incidence properties resembling those of the system $L_{n}$ ($L_{n}$ denotes the lattice of all linear subsets of an $n - 1$ dimensional projective geometry), but its dimension function assumed as values all real numbers in the interval (0, 1).

Von Neumann set out to formulate suitable axioms to characterise the new structure. It happened that just previously, K Menger and G Birkhoff had characterised $L_{n}$ by lattice-type axioms; in particular, Birkhoff had shown the structures $L_{n}$ could be characterised as the complemented modular irreducible lattices which satisfy a chain condition. Von Neumann dropped the chain condition and replaced it by two of its weak consequences: (i) order completeness of the lattice, and (ii) continuity of the lattice operations. Lattices which are complemented, modular, irreducible, satisfy (i) and (ii), but do not satisfy a chain condition, were called by von Neumann: continuous geometries (reducible continuous geometries were considered later, in Part Ill ). It is easy to see that in a continuous geometry there can be no minimal element - that is, no atomic element or point.

The structure previously discovered by Murray and von Neumann in their research on rings of operators was an example, the first, of a continuous geometry.

Von Neumann's first fundamental result was the construction, for an arbitrary continuous geometry, of a dimension function with values ranging over the interval (0, 1). The construction was based on the definition: $x$ and $y$ are to be called equidimensional if $x$ and $y$ are in perspective relation, that is: for some w the lattice join and meet of $x$ with $w$ are identical with those of $y$ with $w$. The essential difficulty is to prove that the perspective relation is transitive. ... Von Neumann reviewed his previous work on continuous geometries in four colloquium lectures delivered before the American Mathematical Society, in September, 1937, at Pennsylvania State College, State College. Pa., U.S.A. Later, he began to write a systematic account of his research on continuous geometry which he planned to publish as a book in the American Mathematical Society Colloquium Series. But his work in the theory of games, other interests, and the war, intervened . As the years went by, he finally decided that the Princeton lecture notes, a t least, should be reproduced. This is now accomplished with the publication of the present book.

8.2. Review by: Frank Smithies.
The Mathematical Gazette 47 (360) (1963), 168-169.

This book reproduces notes of lecture courses given by von Neumann at Princeton in 1935-37. They have been edited by Professor Halperin, who has made some corrections and improvements and has added notes on later work.

Shortly before these lectures were given K Menger and G Birkhoff had shown that the linear manifolds in a projective geometry form an irreducible complemented modular lattice satisfying the ascending and descending chain conditions; apart from some minor exceptions, the converse also holds. In his work on lattices of subspaces associated with operator algebras in Hilbert space, von Neumann discovered structures similar to projective geometries, the chain conditions being replaced by weaker restrictions concerning completeness and continuity. These are the "continuous geometries".

Part I of these lectures establishes the existence of a numerical dimension function in a continuous geometry. ... In Part II, von Neumann carries through a construction analogous to the introduction of coordinates in a projective geometry. ... The concluding Part III of the notes, which has remained unfinished, is concerned with dimensionality in generalised continuous geometries not satisfying the irreducibility condition.

Von Neumann always hoped that the continuous geometries associated with operator algebras would find application in the foundations of quantum mechanics, but this aspiration has not yet been achieved.

8.3. Review by: Fumitomo Maeda.
Mathematical Reviews MR0120174 (22 #10931).

This book reproduces the lecture notes of the author. Part I of this book discusses the axioms for continuous geometry and gives the construction of the numerical dimension function for the irreducible case. I Kaplansky observed that almost all results of Chapter V hold in the complemented modular complete lattices without the continuity axioms of lattice operations. The main contents of Part II of this book are the properties of regular rings and the proof of the coordinatization theorem. The author showed that every complemented modular lattice of order ≥ 4 can be represented as the lattice of all principal right ideals of a regular ring.
...
In printing this book, the original notes are carefully read, and many parts are slightly altered. These changes, together with some comments, are listed at the back of the book. The lecture notes reproduced in the present book do not cover all of the author's work on continuous geometry and its related topics, but a survey of his other published and unpublished work on these topics is given in a foreword by I Halperin. At the back of the present book, I Halperin gives a few comments about the articles of other authors on continuous geometry and related topics.

8.4. Review by: Garrett Birkhoff.
Quarterly of Applied Mathematics 19 (2) (1961), 136.

This is a carefully edited and corrected reprint of lecture notes multigraphed in 1935-37; it provides a definitive technical account of a fascinating generalisation of projective geometry, invented by von Neumann. This generalisation has applications to operators on Hilbert space, and hence it sheds some light on the foundations of quantum mechanics.

No discussion of these applications is given in this book. However, as a display of brilliant algebraic techniques and incisive axiomatic analysis, by one of the most versatile pure and applied mathematicians of this century, the book is still extremely stimulating. In particular, it is one of the most penetrating studies available of the abstract algebra of subspaces of function space under intersection and union, and of the related algebra of semisimple and "regular" rings.

9.1. Titles of works included.

The titles of the works included in Volume I are as follows: The Mathematician. Über die Lage der Nullstellen gewisser Minimumpolynome. Zur Einführung der transfiniten Zahlen. Eine Axiomatisierung der Mengenlehre. Egyenletesen sürü szamsorozatok. Zur Prüferschen Theorie der idealen Zahlen. Über die Grundlagen der Quantenmechanik. Zur Theorie der Darstellungen kontinuierlicher Gruppen. Mathematische Begründung der Quantenmechanik. Wahrscheinlichkeitstheoretischer Aufbau der Quantenmechanik. Thermodynamik quantenmechanischer Gesamtheiten. Zur Hilbertschen Beweistheorie. Die Zerlegung eines Intervalles in abzählbar viele kongruente Teilmengen. Ein System algebraisch unabhängiger Zahlen. Über die Definition durch transfinite Induktion und verwandte Fragen der allgemeinen Mengenlehre. Die Axiomatisierung der Mengenlehre. Einige Bemerkungen zur Diracschen Theorie des Drehelektrons. Zur Erklärung einiger Eigenschaften der Spektren aus der Quantenmechanik des Drehelektrons, I. Zur Erklärung einiger Eigenschaften der Spektren aus der Quantenmechanik des Drehelektrons, II. Zur Erklärung einiger Eigenschaften der Spektren aus der Quantenmechanik des Drehelektrons, III. Über eine Widerspruchfreiheitsfrage der axiomatischen Mengenlehre. Über die analytischen Eigenschaften von Gruppen linearer Transformationen und ihrer Darstellungen. Über merkwürdige diskrete Eigenwerte. Über das Verhalten von Eigenwerten bei adiabatischen Prozessen. Beweis des Ergodensatzes und des H-Theorems in der neuen Mechanik. Zur allgemeinen Theorie des Masses. Zusatz zur Arbeit "Zur allgemeinen Theorie des Masses".

9.2. Review by: Reuben Louis Goodstein.
The Mathematical Gazette 47 (359) (1963), 64-65.

It is the hallmark of a great mathematician that his output is prodigious and von Neumann was indeed a great mathematician. This first of six huge volumes contains his papers on Mathematical Logic, Set Theory and Quantum Mechanics and includes such fundamental works as the 1925 axiomatisation of set theory, the 1927 paper on the foundations of quantum mechanics, the subdivision of an interval into denumerably many congruent parts (1928), and the 1929 paper on analytic properties of groups of linear transformations. The articles have not been reset but have been reproduced from the original publications by photolithography, (so that some type faces appear unnaturally large, and others very small) but the volume has been well produced in a very attractive blue cloth cover. It is greatly to the credit of both Editor and Publisher that the production of this set of collected papers has been accomplished in so short a time.

