1m Zusammenhang mit Vorarbeiten zu einer Biographie uber Heinz Hopf sind wir vor einigen Jahren im Archiv des Schweizerischen Schulrates auf bisher unbekannte Dokumente aus dem Jahre 1930 gestossen, wel- che die N achfolgeregelung von Hermann Weyl an der ETH betreffen und die in mehrfacher Hinsicht Interesse verdienen. Dies hat uns veran- lasst, an der ETH systematisch nach weiteren Dokumenten zu Hermann Weyl und zur Mathematik an der ETH aus der Zeit seiner Tiitigkeit in Zurich zu suchen. Versehen mit einem Rahmentext veroffentlichen wir hier eine Zusammenstellung dieser Dokumente, die bis anhin nur schwer oder uberhaupt nicht zugiinglich waren. Hermann Weyl bezeichnet im Ruckblick die 17 Jahre seiner Tatigkeit in Zurich als die "wohl wichtigsten und produktivsten" seines Lebens. In der Tat sind von ihm zwischen 1913 und 1930 acht Bucher und rund siebzig Arbeiten erschienen. In Zurich erreichten ihn auch zahlreiche Berufun- gen aus Deutschland und den USA. 1m Ruckblick spricht er von ihnen als von der "schlimmste[n] Plage" wiihrend dieser Zeit. Es schien uns eine reizvolle Aufgabe zu sein, die iiusseren Lebensumstiinde Hermann Weyls in Zurich zu verfolgen, die ihm eine so erfolgreiche Tiitigkeit ermoglicht haben. Die aufgefundenen Dokumente fugen sich dariiber hinaus auch zu einer Darstellung der personellen Entwicklung der Mathematik (und der theoretischen Physik) an der ETH in den Jahren 1913 bis 1930.

Dies betrifft 0.11,24,26,27,31 und 0.111,10 (cf. den Verlags- prospekt Birkhauser 1982: Leonhard Euler, Opera omnia). - Eine kurze Geschichte der Euler-Ausgabe mit chronologischen Editions- tabellen findet sich in Leonhard Euler 1707-1783, Beitrage zu Leben und Werk. Gedenkband des Kantons Basel-Stadt, Birkhauser, Basel 1983, K.-R.Biermann:1783-1907, J.J.Burckhardt:1907-1983. Dieser Band wird im folgenden kurz als EGB 83 zitiert. 2 Der 1975 erschienene Band O. IV A, l (Birkhauser, Basel) gibt eine Uebersicht sowie Resumes aller ca. 3000 erhaltenen Briefe von Eulers Korrespondenz. AIle in der vorliegenden Abhandlung heran- gezogenen Briefe werden gemass IV A, l mit ihren Resume-Nummern mit vorangestelltem R gekennzeichnet. Der erste erschienene eigentliche Korrespondenzband ist 0.IVA,5. Er enthalt Eulers Briefwechsel mit Clairaut, d'Alembert und Lagrange (ed. A.P.Jukevic und R.Taton). Erschienen 1980. 3 1m Interesse der Transparenz der genealogischen Verhaltnisse sei ein Stammbaum der Mathematiker Bernoulli wiedergegeben (Aus EGB 83, p.80). Darin mage auch Leonhard Euler als geistiger Sohn Johann Bernoullis Platz finden. Niklau, d.l. Maler r-- --., I Daniel II I I 1751 1834 I L _____ -! 4 Cf. G.Enestram, Der Briefwechsel zwischen Leonhard Euler und Johann I Bernoulli, Bibliotheca Mathematica (3) 4, 1903; (3) 5, 1904; (3) 6, 1905. - Zu Eulers Leistungen auf diesen Gebieten cf. EGB 83 passim.

