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)

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)

Philosophie als -sterben wollen- (Platon) bestimmt die Tradition bis hin zur -Geworfenheit zum Tode- (Heidegger). Die Welt fuhrt sich mit diesem Menschenbild entsprechend auf. Sich absolut setzende Wahrheiten blenden die Wahrnehmung aus, dass wir in einer falschen Welt mit falschem Bewusstsein leben. Der Poet Schiller mokiert sich in seinem Gedicht "Die Taten der Philosophen" (1795) hierzu. In der Schlusszeile spricht er die Krafte an, die aus dem Leben alle Wesen und in besonderer Weise den Menschen bestimmen: HUNGER und LIEBE. Vorliegende (...) Arbeit ist der Versuch einer Theorie des poetischen Menschen aus diesen Urkraften des Lebens. Es ist der Versuch der Wiedervereinigung von KUNST und Mensch und Absage an alle Wahrheits-Ideologien. Der poetische Mensch kennt nur Wahrheiten -auf Zeit-. Einzig im Willen zum Leben kann eine neue Welt gedichtet werden...". (shrink)

Der Band enthält Beiträge von Medizinern, Juristen, Philosophen und Naturwissenschaftlern aus Estland, Polen, den Niederlanden und Deutschland zu Themen der Ethik und Wissenschaftstheorie der Medizin. Diese Beiträge sind im Rahmen des «Arbeitskreises für Ethik und Wissenschaftstheorie der Medizin in Ostmitteleuropa» entstanden, der auf einer Kooperationsvereinbarung des Interdisziplinären Zentrums für Ethik der Europa-Universität Viadrina Frankfurt (Oder) und des Instituts für Geschichte und Ethik der Medizin der Universität Halle-Wittenberg beruht. Sie befassen sich u. a. mit der Gesundheitsreform in Polen, Tschechien und Ungarn, (...) mit ethischen Problemen bei Abtreibung, HIV-Infektion, Drogenkonsum, Sterbehilfe und Gentechnik sowie dem Theorie-Praxis-Verhältnis in der Medizin. (shrink)

Der Band enthält Beiträge von Juristen, Medizinern und Philosophen aus Australien, Estland, Polen, Rußland, Tschechien, den U.S.A. und Deutschland zu Themen der Ethik und Wissenschaftstheorie der Medizin. Die Mehrzahl der Beiträge sind von Nachwuchswissenschaftlern des College for Advanced Central European Studies an der Europa-Universität Viadrina erarbeitet worden. Sie wurden im Rahmen der Jahrestagung des Arbeitskreises für Ethik und Wissenschaftstheorie der Medizin in Ostmitteleuropa neben weiteren Beiträgen, die hier zum Abdruck kommen, zur Diskussion gestellt. Der Arbeitskreis beruht auf einer Kooperationsvereinbarung des (...) Interdisziplinären Zentrums für Ethik der Europa-Universität Viadrina Frankfurt (Oder) und des Instituts für Geschichte und Ethik der Medizin der Universität Halle-Wittenberg. Die Beiträge befassen sich u. a. mit der Gesundheitsreform in Polen, mit ethischen und rechtlichen Problemen bei der künstlichen Befruchtung, der Abtreibung, medizinischen Experimenten an Menschen, der Gentechnik, der Patientenautonomie, Außenseitermethoden in der Medizin, dem Patiententestament, dem Todesbegriff, Drogenkonsum, der Sterbehilfe und der Gentechnik. (shrink)

Erscheint anlasslich des 15. Weltkongresses der Internationalen Vereinigung fur Rechts- und Sozialphilosophie (IVR) vom 18. bis 24. August 1991 in Gottingen und enthalt eine Selbstdarstellung der deutschen Sektion der IVR. Die Autoren berichten uber ihre eigene theoretische Position oder uber neuere Diskussionszusammenhange, die ihnen wichtig erscheinen. Aus dem Inhalt: K. Adomeit: Menschenrechte und Rechtsphilosophie u H. Albert: Erkenntnis, Recht und soziale Ordnung u R. Alexy: Idee und Struktur eines vernunftigen Rechtssystems u O. Ballweg: Analytische Rhetorik als juristische Grundlagenforschung u R. (...) Dreier: Zur gegenwartigen Diskussion des Verhaltnisses von Recht und Moral u W. Eichhorn: Wege und Irrwege einer Transformation u M. Frommel: Mannliche Gerechtigkeitsmathematik versus weiblicher Kontextualismus? u H. Garrn: Zur rechtspraktischen Bedeutung einer Theorie der juristischen Rhetorik u G. Haney: Das Widerspruchsvolle des Rechtsbegriffs u W. Hassemer: Rechtsphilosophie, Rechtswissenschaft, Rechtspolitik - am Beispiel des Strafrechts u A. Kaufmann: 45 Jahre erlebte Jahre Rechtsphilosophie u H. Klenner: Die ontologische und gnoseologische Dimension der Sein/Sollen-Problematik u U. Klug: Multidimensionale Rechtsphilosophie u H.-J. Koch/H. Russmann: Juristische Methodenlehre und analytische Philosophie u M. Kriele: Die demokratische Weltrevolution u K. Kuhl: Rehabilitierung und Aktualisierung des kantischen Vernunftrechts u E.-J. Lampe: Moderne Rechtsanthropologie u K.A. Mollnau: Vom Beginn der Niedergangsjurisprudenz in der DDR u U. Neumann: Sprachphilosophische Ansatze in der neueren Rechtsphilosophie u u.a. (Franz Steiner 1991). (shrink)

