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»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.
We bring into full light some excerpts on musical subjects which were until now scattered throughout the most famous scientific texts. The main scientific and musical cultures outside of Europe are also taken into consideration. The first and most important property to underline in the scientific texts examined here is the language they are written in. This means that our multicultural history of the sciences necessarily also becomes a review of the various dominant languages used in the different historical contexts. In this volume, the history of the development of the sciences is told as it happened in real contexts, not in an alienated ideal world.
We bring into full light some excerpts on musical subjects which were until now scattered throughout the most famous scientific texts. The main scientific and musical cultures outside of Europe are also taken into consideration. The first and most important property to underline in the scientific texts examined here is the language they are written in. This means that our multicultural history of the sciences necessarily also becomes a review of the various dominant languages used in the different historical contexts. In this volume, the history of the development of the sciences is told as it happened in real contexts, not in an alienated ideal world.
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.
This book presents new insights into Leibniz's research on planetary theory and his system of pre-established harmony. It also provides an in-depth discussion of Kepler's influence on Leibniz's planetary theory and more generally, on Leibniz's concept of pre-established harmony.
The main focus of this book is on the interconnection of two unorthodox scientific ideas, the varying-gravity hypothesis and the expanding-earth hypothesis.
The similarities are remarkable, but there is also much in l'Hopital's book that is original and innovative.This book offers the first English translation of Bernoulli's notes, along with the first faithful English translation of l'Hopital's text, complete with annotations and commentary.
The book provides new material on the history of non-Euclidean geometry, stressing the previously neglected role of Leibniz in these developments.This volume will be of interest to historians in mathematics, philosophy or logic, as well as mathematicians interested in non-Euclidean geometry.
This book presents a history of classical mechanics, documenting methods of study from Greece in the fourth century BC to late nineteenth-century Europe. The text assesses effectiveness of forces based on the virtual motion of their points of application.
The aim of this book is to present and analyze philosophical conceptions concerning mathematics and logic as formulated by Polish logicians, mathematicians and philosophers in the 1920s and 1930s. It was a remarkable period in the history of Polish science, in particular in the history of Polish logic and mathematics. Therefore, it is justified to ask whether and to what extent the development of logic and mathematics was accompanied by a philosophical reflection. We try to answer those questions by analyzing both works of Polish logicians and mathematicians who have a philosophical temperament as well as their research practice. Works and philosophical views of the following Polish scientists will be analyzed: Waclaw Sierpinski, Zygmunt Janiszewski, Stefan Mazurkiewicz, Stefan Banach Hugo Steinhaus, Eustachy Zylinsk and Leon Chwistek, Jan Lukasiewicz, Zygmunt Zawirski, Stanislaw Lesniewski, Tadeusz Kotarbinski, Kazimierz Ajdukiewicz, Alfred Tarski, Andrzej Mostowski and Henryk Mehlberg, Jan Sleszynski, Stanislaw Zaremba and Witold Wilkosz. To indicate the background of scientists being active in the 1920s and 1930s we consider in Chapter 1 some predecessors, in particular: Jan Sniadecki, Jozef Maria Hoene-Wronski, Samuel Dickstein and Edward Stamm.
The chief purpose of the book is to present, in detail, a compilation of proofs of the Cantor-Bernstein Theorem (CBT) published through the years since the 1870's. Over thirty such proofs are surveyed.The book comprises five parts. In the first part the discussion covers the role of CBT and related notions in the writings of Cantor and Dedekind. New views are presented, especially regarding the general proof of CBT obtained by Cantor, his proof of the Comparability Theorem, the ruptures in the Cantor-Dedekind correspondence and the origin of Dedekind's proof of CBT.The second part covers the first CBT proofs published (1896-1901). The works of the following mathematicians is considered in detail: Schroder, Bernstein, Bore, Schoenflies and Zermelo. Here a subtheme of the book is launched; it concerns the research project following Bernstein's Division Theorem (BDT).In its third part the book covers proofs that emerged during the period when the logicist movement was developed (1902-1912). It covers the works of Russell and Whitehead, Jourdain, Harward, Poincare, J. Konig, D. Konig (his results in graph theory), Peano, Zermelo, Korselt. Also Hausdorff's paradox is discussed linking it to BDT.In the fourth part of the book are discussed the developments of CBT and BDT (including the inequality-BDT) in the hands of the mathematicians of the Polish School of Logic, including Sierpinski, Banach, Tarski, Lindenbaum, Kuratowski, Sikorski, Knaster, the British Whittaker, and Reichbach.Finally, in the fifth part, the main discussion concentrates on the attempts to port CBT to intuitionist mathematics (with results by Brouwer, Myhill, van Dalen and Troelstra) and to Category Theory (by Trnkova and Koubek).The second purpose of the book is to develop a methodology for the comparison of proofs. The core idea of this methodology is that a proof can be described by two descriptors, called gestalt and metaphor. It is by comparison of their descriptors that the comparison of proofs is obtained. The process by which proof descriptors are extracted from a proof is named 'proof-processing', and it is conjectured that mathematicians perform proof-processing habitually, in the study of proofs.