This important book by a major American philosopher brings together eleven essays treating problems in logic and the philosophy of mathematics. A common point of view, that mathematical thought is central to our thought in general, underlies the essays. In his introduction, Parsons articulates that point of view and relates it to past and recent discussions of the foundations of mathematics.Mathematics in Philosophy is divided into three parts. Ontology the question of the nature and extent of existence assumptions in mathematics is the subject of Part One and recurs elsewhere. Part Two consists of essays on two important historical figures, Kant and Frege, and one contemporary, W. V. Quine. Part Three contains essays on the three interrelated notions of set, class, and truth."

Zum Anlass des 100. Geburtstages der Deutschen Mathematiker-Vereinigung erscheint diese Festschrift, bestehend aus neunzehn Beitragen, in denen anerkannte Fachwissenschaftler die Entwicklung ihres jeweiligen mathematischen Fachgebietes beschreiben und dabei auch kritische Ruckschau auf die Geschichte der Deutschen Mathematiker-Vereinigung seit ihrer Grundung 1890 halten. Insbesondere der erste Beitrag setzt sich intensiv mit der Historie der Mathematik und der Mathematiker im Dritten Reich auseinander."Mit diesem Band wird ein wichtiger Beitrag zur bisher wenig entwickelten Geschichtsschreibung der neueren Mathematik geleistet. (R. Siegmund-Schultze in "Deutsche Literatur-Zeitung" 1,2/1992, Bd. 113)

This introduction to the philosophy of mathematics focuses on contemporary debates in an important and central area of philosophy. The reader is taken on a fascinating and entertaining journey through some intriguing mathematical and philosophical territory, including such topics as the realism/anti-realism debate in mathematics, mathematical explanation, the limits of mathematics, the significance of mathematical notation, inconsistent mathematics and the applications of mathematics. Each chapter has a number of discussion questions and recommended further reading from both the contemporary literature and older sources. Very little mathematical background is assumed and all of the mathematics encountered is clearly introduced and explained using a wide variety of examples. The book is suitable for an undergraduate course in philosophy of mathematics and, more widely, for anyone interested in philosophy and mathematics.

Wahrend einer Konferenz zum "Jiidischen Nietzscheanismus" 1995 in Greifs wald hatte mich EGBERT BRIESKORN eingeladen, in der Edition der Gesam melten Werke FELIX HAUSDORFFS dessen philosophische Schriften mit einer Einleitung herauszugeben. FELIX HAUSDORFF hatte darin eng an NIETZSCHE angeschlossen, und er hatte in Greifswald sein erstes Ordinariat fUr Mathematik erhalten - ich sagte spontan und, wie sich bald herausstellen soUte, leichtsinnig ja. Statt nur mit einer kurzen Einleitung hatte ich es bald auch mit langwieri gen Erschlief&ungen des Werks und seiner Kommentierung zu tun. Doch je mehr ich mich in FELIX HAUSDORFFS Schriften einarbeitete, desto mehr notigten sie mir Respekt ab: in ihrer Klarheit, ihrer Redlichkeit, ihrer vornehmen Beschei denheit, ihrer gedanklichen Selbstandigkeit und vor allem in ihrer erstaunlichen Aktualitat. Vielleicht ist nach iiber hundert Jahren nun die Zeit gekommen, in der sie fiir die philosophische Orientierung so fruchtbar werden konnen, wie sie es verdienen. Bei der Kommentierung haben viele helfende Hande mitgewirkt. Mein Dank gilt zuerst den studentischen und wissenschaftlichen Hilfskraften: MIRKO GRON DER und KATRIN STELTER haben die Hauptarbeit in der Recherchierung der Belege iibernommen, JUDITH KARLA und TANJA SCHMIDT eine Vielzahl von Nachweisen beigesteuert, WOLFGANG SCHNEIDER und RALF WITZLER an den Vorarbeiten mitgewirkt. Doz. Dr. REINHARD PESTER (friiher Greifswald, jetzt Berlin) hat uns bei den Nachweisen zu LOTZE, Prof. Dr. MARTIN HOSE (frii her Greifswald, jetzt Miinchen) bei Zitaten aus der griechischen Literatur, Prof. Dr. GISELA FEBEL (friiher Stuttgart, jetzt Bremen) bei Zitaten aus der franzosischen Literatur, Prof. Dr. WALTER ERHART, Prof. Dr."

