Bøger i Science Networks. Historical Studies serien

»Philosophy of Mathematics« is understood, in this book, as an effort to clarify such questions that mathematics itself raises but cannot answer with its own methods. These include, for example, questions about the ontological status of mathematical objects (e.g., what is the nature of mathematical objects?) and the epistemological status of mathematical theorems (e.g., from what sources do we draw when we prove mathematical theorems?). The answers given by Plato, Aristotle, Euclid, Descartes, Locke, Leibniz, Kant, Cantor, Frege, Dedekind, Hilbert and others will be studied in detail. This will lead us to deep insights, not only into the history of mathematics, but also into the conception of mathematics as it is commonly held in the present time.The book is a translation from the German, however revised and considerably expanded. Various chapters have been completely rewritten.

To attempt to compile a relatively complete bibliography of the theory of functions of a real variable with the requisite bibliographical data, to enumer­ ate the names of the mathematicians who have studied this subject, exhibit their fundamental results, and also include the most essential biographical data about them, to conduct an inventory of the concepts and methods that have been and continue to be applied in the theory of functions of a real variable ... in short, to carry out anyone of these projects with appropriate completeness would require a separate book involving a corresponding amount of work. For that reason the word essays occurs in the title of the present work, allowing some freedom in the selection of material. In justification of this selection, it is reasonable to try to characterize to some degree the subject to whose history these essays are devoted. The truth of the matter is that this is a hopeless enterprise if one requires such a characterization to be exhaustively complete and concise. No living subject can be given a final definition without provoking some objections, usually serious ones. But if we make no such claims, a characterization is possible; and if the first essay of the present book appears unconvincing to anyone, the reason is the personal fault of the author, and not the objective necessity of the attempt.

While it is well known that the Delian problems are impossible to solve with a straightedge and compass ¿ for example, it is impossible to construct a segment whose length is cube root of 2 with these instruments ¿ the discovery of the Italian mathematician Margherita Beloch Piazzolla in 1934 that one can in fact construct a segment of length cube root of 2 with a single paper fold was completely ignored (till the end of the 1980s). This comes as no surprise, since with few exceptions paper folding was seldom considered as a mathematical practice, let alone as a mathematical procedure of inference or proof that could prompt novel mathematical discoveries. A few questions immediately arise: Why did paper folding become a non-instrument? What caused the marginalisation of this technique? And how was the mathematical knowledge, which was nevertheless transmitted and prompted by paper folding, later treated and conceptualised?Aiming to answer these questions, this volume provides, for the first time, an extensive historical study on the history of folding in mathematics, spanning from the 16th century to the 20th century, and offers a general study on the ways mathematical knowledge is marginalised, disappears, is ignored or becomes obsolete.In doing so, it makes a valuable contribution to the field of history and philosophy of science, particularly the history and philosophy of mathematics and is highly recommended for anyone interested in these topics.

In this book the author presents a comprehensive study of Diophantos' monumental work known as Arithmetika, a highly acclaimed and unique set of books within the known Greek mathematical corpus. Its author, Diophantos, is an enigmatic figure of whom we know virtually nothing. Starting with Egyptian, Babylonian and early Greek mathematics the author paints a picture of the sources the Arithmetika may have had. Life in Alexandria, where Diophantos lived, is described and, on the basis of the limited available evidence, his biography is outlined. Of Arithmetika's 13 books only 6 survive in Greek. It was not until 1971 that these were complemented by the discovery of 4 other books in an Arab translation. This allows the author to describe the structure, the contents and the mathematics of the Arithmetika in detail. Furthermore it is shown that Diophantos had a remarkable skill to solve higher degree equations. In the second part, the author draws our attention to the survival of Diophantos' work in both Arab and European mathematical cultures. Once Xylander's critical 1575 edition reached its European public, the fame of the Arithmetika grew. It was studied, translated and modified by such authors as Bombelli, Stevin and Vite. It reached its pinnacle of fame in 1621 with the publication of Bachet's translation into Latin. The marginal notes by Fermat in his copy of Diophantos, including his famous "e;Last Theorem"e;, were the starting point of a whole new research subject: the theory of numbers.