9.3. Review by: Editors.
Mathematical Reviews MR0157871 (28 #1100).

These six volumes contain a reprinting of all the articles published by von Neumann, together with some of his reports to government agencies and other groups, and, perhaps even more important, an indication of unpublished manuscripts found in his files. The published papers, especially the earlier ones, follow essentially the chronological order of publication, although exceptions have been made in order that a certain unity could be preserved in certain fields.

One of the most useful functions of these well-edited volumes lies in their indication of the existence and contents of a number of unpublished manuscripts of von Neumann, because these manuscripts contain partial results or techniques of wide interest, even though the author probably felt that they were not yet suitable for publication at the time of his death. These manuscripts, together with certain other unfinished material, have been placed in the library of the Institute for Advanced Study at Princeton, and short descriptions (or reviews) of their contents are included in these six volumes. Each volume contains a complete bibliography of the scientific works included in this collection.

10. Collected works. Vol. II: Operators, ergodic theory and almost periodic functions in a group (1961), by John von Neumann.

Contents: Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren. Zur Algebra der Funktionaloperatoren und Theorie der normalen Operatoren. Zur Theorie der unbeschränkten Matrizen. Über einen Hilfssatz der Variationsrechung. Über Funktionen von Funktionaloperatoren. Algebraische Repräsentanten der Funktionen "bis auf eine Menge vom Masse Null". Die Eindeutigkeit der Schrödingerschen Operatoren. Bemerkungen zu den Ausführungen von Herrn St. Lesniewski über meine Arbeit "Zur Hilbertschen Beweistheorie". Die formalistische Grundlegung der Mathematik. Zum Beweise des Minkowskischen Satzes über Linearformen. Über adjungierte Funktional-operatoren. Proof of the quasi-ergodic hypothesis. Physical applications of the ergodic hypothesis. Dynamical systems of continuous spectra. Über einen Satz von Herrn M. H. Stone. Einige Sätze über messbare Abbildungen. Zur Operatorenmethode in der klassischen Mechanik. Zusätze zur Arbeit "Zur Operatorenmethode in der klassischen Mechanik". Die Einführung analytischer Parameter in topologischen Gruppen. A koordináta-mérés pontosságának határai az elektron Dirac-féle elméletében (Über die Grenzen der Koordinatenmessungs-Genauigkeit in der Diracschen Theorie des Elektrons). On an algebraic generalization of the quantum mechanical formalism. Zum Haarschen Mass in topologischen Gruppen. Almost periodic functions in a group. I. The Dirac equation in projective relativity. On complete topological spaces. Almost periodic functions in groups. II. Comparison of cells.

11.1. Titles of works included.

Contents: On a certain topology for rings of operators. On rings of operators. On rings of operators, II. On rings of operators, III. On rings of operators, IV. On infinite direct products. On rings of operators; Reduction theory. On some algebraical properties of operator rings. On an algebraic generalization of the quantum mechanical formalism (Part I). Characterization of factors of type II_1.

12.1. Titles of works included.

Contents: On compact solutions of operational-differential equations. I. Charakterisierung des Spektrums eines Integraloperators. On normal operators. On inner products in linear, metric spaces. The determination of representative elements in the residual classes of a Boolean algebra. The uniqueness of Haar's measure. The logic of quantum mechanics. Continuous geometry. Examples of continuous geometries. On regular rings. Algebraic theory of continuous geometries. Continuous rings and their arithmetics. On the transitivity of perspective mappings. Non-isomorphism of certain continuous rings (with introduction by I Kaplansky). Independence of $F_{∞}$ of the sequence $\nu$. Continuous geometries with a transition probability. Quantum logics (Strict- and probability-logics). Lattice abelian groups. On some analytic sets defined by transfinite induction. Some matrix-inequalities and metrization of matrix-space. Minimally almost periodic groups. Fourier integrals and metric geometry. Operator methods in classical mechanics, II. Approximative properties of matrices of high finite order. A theorem on unitary representations of semisimple Lie groups. Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes. Significance of Loewner's theorem in the quantum theory of collisions. On the permutability of self-adjoint operators. The cross-space of linear transformations, II. The cross-space of linear transformations, III. Measure in functional spaces. Representation of certain linear groups by unitary operators in Hilbert space. The mean square successive difference. Distribution of the ratio of the mean square successive difference to the variance. A further remark concerning the distribution of the ratio of the mean square successive difference to the variance. Tabulation of the probabilities for the ratio of the mean square successive difference to the variance. Optimum aiming at an imperfectly located target.

13.1. Titles of works included.

Contents: On the principles of large scale computing machines. Preliminary discussion of the logical design of an electronic computing instrument. Part I, Vol. I. Planning and coding of problems for an electronic computing instrument. Part II, Vol. I. Planning and coding of problems for an electronic computing instrument. Part II, Vol. II. Planning and coding of problems for an electronic computing instrument. Part II, Vol. III. The future of high-speed computing. The NORO and problems in high-speed computing. Entwicklung und Ausnutzung neuerer mathematischer Maschinen. The general and logical theory of automata. Probabilistic logics and the synthesis of reliable organisms from unreliable components. Non-linear capacitance or inductance switching, amplifying and memory devices. Notes on the photon-disequilibrium-amplification scheme. Solution of linear systems of high order. Numerical inverting of matrices of high order. Numerical inverting of matrices of high order, II. The Jacobi method for real symmetric matrices. A study of a numerical solution to a two-dimensional hydrodynamical problem. On the numerical solution of partial differential equations of parabolic type. First report on the numerical calculation of flow problems. Second report on the numerical calculation of flow problems. Statistical methods in neutron diffusion. Statistical treatment of values of first 2000 decimal digits of $e$ and of $\pi$ calculated on the ENIAC. Various techniques used in connection with random digits. A numerical study of a conjecture of Kummer. Continued fraction expansion of $L_{n}$ .

13.2. Review by: I O.
Journal of the American Statistical Association 59 (307) (1964), 981.

This is one of six volumes of the Collected Works of John von Neumann (edited by A H Taub). Volume V consists of 25 papers devoted to von Neumann's work in Design of Computers, Theory of Automata, and Numerical Analysis. In this volume one senses the beginnings and development of areas of research which have occupied the mathematical community for many years.

The following sections may be of particular interest to the statistician. In "Statistical Methods in Neutron Diffusion" there is some correspondence dated 1947 in which is discussed the "method of solving diffusion problems in which data are chosen at random to represent a number of neutrons in a chain-reacting system." In 1950, there is the statistical treatment of the first zero decimal digits of $e$ and of $\pi$ in which tests for randomness for $\pi$ and $e$ were performed. In "Various Techniques Used in Connection with Random Digits" there are remarks on how to generate random deviates from a given population.

14.1. Titles of works included.

Contents: Zur Theorie der Gesellschaftsspiele. Communication on the Borel Notes. A model of general economic equilibrium. Solutions of games by differential equations. A certain zero-sum two-person game equivalent to the optimal assignment problem. Two variants of poker. A numerical method to determine optimum strategy. Discussion of a maximum problem. Numerical method for determination of value and best strategies of 0-sum, 2-person game with large number of strategies. Symmetric solutions of some general N person games. The impact of recent developments in science on the economy and on economics. The statistics of the gravitational field arising from a random distribution of stars, I. The statistics of the gravitational field arising from a random distribution of stars, II. Static solution of Einstein field equation for perfect fluid with $T_{\rho}^{\rho}= 0$. On the relativistic gas-degeneracy and the collapsed configuration of stars. The point source model. The point source solution, assuming a degeneracy of the semi-relativistic type $p = K\rho^{4/3}$ over the entire star. Discussion of de Sitter's space and of Dirac's equation in it. Theory of shock waves. Theory of detonation waves. The point source solution. Oblique reflection of shocks. Refraction, intersection and reflection of shock waves. The Mach effect and the height of burst. Discussion on the existence and uniqueness or multiplicity of solutions of the aerodynamical equations. Proposal and analysis of a numerical method for the treatment of hydrodynamical shock problems. A method for the numerical calculation of hydrodynamic shocks. Blast wave calculation. Numerical integration of the barotropic vorticity equation. Taylor instability at the boundary of two incompressible liquids. Taylor instability problem. Recent theories of turbulence. Description of the conformal mapping method for the integration of partial differential equation systems with 1+2 independent variables. The role of mathematics in the sciences and in society. Method in the physical sciences. Can we survive technology? Impact of atomic energy on the physical and chemical sciences. Defense in atomic war. Discussion remark concerning paper of C S Smith entitled, "Grain shapes and other metallurgical applications of topology".