ZusammenfassungDer vorliegende Aufsatz verfolgt das Ziel, die Entwicklung der Galois-Theorie etwa von 1850 bis 1960 zu skizzieren. Hervorgehoben werden dabei diejenigen Entwicklungslinien, die mit der Funktionentheorie, der algebraischen Topologie und der Verallgemeinerung des Separabilitäts-Begriffs verknüpft sind. Es wird ein Ausblick auf die Galois-Theorie der kommutativen Ringe (nach M. Auslander & O. Goldman) und der Schemata (nach A. Grothendieck) gegeben.

In a series of lectures written around 1952, Schrödinger refers to von Neumann’s account of measurement in quantum mechanics as follows:I said quantum physicists bother very little about accounting, according to the accepted law, for the supposed change of the wave-function by measurement. I know of only one attempt in this direction, to which Dr. Balazs recently directed my attention. You find it in John von Neumann’s well-known book. With great acuity he constructs one analytical example. It does (...) not refer to any actual experiment, it is purely analytical. He indicates in a simple case a supplementary operator which, when added to the internal wave operator, would with any desired approximation turn the wave function as time goes on into an eigenfunction of the observable that is measured. He found it necessary to show that such a mechanism is analytically possible. The idea has not been taken up and worked out since — in about twenty years or more. Indeed I do not think it would pay. I do not believe any real measuring device is of this kind. (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.

Die finanziellen Ressourcen im deutschen Gesundheitssystem sind begrenzt. Diese Mittelknappheit führt im Rahmen der ärztlichen Tätigkeit zu medizinischen, ökonomischen, juristischen und ethischen Problemen, welche sich in den kommenden Jahren weiter verschärfen dürften. Aus ethischer Perspektive sind die Probleme einer gerechten Verteilung knapper Ressourcen sowie mögliche Rollen- oder Gewissenskonflikte der ärztlichen Profession besonders relevant. Mit Hilfe von qualitativen Interviewstudien lässt sich der aktuelle ärztliche Umgang mit der Mittelknappheit in der klinischen Versorgung in seiner Komplexität und seinen ethisch relevanten Aspekten untersuchen. An (...) fünf Kliniken mit unterschiedlichem Versorgungsauftrag wurden Tiefeninterviews mit fünfzehn leitenden Ärzten in den Bereichen Intensivmedizin und Kardiologie durchgeführt. Die auf Tonband aufgezeichneten Interviews wurden nach dem Verfahren der Grounded Theory ausgewertet. Die Ärzte berichten ein heterogenes Spektrum von Einflussfaktoren, Konsequenzen und Reaktionen im Zusammenhang mit ihrem Handeln bei Mittelknappheit. Zuteilungskriterien wie der medizinische Nutzen oder die Kosteneffektivität werden von den befragten Ärzten sehr unterschiedlich interpretiert und führen damit möglicherweise zu inkonsistenten Allokationsentscheidungen. Gleichzeitig bestätigen die Untersuchungen aber auch, dass die Ärzte nicht leichtfertig mit den Allokationsentscheidungen umgehen, im Gegenteil: Die Mittelknappheit und die dadurch hervorgerufenen Verteilungsprobleme können zu Gewissenskonflikten, emotionalem Stress und Gefühlen der Überforderung führen. Um Lösungskonzepte für ethische Probleme im Umgang mit der Mittelknappheit in der klinischen Praxis entwickeln und erfolgreich implementieren zu können, müssen die in dieser Arbeit systematisch zusammengestellten Verständnisweisen, Einflussfaktoren und Konsequenzen im Zusammenhang mit dem Kostendruck berücksichtigt werden. (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)