1 Aim and General Description of the Anthology The purpose of this anthology is to introduce the English speaking public to the wide spectrum of texts authored predominently by physicists portraying the ac­ tual and perceived role of physics in the Nazi state. Up to now no broad and well­ balanced documentation of German physics during this time has been available in English, despite the significant role physics has played both politically (e. g. , in weaponry planning) and ideologically (e. g. , in the controversy over the value of theoretical ('Jewish') vs. experimental ('Aryan') physics), and even though prominent figures like the scientist-philosopher and emigre Albert Einstein and the controversial nuclear physicist Werner Heisenberg have become household names. This anthology will attempt to bridge this gap by presenting contempo­ rary documents and eye-witness accounts by the physicists themselves. Authors were chosen to represent the various political opinions and specialties within the physics community, omitting some of the more readily accessible texts by leading physicists (e. g. , Einstein, Heisenberg, Lenard) in favor of those by less well-known but nonetheless important figures (e. g. , Finkelnburg, Max Wien, Ramsauer). In this way we hope not only to circumvent the constricted 'Great Men' approach to history but also to offer a broader picture of the activities and conflicts within the field and the effects of the political forces exerted upon them.

In 1920s, a long-lasting controversy on the interpretation of nuclear beta spectrum arose between Lise Meitner and Charles Drummond Ellis. A new dispute, this time between Niels Bohr and Wolfgang Pauli, broke out. Pauli argued for a new particle, and Bohr for a new theory;

To attempt to compile a relatively complete bibliography of the theory of functions of a real variable with the requisite bibliographical data, to enumer­ ate the names of the mathematicians who have studied this subject, exhibit their fundamental results, and also include the most essential biographical data about them, to conduct an inventory of the concepts and methods that have been and continue to be applied in the theory of functions of a real variable ... in short, to carry out anyone of these projects with appropriate completeness would require a separate book involving a corresponding amount of work. For that reason the word essays occurs in the title of the present work, allowing some freedom in the selection of material. In justification of this selection, it is reasonable to try to characterize to some degree the subject to whose history these essays are devoted. The truth of the matter is that this is a hopeless enterprise if one requires such a characterization to be exhaustively complete and concise. No living subject can be given a final definition without provoking some objections, usually serious ones. But if we make no such claims, a characterization is possible; and if the first essay of the present book appears unconvincing to anyone, the reason is the personal fault of the author, and not the objective necessity of the attempt.

FOREWORD This book came about as a result of two events: an exhibition on the Solvay Physics Councils, held in Brussels in May 1995, and a conference on the same theme which took place at the Free University of Brussels (ULB) on May 1Oth 1995.

In this volume, a distinguished set of international scholars examine the nature of collaboration between life partners in the sciences, with particular attention to the ways in which personal and professional dynamics can foster or inhibit scientific practice.

While it is well known that the Delian problems are impossible to solve with a straightedge and compass - for example, it is impossible to construct a segment whose length is cube root of 2 with these instruments - the discovery of the Italian mathematician Margherita Beloch Piazzolla in 1934 that one can in fact construct a segment of length cube root of 2 with a single paper fold was completely ignored (till the end of the 1980s). This comes as no surprise, since with few exceptions paper folding was seldom considered as a mathematical practice, let alone as a mathematical procedure of inference or proof that could prompt novel mathematical discoveries. A few questions immediately arise: Why did paper folding become a non-instrument? What caused the marginalisation of this technique? And how was the mathematical knowledge, which was nevertheless transmitted and prompted by paper folding, later treated and conceptualised?Aiming to answer these questions, this volume provides, for the first time, an extensive historical study on the history of folding in mathematics, spanning from the 16th century to the 20th century, and offers a general study on the ways mathematical knowledge is marginalised, disappears, is ignored or becomes obsolete.In doing so, it makes a valuable contribution to the field of history and philosophy of science, particularly the history and philosophy of mathematics and is highly recommended for anyone interested in these topics.

This second volume of Gyllenbok's encyclopaedia of historical metrology comprises the first part of the compendium of measurement systems and currencies of all sovereign states of the modern World (A-I). Units of measurement are of vital importance in every civilization through history.

