Cet ouvrage traite de la transformation fondamentale survenue dans la pensee mathematique a la suite de la decouverte de la geometrie non euclidienne. Cette transformation a eu comme consequence celle d'admettre que, non seulement pouvaient exister plusieurs geometries, mais encore plusieurs espaces mathematiques et plusieurs espaces physiques differents. La recherche s'attache en grande partie a analyser les etapes qui ont conduit a cette nouvelle conception et aux idees mathematiques qui en sont le fondement. Le livre cherche egalement a en elucider la signification epistemologique et a mettre en evidence la nature et le role de l'espace dans la constitution de certaines theories mathematiques et dans la recherche des principes essentiels de la physique.

"Elliptic Tales" describes the latest developments in number theory by looking at one of the most exciting unsolved problems in contemporary mathematics--the Birch and Swinnerton-Dyer Conjecture. In this book, Avner Ash and Robert Gross guide readers through the mathematics they need to understand this captivating problem.

The key to the conjecture lies in elliptic curves, which may appear simple, but arise from some very deep--and often very mystifying--mathematical ideas. Using only basic algebra and calculus while presenting numerous eye-opening examples, Ash and Gross make these ideas accessible to general readers, and, in the process, venture to the very frontiers of modern mathematics.

These two volumes present the collected works of James Serrin. He did seminal work on a number of the basic tools needed for the study of solutions of partial differential equations. Many of them have been and are being applied to solving problems in science and engineering. Among the areas which he studied are maximum principle methods and related phenomena such as Harnack's inequality, the compact support principle, dead cores and bursts, free boundary problems, phase transitions, the symmetry of solutions, boundary layer theory, singularities and fine regularity properties. The volumes include commentaries by leading mathematicians to indicate the significance of the articles and to discuss further developments along the lines of these articles.

This volume provides a unique primary source on the history and philosophy of mathematics and science from the mediaeval Arab world. The fourth volume of "A History of Arabic Sciences and Mathematics" is complemented by three preceding volumes which focused on infinitesimal determinations and other chapters of classical mathematics.

This book includes five main works of the polymath Ibn al-Haytham (Alhazen) on astronomy, spherical geometry and trigonometry, plane trigonometry and studies of astronomical instruments on hour lines, horizontal sundials and compasses for great circles.

In particular, volume four examines:

• the increasing tendency to mathematize the inherited astronomy from Greek sources, namely Ptolemy's "Almagest";
• the development of celestial kinematics;
• new research in spherical geometry and trigonometry required by the new kinematical theory;
• the study on astronomical instruments and its impact on mathematical research. These new historical materials and their mathematical and historical commentaries contribute to rewriting the history of mathematical astronomy and mathematics from the 11th century on.

Including extensive commentary from one of the world s foremost authorities on the subject, this fundamental text is essential reading for historians and mathematicians at the most advanced levels of research."

Kurt Godel (1906 - 1978) was the most outstanding logician of the twentieth century, famous for his hallmark works on the completeness of logic, the incompleteness of number theory, and the consistency of the axiom of choice and the continuum hypothesis. He is also noted for his work on constructivity, the decision problem, and the foundations of computability theory, as well as for the strong individuality of his writings on the philosophy of mathematics. He is less well known for his discovery of unusual cosmological models for Einstein's equations, in theory permitting time travel into the past.
The Collected Works is a landmark resource that draws together a lifetime of creative thought and accomplishment. The first two volumes were devoted to Godel's publications in full (both in original and translation), and the third volume featured a wide selection of unpublished articles and lecture texts found in Godel's Nachlass. The final two volumes contain Godel's correspondence of logical, philosophical, and scientific interest. Volume IV, published for the first time in paperback, covers A to G, with H to Z in volume V; in addition, Volume V contains a full inventory of Godel's Nachlass. All volumes include introductory notes that provide extensive explanatory and historical commentary on each body of work, English translations of material originally written in German (some transcribed from the Gabelsberger shorthand), and a complete bibliography of all works cited.
Kurt Godel: Collected Works is designed to be useful and accessible to as wide an audience as possible without sacrificing scientific or historical accuracy. The only comprehensive edition of Godel's work available, it will be an essential part of the working library of professionals and students in logic, mathematics, philosophy, history of science, and computer science and all others who wish to be acquainted with one of the great minds of the twentieth century.