14.2. Review by: I O.
Journal of the American Statistical Association 59 (307) (1964), 981.

This is one of six volumes of the Collected Works of John von Neumann (edited by A H Taub). Volume VI consists of 41 papers devoted to his work on Theory of Games, Astrophysics, Hydrodynamics, and Meteorology. In this volume one senses the beginnings and development of areas of research which have occupied the mathematical community for many years.

Volume VI deals mostly with technical aspects of game theory and problems of physics. Two papers (with S Chandrasekhar) are concerned with "a general analysis of the statistical aspects of the fluctuating gravitational field arising from a random distribution of mass centres may be expected to provide the necessary basis for several problems of stellar dynamics." There are also included a number of speeches and papers of general interest. These deal with his view concerning the role of mathematics, methodology in the physical sciences, and various aspects of atomic energy.

15.1. From the Preface by Arthur W Burks.

In the late 1940's John von Neumann began to develop a theory of automata. He envisaged a systematic theory which would be mathematical and logical in form and which would contribute in an essential way to our understanding of natural systems (natural automata) as well as to our understanding of both analogue and digital computers (artificial automata).

To this end von Neumann produced five works, in the following order:

(1) "The General and Logical Theory of Automata." Read at the Hixon Symposium in September, 1948; published in 1951. Collected Works 5, 288-328.

(2) "Theory and Organization of Complicated Automata." Five lectures delivered at the University of Illinois in December, 1949. This is Part 1 of the present volume.

(3) "Probabilistic Logics and the Synthesis of Reliable Organisms from Unreliable Components." Lectures given at the California Institute of Technology in January, 1952. Collected Works 5, 329-378.

(4) "The Theory of Automata: Construction, Reproduction, Homogeneity." von Neumann started this manuscript in the fall of 1952 and continued working on it for about a year. This is Part II of the present volume.

(5) The Computer and the Brain. Written during 1955 and 1956; published in 1958.

The second and fourth of these were left at his death in a manuscript form which required extensive editing. As edited they constitute the two parts of the present volume, which thus concludes von Neumann's work on the theory of automata.

15.2. Review by: Jacob T Schwartz.
Mathematics of Computation 21 (100) (1967), 745.

This volume consists of a meticulously edited version of a series of five lectures on basic computer theory given by von Neumann at the University of Illinois in December 1949, together with an extensive but unfinished manuscript on computer self-reproduction written by von Neumann in 1952-1953. The Illinois lectures are particularly interesting for von Neumann's digressions on the future and general significance of computers as seen by him in 1949. Some of his comments on the problem of complexity are still highly apropos, perhaps more in connection with software than with hardware.

The manuscript forming the second part of the book is a good example of von Neumann's very brilliant mathematical style, but is perhaps somewhat disappointing in the result which it presents. Consider an infinite set of small Turing machines, all but one initially in a wait state and with blank tapes, and each capable of writing onto the tape of its neighbours and of putting its neighbours into an initial active state. It is then reasonably clear that by copying its own tape onto the tape of one of its neighbours and starting this neighbour, a Turing machine is able to initiate a process of self-reproduction of a suitable given set of tapes. Von Neumann's paper expands upon this observation, showing by explicit construction that both the basic Turing machines and their tapes can be simulated in a hypothetical crystal medium, each of whose points is an elementary 29-state automaton.

The editor provides a well-written and instructive historical account of von Neumann's work with computers.

15.3. Review by: Anatol Rapoport.
BioScience 17 (9) (1967), 659-660.

The book is an extensively edited collection of lectures and uncompleted chapters of a work interrupted by von Neumann's death in 1957 at the age of 53. The editor undertook the formidable task of reconstructing the outlines of the theory from the fragments. Professor Burks' profound knowledge of the field and his intimate acquaintance with the content and thrust of von Neumann's ideas has enabled him, by the use of extensive interpolated comments end elucidations, to weld this unfinished testament into a form which will doubtless inspire others to continue these remarkable investigations.

An automaton is a device whose behaviour at any given moment is determined by its inner state and by the input impinging upon it. The input is usually thought of as being "fed" into the automaton on a tape. The "behaviour" of the automaton is its input, which can include making changes in the tape. Thus an automaton "takes cognisance" of the environment (i.e., "reads" a portion of the tape), "makes decisions" (according to the appropriate rule), and produces a signal (which may affect both the environment and future inputs). The physical features of the situation (i.e., the way the automaton perceives the inputs, the physical events instigated by the output signal) are not central in the theory of automata. The central problems deal with the logical relationships between the inputs and the outputs, in particular, with the degree of complexity of behaviour of which a given automaton or a class of automata is capable. Thus it is known that any pattern of behaviour specifiable by propositions involving the fundamental logical connectives, "or," "if ... then," "not," and their combinations can be realised by a network of quite simple automata.
...

It would seem that the insights derived from cybernetics and information theory, the sciences stimulated by the new technology, have brought us nearer to incorporating biology in the expanding domain of the exact sciences, since information processing and the exercise of control are precisely the features which distinguish the living from the non-living. The distance from these mathematical formulations to concrete theories of brain function and reproduction is, however, still immense. Only if the biologist is intrigued by the possibility that the gap will some day be bridged will it be worth his while to push his way through the labyrinths of Part II of the present volume (including 56 diagrams), in which the design of a self-reproducing automaton is developed together with the numerous problems which are spawned in the process. Those who cannot follow the maze of instructions and deductions will do better to read Part I, "Theory and Organization of Complicated Automata" (which is good, clear prose), Chapter 1 of Part II, which contains the central questions in the theory of automata, and Burks' concluding remarks, wherein he summarises the answers which von Neumann obtained to these questions. Perhaps these intriguing questions and answers will be an additional stimulus to read the remainder of Part II, "The Theory of Automata: Construction, Reproduction, and Homogeneity."

15.4. Review by: Jay Goldman.
American Scientist 55 (4) (1967), 497A.

This volume completes the publication of John Von Neumann's lectures and manuscripts on a theory of automata he was developing during the last years of his life.

The book is divided into three parts. First we have an introduction by the editor describing Von Neumann's work on computers and automata theory. The second part reproduces a set of lectures delivered in 1949 entitled "Theory and Organization of Complicated Automata." Some topics discussed include: computing machines in general, conceptual and numerical methods in mathematics, analogue and digital procedures, rigorous theories of control and information, statistical theories of information, thermodynamical aspects, the role of complexity, computing machines and the nervous system, size of automata, the problem of reliability, self-reproduction.

It is particularly interesting to compare these lectures with Von Neumann's later book The Computer and the Brain, and to see the development of many of his ideas over a period of four or five years.

The third part of this book, comprising more than half the volume, a manuscript entitled "The Theory of Automata: Construction, Reproduction, Homogeneity," describes the construction and properties of a specific self-producing automaton.