ZusammenfassungDie finanziellen Ressourcen im deutschen Gesundheitssystem sind begrenzt. Diese Mittelknappheit führt im Rahmen der ärztlichen Tätigkeit zu medizinischen, ökonomischen, juristischen und ethischen Problemen, welche sich in den kommenden Jahren weiter verschärfen dürften. Aus ethischer Perspektive sind die Probleme einer gerechten Verteilung knapper Ressourcen sowie mögliche Rollen- oder Gewissenskonflikte der ärztlichen Profession besonders relevant. Mit Hilfe von qualitativen Interviewstudien lässt sich der aktuelle ärztliche Umgang mit der Mittelknappheit in der klinischen Versorgung in seiner Komplexität und seinen ethisch relevanten Aspekten untersuchen. An (...) fünf Kliniken mit unterschiedlichem Versorgungsauftrag wurden Tiefeninterviews mit fünfzehn leitenden Ärzten in den Bereichen Intensivmedizin und Kardiologie durchgeführt. Die auf Tonband aufgezeichneten Interviews wurden nach dem Verfahren der Grounded Theory ausgewertet. Die Ärzte berichten ein heterogenes Spektrum von Einflussfaktoren, Konsequenzen und Reaktionen im Zusammenhang mit ihrem Handeln bei Mittelknappheit. Zuteilungskriterien wie der medizinische Nutzen oder die Kosteneffektivität werden von den befragten Ärzten sehr unterschiedlich interpretiert und führen damit möglicherweise zu inkonsistenten Allokationsentscheidungen. Gleichzeitig bestätigen die Untersuchungen aber auch, dass die Ärzte nicht leichtfertig mit den Allokationsentscheidungen umgehen, im Gegenteil: Die Mittelknappheit und die dadurch hervorgerufenen Verteilungsprobleme können zu Gewissenskonflikten, emotionalem Stress und Gefühlen der Überforderung führen. Um Lösungskonzepte für ethische Probleme im Umgang mit der Mittelknappheit in der klinischen Praxis entwickeln und erfolgreich implementieren zu können, müssen die in dieser Arbeit systematisch zusammengestellten Verständnisweisen, Einflussfaktoren und Konsequenzen im Zusammenhang mit dem Kostendruck berücksichtigt werden. (shrink)

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)

Von Neumann's theory of measurement in quantum mechanics is reinterpreted so that the experimental arrangement specifies the location of the “cut” by calling for the separate observation of the object and the measuring apparatus after the initial measurement interaction. The measurement ascertains which element of the mixture describing the final state of the apparatus is actually present. The relevance and feasibility of observing the final coherent state of the object plus apparatus is criticized and the paradoxes of “Schrödinger's (...) cat” and “Wigner's friend” are discussed. (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)

Around 1989, a striking letter written in March 1956 from Kurt Gödel to John von Neumann came to light. It poses some problems about the complexity of algorithms; in particular, it asks a question that can be seen as the first formulation of the P=?NP question. This paper discusses some of the background to this letter, including von Neumann's own ideas on complexity theory. Von Neumann had already raised explicit questions about the complexity of Tarski's decision (...) procedure for elementary algebra and geometry in a letter of 1949 to J. C. C. McKinsey. The paper concludes with a discussion of why theoretical computer science did not emerge as a separate discipline until the 1960s. (shrink)

According to a popular narrative, in 1932 von Neumann introduced a theorem that intended to be a proof of the impossibility of hidden variables in quantum mechanics. However, the narrative goes, Bell later spotted a flaw that allegedly shows its irrelevance. Bell’s widely accepted criticism has been challenged by Bub and Dieks: they claim that the proof shows that viable hidden variables theories cannot be theories in Hilbert space. Bub’s and Dieks’ reassessment has been in turn challenged by Mermin (...) and Schack. Hereby I critically assess their reply, with the aim of bringing further clarification concerning the meaning, scope and relevance of von Neumann’s theorem. I show that despite Mermin and Schack’s response, Bub’s and Dieks’ reassessment is quite correct, and that this reading gets strongly reinforced when we carefully consider the connection between von Neumann’s proof and Gleason’s theorem. (shrink)

In this paper, I shall discuss the heuristic role of symmetry in the mathematical formulation of quantum mechanics. I shall first set out the scene in terms of Bas van Fraassen’s elegant presentation of how symmetry principles can be used as problem-solving devices (see van Fraassen [1989] and [1991]). I will then examine in what ways Hermann Weyl and John von Neumann have used symmetry principles in their work as a crucial problem-solving tool. Finally, I shall explore one consequence (...) of this situation to recent debates about structural realism (SR) and empiricism in physics (Worrall [1989], Ladyman [1998], and French [1999]). (shrink)