Kurt Godel (1906 - 1978) was the most outstanding logician of the twentieth century, famous for his hallmark works on the completeness of logic, the incompleteness of number theory, and the consistency of the axiom of choice and the continuum hypothesis. He is also noted for his work on constructivity, the decision problem, and the foundations of computability theory, as well as for the strong individuality of his writings on the philosophy of mathematics. He is less well known for his discovery of unusual cosmological models for Einstein's equations, in theory permitting time travel into the past.
The Collected Works is a landmark resource that draws together a lifetime of creative thought and accomplishment. The first two volumes were devoted to Godel's publications in full (both in original and translation), and the third volume featured a wide selection of unpublished articles and lecture texts found in Godel's Nachlass. These long-awaited final two volumes contain Godel's correspondence of logical, philosophical, and scientific interest. Volume V, published for the first time in paperback, includes H to Z as well as a full inventory of Godel's Nachlass, while Volume IV covers A to G. All volumes include introductory notes that provide extensive explanatory and historical commentary on each body of work, English translations of material originally written in German (some transcribed from the Gabelsberger shorthand), and a complete bibliography of all works cited.
Kurt Godel: Collected Works is designed to be useful and accessible to as wide an audience as possible without sacrificing scientific or historical accuracy. The only comprehensive edition of Godel's work available, it will be an essential part of the working library of professionals and students in logic, mathematics, philosophy, history of science, and computer science and all others who wish to be acquainted with one of the great minds of the twentieth century.

The German philosopher and mathematician Gottlob Frege (1848-1925) was the father of analytic philosophy and to all intents and purposes the inventor of modern logic. Basic Laws of Arithmetic, originally published in German in two volumes (1893, 1903), is Freges magnum opus. It was to be the pinnacle of Freges lifes work. It represents the final stage of his logicist project the idea that arithmetic and analysis are reducible to logic and contains his mature philosophy of mathematics and logic. The aim of Basic Laws of Arithmetic is to demonstrate the logical nature of mathematical theorems by providing gapless proofs in Frege's formal system using only basic laws of logic, logical inference, and explicit definitions. The work contains a philosophical foreword, an introduction to Frege's logic, a derivation of arithmetic from this logic, a critique of contemporary approaches to the real numbers, and the beginnings of a logicist treatment of real analysis. As is well-known, a letter received from Bertrand Russell shortly before the publication of the second volume made Frege realise that his basic law V, governing the identity of value-ranges, leads into inconsistency. Frege discusses a revision to basic law V written in response to Russells letter in an afterword to volume II. The continuing importance of Basic Laws of Arithmetic lies not only in its bearing on issues in the foundations of mathematics and logic but in its model of philosophical inquiry. Frege's ability to locate the essential questions, his integration of logical and philosophical analysis, and his rigorous approach to criticism and argument in general are vividly in evidence in this, his most ambitious work. Philip Ebert and Marcus Rossberg present the first full English translation of both volumes of Freges major work preserving the original formalism and pagination. The edition contains a foreword by Crispin Wright and an extensive appendix providing an introduction to Frege's formal system by Roy T. Cook.

In the folklore of mathematics, James Joseph Sylvester (1814-1897) is the eccentric, hot-tempered, sword-cane-wielding, nineteenth-century British Jew who, together with the taciturn Arthur Cayley, developed a theory and language of invariants that then died spectacularly in the 1890s as a result of David Hilbert's groundbreaking, 'modern' techniques. This, like all folklore, has some grounding in fact but owes much to fiction. The present volume brings together for the first time 140 letters from Sylvester's correspondence in an effort to establish the true picture. It reveals - through the letters as well as through the detailed mathematical and historical commentary accompanying them - Sylvester the friend, man of principle, mathematician, poet, professor, scientific activist, social observer, traveller. It also provides a detailed look at Sylvester's thoughts and thought processes as it shows him acting in both personal and professional spheres over the course of his eighty-two year life. The Sylvester who emerges from this analysis - unlike the Sylvester of the folkloric caricature - offers deep insight into the development of the technical and social structures of mathematics.