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.
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.
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.
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 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.
This monograph presents a groundbreaking scholarly treatment of the German mathematician Jost Bürgi's original work on logarithms, Arithmetische und Geometrische Progreß Tabulen. It provides the first-ever English translation of Bürgi's text and illuminates his role in the development of the conception of logarithms, for which John Napier is traditionally given priority. High-resolution scans of each page of the his handwritten text are reproduced for the reader and as a means of preserving an important work for which there are very few surviving copies.The book begins with a brief biography of Bürgi to familiarize readers with his life and work, as well as to offer an historical context in which to explore his contributions. The second chapter then describes the extant copies of the Arithmetische und Geometrische Progreß Tabulen, with a detailed description of the copy that is the focus of this book, the 1620 "Graz manuscript". A complete facsimile of the text is included in the next chapter, along with a corresponding transcription and an English translation; a transcription of a second version of the manuscript (the "Gdansk manuscript") is included alongside that of the Graz edition so that readers can easily and closely examine the differences between the two. The final chapter considers two important questions about Bürgi's work, such as who was the copyist of the Graz manuscript and what the relationship is between the Graz and Gdansk versions. Appendices are also included that contain a timeline of Bürgi's life, the underlying concept of Napier's construction of logarithms, and scans of all 58 sheets of the tables from Bürgi's text.Anyone with an appreciation for the history of mathematics will find this book to be an insightful and interesting look at an important and often overlooked work. It will also be a valuable resource for undergraduates taking courses in the history of mathematics, researchers of the history of mathematics, and professors of mathematics education who wish to incorporate historical context into their teaching.
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.
The tremendous success of indivisibles methods in geometry in the seventeenth century, responds to a vast project: installation of infinity in mathematics. The pathways by the authors are very diverse, as are the characterizations of indivisibles, but there are significant factors of unity between the various doctrines of indivisible; the permanence of the language used by all authors is the strongest sign.These efforts do not lead to the stabilization of a mathematical theory (with principles or axioms, theorems respecting these first statements, followed by applications to a set of geometric situations), one must nevertheless admire the magnitude of the results obtained by these methods and highlights the rich relationships between them and integral calculus.The present book aims to be exhaustive since it analyzes the works of all major inventors of methods of indivisibles during the seventeenth century, from Kepler to Leibniz. It takes into account the rich existingliterature usually devoted to a single author. This book results from the joint work of a team of specialists able to browse through this entire important episode in the history of mathematics and to comment it.The list of authors involved in indivisibles¿ field is probably sufficient to realize the richness of this attempt; one meets Kepler, Cavalieri, Galileo, Torricelli, Gregoire de Saint Vincent, Descartes, Roberval, Pascal, Tacquet, Lalouvère, Guldin, Barrow, Mengoli, Wallis, Leibniz, Newton.
This monograph presents a groundbreaking scholarly treatment of the German mathematician Jost Burgi's original work on logarithms, Arithmetische und Geometrische Progress Tabulen.
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.
His most famous work is the three-volume Traite du calcul differentiel et du calcul integral, which is an encyclopedic appraisal of 18th-century calculus that remained the standard reference on the subject through much of the 19th century.
This book reconstructs, from both historical and theoretical points of view, Leibniz's geometrical studies, focusing in particular on the research Leibniz carried out in his final years. This is the first ever, comprehensive historical reconstruction of Leibniz's geometry.
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.
An anthology of fully annotated translations of texts by physicists, written both before, during and after the Nazi period of rule in Germany. There is also an introductory section giving an overview of the impact of Nazi ideology on physics during this period.
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.
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.
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