Since the analysis by John Bell in 1965, the consensus in the literature is that von Neumann’s ‘no hidden variables’ proof fails to exclude any significant class of hidden variables. Bell raised the question whether it could be shown that any hidden variable theory would have to be nonlocal, and in this sense ‘like Bohm’s theory.’ His seminal result provides a positive answer to the question. I argue that Bell’s analysis misconstrues von Neumann’s argument. What von (...) Neumann proved was the impossibility of recovering the quantum probabilities from a hidden variable theory of dispersion free (deterministic) states in which the quantum observables are represented as the ‘beables’ of the theory, to use Bell’s term. That is, the quantum probabilities could not reflect the distribution of pre-measurement values of beables, but would have to be derived in some other way, e.g., as in Bohm’s theory, where the probabilities are an artefact of a dynamical process that is not in fact a measurement of any beable of the system. (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)

In this paper we analyse in the framework of constructive mathematics (BISH) the validity of Farkas' lemma and related propositions, namely the Fredholm alternative for solvability of systems of linear equations, optimality criteria in linear programming, Stiemke's lemma and the Superhedging Duality from mathematical finance, and von Neumann's minimax theorem with application to constructive game theory.

According to what has become a standard history of quantum mechanics, von Neumann in 1932 succeeded in convincing the physics community that he had proved that hidden variables were impossible as a matter of principle. Subsequently, leading proponents of the Copenhagen interpretation emphatically confirmed that von Neumann's proof showed the completeness of quantum mechanics. Then, the story continues, Bell in 1966 finally exposed the proof as seriously and obviously wrong; this rehabilitated hidden variables and made serious foundational research (...) possible. It is often added in recent accounts that von Neumann's error had been spotted almost immediately by Grete Hermann, but that her discovery was of no effect due to the dominant Copenhagen Zeitgeist. We shall attempt to tell a more balanced story. Most importantly, von Neumann did not claim to have shown the impossibility of hidden variables tout court, but argued that hidden-variable theories must possess a structure that deviates fundamentally from that of quantum mechanics. Both Hermann and Bell appear to have missed this point; moreover, both raised unjustified technical objections to the proof. Von Neumann's conclusion was basically that hidden-variables schemes must violate the "quantum principle" that all physical quantities are to be represented by operators in a Hilbert space. According to this conclusion, hidden-variables schemes are possible in principle but necessarily exhibit a certain kind of contextuality. As we shall illustrate, early reactions to Bohm's theory are in agreement with this account. Leading physicists pointed out that Bohm's theory has the strange feature that particle properties do not generally reveal themselves in measurements, in accordance with von Neumann's result. They did not conclude that the "impossible was done" and that von Neumann had been shown wrong. (shrink)

It is a widespread belief that the Kochen-Specker theorem imposes a contextuality constraint on the ontology of beables in quantum hidden variables theories. On the other hand, after Bell’s influential critique, the importance of von Neumann’s wrongly called ‘impossibility proof’ has been severely questioned. However, Max Jammer, Jeffrey Bub and Dennis Dieks have proposed insightful reassessments of von Neumann’s theorem: what it really shows is that hidden variables theories cannot represent their beables by means of Hermitian operators in (...) Hilbert space. Hereby I show that i) the very same constraint can be derived from Gleason’s theorem, and that ii) if we consider the import of von Neumann’s and Gleason’s theorems, the relevance of the Kochen-Specker theorem for hidden variables theories gets substantially weakened: it does not force them to be contextual in any interesting sense of the term. (shrink)

This paper offers a modified version of the certainty equivalence (CE) theory of utility for uncertain prospects and a new set of axioms as its basis. It shows that the CE and the von Neumann-Morgenstern (NM) approaches to uncertainty are opposite in spirit: The CE approach represents a flight from the world of uncertainty to the rules of certainty while the NM approach represents a flight from the world of certainty to one of uncertainty. The two approaches differ (...) even in their treatment of compound prospects and their actuarially identical simple counterparts. (shrink)

The paper discusses some of the poorly explored links between the conceptual systems of logic in Kurt Gödel, the theory of automata in Alan Turing, and the theory of self-reproducing automata in John von Neumann. Traditional controversies are left aside (especially the opposition of Gödel and Turing in the view of mind) and attention is focused on the similarities between all three authors. In individual chapters, the text deals with: the form of differentiation of syntax and semantics (...) in formal system in Gödel, Turing and von Neumann; von Neumann’s variant of Gödel’s theorem and von Neumann’s and Gödel’s conception of Turing machine; and finally the same basis of the view of the relation between mind and automaton in all three authors. (shrink)

Book Reviews Paul Erickson, Economics and Philosophy, FirstView Article.