This introduction to the philosophy of mathematics focuses on contemporary debates in an important and central area of philosophy. The reader is taken on a fascinating and entertaining journey through some intriguing mathematical and philosophical territory, including such topics as the realism/anti-realism debate in mathematics, mathematical explanation, the limits of mathematics, the significance of mathematical notation, inconsistent mathematics and the applications of mathematics. Each chapter has a number of discussion questions and recommended further reading from both the contemporary literature and older sources. Very little mathematical background is assumed and all of the mathematics encountered is clearly introduced and explained using a wide variety of examples. The book is suitable for an undergraduate course in philosophy of mathematics and, more widely, for anyone interested in philosophy and mathematics.

Das Unendliche hat wie keine andere Frage von jeher so tief das Gemut der Menschen bewegt," das Unendliche hat wie kaum eine andere Idee auf den Verstand so an- regend und fruchtbar gewirkt," das Unendliche ist aber auch wie kein anderer Begriff so der Aufklarung bedurftig. HILBERT [226, p. 163] Etwas mehr als 100 Jahre sind vergangen, seit in den Mathemati- schen Annalen der sechste und letzte Teil von CANTORS fundamenta- ler Arbeit UEber unendliche lineare Punktmannichfaltigkeiten erschie- nen ist. Damit war die Mengenlehre geboren und mit ihr eine prinzipiell neue Auffassung des Unendlichen in der Mathematik, verkoerpert in CANTORS Theorie der transfiniten Zahlen. Diese Theo- rie hat HILBERT als "die bewundernswerteste Blute mathematischen Geistes und uberhaupt eine der hoechsten Leistungen rein verstandes- massiger menschlicher Tatigkeit" bezeichnet. Anfangs unbeachtet oder abgelehnt, zu Ende des vorigen Jahrhunderts zunehmend anerkannt und verwendet, durch die Ent- deckung der Antinomien erneut erschuttert, ist die Mengenlehre in ihrer heutigen axiomatisierten Gestalt eines der Fundamente der Mathematik. Die Tatsache, dass alle mathematischen Begriffe auf mengentheoretische Begriffe zuruckgefuhrt werden koennen, hat ei- nige Autoren sogar zu der Behauptung veranlasst, die gesamte Ma- thematik sei letztendlich mit der Mengenlehre identisch. Wenn uns allerdings eine solche Ansicht als eine ungerechtfertigte UEberbeto- nung des Formalen gegenuber dem Inhaltlichen erscheint, so ist doch unbestritten, dass die mengentheoretische Durchdringung der Mathematik neben der Entstehung des strukturellen Denkens und der Verwendung der axiomatischen Methode ein Wesenszug der mo- dernen Mathematik ist. Das hat in zahlreichen Landern bis in den Schulunterricht hinein gewirkt.

How has computer science changed mathematical thinking? In this first ever comprehensive survey of the subject for popular science readers, Arturo Sangalli explains how computers have brought a new practicality to mathematics and mathematical applications. By using fuzzy logic and related concepts, programmers have been able to sidestep the traditional and often cumbersome search for perfect mathematical solutions to embrace instead solutions that are "good enough." If mathematicians want their work to be relevant to the problems of the modern world, Sangalli shows, they must increasingly recognize "the importance of being fuzzy."

As Sangalli explains, fuzzy logic is a technique that allows computers to work with imprecise terms--to answer questions with "maybe" rather than just "yes" and "no." The practical implications of this flexible type of mathematical thinking are remarkable. Japanese programmers have used fuzzy logic to develop the city of Sendai's unusually energy-efficient and smooth-running subway system--one that does not even require drivers. Similar techniques have been used in fields as diverse as medical diagnosis, image understanding by robots, the engineering of automatic transmissions, and the forecasting of currency exchange rates. Sangalli also explores in his characteristically clear and engaging manner the limits of classical computing, reviewing many of the central ideas of Turing and Godel. He shows us how "genetic algorithms" can solve problems by an evolutionary process in which chance plays a fundamental role. He introduces us to "neural networks," which recognize ill-defined patterns without an explicit set of rules--much as a dog can be trained to scent drugs without ever having an exact definition of "drug." Sangalli argues that even though "fuzziness" and related concepts are often compared to human thinking, they can be understood only through mathematics--but the math he uses in the book is straightforward and easy to grasp.