This volume is truly a sourcebook of ideas and, although it represents Von Neumann's early thoughts on this subject, it will undoubtedly influence the development of automata theory for many years to come.

In the editor's words, "The manuscripts for both parts of the present volume were unfinished; indeed, they were both, in a sense, first drafts." The editor has completed a very difficult task in a masterful fashion. This reconstruction of ambiguous parts of the manuscripts and his clarifying comments have helped to produce an out standing work. Again in his words "There is one compensation in this (i.e., the early draft): one can see Von Neumann's powerful mind at work."

It is wonderful to see this and I recommend it to all.

15.5. Review by: Homer Jacobson.
The Quarterly Review of Biology 42 (4) (1967), 521-523.

The long-awaited von Neumann manuscript on the two-dimensional cell model of self-reproduction, heavily edited, expatiated, corrected, and completed by A. W. Burks, has appeared. I read it with a suspense that a dozen years had failed to extinguish. It was publicly described in 1955 in a tantalisingly incomplete Scientific American article, and had been mentioned in trade grapevine circles; the whole story is nevertheless most welcome. Von Neumann's grasp, as early as 1948, of the possibility of using a Turing machine as a self-reproducing analogue, showed a pioneering awareness of the possibility of designing one, at least in abstracto.

The work itself is the usual imaginative von Neumann, especially in the light of the fact that it was conceived in the early fifties. It is divided into two parts, an essentially verbal one and a mathematical one - very strongly reminiscent of Wiener's Cybernetics in organisation. The first sections of both read nearly like novels, and both say relatively little here. This first section contains material which is qualitative, dated, most general, and not likely to be helpful to workers in the field of automata, except as history. Most of the circuitry is geared to a vacuum tube technology, and mention of our newer speculations on the neurophysiology of memory is naturally absent. A surprisingly prescient reference to the likelihood of a chemical memory storage, on quantitative informational grounds, was made, however. The body of worthwhile content lies, as in Cybernetics, in the second section. While Wiener's comments brook no compromise with his desire to communicate only with the cognoscenti, this work shows von Neumann's interest, and Burks' care, in presenting the results intelligibly. Much of this material is perforce turgid, as it must include a sort of super-appendix of needful grubby details. Its primary message is that a particular properly designed model, consisting of identical two-dimensional cells, filling a two-dimensional lattice-universe of discourse, will model the essentials of the process of self-reproduction and of the related process of universal duplication. This message will carry to those who invest the time needed to understand the details of the model. The details of the cells themselves are well worth considering; one has the feeling that these might be used for something other than self-reproducing machines. This, plus some perusal of the block diagrams of the overall model, will give the casual reader the essentials. Construction of the readout and effector organs is rather complex, and of most interest to those who take a primary interest in analogue circuitry.

Unfortunately, the publication rate of designers of self-reproduction models has been reciprocal to their conception rates. The long-past actual constructions of the Penrose and the Jacobson models have demonstrated von Neumann's early thesis to the body scientific. We nevertheless welcome his designs, and their completion by Burks, as an elegant and different way of looking at the problem.

15.6. Review by: John G Kemeny.
Science New Series 157 (3785) (1967), 180.

The scientific community is indebted to Arthur W Burks for completing this unfinished work of the late John von Neumann. An immense effort must have gone into the collection of the fragmentary manuscripts, obtaining information from friends and co-workers of von Neumann, and completing the project without doing violence to the intentions of the author.

The volume begins with an editor's introduction, which consists of a very interesting discussion of the contributions von Neumann made to the development of modern computers. Part 1 consists of a series of lectures which were delivered at the University of Illinois in 1949 and which are now somewhat dated. But part 2 contains a complete design for an automaton that can reproduce itself, and it is this part that makes the volume a significant contribution to the scientific literature. I shall restrict my comments to this part.

The theory of computers must be traced back to a fundamental result by A M Turing, 30 years ago, establishing the possibility of a computer that is "universal." Such a computer, given appropriate instructions, can do anything any computer can do. Von Neumann wished to prove a companion result, to show that there exist universal construction automata. Indeed, the book concludes by showing the existence of an automaton that is a universal computer, a universal constructor, and able to reproduce itself.

For simplicity the model of the automaton is constructed in two dimensions. It is assumed that the plane is divided into square cells, all of which are alike. Each cell can be in one of 29 states, and construction is accomplished by changing the states of remote cells. A self-reproducing automaton imbedded in inert surroundings will through a long sequence of steps change the surroundings until a copy of itself exists at a remote location.

Sixteen of the states are used for transmission of information. They can be hooked up to build lines through which information travels in any desired direction. They transmit two types of signals, one of which is for ordinary transmission and the other is used to "kill," that is, to re turn a state to its inert status. Four states do most of the work, containing the logic of the system. Eight states may be viewed as embryonic, transitional states in the changing of an inert cell to an active one. And the 29th state is the inert one. The process has a discrete time-structure built into it. The state of any cell in the next moment is determined by its own state at the present, and by the state of its four neighbouring cells. That is all the machinery that von Neumann requires.

He first shows how, with ingenuity, a wide variety of specialised organs may be built of these simple cells. Then he uses the components to build his universal automaton. The latter consists of three major parts. One is the construction unit (CD), which uses information supplied by its memory to construct a remote automaton. This is quite similar to a universal Turing machine. Indeed it can be constructed so that it also serves as a universal computer. The second part is the memory control unit (MC), which extracts information from the memory and changes this information. The memory itself is contained in a linear array (L), in a simple code. The array is manipulated by extendable arms under the control of MC.

The basic idea is really very simple: A universal construction automaton is designed to follow instructions, coded in its memory, to construct an automaton. It also has the ability to copy its own instructions. When such a unit is given its own description, it will copy itself together with these instructions, and will thus have reproduced itself. If the newly horn automaton is then sent a starting signal, it will in tum reproduce itself. While this is a simple and ingenious idea, its complete implementation is a monumental task.

Burks deserves credit for the editing of a manuscript that was often no more than a sketch, for the correction of various minor errors, and for the completion of the design of the self-reproducing automaton . He has carefully preserved as much of the original manuscript as possible, and sections written by him are specially marked. Indeed, my one criticism is that the editor has sacrificed readability for historical accuracy.

An interesting philosophical question may be raised in connection with the model constructed. It is capable of parallel processing in the strongest sense - that is, a change may occur in every cell at every moment. Yet only a minute fraction of this capability is used in the construction, just as our computers do parallel processing only in a most trivial sense. Wouldn't a computer all of whose memory cells could be changed simultaneously be a vastly more powerful tool than our present machines? And does this furnish a clue to why the human mind is so incredibly efficient compared with computers?

16.1. From the Publisher.

This volume brings together for the first time John von Neumann's long-out-of-print articles on computer architecture, programming, large-scale computing, and automata theory. A number of significant papers in these areas that were not included in the multivolume John von Neumann. Collected Works (1963) have now been reprinted here. These pioneering articles - written between the mid-1940s and the mid-1950s - are of enduring value not only to computer historians but to computer scientists at the vanguard of current research. Most of today's computers are still constructed in accordance with the "von Neumann architecture," and his technique of flow charting remains basic in the domain. Papers of John von Neumann on Computers and Computer Theory is volume 12 in the Charles Babbage Institute Reprint Series for the History of Computing.