Much of the recent discussion of problematic aspects of quantum-mechanical measurement centers around that feature of quantum theory which is called "the projection postulate." This is roughly the claim that a change of a certain sort occurs in the state of a physical system when a measurement is made on the system. In this paper an argument for the projection postulate due to von Neumann is considered. Attention is focused on trying to provide an understanding of the notion (...) of "the state of a physical system" which is compatible with the argument von Neumann offers. An attempt is made to formulate the argument in terms of an objectivistic interpretation of probability concepts. It is seen that such an interpretation does not provide a suitable way of understanding the argument. An attempt is made to illustrate the source of this failure in terms of a non-quantum-mechanical example. (shrink)

For a finite von Neumann algebra factor M, the projections form a modular ortholattice L(M). We show that the equational theory of L(M) coincides with that of some resp. all L(ℂ n × n ) and is decidable. In contrast, the uniform word problem for the variety generated by all L(ℂ n × n ) is shown to be undecidable.

Sequential von Neumann–Morgernstern (VM) games are a very general formalism for representing multi-agent interactions and planning problems in a variety of types of environments. We show that sequential VM games with countably many actions and continuous utility functions have a sound and complete axiomatization in the situation calculus. This axiomatization allows us to represent game-theoretic reasoning and solution concepts such as Nash equilibrium. We discuss the application of various concepts from VM game theory to the theory of (...) planning and multi-agent interactions, such as representing concurrent actions and using the Baire topology to define continuous payoff functions. (shrink)

1. The theory of classes presented in this paper is a simplification of that presented by J. von Neumann in his paper Die Axiomatisierung der Mengenlehre. However, this paper is written so that it can be read independently of von Neumann's. The principal modifications of his system are the following.(1) The idea of ordered pair is defined in terms of the other primitive concepts of the system. (See Axiom 4.3 below.)(2) A much simpler proof of the well-ordering (...) theorem, based on von Neumann's equivalence axiom (Axiom 2.2 below) is given. (More exactly, the theory of ordinal numbers, on which the proof of the well-ordering theorem is based, is simplified—see §8.)(3) Functions are not assumed to be defined for all arguments. In place of “ = A” we have “is undefined.” This makes possible a constructive interpretation of the system. (See §2.)2. We wish first to give a rough picture of the system which we are trying to construct. Suppose that we have the idea “class,” but no material from which to construct classes. Nevertheless, we can construct the class 0 having no elements. And then the class 1 having 0 as its only element. And then the classes {1}, and {0, 1}, etc., using always the previously constructed classes as elements. To extend this method to infinite classes, we must give rules telling what elements are to be included. Since the nature of the rules which we can use is not altogether clear, we try to formalize the whole system; that is, to set up a system of axioms which seems to characterize this system of classes. Now in this set of axioms we find it necessary to use the idea of function (in the “Axiom of Replacement”). (shrink)

In early 1927, Pascual Jordan published his version of what came to be known as the Dirac-Jordan statistical transformation theory. Later that year and partly in response to Jordan, John von Neumann published the modern Hilbert space formalism of quantum mechanics. Central to both formalisms are expressions for conditional probabilities of finding some value for one quantity given the value of another. Beyond that Jordan and von Neumann had very different views about the appropriate formulation of problems (...) in the new theory. For Jordan, unable to let go of the analogy to classical mechanics, the solution of such problems required the identification of sets of canonically conjugate variables, i.e., p’s and q’s. Jordan ran into serious difficulties when he tried to extend his approach from quantities with fully continuous spectra to those with wholly or partly discrete spectra. For von Neumann, not constrained by the analogy to classical physics and aware of the daunting mathematical difficulties facing the approach of Jordan ), the solution of a problem in the new quantum mechanics required only the identification of a maximal set of commuting operators with simultaneous eigenstates. He had no need for p’s and q’s. Related to their disagreement about the appropriate general formalism for the new theory, Jordan and von Neumann stated the characteristic new rules for probabilities in quantum mechanics somewhat differently. Jordan was the first to state those rules in full generality, von Neumann rephrased them and then sought to derive them from more basic considerations. In this paper we reconstruct the central arguments of these 1927 papers by Jordan and von Neumann and of a paper on Jordan’s approach by Hilbert, von Neumann, and Nordheim. We highlight those elements in these papers that bring out the gradual loosening of the ties between the new quantum formalism and classical mechanics. (shrink)