This is the classic work upon which modern-day game theory is based. What began as a modest proposal that a mathematician and an economist write a short paper together blossomed, 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 organization, based on a theory of games of strategy. Not only would this revolutionize economics, but the entirely new field of (...) scientific inquiry it yielded--game theory--has since been widely used to analyze 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. (shrink)

This book represents the views of one of the greatest mathematicians of the twentieth century on the analogies between computing machines and the living human brain.

The invitation of the Organizing Committee for me to speak about “Unsolved problems in mathematics” fills me as it should with considerable trepidation and a prevailing feeling of personal inadequacy. Hilbert gave a talk on this subject at the similar congress about 50 years ago and this is a very formidable precedent. He stated about a dozen unsolved problems in another widely separated areas of mathematics, and they proved to be prototypical for much of the development that followed in the (...) next decades. It would be absolutely foolish, if I tried to emulate this quite singular feat. In addition I do not know the future and the future at any rate can only be predicted ex post with any degree of reliability. I will, therefore, define what I am trying to do in a much more narrow way, hoping that in this manner I have a better chance of not failing. I will limit myself to a particular area of mathematics which I think I know and I will talk about it and about what its open ends appear to be, particularly in some directions which are not the ones that the evolution so far has mainly emphasized and which are, I think, quite important. (shrink)

I wish to discuss some rather incomplete ideas concerning difficulties that arise in some parts of quantum mechanics. In general there have been no serious difficulties when we are dealing with a finite number of particles, but very essential difficulties arise as soon as we treat a system having an infinite number of degrees of freedom; for example, the theory of holes, which, because of the pair generation, requires an indefinite number of particles; also the Dirac non-relativistic theory of light (...) and the Pauli-Heisenberg relativistic quantum electro-dynamics, these being equivalent to systems consisting of an infinite number of particles. (shrink)

John von Neumann (1903-1957) was undoubtedly one of the scientific geniuses of the 20th century. The main fields to which he contributed include various disciplines of pure and applied mathematics, mathematical and theoretical physics, logic, theoretical computer science, and computer architecture. Von Neumann was also actively involved in politics and science management and he had a major impact on US government decisions during, and especially after, the Second World War. There exist several popular books on his personality (...) and various collections focusing on his achievements in mathematics, computer science, and economy. Strangely enough, to date no detailed appraisal of his seminal contributions to the mathematical foundations of quantum physics has appeared. Von Neumann's theory of measurement and his critique of hidden variables became the touchstone of most debates in the foundations of quantum mechanics. Today, his name also figures most prominently in the mathematically rigorous branches of contemporary quantum mechanics of large systems and quantum field theory. And finally - as one of his last lectures, published in this volume for the first time, shows - he considered the relation of quantum logic and quantum mechanical probability as his most important problem for the second half of the twentieth century. The present volume embraces both historical and systematic analyses of his methodology of mathematical physics, and of the various aspects of his work in the foundations of quantum physics, such as theory of measurement, quantum logic, and quantum mechanical entropy. The volume is rounded off by previously unpublished letters and lectures documenting von Neumann's thinking about quantum theory after his 1932 Mathematical Foundations of Quantum Mechanics. The general part of the Yearbook contains papers emerging from the Institute's annual lecture series and reviews of important publications of philosophy of science and its history. (shrink)

Shortly after Kurt Gödel had announced an early version of the 1st incompleteness theorem, John von Neumann wrote a letter to inform him of a remarkable discovery, i.e. that the consistency of a formal system containing arithmetic is unprovable, now known as the 2nd incompleteness theorem. Although today von Neumann’s proof of the theorem is considered lost, recent literature has explored many of the issues surrounding his discovery. Yet, one question still awaits a satisfactory answer: how did (...) von Neumann achieve his result, knowing as little as he seemingly did about the 1st incompleteness theorem? In this article, I shall advance a conjectural argument to answer this question, after having rejected the argument widely shared in the literature and having analyzed the relevant documents surrounding his discovery. The argument I shall advance strictly links two of the three letters written by von Neumann to Gödel in the late 1930 and early 1931 (i.e. respectively that of November 20, 1930 and that of January 12, 1931) and finds the key for von Neumann’s discovery in his prompt understanding of the Gödel sentence A – as the documents refer to it – as expressing consistency for a formal system that contains arithmetic. (shrink)

