Foundations of Set Theory discusses the reconstruction undergone by set theory in the hands of Brouwer, Russell, and Zermelo. Only in the axiomatic foundations, however, have there been such extensive, almost revolutionary, developments. This book tries to avoid a detailed discussion of those topics which would have required heavy technical machinery, while describing the major results obtained in their treatment if these results could be stated in relatively non-technical terms. This book comprises five chapters and begins with a discussion of (...) the antinomies that led to the reconstruction of set theory as it was known before. It then moves to the axiomatic foundations of set theory, including a discussion of the basic notions of equality and extensionality and axioms of comprehension and infinity. The next chapters discuss type-theoretical approaches, including the ideal calculus, the theory of types, and Quine's mathematical logic and new foundations; intuitionistic conceptions of mathematics and its constructive character; and metamathematical and semantical approaches, such as the Hilbert program. This book will be of interest to mathematicians, logicians, and statisticians. (shrink)

Bibliography of A. A. Fraenkel (p. ix-x)--Axiomatic set theory. Zur Frage der Unendlichkeitsschemata in der axiomatischen Mengenlehre, von P. Bernays.--On some problems involving inaccessible cardinals, by P. Erdös and A. Tarski.--Comparing the axioms of local and universal choice, by A. Lévy.--Frankel's addition to the axioms of Zermelo, by R. Mantague.--More on the axiom of extensionality, by D. Scott.--The problem of predicativity, by J. R. Shoenfield.--Mathematical logic. Grundgedanken einer typenfreien Logik, von W. Ackermann.--On the use of Hilbert's [epsilon]-operator in scientific (...) theories, by R. Carnap.--Basic verifiability in the combinatory theory of restricted generality, by H. B. Curry.--Uniqueness ordinals in constructive number classes, by H. Putnam.--On the construction of models, by A. Robinson.--Interpretation of mathematical theories in the first order predicate calculus, by T. Skolem.--The elementary character of two notions from general algebra, by R. Vaught.--Foundations of arithmetic and analysis. Axiomatic method and intuitionism, by A. Heyting.--On rank-decreasing functions, by G. Kurepa.--On non-standard models for number theory, by E. Mendelson.--Concerning the problem of axiomatizability of the field of real numbers in the weak second order logic, by A. Mostowski.--Non-standard models and independence of the induction axiom, by M. O. Rabin.--Sur les ensembles raréfiés de nombres naturels, par W. Sierpinski.--Philosophy of logic and mathematics. Remarks on the paradoxes of logic and set theory, by E. W. Beth.--Logique formalisée et raisonnement juridique, par R. Feys.--Im Umkreis der sogenannten Raumprobleme, von H. Freudenthal.--Process and existence in mathematics, by H. Wang. (shrink)

The main purpose of this article is to present new material concerning Leibniz's use of indivisibles and infinitesimals in his early mathematical texts. Most of these texts are contained in hitherto unpublished manuscripts and are soon to be printed in volume VII, 4 of the Academy Edition. They present examples which illustrate how Leibniz operated with concepts such as indivisibles and infinitesimals in that period of his development.

The Dual State, first published in 1941, remains one of the most erudite books on the legal origins of democracy and dictatorship. It provided the first comprehensive analysis of the rise and nature of National Socialism, and was the only such analysis written from within Hitler's Germany. Fraenkel's concept of the dual state, being the normative state and the the prerogative state. It retains its vital relevance for the theory of democracy in the twenty-first century. The Dual State considerably (...) influenced scholars studying and working on questions of political justice in the period following World War II, particularly in the context of political and legal theory; in the domain of legal history; in the area of constitutional theory; in the context of comparative politics; and in what has become known as the field of comparative constitutional law. This republication of Fraenkel's classic work makes it once again widely available to scholars and students in the field. It includes both Fraenkel's 1974 introduction to the German second edition, never before published in English, and a new introduction by Dr Jens Meierhenrich, examining the world in which The Dual State was originally published, and the lasting legacy of this classic work. (shrink)

Physics was the leading science of the twentieth century and the book retraces important discoveries, made between 1895 and 2001, in 100 self-contained episodes. Each is a short story of the scientists involved, their time and their work. The book is richly illustrated byabout 600 portraits, photographs, and figures.