Of equal appeal to specialists and the general reader, "The Importance of Being Fuzzy" reveals how computer science is changing both the nature of mathematical practice and the shape of the world around us.

Karl Weierstrass (1815-1897) was among the leading mathematical figure of the 19th century, a man who had a decisive influence on the way we view analysis today. The centrepiece of this book is the reproduction of a photo album given to Weierstrass in 1885 as a 70th birthday present. The album, which lay hidden in a Berlin museum for over 70 years, contains the portraits of more than 300 students, friends and colleagues from all over Europe, and forms an extraordinary document of the admiration and appreciation shown to him. In an accompanying text, Reinhard Bolling gives interesting details of Weierstrass' life, the lives of those involved in the preparations for his birthday celebrations, and the story of how the album came about."

This collection of specially-commissioned essays by leading scholars presents new research on Isaac Newton and his main philosophical interlocutors and critics. The essays analyze Newton's relation to his contemporaries, especially Barrow, Descartes, Leibniz and Locke, and discuss the ways in which a broad range of figures, including Hume, Maclaurin, Maupertuis, and Kant, reacted to his thought. The wide range of topics discussed includes the laws of nature, the notion of force, the relation of mathematics to nature, Newton's argument for universal gravitation, his attitude toward philosophical empiricism, his use of fluxions, his approach toward measurement problems, and his concept of absolute motion, together with new interpretations of Newton's matter theory. The volume concludes with an extended essay that analyzes the changes in physics wrought by Newton's Principia. A substantial introduction and bibliography provide essential reference guides.

This anthology brings together the year's finest mathematics writing from around the world. Featuring promising new voices alongside some of the foremost names in the field, "The Best Writing on Mathematics 2011" makes available to a wide audience many articles not easily found anywhere else--and you don't need to be a mathematician to enjoy them. These writings offer surprising insights into the nature, meaning, and practice of mathematics today. They delve into the history, philosophy, teaching, and everyday occurrences of math, and take readers behind the scenes of today's hottest mathematical debates. Here Ian Hacking discusses the salient features that distinguish mathematics from other disciplines of the mind; Doris Schattschneider identifies some of the mathematical inspirations of M. C. Escher's art; Jordan Ellenberg describes compressed sensing, a mathematical field that is reshaping the way people use large sets of data; Erica Klarreich reports on the use of algorithms in the job market for doctors; and much, much more.

In addition to presenting the year's most memorable writings on mathematics, this must-have anthology includes a foreword by esteemed physicist and mathematician Freeman Dyson. This book belongs on the shelf of anyone interested in where math has taken us--and where it is headed.

Gottlob Frege (1848 1925) was unquestionably one of the most important philosophers of all time. He trained as a mathematician, and his work in philosophy started as an attempt to provide an explanation of the truths of arithmetic, but in the course of this attempt he not only founded modern logic but also had to address fundamental questions in the philosophy of language and philosophical logic. Frege is generally seen (along with Russell and Wittgenstein) as one of the fathers of the analytic method, which dominated philosophy in English-speaking countries for most of the twentieth century. His work is studied today not just for its historical importance but also because many of his ideas are still seen as relevant to current debates in the philosophies of logic, language, mathematics and the mind. The Cambridge Companion to Frege provides a route into this lively area of research.