This is the fascinating story of two women who lives were guided by a passion for mathematics and an insatiable curiosity to know and understand the world around them - the beautiful, outrageous Emilie du Chatelet and the charmingly subversive Mary Somerville. Against great odds, Emilie and Mary taught themselves mathematics, and did it so well that they each became a world authority on Newtonian mathematical physics. Seduced by Logic begins with Emilie du Chatelet, an 18th-century French aristocrat, intellectual, and Voltaire's lover, whose true ambition was to be a mathematician. She strove not only to further Newton's ideas in France, but to prove that they had French connections, including to the work of Descartes, whom Newton had read. She translated the great Principia Mathematica into French, in what became the accepted French version of Newton's work, and was instrumental in bringing Newton's revolutionary opus to a Continental audience. A century later, in Scotland, Mary Somerville taught herself mathematics and rose from genteel poverty to become a figure of authority on Newtonian physics. Living in France, she became acquainted with the work of one of Newton's proteges, Pierre Simon Laplace, and translated his six-volume Celestial Mechanics into English. It remained the standard astronomy text for the next century, and was considered the most influential work since Principia. Connected by their love for mathematics, Emilie and Mary bring to life a period of remarkable political and scientific change. Combining biography and history of science, Robyn Arianrhod's book explores the roles both women played in bringing Newton's Principia to a wider audience, and reveals the intimate links between the unfolding Newtonian revolution and the origins of intellectual and political liberty.

Containing many previously unpublished letters, this third volume of a six volume collection of the complete correspondence of John Wallis (1616-1703), documents an important period in the history of the Royal Society and the University of Oxford. By providing access to these letters, this painstakingly crafted edition will enable readers to gain a deeper and richer awareness of the intellectual culture on which the growth of scientific knowledge in early modern Europe was based.
Wallis was Savilian Professor of Geometry of Oxford from 1649 until his death, and was a founding member of the Royal Society and a central figure in the scientific and intellectual history of England. In the period covered Wallis is engaged in scientific debates on techniques for determining areas contained by curves (quadratures) and figures (cubatures), as well as on the theory of motion and the nature of the tides. He also continues to attack the mathematical undertakings of Thomas Hobbes and to respond to attacks which the philosopher in turn levels against him. We also find evidence for the consolidation of mathematics as an academic discipline in the University of Oxford just fifty years after the establishment of the first mathematical lecturerships. Wallis is called upon more than once to deliver ceremonial lectures on mathematical topics to foreign dignitaries visiting the University.
At the same time the volume allows us to witness the beginnings of a remarkable development in mathematical publishing. Many of Wallis's letters to Henry Oldenburg, secretary of the Royal Society, on a variety of topics in the mathematical and physical sciences, are transformed into articles and published in Oldenburg's journal, the Philosophical Transactions. Part of the reason for this development also becomes clear in the letters: the long and costly process of publishing mathematical books such as Wallis's three part Mechanica: sive de motu. This volume not only signals the modernization of mathematics in the second half of the seventeenth century but we also see two new figures emerge for the first time, whose careers are in different ways closely associated with Wallis: Isaac Newton and Gottfried Wilhelm Leibniz.

Procreare iucundum, sed parturire molestum. (Gauss, sec. Eisenstein) The plan of this book was first conceived eight years ago. The manuscript developed slowly through several versions until it attained its present form in 1979. It would be inappropriate to list the names of all the friends and advisors with whom I discussed my various drafts but I should like to mention the name of Mr. Gary Cornell who, besides discussing with me numerous details of the manuscript, revised it stylistically. There is much interest among mathematicians to know more about Gauss's life, and the generous help I received has certainly more to do with this than with any individual, positive or negative, aspect of my manuscript. Any mistakes, errors of judgement, or other inadequacies are, of course, the author's responsi bility. The most incisive and, in a way, easiest decisions I had to make were those of personal taste in the choice and treatment of topics. Much had to be omitted or could only be discussed in a cursory way."

Das Buch ist eine unterhaltsame und formelfreie Darstellung der modernen Physik vom 19. Jahrhundert bis zur Gegenwart. Das Leben Albert Einsteins und seine wissenschaftlichen Leistungen ziehen sich als roter Faden durch den Text. Der Autor erlautert zentrale Begriffe und Ergebnisse der modernen Physik in popularwissenschaftlicher Form aus der historischen Perspektive. Der Leser erfahrt auf amusante Weise, wie sich die moderne Physik entwickelt hat.