16.2. Review by: David K Allison.
Isis 78 (4) (1987), 603.

John von Neumann was a central figure in the early history of the electronic digital computer. As a mathematician, he developed coherent logical structures for computer design. As an innovator, he pioneered both technical advances and computer applications in many different areas. As a leader in academic, governmental, and business circles, he fostered institutional support. His involvement not only bestowed legitimacy, but also brought intellectual excitement to an area that in the 1940s and 1950s might have tended toward "mere instrumentation" in the service of such fields as atomic physics and aerodynamics. In this collection William Aspray and Arthur Burks provide a comprehensive, although incomplete, compendium of von Neumann's major writings related to computers. Most of the papers were actually collaborative efforts of von Neumann with Herman Goldstine and Burks himself, both of whom became associated with von Neumann during World War II.
...
The historical analysis here is limited to a brief biographical sketch and technical context for the papers. The editors make no serious attempt to relate von Neumann's intellectual contributions to the broader social and institutional context in which he worked. Nonetheless, the collection delivers what it promises and will be an essential source for anyone with a deep interest in von Neumann or in the early conceptual history of the computer.

17.1. From the Publisher.

After three decades since the first nearly complete edition of John von Neumann's papers, this book is a valuable selection of those papers and excerpts of his books that are most characteristic of his activity, and reveal that of his continuous influence. The results receiving the 1994 Nobel Prizes in economy deeply rooted in Neumann's game theory are only minor traces of his exceptionally broad spectrum of creativity and stimulation. The book is organised by the specific subjects - quantum mechanics, ergodic theory, operator algebra, hydrodynamics, economics, computers, science and society. In addition, one paper which was written in German will be translated and published in English for the first time. The sections are introduced by short explanatory notes with an emphasis on recent developments based on von Neumann's contributions. An overall picture is provided by Ulam's, one of his most intimate partners in thinking, 1958 memorial lecture. Facsimilae and translations of some of his personal letters and a newly completed bibliography based on von Neumann's own careful compilation are added.

17.2. Review by: Matthew Donald.
Mathematical Reviews MR1355812 (98b:01041).

This collection is a fascinating introduction to the work of John von Neumann. It consists mainly of reprints of his most fundamental contributions to a number of fields. We are also given Ulam's extensive biographical notice from the 1958 AMS Bulletin, a selection of von Neumann's speeches and papers on general problems of science and society, and a newly-revised bibliography. Set theory is the most significant area of von Neumann's mathematical work which is not represented. Each section of the compendium begins with a modern introductory note. These are all short, except for the note by Petz and Rédei on operator algebras, in which they sketch some of the significant developments of von Neumann's legacy in this area. It should be noted that, in the operator algebras section, only the tables of contents and introductions are provided for those papers which are marked as excerpts.

The papers collected here demonstrate that, as well as being a great scientist with an awesome range, von Neumann was also a master of exposition. He presented his ideas clearly and thoughtfully and paid much attention to the wider implications of his work. This means that this compendium has much to offer even to the casual browser. It will also be relevant and interesting to those working today in the fields on which von Neumann had such enormous influence.

18.1. From the Publisher.

Mathematical Foundations of Quantum Mechanics was a revolutionary book that caused a sea change in theoretical physics. Here, John von Neumann, one of the leading mathematicians of the twentieth century, shows that great insights in quantum physics can be obtained by exploring the mathematical structure of quantum mechanics. He begins by presenting the theory of Hermitian operators and Hilbert spaces. These provide the framework for transformation theory, which von Neumann regards as the definitive form of quantum mechanics. Using this theory, he attacks with mathematical rigour some of the general problems of quantum theory, such as quantum statistical mechanics as well as measurement processes. Regarded as a tour de force at the time of publication, this book is still indispensable for those interested in the fundamental issues of quantum mechanics.

18.2. Review by: Stanley P Gudder.
Mathematical Reviews MR1435976 (98b:81006).

Many mathematics researchers believe that John von Neumann and David Hilbert were the two greatest mathematicians of the twentieth century. Even those who have other favourite candidates will probably agree that von Neumann is among the top five of such a list. Unlike most mathematics investigators, we can easily gain a glimpse of von Neumann's personality from his writings. Here was a playful man, a man who was bold and fearless. He was not afraid to experiment, to calculate and to reveal his thought processes in public. In the late 1950s and early 1960s when the reviewer and some of his colleagues were in graduate school, von Neumann's book was the only one available (in English) that presented quantum mechanics in a rigorous mathematical fashion. And what a delight it was. It was insightful, motivated, intuitive and understandable physics and it was mathematically consistent. When we took courses in functional analysis, we found that we were seeing the same things over again. Quantum mechanics was Hilbert space analysis and, conversely, much of Hilbert space analysis was quantum mechanics. It was exciting to see that such an elegant, abstract field of mathematics was applicable to revolutionary studies of the natural world. Quantum mechanics was the greatest scientific revolution since Newtonian mechanics. Hilbert space theory was to quantum mechanics as the calculus was to Newtonian mechanics, and von Neumann held the leading beacon of light.

19.1. From the Preface.

In 1940-1941 von Neumann lectured on invariant measures at the Institute for Advanced Study. This book is essentially a written version of what he said.

The lectures began with general measure theory and went on to Haar measure and some of its generalisations. Shizuo Kakutani was at the Institute that year, and he and von Neumann had many conversations on the subject. The conversations revealed facts and produced proofs - quite a bit of the content of the course, especially toward the end, was discovered just a week or two or three before it appeared on the blackboard. The original version of these notes was prepared by Paul Halmos, von Neumann's assistant that year. Von Neumann read the handwritten version before it went to the typist, and sometimes scribbled comments on the margins. On Chapter VI, the last one, he did more than scribble - he himself wrote most of it.

The notes were typed. Two or three copies were kept in the Institute - von Neumann had one and the Institute library had another. Since then a few photocopies have been made, but until now the notes have never been published in any proper sense of the word.

20.1. Review by: Gualtiero Piccinini.
Minds and Machines 13 (2) (2003), 327-332.

When John von Neumann turned his interest to computers, he was one of the leading mathematicians of his time. In the 1940s, he helped design two of the first stored-program digital electronic computers. He authored reports explaining the functional organisation of modern computers for the first time, thereby influencing their construction worldwide (von Neumann, 1945; Burks et al., 1946). In the first of these reports, von Neumann described the computer as analogous to a brain, with an input "organ" (analogous to sensory neurons), a memory, an arithmetical and a logical "organ" (analogous to associative neurons), and an output "organ" (analogous to motor neurons).

His experience with computers convinced him that brains and computers, both having to do with the processing of information, should be studied by a new discipline - automata theory. In fact, according to von Neumann, automata theory would cover not only computers and brains, but also any biological or artificial systems that dealt with information and control, including robots and genes. Von Neumann never formulated a full-blown mathematical theory of automata, but he wrote several important exploratory papers (von Neumann, 1951, 1956, 1966). Meanwhile, besides designing hardware, he developed some of the first programs, programming languages, programming techniques, and numerical methods for solving mathematical problems using computers. (Much of his work on computing is reprinted in Aspray and Burks, 1987.) Shortly before his death in 1956, he wrote an informal synthesis of his views about brains. Though von Neumann left his manuscript sketchy and unfinished, Yale University Press published it as The Computer and the Brain in 1958. The 2000 reprint of this small but informative book is an opportunity to learn, or be reminded of, von Neumann's thoughts on the computational organisation of the mind-brain.
...
Von Neumann was one of the top experts in all aspects of computing - hardware, programming, numerical analysis, and computability theory - and one of the most rigorous minds ever to discuss the computational organisation of brains. His last book presents one of the most sophisticated comparisons ever made between computers and brains, with the hope that a new discipline, automata theory, would soon develop to explain them both. Automata theory never materialised in the form that von Neumann dreamt: mathematicians developed A Turing's computability theory in a way that applies to artificial computers; engineers went on to use C Shannon's theory of information for their communication problems; while neuroscientists studied neural mechanisms by their own means. But the idea that brains are, somehow, computers has remained a fundamental assumption of many psychologists, philosophers, and neuroscientists of today. The Computer and the Brain is an important historical document, from the era in which the computational theory of the mind-brain was introduced and explored in detail. It's a landmark in the history of computing, psychology, and neuroscience, and it's required reading for anyone interested in the foundations of those disciplines.