Matrices offer some of the most powerful techniques in modem mathematics. In the social sciences they provide fresh insights into an astonishing variety of topics. Dominance matrices can show how power struggles in offices or committees develop; Markov chains predict how fast news or gossip will spread in a village; permutation matrices illuminate kinship structures in tribal societies. All these invaluable techniques and many more are explained clearly and simply in this wide-ranging book. Originally published in 1986. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Professor Morgenstern's deep interests in economic time series and problems of measurement are represented by path-breaking articles devoted to the application of modern statistical analysis to temporal economic data. Originally published in 1967. The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Mathematical Explorations follows on from the author's previous book, Creative Mathematics, in the same series, and gives the reader experience in working on problems requiring a little more mathematical maturity. The author's main aim is to show that problems are often solved by using mathematics that is not obviously connected to the problem, and readers are encouraged to consider as wide a variety of mathematical ideas as possible. In each case, the emphasis is placed on the important underlying ideas rather than on the solutions for their own sake. To enhance understanding of how mathematical research is conducted, each problem has been chosen not for its mathematical importance, but because it provides a good illustration of how arguments can be developed. While the reader does not require a deep mathematical background to tackle these problems, they will find their mathematical understanding is enriched by attempting to solve them.

Information is a central topic in computer science, cognitive science, and philosophy. In spite of its importance in the "information age," there is no consensus on what information is, what makes it possible, and what it means for one medium to carry information about another. Drawing on ideas from mathematics, computer science, and philosophy, this book addresses the definition and place of information in society. The authors, observing that information flow is possible only within a connected distribution system, provide a mathematically rigorous, philosophically sound foundation for a science of information. They illustrate their theory by applying it to a wide range of phenomena, from file transfer to DNA, from quantum mechanics to speech act theory.

Alfred Tarski, one of the greatest logicians of all time, is widely thought of as 'the man who defined truth'. His mathematical work on the concepts of truth and logical consequence are cornerstones of modern logic, influencing developments in philosophy, linguistics and computer science. Tarski was a charismatic teacher and zealous promoter of his view of logic as the foundation of all rational thought, a bon-vivant and a womanizer, who played the 'great man' to the hilt. Born in Warsaw in 1901 to Jewish parents, he changed his name and converted to Catholicism, but was never able to obtain a professorship in his home country. A fortuitous trip to the United States at the outbreak of war saved his life and turned his career around, even while it separated him from his family for years. By the war's end he was established as a professor of mathematics at the University of California, Berkeley. There Tarski built an empire in logic and methodology that attracted students and distinguished researchers from all over the world. From the cafes of Warsaw and Vienna to the mountains and deserts of California, this first full length biography places Tarski in the social, intellectual and historical context of his times and presents a frank, vivid picture of a personally and professionally passionate man, interlaced with an account of his major scientific achievements.

Lo scibile matematico si espande a un ritmo vertiginoso. Nel corso degli ultimi cinquant'anni sono stati dimostrati piu teoremi che nei precedenti millenni della storia umana. Per illustrare la ricchezza della matematica del Novecento, il presente volume porta sulla ribalta alcuni dei protagonisti di questa straordinaria impresa intellettuale, che ha messo a nostra disposizione nuovi e potenti strumenti per indagare la realta che ci circonda.

Presentando matematici famosi accanto ad altri meno noti al grande pubblico - da Hilbert a Godel, da Turing a Nash, da De Giorgi a Wiles - i ritratti raccolti in questo volume ci presentano personaggi dal forte carisma personale, dai vasti interessi culturali, appassionati nel difendere l'importanza delle proprie ricerche, sensibili alla bellezza, attenti ai problemi sociali e politici del loro tempo.

Ne risulta un affresco che documenta la centralita della matematica nella cultura, non solo scientifica ma anche filosofica, artistica e letteraria, del nostro tempo, in un continuo gioco di scambi e di rimandi, di corrispondenze e di suggestioni.

This collection of new essays offers a "state-of-the-art" conspectus of major trends in the philosophy of logic and philosophy of mathematics. A distinguished group of philosophers addresses issues at the center of contemporary debate: semantic and set-theoretic paradoxes, the set/class distinction, foundations of set theory, mathematical intuition and many others. The volume includes Hilary Putnam's 1995 Alfred Tarski lectures published here for the first time. The essays are presented to honor the work of Charles Parsons.