John von Neumann's proof that quantum mechanics is logically incompatible with hidden varibales has been the object of extensive study both by physicists and by historians. The latter have concentrated mainly on the way the proof was interpreted, accepted and rejected between 1932, when it was published, and 1966, when J.S. Bell published the first explicit identification of the mistake it involved. What is proposed in this paper is an investigation into the origins of the proof rather than (...) the aftermath. In the first section, a brief overview of the his personal life and his proof is given to set the scene. There follows a discussion on the merits of using here the historical method employed elsewhere by Andrew Warwick. It will be argued that a study of the origins of von Neumann's proof shows how there is an interaction between the following factors: the broad issues within a specific culture, the learning process of the theoretical physicist concerned, and the conceptual techniques available. In our case, the ‘conceptual technology’ employed by von Neumann is identified as the method of axiomatisation. (shrink)

This paper surveys John von Neumann's work on the mathematical foundations of quantum theories in the light of Hilbert's Sixth Problem concerning the geometrical axiomatization of physics. We argue that in von Neumann's view geometry was so tied to logic that he ultimately developed a logical interpretation of quantum probabilities. That motivated his abandonment of Hilbert space in favor of von Neumann algebras, specifically the type II1II1 factors, as the proper limit of quantum mechanics in infinite (...) dimensions. Finally, we present the reasons why his axiomatic program remained an “unsolved problem” in mathematical physics. A recent unpublished result by Huzimiro Araki, proving that no algebra with a tracial state defined on it, such as the type II1II1 factors, can support any (regular) representation of the canonical commutation relations, is also reviewed and its consequences for von Neumann's projects are discussed. (shrink)

Papers advocating a hidden-variable interpretation of quantum mechanics typically begin by emphasizing that John von Neumann’s no-go theorem does not apply to them. If authors are ontologically minded, their criticism also takes aim at his theory of measurement as expressed in his seminal 1932 book Mathematical Foundations of Quantum Mechanics Additionally, David Bohm and Basil Hiley have recently argued that “in so far as von Neumann effectively gave the quantum state a certain ontological significance, the net result (...) was to produce a confused and unsatisfactory ontology. This ontology is such that the collapse of the wave function must also have an ontological significance.”. (shrink)

This paper interprets John von Neumann's views on the proper role of mathematics in science, with attention to its implications for economics. The evolution of von Neumann's thought over his lifespan, an examination of which is greatly aided by some self-conscious appraisals of von Neumann himself, suggests that von Neumann ended by taking a very pragmatic approach to the use of mathematics. One can almost characterize it as an engineering approach to the philosophy of mathematics (...) since von Neumann takes mathematics to be justified by its applications. This approach, which is surprisingly non-technical and hence accessible to general readers, is then placed in the context of his wider world-view. (shrink)

We report on an unpublished and previously unknown manuscript of John von Neumann and contextualize it within the development of the theory of shock waves and detonations during the nineteenth and twentieth centuries. Von Neumann studies bombs comprising a primary explosive charge along with explosive booster material. His goal is to calculate the minimal amount of booster needed to create a sustainable detonation, presumably because booster material is often more expensive and more volatile. In service of this (...) goal, he formulates and analyzes a partial differential equation-based model describing a moving shock wave at the interface of detonated and undetonated material. We provide a complete transcription of von Neumann’s work and give our own accompanying explanations and analyses, including the correction of two small errors in his calculations. Today, detonations are typically modeled using a combination of experimental results and numerical simulations particular to the shape and materials of the explosive, as the complex three dimensional dynamics of detonations are analytically intractable. Although von Neumann’s manuscript will not revolutionize our modern understanding of detonations, the document is a valuable historical record of the state of hydrodynamics research during and after World War II. (shrink)