Wir begegnen Poincare, Lorentz und Hilbert, Boltzmann und Bohr, Minkowski, Planck, de Broglie, Hubble und Weyl, Gamow, Hahn und Meitner, Kapiza und Landau, Fermi und vielen anderen beruhmten Wissenschaftlern. Was hatte Eddington gegen Chandrasekhar und was hatte Einstein gegen Schwarze Locher? Warum sollten Raumtouristen, Traumtouristen und Weltraumtraumtouristen nicht am Loch Ness, sondern auf der sicheren Seite eines Schwarzen Loches Urlaub machen? Warum wetterte Pauli gegen Einstein? Stimmt die Sache mit der Atombombenformel? Vermatschte Materie, Urknall und kosmische Hintergrundstrahlung, Gravitationswellen und Doppelpulsare, die kosmologische Konstante und die Expansion des Universums sind weitere Themen, die den Leser in Atem halten und kein geistiges Vakuum aufkommen lassen."

Providing the first comprehensive account of the widely unknown cooperation and friendship between Emmy Noether and Helmut Hasse, this book contains English translations of all available letters which were exchanged between them in the years 1925-1935. It features a special chapter on class field theory, a subject which was completely renewed in those years, Noether and Hasse being among its main proponents. These historical items give evidence that Emmy Noether's impact on the development of mathematics is not confined to abstract algebra but also extends to important ideas in modern class field theory as part of algebraic number theory. In her letters, details of proofs appear alongside conjectures and speculations, offering a rich source for those who are interested in the rise and development of mathematical notions and ideas. The letters are supplemented by extensive comments, helping the reader to understand their content within the mathematical environment of the 1920s and 1930s.

How did Pierre Fatou and Gaston Julia create what we now call Complex Dynamics, in the context of the early twentieth century and especially of the First World War? The book is based partly on new, unpublished sources. Who were Pierre Fatou, Gaston Julia, Paul Montel? New biographical information is given on the little known mathematician that was Pierre Fatou. How did the injury of Julia during WW1 influence mathematical life in France? From the reviews of the French version: "Audin's book is ! filled with marvelous biographical information and analysis, dealing not just with the men mentioned in the book's title but a large number of other players, too ! [It] addresses itself to scholars for whom the history of mathematics has a particular resonance and especially to mathematicians active, or even with merely an interest, in complex dynamics. ! presents it all to the reader in a very appealing form." (Michael Berg, The Mathematical Association of America, October, 2009)

This volume presents a selection of 434 letters from and to the Dutch physicist and Nobel Prize winner Hendrik Antoon Lorentz (1853-1928), covering the period from 1883 until a few months before his death in February 1928. The sheer size of the available correspondence (approximately 6000 letters from and to Lorentz) preclude a full publication. The letters included in this volume have been selected according to various criteria, the most important of which is scientific importance. A second criterion has been the availability of letters both from and to Lorentz, so that the reader can follow the exchange between Lorentz and his correspondent. Within such correspondences a few unimportant items, dealing with routine administrative or organizational matters, have been omitted. An exception to the scientific criterion is the exchange of letters between Lorentz and Albert Einstein, Max Planck, Woldemar Voigt, and Wilhelm Wien during World War I: these letters have been included because they shed important light on the disruption of the scientific relations during the war and on the political views of these correspondents as well as of Lorentz. similar reasons the letters exchanged with Einstein and Planck on post-war political issues have been included. Biographical sketch Hendrik Antoon Lorentz was born on July 18, 1853 in the Dutch town of Arnhem. He was the son of a relatively well-to-do owner of a nursery.

Drawing on entirely new evidence, The English Renaissance Stage: Geometry, Poetics, and the Practical Spatial Arts 1580-1630 examines the history of English dramatic form and its relationship to the mathematics, technology, and early scientific thought during the Renaissance period. The book demonstrates how practical modes of thinking that were typical of the sixteenth century resulted in new genres of plays and a new vocabulary for problems of poetic representation. In the epistemological moment the book recovers, we find new ideas about form and language that would become central to Renaissance literary discourse; in this same moment, too, we find new ways of thinking about the relationship between theory and practice that are typical of modernity, new attitudes towards spatial representation, and a new interest in both poetics and mathematics as distinctive ways of producing knowledge about the world. By emphasizing the importance of theatrical performance, the book engages with continuing debates over the cultural function of the early modern stage and with scholarship on the status of modern authorship. When we consider playwrights in relation to the theatre rather than the printed book, they appear less as 'authors' than as figures whose social position and epistemological presuppositions were very similar to the craftsmen, surveyors, and engineers who began to flourish during the sixteenth century and whose mathematical knowledge made them increasingly sought after by men of wealth and power.