The correspondence between Richard von Mises and George Pólya of 1919/20 contains reflections on two well-known articles by von Mises on the foundations of probability in the Mathematische Zeitschrift of 1919, and one paper from the Physikalische Zeitschrift of 1918. The topics touched on in the correspondence are: the proof of the central limit theorem of probability theory, von Mises' notion of randomness, and a statistical criterion for integer-valuedness of physical data. The investigation will hint at both the fruitfulness and (...) the limits of several of von Mises' notions such as ``collective'', ``distribution'' and ``complex adjuncts'' (characteristic functions) for further developments in probability theory and in ``directional statistics''. By pointing to the selectiveness of Pólya's criticism, the historical analysis shows the differing expectations of the two men with respect to the further development of the theory of probability and its applications. The paper thus gives a glimpse of the provisional state of the theory around 1920, before others such as P. Lévy (1886–1971) and A. N. Kolmogorov (1903–1987) stepped in and created a new paradigm for probability theory. (shrink)

"Der geschichtliche Sinn, welcher schon das Altertum auszeichnete, bewahrte sich auch auf unserm Gebiete, und es hat schon in fruher Zeit nicht an Versuchen gefehlt, den Entwicklungsgang des Wissens bis zu einem gewissen Termine zu schildern. Nach Diogenes von Laerte, einem freilich nicht immer vollig zuverlassigen Gewahrsmann, hat schon ein Schuler des Aristoteles, der als Naturforscher bekanntere Theophrast, Geschichtswerke uber Arithmetik, Geometrie und Astronomie verfasst, von denen sich jedoch hochstens insofern eine Spur erhalten hat, als nach den ausgezeichneten Untersuchungen von (...) Diels bei einer ganzen Reihe spatlateinischer und spatgriechischer Autoren Anklange an historische Angaben Theophrasts zu finden sind. Ahnliches gilt von der angeblich geometrischen Schrift des Xenokrates, deren ebenfalls Diogenes Erwahnung tut." [...] Der bedeutende Philosoph und Philosophiehistoriker Dr. Wilhelm Windelband (1848-1915) gilt als Begrunder der sudwestdeutschen Schule (auch Badische Schule ), die eine Richtung des Neukantianismus darstellt. Windelbands Hauptaugenmerk galt der Abgrenzung von Natur- und Kulturwissenschaften. Sein wichtigster Schuler war Heinrich Rickert. Sigmund Gunther lebte von 1848 - 1923 und gilt als einer der wichtigsten Mathematikhistoriker. Er wurde mit der Dissertation "Studien zu theoretischen Photometrie" promoviert. In dem vorliegenden Werk gehen beide Autoren ausfuhrlich auf die geschichtlichen Hintergrunde der Naturwissenschaften und der Philosophie ein. Das Buch erschien ursprunglich als funfter Band im "Handbuch der klassischen Altertums-Wissenschaft" und wurde 1888 erstmals veroffentlicht. Der Verlag der Wissenschaften verlegt historische Literatur bekannter und unbekannter wissenschaftlicher Autoren. Dem interessierten Leser werden so teilweise langst nicht mehr verlegte Werke wieder zugangig gemacht. Das vorliegende Buch ist ein unveranderter Nachdruck der Originalausgabe.". (shrink)

Noise has become an increasingly noticeable and significant symptom of our civilization. Fundamentally an acoustic phenomenon, noise has wider implications. It is the legitimate object of scientific investigations in the fields of psychology and physiology. It can be properly evaluated by its role in music and in general aesthetics. It leads to basic questions of sociology. We shall pursue the implications in these various fields one by one. In this process, as elsewhere, music provides the bridge from facts to commitments (...) . Siegmund Levarie is professor of music at the City University of New York. The author of books on Mozart, Guillaume de Machaut, harmony, and Italian music, he has also collaborated with Ernst Levy on Tone: A Study in Musical Acoustics and the forthcoming A Dictionary of Musical Morphology. (shrink)