This third volume of Gyllenbok's encyclopaedia of historical metrology comprises the second part of the compendium of measurement systems and currencies of all sovereign states of the modern World (J-Z). Units of measurement are of vital importance in every civilization through history.

George Boole (1815-1864) is well known to mathematicians for his research and textbooks on the calculus, but his name has spread world-wide for his innovations in symbolic logic and the development and applications made since his day.

Category theory is a general mathematical theory of structures and of structures of structures.

This book provides a scientific and intellectual biography of John von Neumann, who proposed a conception of the world as a mathematical game, one globally governed by a universal logic in which individual consciousness moved following different strategies.

This first of three volumes starts with a short introduction to historical metrology as a scientific discipline and goes on with an anthology of acient and modern measurement systems of all kind, scientific measures, units of time, weights, currencies etc.

The Latin "Version II", till now attributed to Adelard of Bath, is edited here for the first time. But a comparison of the text of version II with those of versions I and III yields little or no reason to assume that Adelard was the author of version II. Version II must have been written later than version I and before version III;

Die Entstehung des chemischen Verbindungsbegriffs, ein Fundamentalbegriff der neuzeitlichen Chemie, ist kaum untersucht, weil er für eine blosse Präzisierung des naturphilosophischen Atomkonzepts gehalten wurde. Das Buch widerlegt diese Ansicht als ideengeschichtliches Vorurteil und weist minutiös nach, dass der Verbindungsbegriff als Resultat der begrifflichen Strukturierung des empirischen Wissens entstand, das aus der gewerblichen chemischen Praxis des 16. und 17. Jahrhunderts stammte. Damit rekonstruiert die Autorin nicht allein den Beginn der wissenschaftlichen Chemie in einer grundlegend neuen historischen, begrifflichen und methodischen Ordnung. Sie zeigt darüber hinaus, wie diese "chemische Revolution" in den Kontext der wissenschaftlichen Revolution des 17. Jahrhunderts einzuordnen ist, nämlich als die einer "baconischen Wissenschaft" (Kuhn), die das Erfahrungswissen einer neuartigen gewerblichen Praxis ohne Rückgriffsmöglichkeiten auf brauchbare Theorietraditionen in eine wissenschaftliche Theorie überführte. Das Buch ist so zugleich ein substanzieller Beitrag zum Verständnis wissenschaftlicher Revolutionen, der insbesondere zum Überdenken der wissenschaftstheoretischen Annahme einer Theoriedominanz bei der Entstehung und Entwicklung der Wissenschaften nötigt. Es ist deswegen nicht nur für Chemiehistoriker, für die es zum neuen Standardwerk über die Chemie des 17. Jahr- hunderts werden dürfte, von Interesse, sondern ebenso für Wissenschaftshistoriker allgemein sowie für Wissenschaftstheoretiker, Philosophen und Kulturhistoriker.

Volterra's interest in the application of mathematics to the non physical sciences, and to biology and economics in particular, dates back to the turn of the century and was expressed in his inaugural address at the University of Rome for the academic year 1900/01 (VOLTERRA 1901).

The significance of foundational debate in mathematics that took place in the 1920s seems to have been recognized only in circles of mathematicians and philosophers. A period in the history of mathematics when mathematics and philosophy, usually so far away from each other, seemed to meet.

The significance of foundational debate in mathematics that took place in the 1920s seems to have been recognized only in circles of mathematicians and philosophers. A period in the history of mathematics when mathematics and philosophy, usually so far away from each other, seemed to meet.

The Latin "Version II", till now attributed to Adelard of Bath, is edited here for the first time. But a comparison of the text of version II with those of versions I and III yields little or no reason to assume that Adelard was the author of version II. Version II must have been written later than version I and before version III;

The controversy between the wave theory and the emission theory of light early in the nineteenth century has been a subject of numerous studies. It examines the impact of the concept of interference of light on the development of the early nineteenth century optics in general, and the theory of light, in particular.