Between inventing the concept of a universal computer in 1936 and breaking the German Enigma code during World War II, Alan Turing (1912-1954), the British founder of computer science and artificial intelligence, came to Princeton University to study mathematical logic. Some of the greatest logicians in the world--including Alonzo Church, Kurt Godel, John von Neumann, and Stephen Kleene--were at Princeton in the 1930s, and they were working on ideas that would lay the groundwork for what would become known as computer science. This book presents a facsimile of the original typescript of Turing's fascinating and influential 1938 Princeton PhD thesis, one of the key documents in the history of mathematics and computer science. The book also features essays by Andrew Appel and Solomon Feferman that explain the still-unfolding significance of the ideas Turing developed at Princeton.

A work of philosophy as well as mathematics, Turing's thesis envisions a practical goal--a logical system to formalize mathematical proofs so they can be checked mechanically. If every step of a theorem could be verified mechanically, the burden on intuition would be limited to the axioms. Turing's point, as Appel writes, is that "mathematical reasoning can be done, and should be done, in mechanizable formal logic." Turing's vision of "constructive systems of logic for practical use" has become reality: in the twenty-first century, automated "formal methods" are now routine.

Presented here in its original form, this fascinating thesis is one of the key documents in the history of mathematics and computer science."

Nearly a century before Mondrian made geometrical red, yellow, and blue lines famous, 19th-century mathematician Oliver Byrne employed the color scheme for his 1847 edition of Euclid's mathematical and geometric treatise Elements. Byrne's idea was to use color to make learning easier and "diffuse permanent knowledge." The result has been described as one of the oddest and most beautiful books of the 19th century. The facsimile of Byrne's vivid publication is now available in a beautiful new edition. A masterwork of art and science, it is as beautiful in the boldness of its red, yellow, and blue figures and diagrams as it is in the mathematical precision of its theories. In the simplicity of forms and colors, the pages anticipate the vigor of De Stijl and Bauhaus design. In making complex information at once accessible and aesthetically engaging, this work is a forerunner to the information graphics that today define much of our data consumption.

Writings by early mathematicians feature language and notations that are quite different from what we're familiar with today. Sourcebooks on the history of mathematics provide some guidance, but what has been lacking is a guide tailored to the needs of readers approaching these writings for the first time. "How to Read Historical Mathematics" fills this gap by introducing readers to the analytical questions historians ask when deciphering historical texts.

Sampling actual writings from the history of mathematics, Benjamin Wardhaugh reveals the questions that will unlock the meaning and significance of a given text--Who wrote it, why, and for whom? What was its author's intended meaning? How did it reach its present form? Is it original or a translation? Why is it important today? Wardhaugh teaches readers to think about what the original text might have looked like, to consider where and when it was written, and to formulate questions of their own. Readers pick up new skills with each chapter, and gain the confidence and analytical sophistication needed to tackle virtually any text in the history of mathematics.Introduces readers to the methods of textual analysis used by historians Uses actual source material as examples Features boxed summaries, discussion questions, and suggestions for further reading Supplements all major sourcebooks in mathematics history Designed for easy reference Ideal for students and teachers

Paris of the year 1900 left two landmarks: the Tour Eiffel, and David Hilbert's celebrated list of twenty-four mathematical problems presented at a conference opening the new century. Kurt Goedel, a logical icon of that time, showed Hilbert's ideal of complete axiomatization of mathematics to be unattainable. The result, of 1931, is called Goedel's incompleteness theorem. Goedel then went on to attack Hilbert's first and second Paris problems, namely Cantor's continuum problem about the type of infinity of the real numbers, and the freedom from contradiction of the theory of real numbers. By 1963, it became clear that Hilbert's first question could not be answered by any known means, half of the credit of this seeming faux pas going to Goedel. The second is a problem still wide open. Goedel worked on it for years, with no definitive results; The best he could offer was a start with the arithmetic of the entire numbers. This book, Goedel's lectures at the famous Princeton Institute for Advanced Study in 1941, shows how far he had come with Hilbert's second problem, namely to a theory of computable functionals of finite type and a proof of the consistency of ordinary arithmetic. It offers indispensable reading for logicians, mathematicians, and computer scientists interested in foundational questions. It will form a basis for further investigations into Goedel's vast Nachlass of unpublished notes on how to extend the results of his lectures to the theory of real numbers. The book also gives insights into the conceptual and formal work that is needed for the solution of profound scientific questions, by one of the central figures of 20th century science and philosophy.