Die Abgrenzung des Designs von der Kunst, welche die ästhetische Theorie im 20. Jahrhundert vornahm, hat uns implizit eine Definition von Design und designerischem Handeln vorgegeben, die diesem nicht in Gänze gerecht wird. Mit der Auszeichnung der Kunst fand eine Abwertung gestalterischen Handelns statt. Die angebliche Distanz des Künstlerischen zum Funktionalen und Ökonomischen steht heute mehr denn je in Frage, wie sich auch bemerken lässt, dass Design seit seiner Entstehung vielfach durch künstlerische Einflüsse geprägt worden ist. Um heute Design adäquat (...) theoretisch zu fassen, muss seine kategoriale Abgrenzung von den. (shrink)

This volume draws a balanced picture of the Rationalists by bringing their intellectual contexts, sources and full range of interests into sharper focus, without neglecting their core commitment to the epistemological doctrine that earned ...

ArgumentThe paper is based on a hitherto unexplored document (audiotape of an interview accompanied by a German transcript) from 1953, located in the Freud Papers at the Library of Congress. It contributes to a better understanding of the impact of Freud and of Psychoanalysis on personalities from the exact sciences, here represented by the noted applied mathematicians Richard von Mises and Hilda Geiringer from Vienna. The detailed discussion of the interview sheds some new light on the different roles of Kraus (...) and Freud in the Vienna culture, on the Vienna Jugendkulturbewegung (youth culture movement) during WWI in which Geiringer was involved, on Freud’s and Siegfried Bernfeld’s standing around 1930 among German philosophers and psychologists, and on Wilhelm Fließ’ theory of periodicity, which von Mises—based on his attitude as an applied mathematician—defended against superficial accusations. Finally, new biographical material is provided for von Mises and the remotely related Freud family, and for Geiringer’s and von Mises’ early lives. The interview, which was taken during the Cold War, also allows conclusions as to how politics influenced the memories and views of the participants. Part of the aim of the paper is historical documentation of unknown material (letters by Karl Kraus and Wolfgang Köhler, one book review by Wilhelm Ostwald, a file on Richard Pfennig), including some correction of erroneous information in the literature. (shrink)

Der Beitrag beschäftigt sich aus philosophischer Perspektive mit der kontroversen öffentlichen Diskussion zur documenta fifteen. Einigen Stellungnahmen in dieser Debatte liegen implizit Prämissen einer kantisch geprägten ästhetischen Theorie zugrunde; ihnen stehen anders situierte Diskussionsstrategien aus außereuropäischen Kontexten gegenüber. Im Text wird der kantische universalistische Gedanke eines ästhetischen Gemeinsinns kritisch im Sinne einer postkolonialen Kritik gelesen. Gibt es andere universale Annäherungen an künstlerisches Arbeiten, die stärker von einer Haltung des Zuhörens geprägt sind? Entsprechend den in diesem Feld möglichen Differenzen variiert auch (...) das Verständnis, was es bedeutet, in einem Künstler:innenkollektiv zu arbeiten. (shrink)

Many pagan, Jewish, Christian and Muslim philosophers from Antiquity to the Enlightenment made no meaningful distinction between philosophy and religion. Instead they advocated a philosophical religion, arguing that God is Reason and that the historical forms of a religious tradition serve as philosophy's handmaid to promote the life of reason among non-philosophers. Carlos Fraenkel provides the first account of this concept and traces its history back to Plato. He shows how Jews and Christians appropriated it in Antiquity, follows it (...) through the Middle Ages in both Islamic and Jewish forms and argues that it underlies Spinoza's interpretation of Christianity. The main challenge to a philosophical religion comes from the modern view that all human beings are equally able to order their lives rationally and hence need no guidance from religion. Fraenkel's wide-ranging book will appeal to anyone interested in how philosophy has interacted with Jewish, Christian, and Muslim religious traditions. (shrink)