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)

David Hilbert was arguably the leading mathematician of his generation. He was among the few mathematicians who could reshape mathematics, and was able to because he brought together an impressive technical power and mastery of detail with a vision of where the subject was going and how it should get there. This was the unique combination which he brought to the setting of his famous 23 Problems. Few problems in mathematics have the status of those posed by David Hilbert in (...) 1900. Mathematicians have made their reputations by solving individual ones such as Fermat's last theorem, and several remain unsolved including the Riemann hypotheses, which has eluded all the great minds of this century. A hundred years on, it is timely to take a fresh look at the problems, the man who set them, and the reasons for their lasting impact on the mathematics of the twentieth century. In this fascinating new book, Jeremy Gray and David Rowe consider what has made this the pre-eminent collection of problems in mathematics, what they tell us about what drives mathematicians, and the nature of reputation, influence and power in the world of modern mathematics. The book is written in a clear and lively manner and will appeal both to the general reader with an interest in mathematics and to mathematicians themselves. (shrink)

This is one of those cases to which Dr. 8 oodhouse's remark applies with all its force, that a method which leads to true results must have its logic — H.S Smith (" On Some of the Methods at Present in Use in Pure Geometry," p. 6) A goodly amount of modern metaphysics has concerned itself, in one form or another, with the question: what attitude should we take in regard to a language whose semantic underpinnings seem less than certain? (...) Since much of standard English is surely of this type, many philosophers have felt driven to extreme and unpalatable resolutions — wholesale rejection of attractive logical rules, radical "antirealism", "minimalism" about truth, and so forth, Scientists, however, have long dealt with analogous foundational insecurities and, under the duress of practicality, have framed a set of shifting attitudes towards language and reasoning that seem far more commonsensical than the extreme positions to which pure philosophy has often been led. Following this lead, this essay will recommend a more nuanced approach to linguistic insecurity that is neither naively optimistic nor mired in the gloomy sloughs of antirealism. (shrink)

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)

The paper deals with some of the developments in analysis against the background of Hilbert's contributions to the Calculus of Variations. As a starting point the transformation is chosen that took place at the end of the 19th century in the Calculus of Variations, and emphasis is placed on the influence of Dirichlet's principle. The proof of the principle (the resuscitation ) led Hilbert to questions arising in the 19th and 20th problems of his famous Paris address in 1900: theexistence (...) in a generalized sense, and theregularity of solutions of elliptic partial differential equations. By this new concept Hilbert pointed out two very important issues the history of which is closely tied up with the rise of modern analysis. Further on, Hilbert'stheorem of independence of the Calculus of Variations, an important contribution to its formal apparatus as well as to its field theory, is briefly discussed. But even as Hilbert published his first essential results, he was turning his attention to another area of mathematics that differs in an important respect from the Calculus of Variations:integral equations. Hilbert did so because he recognized that in this branch by its flexibility he was coming closer to his goal of an unifying methodological approach to analysis. (shrink)

Only recently has David Hilbert’s program to axiomatize the sciences according to the pattern of geometry left the shade of his formalist program in the foundations of mathematics.1 This relative neglect — which is surprising in view of the enormous efforts Hilbert himself had devoted to it — was certainly influenced by Logical Empiricists’ almost exclusively focusing on his contributions to the foundational debates. Ulrich Majer puts part of the blame for this neglect on Hilbert himself because “he failed to (...) make his position sufficiently clear, and he did not take much effort to promote his views beyond the narrow circle of mathematical physics in Göttingen.”. (shrink)

John Norton has recently argued that Newtonian gravitation theory (at least as applied to cosmological contexts where one envisions the possibility of a homogeneous mass distribution throughout all of space) is inconsistent. I am not convinced. Traditional formulations of the theory may seem to break down in cases of the sort Norton considers. But the difficulties they face are only apparent. They are artifacts of the formulations themselves, and disappear if one passes to the so-called "geometrized" formulation of the theory.

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)