Ancient Greek Philosophy routinely relied upon concepts of number to explain the tangible order of the universe. Plotinus' contribution to this tradition, however, has been often omitted, if not ignored. The main reason for this, at first glance, is the Plotinus does not treat the subject of number in the Enneads as pervasively as the Neopythagoreans or even his own successors Lamblichus, Syrianus, and Proclus. Nevertheless, a close examination of the Enneads reveals that Plotinus systematically discusses number in relation to each of his underlying principles of existence--the One, Intellect, and Soul. Plotinus on Number offers the first comprehensive analysis of Plotinus' concept of number, beginning with its origins in Plato and the Neopythagoreans and ending with its influence on Porphyry's arrangement of the Enneads. It's main argument is that Plotinus adapts Plato's and the Neopythagoreans' cosmology to place number in the foundation of the intelligible realm and in the construction of the universe. Through Plotinus' defense of Plato's Ideal Numbers from Aristotle's criticism, Svetla Slaveva-Griffin reveals the founder of Neoplatonism as the first post-Platonic philosopher who purposefully and systematically develops what we may call a theory of number, distinguishing between number in the intelligible realm and number in the quantitative, mathematical realm. Finally, the book draws attention to Plotinus' concept as a necesscary and fundamental linke between Platonic and late Neoplatonic schools of philosophy.

Aimed at graduates and researchers in Mathematics, History of Mathematics and Science, this book examines the development of mathematics from the late 16th Century to the end of the 20th Century. Mathematics has an amazingly long and rich history, it has been practised in every society and culture, with written records reaching back in some cases as far as four thousand years. This book will focus on just a small part of the story, in a sense the most recent chapter of it: the mathematics of western Europe from the sixteenth to the nineteenth centuries. Each chapter will focus on a particular topic and outline its history with the provision of facsimiles of primary source material along with explanatory notes and modern interpretations. Almost every source is given in its original form, not just in the language in which it was first written, but as far as practicable in the layout and typeface in which it was read by contemporaries.This book is designed to provide mathematics undergraduates with some historical background to the material that is now taught universally to students in their final years at school and the first years at college or university: the core subjects of calculus, analysis, and abstract algebra, along with others such as mechanics, probability, and number theory. All of these evolved into their present form in a relatively limited area of western Europe from the mid sixteenth century onwards, and it is there that we find the major writings that relate in a recognizable way to contemporary mathematics.

This open access book collects the historical and medial perspectives of a systematic and epistemological analysis of the complicated, multifaceted relationship between model and mathematics, ranging from, for example, the physical mathematical models of the 19th century to the simulation and digital modelling of the 21st century. The aim of this anthology is to showcase the status of the mathematical model between abstraction and realization, presentation and representation, what is modeled and what models. This book is open access under a CC BY 4.0 license.

Cet ouvrage contient les correspondances actives et passives de Jules Houel avec Joseph-Marie De Tilly, Gaston Darboux et Victor-Amedee Le Besgue ainsi qu'une introduction qui se focalise sur la decouverte de l'impossibilite de demontrer le postulat des paralleles d'Euclide et l'apparition des premiers exemples de fonctions continues non derivables. Jules Houel (1823-1886) a occupe une place particuliere dans les mathematiques en France durant la seconde partie du 19eme siecle. Par ses travaux de traduction et ses recensions, il a vivement contribue a la reception de la geometrie non euclidienne de Bolyai et Lobatchevski ainsi qu'aux debats sur les fondements de l'analyse. Il se situe au centre d'un vaste reseau international de correspondances en lien avec son role de redacteur pour le Bulletin des sciences mathematiques et astronomiques.