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 The volume is rounded off by previously unpublished letters and lectures documenting von Neumann's thinking about quantum theory after his 1932 Mathematical Foundations of Quantum Mechanics.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)

The aim of the paper is this: Instead of presenting a provisional and necessarily insufficient characterization of what mathematical physics is, I will ask the reader to take it just as that, what he or she thinks or believes it is, yet to be prepared to revise his opinion in the light of what I am going to tell. Because this is precisely, what I intend to do. I will challenge some of the received or standard views about mathematical physics (...) and replace them by a more sophisticated picture, which takes into account the methodological and philosophical roots of mathematical physics in Göttingen. (shrink)

In recent years the works of Friedman, Howard and many others have made obvious what perhaps was always self-evident. Namely, that the philosophy of the logical empiricists was shaped primarily by Einstein and his invention of the theory of relativity, whereas Hilbert and his axiomatic approach to the exact sciences had comparatively little impact on the logical empiricists and their understanding of science — if they had any effect at all. This is in one respect quite astonishing, insofar as Einstein (...) himself confessed 1921 in his famous lecture before the Prussian Academy of Science that “without it [the axiomatic point of view] it would have been impossible for me to propound the theory of relativity”.2 Hence the simple question arises: why didn’t the logical empiricists pay more attention to Hilbert and his axiomatic point of view, than they in fact did? It is an aim of this paper to answer this question and in part to correct this one-sided view in the hypothetical or contra-factual sense that the logical empiricists would have done better, if they had paid somewhat more attention to Hilbert and his axiomatic approach to science. (shrink)

The dramatic development of physics in the twentieth century has thrown philosophy into a crisis regarding its self-image, from which today's philosophy has still not fully recovered.1 The crisis had two consequences or complementary manifestations: First, there was a gradual retreat of philosophy from the natural sciences. Because physics turned out to be an autonomous discipline, which aims at cognition of nature apparently totally independent of any kind of philosophy, "natural philosophy" as a particular branch of philosophy seemed superfluous. Second, (...) in order to prevent its own abdication, natural philosophy mutated into a kind of meta-science, called philosophy of science, that avoids any serious .. (shrink)

Der Beitrag der Mathematik zur Naturwissenschaft besteht darin, Erkenntnisse a priori von den (empirischen) Gegenständen der letzteren zu gewinnen und - unter der Voraussetzung, daß es überhaupt Empirisches gibt - , die Existenz solcher Gegenstände a priori zu sichern. Die Aufgabe der Metaphysik ist es, besondere Begriffe ( in bezug auf die körperliche Natur) zu bilden und die Prinzipien für deren mathematische Konstruktion zu entwickeln. Die so "metaphysischkonstruierten" Begriffe werden dann durch Angabe empirischer Modelle physikalisch interpretiert.

Der Beitrag der Mathematik zur Naturwissenschaft besteht darin, Erkenntnisse a priori von den (empirischen) Gegenständen der letzteren zu gewinnen und — unter der Voraussetzung, daß es überhaupt Empirisches g i b t -, die Existenz solcher Gegenstände a priori zu sichern. Die Aufgabe der Metaphysik ist es, besondere Begriffe ( in bezug auf die körperliche Natur) zu bilden und die Prinzipien für deren mathematische Konstruktion zu entwickeln. Die so "metaphysischkonstruierten" Begriffe werden dann durch Angabe empirischer Modelle physikalisch interpretiert.

The traditional view regarding the philosophy of mathematics in the twentieth century is the dogma of three schools: Logicism, Intuitionism and Formalism. The problem with this dogma is not, at least not first and foremost, that it is wrong, but that it is biased and essentially incomplete. 'Biased' because it was formulated by one of the involved parties, namely the logical empiricists - if I see it right - in order to make their own position look more agreeable with Intuitionism (...) and Formalism. 'Essentially incomplete' because there was - and still exists - beside Frege's Logicism, Brouwer's Intuitionism and Hilbert's Formalism at least one further position, namely Husserl's phenomenological approach to the foundations of arithmetic, which is also philosophically interesting. In what follows, I want to do two things: First, I will show that the standard dogma regarding the foundations of mathematics is not only incomplete, but also inaccurate, misleading and basically wrong with respect to the three schools themselves. In doing this I hope to make room for Husserl and his phenomenological approach as a viable alternative in the foundations of arithmetic. Second, I will show how Husserl's phenomenological point of view is a position that fits exactly in between Frege's "logicism", properlyunderstood, and Hilbert's mature proof theory, in which his so called "formalism" turns out to be only a means to an end and not a goal in itself. (shrink)