George Boole (1815-1864) is well known to mathematicians for his research and textbooks on the calculus, but his name has spread world-wide for his innovations in symbolic logic and the development and applications made since his day.

Volterra's interest in the application of mathematics to the non physical sciences, and to biology and economics in particular, dates back to the turn of the century and was expressed in his inaugural address at the University of Rome for the academic year 1900/01 (VOLTERRA 1901).

Matvei Bronstein was one of those for whom the vast territory of theoretical physics was as familiar as his own home: he worked in cosmology, nuclear physics, gravitation, semiconductors, atmospheric physics, quantum electrodynamics, astro physics and the relativistic quantum theory.

Die Relativitatstheorien (RT) Einsteins gehoren zu den meistdiskutierten Theorien der Physik des zwanzigsten Jahrhunderts. Nach der Formulie­ rung der sog. 'speziellen Relativitatstheorie' (SRT) im Jahr 1905 nah­ men zunachst nur einige Spezialisten von ihr Kenntnis, bis mit ungefiihr fiinf Jahren Verspatung dann auch zunehmend Nicht-Physiker sich mit ihr zu beschaftigen begannen, angeregt durch populiirwissenschaftliche, all­ gemeinverstiindliche 'Einfiihrungen' von Kollegen Einsteins wie z. B. Paul Langevin in Frankreich oder Max von Laue in Deutschland. Diese Pha­ senverschiebung zwischen fachwissenschaftlichem Ausbau der Theorie und offentlicher Notiznahme wiederholte sich bei Einsteins 'allgemeiner Theorie der Relativitat und Gravitation' (ART). Zwischen 1913 und 1915 in ihren wesentlichen Ziigen ausformuliert, wurde sie erst nach einer spektakuliiren experimentellen Bestatigung im Jahr 1919 einem breiterem Publikum be­ kannt. In meiner Arbeit werde ich den Facettenreichtum der Ausdeutungen, die beide RT erfuhren, zunachst zu referieren und durch repriisentative Zi­ tate aus der Literatur der Zeit zu belegen haben. Der Umfang dieser Arbeit geht wesentlich auf das Konto dieser ausgewahlten Belege - davon verspre­ che ich mir, dafi nicht nur bislang unveroffentlichte Dokumente, sondern auch entlegene Texte hier in ihren zentralen Passagen leicht zuganglich ge­ macht werden. Fernerhin werde ich aber auch zu analysieren haben, warum derartig vielfaltige, einander mitunter diametral entgegengesetzte Interpre­ tationen einer wissenschaftlichen Theorie vorgelegt wurden.

The Twenty-First International Congress of Mathematicians (ICM) was held in Kyoto, Japan, from August 21 through 29, 1990, the first congress that has taken place in the Eastern hemisphere. On this occasion, Japanese historians of mathe­ matics organized the History of Mathematics Symposium which was held at the Sanjo Conference Hall of the University of Tokyo on August 31 and September 1, as one of the related conferences of the Congress. The symposium was officially sponsored by the Executive Committee of the ICM 90, the History of Science So­ ciety of Japan, and the International Commission on the History of Mathematics. The Executive Committee consisted of Murata Tamotsu (Chairperson, Momoyama Gakuin University), Sugiura Mitsuo (Vice-Chairperson, Tsuda College), Sasaki Chikara (Secretary, The University of Tokyo), Adachi Norio (Waseda University), Nagaoka Ryosuke (Tsuda College until 1990, now Daito Bunka University), and Hirano Yoichi (Treasurer, Tokai University). The symposium emphasized the following three fields of study: (1) mathe­ matical traditions in the East, (2) the history of modern European mathematics, and (3) interaction between mathematical research and the history of mathemat­ ics. These fields were chosen mainly because, first, the symposium was related to the ICM, the most important congress of working mathematicians, and, second, the Kyoto ICM was held in a non-Western country for the first time. The sym­ posium consisted of the two Sessions: Session A for invited speakers and Session B for short communications.

Philanthropic societies funded by the Rockefeller family were prominent in the social history of the twentieth century, for their involvement in medicine and applied science. This book provides the first detailed study of their relatively brief but nonetheless influential foray into the field of mathematics.