Ernst Zermelo (1871-1953) is regarded as the founder of axiomatic set theory and best-known for the first formulation of the axiom of choice. However, his papers include also pioneering work in applied mathematics and mathematical physics. This edition of his collected papers will consist of two volumes. Besides providing a biography, the present Volume I covers set theory, the foundations of mathematics, and pure mathematics and is supplemented by selected items from his Nachlass and part of his translations of Homer's Odyssey. Volume II will contain his work in the calculus of variations, applied mathematics, and physics. The papers are each presented in their original language together with an English translation, the versions facing each other on opposite pages. Each paper or coherent group of papers is preceded by an introductory note provided by an acknowledged expert in the field which comments on the historical background, motivations, accomplishments, and influence.
Ernst Zermelo (1871-1953) is regarded as the founder of axiomatic set theory and best-known for the first formulation of the axiom of choice. However, his papers include also pioneering work in applied mathematics and mathematical physics. This edition of his collected papers will consist of two volumes. Besides providing a biography, the present Volume I covers set theory, the foundations of mathematics, and pure mathematics and is supplemented by selected items from his Nachlass and part of his translations of Homer's Odyssey. Volume II will contain his work in the calculus of variations, applied mathematics, and physics. The papers are each presented in their original language together with an English translation, the versions facing each other on opposite pages. Each paper or coherent group of papers is preceded by an introductory note provided by an acknowledged expert in the field which comments on the historical background, motivations, accomplishments, and influence.
Ernst Zermelo (1871-1953) is regarded as the founder of axiomatic set theory and is best-known for the first formulation of the axiom of choice. However, his papers also include pioneering work in applied mathematics and mathematical physics. This edition of his collected papers consists of two volumes. The present Volume II covers Ernst Zermelo’s work on the calculus of variations, applied mathematics, and physics. The papers are each presented in their original language together with an English translation, the versions facing each other on opposite pages. Each paper or coherent group of papers is preceded by an introductory note provided by an acknowledged expert in the field who comments on the historical background, motivation, accomplishments, and influence.
Ernst Zermelo (1871-1953) is regarded as the founder of axiomatic set theory and best-known for the first formulation of the axiom of choice. However, his papers include also pioneering work in applied mathematics and mathematical physics. This edition of his collected papers will consist of two volumes. Besides providing a biography, the present Volume I covers set theory, the foundations of mathematics, and pure mathematics and is supplemented by selected items from his Nachlass and part of his translations of Homer's Odyssey. Volume II will contain his work in the calculus of variations, applied mathematics, and physics. The papers are each presented in their original language together with an English translation, the versions facing each other on opposite pages. Each paper or coherent group of papers is preceded by an introductory note provided by an acknowledged expert in the field which comments on the historical background, motivations, accomplishments, and influence.
Ernst Zermelo (1871-1953) is regarded as the founder of axiomatic set theory and is best-known for the first formulation of the axiom of choice. However, his papers also include pioneering work in applied mathematics and mathematical physics. This edition of his collected papers consists of two volumes. The present Volume II covers Ernst Zermelo’s work on the calculus of variations, applied mathematics, and physics. The papers are each presented in their original language together with an English translation, the versions facing each other on opposite pages. Each paper or coherent group of papers is preceded by an introductory note provided by an acknowledged expert in the field who comments on the historical background, motivation, accomplishments, and influence.
The Symbolic Forms has long been considered the greatest of Cassirer's works. Into it he poured all the resources of his vast learning about language and myth, religion, art, and science--the various creative symbolizing activities and constructions through which man has expressed himself and given intelligible objective form to this experience. "These three volumes alone (apart from Cassirer's other papers and books) make an outstanding contribution to epistemology and to the human power of abstraction. It is rather as if 'The Golden Bough' had been written in philosophical rather than in historical terms."--F.I.G. Rawlins, Nature
This textbook addresses the mathematical description of sets, categories, topologies and measures, as part of the basis for advanced areas in theoretical computer science like semantics, programming languages, probabilistic process algebras, modal and dynamic logics and Markov transition systems. Using motivations, rigorous definitions, proofs and various examples, the author systematically introduces the Axiom of Choice, explains Banach-Mazur games and the Axiom of Determinacy, discusses the basic constructions of sets and the interplay of coalgebras and Kripke models for modal logics with an emphasis on Kleisli categories, monads and probabilistic systems. The text further shows various ways of defining topologies, building on selected topics like uniform spaces, Gödel’s Completeness Theorem and topological systems. Finally, measurability, general integration, Borel sets and measures on Polish spaces, as well as the coalgebraic side of Markov transition kernels along with applications to probabilistic interpretations of modal logics are presented. Special emphasis is given to the integration of (co-)algebraic and measure-theoretic structures, a fairly new and exciting field, which is demonstrated through the interpretation of game logics. Readers familiar with basic mathematical structures like groups, Boolean algebras and elementary calculus including mathematical induction will discover a wealth of useful research tools. Throughout the book, exercises offer additional information, and case studies give examples of how the techniques can be applied in diverse areas of theoretical computer science and logics. References to the relevant mathematical literature enable the reader to find the original works and classical treatises, while the bibliographic notes at the end of each chapter provide further insights and discussions of alternative approaches.
TABLE OF CONTENTS: ALGEBRA, WHAT ELSE?: 1. The Birth of a Masterwork - 2. Commutativity and Left- and Right-Division - 3. Algorithms, Algorithms, Algorithms - 4. Formalism - 5. A Fateful Choice - 6. Overview - 7. A Strange Document - 8. Acknowledgements - 9. Tools - Notes — ON THE FORMAL ELEMENTS OF THE ABSOLUTE ALGEBRA: §. 1. Character des zu behandelnden Problems. Character of the Problem in Issue - §. 2. Einschränkungen der Aufgabe. Restrictions of our Scope - §. 3. Die Fundamentalgleichungen für nur zwei Zahlen. Algorithmen. The Fundamental Equations for only Two Numbers. Algorithms - §. 4. Vertauschungsprincipien. Principles of Permutation - §. 5. Die Fundamentalgleichungen für drei Zahlen. Elementarcyklen und Gruppen. The Fundamental Equations for Three Numbers. Elementary Cycles and Groups - §. 6. Consequenzen der Algorithmen C1; C2; C3 für drei Zahlen. Consequences of the Algorithms C1; C2; C3 for Three Numbers - §. 7. Consequenzen von C0. Consequences of C0 - §. 8. Combination der Ci. Combination of the Ci - §. 9. Das Formelsystem O1 der ordinäre Algebra. The Formal System O1 of the Usual Algebra - §. 10. Untergeordnete Algorithmen von O1: Weitere ermittelte Tragweitezahlen. Subordinate Algorithms of O1: Further Sizes — FIGURES - Notes — APPENDIX - Notes — ILLUSTRATIONS - Bibliography - Index of the Main Concepts - Index of the Illustrations.
NG van Kampen is a well-known theoretical physicist who has had a long and distinguished career. His research covers scattering theory, plasma physics, statistical mechanics, and various mathematical aspects of physics. In addition to his scientific work, he has written a number of papers about more general aspects of science. An indefatigable fighter for intellectual honesty and clarity, he has pointed out repeatedly that the fundamental ideas of physics have been needlessly obscured. As those papers appeared in various journals, partly in Dutch, it was felt that it would be worthwhile to collect them (translating the Dutch material into English) and make them available to a larger audience. This is a book of major importance to scientists and university teachers.
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