21.1. From the Publisher.

This is the classic work upon which modern-day game theory is based. What began more than sixty years ago as a modest proposal that a mathematician and an economist write a short paper together blossomed, in 1944, when Princeton University Press published Theory of Games and Economic Behavior. In it, John von Neumann and Oskar Morgenstern conceived a groundbreaking mathematical theory of economic and social organisation, based on a theory of games of strategy. Not only would this revolutionise economics, but the entirely new field of scientific inquiry it yielded - game theory - has since been widely used to analyse a host of real-world phenomena from arms races to optimal policy choices of presidential candidates, from vaccination policy to major league baseball salary negotiations. And it is today established throughout both the social sciences and a wide range of other sciences.

This sixtieth anniversary edition includes not only the original text but also an introduction by Harold Kuhn, an afterword by Ariel Rubinstein, and reviews and articles on the book that appeared at the time of its original publication in the New York Times, the American Economic Review, and a variety of other publications. Together, these writings provide readers a matchless opportunity to more fully appreciate a work whose influence will yet resound for generations to come.

21.2. Review by: Nicola Giocoli.
History of Economic Ideas 13 (3) (2005), 152-156.

This is a reprint of the classic work that six decades ago established the foundations of the theoretical core of contemporary economics, namely, game theory. One cannot but praise the publisher's decision to produce a new, celebratory edition of such an intellectual milestone for the whole of 20th-century social sciences: indeed, the TGEB (as the Theory of Games and Economic Behavior is usually indicated) is one of the last economics books that truly deserves to stand alongside the old masterpieces, like Adam Smith's Wealth of the Nations, Alfred Marshall's Principles or John Maynard Keynes's General Theory. As is well known, in fact, post-World War II economics has become a science of papers, more than of books, so much so that were Jacob Marschak still alive, he would have serious troubles to single out - as he famously put it - ten more volumes of equivalent importance for the progress of economics published during the last 60 years.

Yet, the present edition is much more than just a reprint. The volume also contains a lot of extra material which is of invaluable interest for the historian of modern economics. This includes an introduction by Harold Kuhn, with some personal recollections of the very first years of the new discipline at Princeton mathematics department, an afterword by Ariel Rubinstein, offering a few stimulating reflections on what game theory is really about, and a reprint of Oskar Morgenstern's 1976 JEL paper on how the TGEB came to be. But the real measure of the new edition is the reprint of eleven of the several reviews and comments that appeared between 1945 and the mid-1960s in academic journals, newspapers and magazines. These pages should give the historian a vivid impression of the deep impact that the TGEB had on a rather broad audience which far exceeded the restricted milieu of economists and mathematicians.

22.1. From the Publisher.

John von Neumann was perhaps the most influential mathematician of the twentieth century. Not only did he contribute to almost all branches of mathematics, he created new fields and was a pioneering influence in the development of computer science.

During and after World War II, he was a much sought-after technical advisor. He served as a member of the Scientific Advisory Committee at the Ballistic Research Laboratories, the Navy Bureau of Ordinance, and the Armed Forces Special Weapons Project. He was a consultant to the Los Alamos Scientific Laboratory and was appointed by U.S. President Dwight D Eisenhower to the Atomic Energy Commission. He received the Albert Einstein Commemorative Award, the Enrico Fermi Award, and the Medal of Freedom.

This collection of about 150 of von Neumann's letters to colleagues, friends, government officials, and others illustrates both his brilliance and his strong sense of responsibility. It is the first substantial collection of his letters, giving a rare inside glimpse of his thinking on mathematics, physics, computer science, science management, education, consulting, politics, and war. With an introductory chapter describing the many aspects of von Neumann's scientific, political, and social activities, this book makes great reading. Readers of quite diverse backgrounds will be fascinated by this first-hand look at one of the towering figures of twentieth century science.

22.2. From the Preface by Miklós Rédei.

The present volume is a collection of selected letters written by John von Neumann (1903, Budapest, Hungary; 1957, Washington, D.C., U.S.A.). Apart from a short volume containing von Neumann's letters (written and published in Hungarian) to Rudolf Ortvay [36], the present volume is the first and only substantial published collection of letters by von Neumann.

While substantial, this is a strongly selected volume since von Neumann had written hundreds more letters. Many of those not selected for the volume are not suitable for publication for one reason or another, many could in principle have been included however. The guiding principle of selection was that the letters published here should contribute to our understanding of John von Neumann as a scientist - broadly interpreted - and as a public figure. The volume also should be interesting for historians of science, especially of mathematics and mathematical physics. Accordingly, letters of exclusively private nature or content are not included in the volume. Also excluded from the collection are letters of reference written on behalf of colleagues - irrespective of whether the persons involved are still alive. Some of von Neumann's letters are still classified hence not accessible for scholarly research, and there may exist letters in private property, not available in archives, which could have been included in the volume had they been known to me. Keeping the volume within a reasonable size also restricted the number of letters that could be included. Thus the selection has been both disciplined and determined by contingent factors.

The letters published here vary in content, style and length. Some are on very general issues that are easy to understand but some concern very technical topics in mathematics and mathematical physics and, naturally, not all technical letters are self-contained. The "Introductory Comments" are intended to facilitate reading the collection, especially the technical letters, by giving brief and concise reviews of the related technical background and by putting some of the letters into perspective. The organisation of the introductory comments is orthogonal to the organisation of the rest of the volume: while von Neumann's letters are arranged in alphabetic order of the addressees (given an addressee the letters are arranged in order of their dates) the comments are thematically grouped.

22.3. From the Foreword by Peter D Lax.

Everybody who ever met von Neumann was astounded by the speed, power, depth and range of his thinking. In a film about him, distributed on VHS by the Mathematical Association of America, the physicist Hans Bethe remarks, only half in jest, that von Neumann's brain seemed like an upward mutation in the human species.

To gain a measure of von Neumann's achievements, consider that had he lived a normal span of years, he would certainly have been a recipient of a Nobel Prize in economics. And if there were Nobel Prizes in computer science and mathematics, he would have been honoured by these, too. So the writer of these letters should be thought of as a triple Nobel laureate or, possibly, a 3 1/2-fo1d winner, for his work in physics, in particular, quantum mechanics.

This collection of letters opens a window on von Neumann's way of thinking, his interests, his relation to people, and his personality. One can make a few general observations. He was exquisitely polite, as shown for instance in his letter to Dixmier. Even in his letter of resignation from the German Mathematical Society, written in 1935 to Blaschke, he maintains an ice-cold politeness. On the other hand, his letter to Johannes Stroux, President of the German Academy in East Berlin, declining an offer of membership, is tinged with genuine regret.

Von Neumann was a dedicated liberal. In his 1949 letter to Lyman Spitzer, then chairman of the Scientists' Committee on Loyalty Problems, von Neumann brands as pernicious a suggestion that clearance be required for unclassified research. He states eloquently that the training of talented people in science should be done with absolutely no consideration for anything other than scientific ability.

Von Neumann had a deep interest in world affairs, especially the coming of World War II . His judgments, expressed in letters to Veblen and Ortvay, are prescient, displaying his great powers of analysis, and profound knowledge of history. Von Neumann wrote with a light touch; his letters are full of levity, tinged occasionally with black humour. In a letter to Schrödinger, he points out that the quantum entanglement of separated particles does not contradict the principle of locality, and illustrates his point with a charming off-colour joke.