The renewed interest in the foundations of quantum statistical mechanics in recent years has led us to study John von Neumann’s 1929 article on the quantum ergodic theorem. We have found this almost forgotten article, which until now has been available only in German, to be a treasure chest, and to be much misunderstood. In it, von Neumann studied the long-time behavior of macroscopic quantum systems. While one of the two theorems announced in his title, the one (...) he calls the “quantum H-theorem,” is actually a much weaker statement than Boltzmann’s classical H-theorem, the other theorem, which he calls the “quantum ergodic theorem,” is a beautiful and very non-trivial result. It expresses a fact we call “normal typicality” and can be summarized as follows: For a “typical” finite family of commuting macroscopic observables, every initial wave function ψ0 from a micro-canonical energy shell so evolves that for most times t in the long run, the joint probability distribution of these observables obtained from ψt is close to their micro-canonical distribution. (shrink)

As interpreted by Pattee, von Neumann’s Theory of Self-Reproducing Automata has proved to be a useful tool for understanding some of the difficulties and paradoxes of molecular biosemiotics. But is its utility limited to molecular systems or is it more generally applicable within biosemiotics? One way of answering that question is to look at the Theory as a model for one particular high-level biosemiotic activity, human language. If the model is not useful for language, then it certainly cannot be (...) generally useful to biosemiotics. Beginning with the Universal Turing Machine and continuing with von Neumann’s Theory and Pattee’s interpretation, the properties of universality, programmability, underspecification, complementarity of description/construction, and open-ended evolutionary potential are shown to be usefully applicable to language, thus opening a new line of inquiry in biosemiotics. (shrink)

Applied mathematics often operates by way of shakily rationalizedexpedients that can neither be understood in a deductive-nomological nor in an anti-realist setting.Rather do these complexities, so a recent paper of Mark Wilson argues, indicate some element in ourmathematical descriptions that is alien to the physical world. In this vein the mathematical opportunistopenly seeks or engineers appropriate conditions for mathematics to get hold on a given problem.Honest mathematical optimists, instead, try to liberalize mathematical ontology so as to include all physicalsolutions. Following (...) John von Neumann, the present paper argues that the axiomatization of a scientifictheory can be performed in a rather opportunistic fashion, such that optimism and opportunism appear as twomodes of a single strategy whose relative weight is determined by the status of the field to beinvestigated. Wilson's promising approach may thus be reformulated so as to avoid precarious talk about a physicalworld that is void of mathematical structure. This also makes the appraisal of the axiomatic method inapplied matthematics less dependent upon foundationalist issues. (shrink)

The debate over the question whether quantum mechanics should be considered as a complete account of microphenomena has a long and deeply involved history, a turning point in which has been certainly the Einstein-Bohr debate, with the ensuing charge of incompleteness raised by the Einstein-Podolsky-Rosen argument. In quantum mechanics, physical systems can be prepared in pure states that nevertheless have in general positive dispersion for most physical quantities; hence in the EPR argument, the attention is focused on the question whether (...) the account of the microphysical phenomena provided by quantum mechanics is to be regarded as an exhaustive description of the physical reality to which those phenomena are supposed to refer, a question to which Einstein himself answered in the negative. However, there is a mathematical side of the completeness issue in quantum mechanics, namely the question whether the kind of states with positive dispersion can be represented as a different, dispersion-free kind of states in a way consistent with the mathematical constraints of the quantum mechanical formalism. From this point of view, the other source of the completeness issue in quantum mechanics is the no hidden variables theorem formulated by John von Neumann in his celebrated book on the mathematical foundations of quantum mechanics, the preface of which already anticipates the program and the conclusion concerning the possibility of ‘neutralizing’ the statistical character of quantum mechanics. (shrink)

The renewed interest in the foundations of quantum statistical mechanics in recent years has led us to study John von Neumann’s 1929 article on the quantum ergodic theorem. We have found this almost forgotten article, which until now has been available only in German, to be a treasure chest, and to be much misunderstood. In it, von Neumann studied the long-time behavior of macroscopic quantum systems. While one of the two theorems announced in his title, the one (...) he calls the “quantum H-theorem,” is actually a much weaker statement than Boltzmann’s classical H-theorem, the other theorem, which he calls the “quantum ergodic theorem,” is a beautiful and very non-trivial result. It expresses a fact we call “normal typicality” and can be summarized as follows: For a “typical” finite family of commuting macroscopic observables, every initial wave function ψ0 from a micro-canonical energy shell so evolves that for most times t in the long run, the joint probability distribution of these observables obtained from ψ is close to.. (shrink)