Le Mathematical Institute (MI) de l'université de Göttingen et la Niedersächsische Staats- und Universitätsbibliothek (SUB) possèdent ensemble une impressionnante collection de documents de David Hilbert, qui est peut-être le mathématicien le plus important de la période 1885-1930, un spécialiste notable de physique mathématique. Ce fonds sera appelé ici le Nachlaß de Hilbert en un sens étendu. Outre des lettres et bien d'autres documents, le Nachlaß contient une grande collection d'autographes et de textes dactylographiés de conférences données par Hilbert tout au (...) long de sa carrière. Les manuscrits des conférences portant sur les fondements des mathématiques et des sciences de la nature forment la base d'une édition (à laquelle nous référons comme l'« Édition Hilbert ») en six volumes : un volume sur les fondements de la géométrie (Vol. 1), deux sur les fondements de la logique et de l'arithmétique (Vol. 2, 3), deux sur les fondements de la physique (Vol. 4, 5), et un volume de sélections des carnets et d'exposés plus généraux de Hilbert. L'Édition contiendra en grande partie des documents inédits, quoique s'y ajoutent par endroits les publications centrales des champs concernés. En publiant ces documents, nous espérons promouvoir une vision renouvelée et plus complète des réalisations scientifiques de Hilbert, comme de son point de vue philosophique. L'article discute les objectifs scientifiques de l'Édition Hilbert, et les critères de sélection ayant présidé au choix des documents. (shrink)

Le Mathematical Institute (MI) de l'université de Göttingen et la Niedersächsische Staats- und Universitätsbibliothek (SUB) possèdent ensemble une impressionnante collection de documents de David Hilbert, qui est peut-être le mathématicien le plus important de la période 1885-1930, un spécialiste notable de physique mathématique. Ce fonds sera appelé ici le Nachlaß de Hilbert en un sens étendu. Outre des lettres et bien d'autres documents, le Nachlaß contient une grande collection d'autographes et de textes dactylographiés de conférences données par Hilbert tout au (...) long de sa carrière. Les manuscrits des conférences portant sur les fondements des mathématiques et des sciences de la nature forment la base d'une édition (à laquelle nous référons comme l'« Édition Hilbert ») en six volumes : un volume sur les fondements de la géométrie (Vol. 1), deux sur les fondements de la logique et de l'arithmétique (Vol. 2, 3), deux sur les fondements de la physique (Vol. 4, 5), et un volume de sélections des carnets et d'exposés plus généraux de Hilbert. L'Édition contiendra en grande partie des documents inédits, quoique s'y ajoutent par endroits les publications centrales des champs concernés. En publiant ces documents, nous espérons promouvoir une vision renouvelée et plus complète des réalisations scientifiques de Hilbert, comme de son point de vue philosophique. L'article discute les objectifs scientifiques de l'Édition Hilbert, et les critères de sélection ayant présidé au choix des documents. (shrink)

ZusammenfassungZunächst wird Dedekinds Charakterisierung der Reihe der natürlichen Zahlen als einfaches unendliches System mittels des Begriffes der Kette vorgestellt und gezeigt, dass diese Charakterisierung einen grossen Vorteil, aber auch einen entscheidenden Nachteil hat: Der Vorteil besteht darin, dass bei Dedekind nicht mehr – wie noch bei Frege – die einzelne Zahl, sondern die ganze, unendliche Reihe der natürlichen Zahlen zum Gegenstand der logischen Analyse gemacht wird. Der Nachteil besteht darin, dass Dedekind die unendliche Reihe der Zahlen als fertig vorliegende Gesamtheit (...) und nicht als etwas im Entstehen befindliches betrachtet. Hieran knüpft Weyls Kritik an. Abschliessend wird argumentiert, dass Dedekind der Weylschen Kritik hätte Rechnung tragen können, ohne den Vorteil der strukturalistischen Kennzeichnung aufgeben zu müssen, wenn er sich an seine eigene frühere, mehr dynamische Betrachtungsweise gehalten hätte, dergemäss die Zahlen von uns erzeugt und nicht vorgefunden werden. (shrink)