Von Neumann liked to avoid generalities and dive right into the details of whatever he was dealing with. In writing to Cirker, President of Dover publications, about an enlarged edition of his book on quantum mechanics, he calculates to three figure accuracy the portion of new material. In a letter to Veblen he estimates carefully the cost of setting up Math Reviews. Then in a letter to T V Moore at Standard Oil he analyses in excruciating detail the efficient management of a fleet of 18 tankers, adding at the end "I realise that the above description is rather sketchy."

Von Neumann was very receptive to ideas of others and was generous with his praise, as in his letter to Tannaka. He had unbounded admiration for Gödel. When he first learned of Gödel's incompleteness theorem, he realised that it implied that the consistency of a system of axioms cannot be proved within the system. Since Gödel had reached the same conclusion, von Neumann didn't publish his own work. Later he was instrumental in rescuing Gödel from Europe and getting a position for him at the Institute for Advanced Study.

Von Neumann was ready to help those who needed it. When his student Halperin was accused of espionage, he came to his defence. There is a touching correspondence with Ortvay about getting high school mathematics texts in Hungarian for an immigrant working man in Chattanooga, Tennessee.

The heart of this collection are the letters about mathematics, such as rings of operators, about quantum mechanics and quantum logic, superconductivity, computing and computers, neurology, etc. Both von Neumann and G D Birkhoff were deeply concerned about ergodicity in statistical mechanics; in some of these letters von Neumann expresses his annoyance with Birkhoff for unfairly scooping him on the ergodic theorem. Looking back, neither von Neumann's mean ergodic theorem nor Birkhoff's pointwise ergodic theorem can be applied in statistical mechanics, for it is very difficult to verify the hypothesis of metric transitivity. One of the difficulties is that many physical systems are not metrically transitive; this can be demonstrated by applying KAM theory, or even by numerical calculations that were not available in the time of Birkhoff and von Neumann. The ingenious theorem of Oxtoby and Ulam, that all measure preserving transformations are ergodic except for a set of first category, is irrelevant. Transformations that describe physical processes are highly differentiable and so form a set of first category in the maximum norm.

So why hasn't statistical mechanics collapsed? Jack Schwartz and Arthur Wightman have pointed out that in statistical mechanics ergodicity is an overkill; it would show that the time average of every continuous function equals its phase-space average. But in statistical mechanics we are not interested in every continuous function, only in those that have thermodynamic significance. This is a very special class of functions; the equality of their time and phase-space average is due to other reasons, such as their high degree of symmetry.

In a footnote to a letter to Rudolf Ortvay in Hungary, written in February 1939, von Neumann remarks that in the U.S. the uranium-barium disintegration is thought to be of great importance. Clearly at this date von Neumann must have been thinking of the theory of the nucleus, not of nuclear weapons.

The tragedy of von Neumann's early death has robbed mathematics and the sciences of a natural leader and an eloquent spokesman. These letters make it possible for the present generation to catch a glimpse of the most scintillating mind of the twentieth century. Those who wish to learn more about von Neumann and about his mathematics should look at the outstanding Memorial Issue of the Bulletin of the American Mathematical Society, vol. 64, no. 8, 1958, and at Norman Macrae's authoritative biography, distributed by the AMS.

22.4. From the Introduction by Marina von Neumann Whitman.

As John von Neumann's daughter, it is a privilege and a pleasure to write some introductory words to this collection of his letters so thoughtfully and insightfully collected and edited by Professor Redei. Those letters - perhaps the majority - that deal with mathematical subjects are beyond my reach. But that leaves many that gave me new insights into my father, glimpses into his life and personality that I was never fully aware of while he was alive.

The letters reminded me forcibly of what a remarkable polymath my father was. I knew, of course, of his contributions not only to mathematics but to physics, economics, and perhaps neurobiology as well. And I remembered that his knowledge of ancient and Byzantine history often left specialists in the field goggle-eyed. But it hadn't dawned on me until I read letters concerning organisational issues at Harvard, the University of Pennsylvania, and the Institute itself, that he had remarkable insights into such practical matters as well. I had forgotten also what a courtly, formal, unfailingly polite European gentleman he was, whether refusing an invitation from the World Affairs Council of Boston, gently informing a proud mother that her son's paper was worthless, or apologising for his tardiness in delivering a promised paper. In light of his disdain for the mathematical apparatus of classical economics, his letter refusing to write a review of Samuelson's Foundations of Economic Analysis reflects an impressive adherence to the old adage: "if you can't say something good about someone, don't say anything at all."

Having lived through a good bit of history myself by now, I am astounded at the prescience my father displayed in his letters of the 1930s regarding the increasing certainty of a war in Europe that would ultimately engulf the entire world, the certainty that the U.S. would intervene on the side of England, and the likely progress and outcome of that war. From the vantage point of the 21st century, his 1939 analysis of the role and motives of the United State is truly remarkable: "I admit that the U.S.A. could be imperialist. I would not be surprised if in 20 years it would become so. But today it is not yet. I think that in Europe the U.S.A. is understood just as little as Europe is understood here." For John von Neumann, his knowledge of history and his coolly logical analysis of current conditions combined to produce a Cassandra-like view of the future that was both a gift and a curse.

The letters also reveal, more or less in passing, how incredibly busy and peripatetic my father was; he seemed always to be about to embark on a trip or just returned from one. This perpetual motion was attributable not only to his cosmopolitan background but to the fact that, throughout much of his career, he led a double life: as an intellectual leader in the ivory tower of pure mathematics and as a man of action, in constant demand as an advisor, consultant and decision-maker to what is sometimes called the military-industrial complex of the United States. My own belief is that these two aspects of his double life, his wide-ranging activities as well as his strictly intellectual pursuits, were motivated by two profound convictions. The first was the overriding responsibility that each of us has to make full use of whatever intellectual capabilities we were endowed with. He had the scientist's passion for learning and discovery for its own sake and the genius's ego-driven concern for the significance and durability of his own contributions. The second was the critical importance of an environment of political freedom for the pursuit of the first, and for the welfare of mankind in general.

I'm convinced, in fact, that all his involvements with the halls of power were driven by his sense of the fragility of that freedom. By the beginning of the 1930s, if not even earlier, he became convinced that the lights of civilisation would be snuffed out all over Europe by the spread of totalitarianism from the right: Nazism and Fascism. So he made an unequivocal commitment to his home in the new world and to fight to preserve and re-establish freedom from that new beachhead.

In the 1940s and 1950s, he was equally convinced that the threat to civilisation now came from totalitarianism on the left, that is, Soviet Communism, and his commitment was just as unequivocal to fighting it with whatever weapons lay at hand, scientific and economic as well as military. It was a matter of utter indifference to him, I believe, whether the threat came from the right or from the left. What motivated both his intense involvement in the issues of the day and his uncompromisingly hardline attitude was his belief in the overriding importance of political freedom, his strong sense of its continuing fragility, and his convict ion that it was in the United States, and the passionate defence of the United States, that its best hope lay.

Now, a little bit about John von Neumann's legacy, from the vantage point of his daughter and only child. In particular, I will focus on his concerns during the last year or two of his life. Especially toward the end, but even before, he became deeply concerned about the question of his ongoing legacy, in two respects. One had to do with the durability of his work, his intellectual contributions; he was surprisingly insecure about whether his work would still be recognised "in a hundred years". Well, the hundred years he had in mind aren't up yet, but he might be reassured to know that the royalties I still receive on books he wrote in the 1930s, 40s, and 50s vastly exceed anything I receive on my own much more recent publications.