Durante la II guerra mondiale hanno avuto luogo numerosi risultati di rilievo nel campo della crittografia militare. Uno dei meno conosciuti e quello usato dal servizio di intelligence svedese, nei confronti del codice tedesco per le comunicazioni strategiche con i comandi dei paesi occupati nel nord Europa, le cui linee passavano per la Svezia. In tal modo, durante la fase piu critica della guerra la direzione politica e militare svedese era in grado di seguire i piani e le disposizioni dei Tedeschi, venendo a conoscenza dei piu arditi progetti per modificare la propria politica, tenendo la Svezia fuori dalla guerra.

La violazione del codice tedesco e narrata in dettaglio, per la prima volta, con elementi che gli permettono di essere un ottima introduzione al campo della crittografia, oltre che un ritratto vitale e umano della societa del tempo: una disperata condizione bellica, l'intrigo politico e spionistico, il genio del matematico Arne Beurling, le difficolta e i trucchi del mestiere, e il lavoro sistematico e oscuro di una folla di decrittatori.

The design inference uncovers intelligent causes by isolating their key trademark: specified events of small probability. Just about anything that happens is highly improbable, but when a highly improbable event is also specified (i.e. conforms to an independently given pattern) undirected natural causes lose their explanatory power. Design inferences can be found in a range of scientific pursuits from forensic science to research into the origins of life to the search for extraterrestrial intelligence. This challenging and provocative 1998 book shows how incomplete undirected causes are for science and breathes new life into classical design arguments. It will be read with particular interest by philosophers of science and religion, other philosophers concerned with epistemology and logic, probability and complexity theorists, and statisticians.

Guicciardini presents a comprehensive survey of both the research and teaching of Newtonian calculus, the calculus of "fluxions", over the period between 1700 and 1810. Although Newton was one of the inventors of calculus, the developments in Britain remained separate from the rest of Europe for over a century. While it is usually maintained that after Newton there was a period of decline in British mathematics, the author's research demonstrates that the methods used by researchers of the period yielded considerable success in laying the foundations and investigating the applications of the calculus. Even when "decline" set in, in mid century, the foundations of the reform were being laid, which were to change the direction and nature of the mathematics community. The book considers the importance of Isaac Newton, Roger Cotes, Brook Taylor, James Stirling, Abraham de Moivre, Colin Maclaurin, Thomas Bayes, John Landen and Edward Waring. This will be a useful book for students and researchers in the history of science, philosophers of science and undergraduates studying the history of mathematics.

Philosophical considerations, which are often ignored or treated casually, are given careful consideration in this introduction. Thomas Forster places the notion of inductively defined sets (recursive datatypes) at the center of his exposition resulting in an original analysis of well established topics. The presentation illustrates difficult points and includes many exercises. Little previous knowledge of logic is required and only a knowledge of standard undergraduate mathematics is assumed.

Questo volume raccoglie lo scambio epistolare tra Cantor e Dedekind, finora edito parte in tedesco e parte in francese. Sara la prima edizione italiana completa di questo fondamentale carteggio, in cui si vedono nascere la nozione di cardinale e ordinale transfiniti, in cui si dimostra la non numerabilita dell'insieme dei numeri reali R e si leggono i primi tentativi e le correzioni alla costruzione di una biiezione tra R e R2, e le discussioni fra Cantor e Dedekind sull'invarianza della nozione di dimensione. "Pochi scritti matematici possono competere - scrive Pietro Nastasi nell'Introduzione - con questa corrispondenza nell'evidenziare il complesso intreccio psicologico che presiede all'invenzione matematica. E nessun lavoro storiografico potrebbe far emergere, meglio di queste lettere, la differenza fra le due personalita implicate: focosa e fantasiosa quella di Cantor, pacata e critica quella del piu anziano amico".