Interestingly enough, despite his prescience in matters of state, my father wasn't a very accurate prophet regarding what turns the practical applications of his pioneering work would take. For example, he clearly expected that the computer would have its impact primarily on scientific research and military work. He was particularly interested in its role in advancing the accuracy of weather forecasting, and ultimately, climate modification. I don't think progress in this area has gone nearly as far or as fast as he hoped and expected. Similarly, I think he anticipated that the theory of games would have a more immediate impact on military and business decision-making than in fact it did. He might have found it a bit ironic that when finally, in 1994, the role of game theory in economics was recognised with a Nobel Prize, the prize went not to the inventors of the basic theory (who were both long-since dead) but to the developers of an important advance - the analysis of equilibria in the theory of non-cooperative games. Unsurprisingly, one of the three winners was another Hungarian, John C Harsanyi.

On the other hand, if anyone had ever told him that the company I used to work for, General Motors, would produce and utilise literally millions of computers each year (each of the roughly eight million vehicles the company produces each year contains several, not to mention the ones in its plants and offices), I believe he would have been start led. The notion of adults fulminating against computers as corrupters of youth in the form of video games would have amused and perhaps secretly pleased the playful, childlike aspect of his personality.

In fact, my father foresaw the inadequacy not only of his own scientific forecasts but of such forecasts in general. In a 1955 article in Fortune magazine, he said: "All experience shows that technical changes profoundly transform political and social relationships. Experience also shows that these transformations are not a priori predictable, and that most contemporary first guesses concerning them are wrong."

The second focus of John von Neumann's concern about his earthly legacy was, to put it simply, me. I was his only offspring and, toward the end of his life, he became acutely conscious that all his eggs were in one basket, genetically speaking (if biological inaccuracy can be forgiven for the sake of metaphor). So he put tremendous pressure on me to perform up to the peak of my abilities, and made clear his displeasure with the path I appeared to be taking. I had married young, right out of college, and he thought that this was a bad beginning, simply because he feared (and it was a reasonable fear, in the 1950s) that a woman who married young was very probably reducing her chances of making a significant intellectual or professional contribution. Statistically he was right, of course, but I like to think that in this particular case he was wrong. I'm no John von Neumann, obviously, but I have had a reasonably successful and highly rewarding career as an academic economist, a presidential adviser, and a corporate executive. And, in all these careers, I have been mindful of his insistence that it is immoral not to make maximum use of one's intellectual capacities.

Beyond me is the next generation, the grandchildren whose accomplishments he couldn't foresee because he died far too early, before they were born or even contemplated. I'm deeply sorry that my father never got to know the grandson who has translated a ten-year-old's dream of "someday finding a cure for cancer" into a career as a molecular biologist/biochemist doing research on the chemistry of intercellular message transmission as a professor at the Harvard Medical School. I'm equally sorry that he could never know the granddaughter who is a physician and a teacher of physicians at the Yale Medical School. It's too early yet to tell about the generation beyond, the great-grandchildren who are only eight and five years old, but there is no question that they have been indoctrinated from birth on the importance of the life of the mind. John von Neumann would have felt reassured and gratified, I believe, by the choices his progeny and his progeny's progeny have made, to do what he considered most important, that is, utilise our intellectual capacities up to their limits to fulfil whatever potential we have. He would have felt both relieved and vindicated that the forces of democracy and freedom, although as ever under threat, have gained ground against the forces of intolerance and totalitarianism, including recapturing the land of his birth, Hungary. He would have been surprised and perhaps amused, as well as disappointed in some areas, at the twists and t urns his contributions have taken in affecting our everyday lives. He would have felt reassured that we are all still here to reflect on such matters in 2005, given the fears he expressed in the 1955 Fortune article I referred to that mankind might destroy itself before 1980. And he would be gratified to know that the Mathematical Societies of the two nations on whose battle against Armageddon he focused his enormous physical and intellectual energies during the Second World War are jointly publishing a collection of letters that will provide remarkable insights into the man and his work for a generation that never knew him.

22.5. Review by: Leo Corry.
Mathematical Reviews MR2210045 (2006m:01020).

This book presents the reader with a selection of letters written by John von Neumann. It is intended as a contribution to a broader understanding of von Neumann the scientist, the public figure, and the man. The letters to 69 different correspondents vary in contents and in style. The spectrum of topics discussed is very broad. They include, among others, very technical topics in mathematics and in mathematical physics, suggestions to a Standard Oil executive on a logistic problem related to worldwide operations of oil tankers, remarks to the editor of the Washington Evening Star on a misleading article published in that journal on aerial bombing methods, and explanations about the kind of practical tasks that high-speed computers could help to solve. Introductory comments written by the editor are highly useful for reading these letters within the proper historical context, and more generally, for getting a broad picture of von Neumann's always astonishing horizon of scientific achievements. A brief introduction by von Neumann's only daughter, Marina von Neumann Whitman, adds an illuminating perspective on the personal side of the story.
...
Of the many interesting examples that could be brought up here, I mention two letters to Kurt Gödel in late 1930-early 1931. Von Neumann had attended the 1930 conference in Königsberg where Gödel announced the proof of his first incompleteness theorem. Unlike many others, von Neumann immediately understood the deep significance of this result and very soon came up with the so-called second Gödel theorem for which he apparently had a proof. From his correspondence we understand that von Neumann realised that Gödel also had reached this result and thus decided not to publish his own. Other letters in the collection indicate that, although von Neumann developed views on the foundations of mathematics that sensibly differed from those of Gödel, his admiration for the man remained unchanged. Von Neumann's very extensive range of interests touched on many different fields of pure and applied mathematics. He was actively involved in very influential institutions, such as the Atomic Energy Commission and the Institute for Advanced Studies. He served as advisor in various ways to the military and political establishment in the USA. His work on the design of the modern digital computer was of lasting impact. No doubt, the story of his life and works offer a most important subject of historical research. Such research is actually just in its early phases. Some historical work on von Neumann and his scientific world has been done so far, and this is duly listed in the bibliography of this collection. The publication of the collection under review here provides, no doubt, an additional, very significant contribution for any future research on von Neumann himself and the impact of his research, as well as on some more general questions about the history of science in the first half of the twentieth century and in the years immediately following World War II. This book will appeal to anyone with an interest in the history of science in the twentieth century.

23.1. From the Publisher.

In this classic work, one of the greatest mathematicians of the twentieth century explores the analogies between computing machines and the living human brain. John von Neumann, whose many contributions to science, mathematics, and engineering include the basic organisational framework at the heart of today's computers, concludes that the brain operates both digitally and analogically, but also has its own peculiar statistical language.

In his foreword to this new edition, Ray Kurzweil, a futurist famous in part for his own reflections on the relationship between technology and intelligence, places von Neumann's work in a historical context and shows how it remains relevant today.

24.1. From the Publisher.

Quantum mechanics was still in its infancy in 1932 when the young John von Neumann, who would go on to become one of the greatest mathematicians of the twentieth century, published Mathematical Foundations of Quantum Mechanics - a revolutionary book that for the first time provided a rigorous mathematical framework for the new science. Robert Beyer's 1955 English translation, which von Neumann reviewed and approved, is cited more frequently today than ever before. But its many treasures and insights were too often obscured by the limitations of the way the text and equations were set on the page. In this new edition of this classic work, mathematical physicist Nicholas Wheeler has completely reset the book in TeX, making the text and equations far easier to read. He has also corrected a handful of typographic errors, revised some sentences for clarity and readability, provided an index for the first time, and added prefatory remarks drawn from the writings of Léon Van Hove and Freeman Dyson. The result brings new life to an essential work in theoretical physics and mathematic