"Alle Formeln und Resultate sind fertig, nur den Weg muss ich noch finden, auf dem ich zu ihnen gelangen werde", soll Gauss einmal gesagt haben. Um den Weg, um die vielen Wege zu den Formeln und Resultaten der Mathematik, geht es in diesem Buch. Geboren aus der Lust am Wissen, genahrt von der Naturphilosophie, begrenzt nur von den Grenzen des Denkens, stellt die Mathematik dessen Werkzeug und Gegenstand dar. Wir folgen ihren Spuren von der Antike bis in unsere Tage. In acht Kapiteln fuhrt das Buch durch zweitausend Jahre Wissenschaft von den Zahlen, den Figuren, den Gleichungen, von Differential und Integral, vom Zufall, von den Raumen, den Mengen und den logischen Schlussen.

This book deals with a topic that has been largely neglected by philosophers of science to date: the ability to refer and analyze in tandem. On the basis of a set of philosophical case studies involving both problems in number theory and issues concerning time and cosmology from the era of Galileo, Newton and Leibniz up through the present day, the author argues that scientific knowledge is a combination of accurate reference and analytical interpretation. In order to think well, we must be able to refer successfully, so that we can show publicly and clearly what we are talking about. And we must be able to analyze well, that is, to discover productive and explanatory conditions of intelligibility for the things we are thinking about. The book's central claim is that the kinds of representations that make successful reference possible and those that make successful analysis possible are not the same, so that significant scientific and mathematical work typically proceeds by means of a heterogeneous discourse that juxtaposes and often superimposes a variety of kinds of representation, including formal and natural languages as well as more iconic modes. It demonstrates the virtues and necessity of heterogeneity in historically central reasoning, thus filling an important gap in the literature and fostering a new, timely discussion on the epistemology of science and mathematics.

This book provides an overview of the confluence of ideas in Turing's era and work and examines the impact of his work on mathematical logic and theoretical computer science. It combines contributions by well-known scientists on the history and philosophy of computability theory as well as on generalised Turing computability. By looking at the roots and at the philosophical and technical influence of Turing's work, it is possible to gather new perspectives and new research topics which might be considered as a continuation of Turing's working ideas well into the 21st century. The Stored-Program Universal Computer: Did Zuse Anticipate Turing and von Neumann?" is available open access under a Creative Commons Attribution 4.0 International License via link.springer.com

This book explores facets of Otto Neugebauer's career, his impact on the history and practice of mathematics, and the ways in which his legacy has been preserved or transformed in recent decades, looking ahead to the directions in which the study of the history of science will head in the twenty-first century. Neugebauer, more than any other scholar of recent times, shaped the way we perceive premodern science. Through his scholarship and influence on students and collaborators, he inculcated both an approach to historical research on ancient and medieval mathematics and astronomy through precise mathematical and philological study of texts, and a vision of these sciences as systems of knowledge and method that spread outward from the ancient Near Eastern civilizations, crossing cultural boundaries and circulating over a tremendous geographical expanse of the Old World from the Atlantic to India.

This book presents, in his own words, the life of Hugo Steinhaus (1887-1972), noted Polish mathematician of Jewish background, educator, and mathematical popularizer. A student of Hilbert, a pioneer of the foundations of probability and game theory, and a contributor to the development of functional analysis, he was one of those instrumental to the extraordinary flowering of Polish mathematics before and after World War I. In particular, it was he who "discovered" the great Stefan Banach. Exhibiting his great integrity and wit, Steinhaus's personal story of the turbulent times he survived - including two world wars and life postwar under the Soviet heel - cannot but be of consuming interest. His recounting of the fearful years spent evading Nazi terror is especially moving. The steadfast honesty and natural dignity he maintained while pursuing a life of demanding scientific and intellectual enquiry in the face of encroaching calamity and chaos show him to be truly a mathematician for all seasons. The present work will be of great interest not only to mathematicians wanting to learn some of the details of the mathematical blossoming that occurred in Poland in the first half of the 20th century, but also to anyone wishing to read a first-hand account of the history of those unquiet times in Europe - and indeed world-wide - by someone of uncommon intelligence and forthrightness situated near an eye of the storm.

This biography of the mathematician, Sophie Germain, paints a rich portrait of a brilliant and complex woman, the mathematics she developed, her associations with Gauss, Legendre, and other leading researchers, and the tumultuous times in which she lived. Sophie Germain stood right between Gauss and Legendre, and both publicly recognized her scientific efforts. Unlike her female predecessors and contemporaries, Sophie Germain was an impressive mathematician and made lasting contributions to both number theory and the theories of plate vibrations and elasticity. She was able to walk with ease across the bridge between the fields of pure mathematics and engineering physics. Though isolated and snubbed by her peers, Sophie Germain was the first woman to win the prize of mathematics from the French Academy of Sciences. She is the only woman who contributed to the proof of Fermat's Last Theorem. In this unique biography, Dora Musielak has done the impossible she has chronicled Sophie Germain's brilliance through her life and work in mathematics, in a way that is simultaneously informative, comprehensive, and accurate.

To Infinity and Beyond explores the idea of infinity in mathematics and art. Eli Maor examines the role of infinity, as well as its cultural impact on the arts and sciences. He evokes the profound intellectual impact the infinite has exercised on the human mind--from the "horror infiniti" of the Greeks to the works of M. C. Escher; from the ornamental designs of the Moslems, to the sage Giordano Bruno, whose belief in an infinite universe led to his death at the hands of the Inquisition. But above all, the book describes the mathematician's fascination with infinity--a fascination mingled with puzzlement.

This book collects more than thirty contributions in memory of Wolfgang Schwarz, most of which were presented at the seventh International Conference on Elementary and Analytic Number Theory (ELAZ), held July 2014 in Hildesheim, Germany. Ranging from the theory of arithmetical functions to diophantine problems, to analytic aspects of zeta-functions, the various research and survey articles cover the broad interests of the well-known number theorist and cherished colleague Wolfgang Schwarz (1934-2013), who contributed over one hundred articles on number theory, its history and related fields. Readers interested in elementary or analytic number theory and related fields will certainly find many fascinating topical results among the contributions from both respected mathematicians and up-and-coming young researchers. In addition, some biographical articles highlight the life and mathematical works of Wolfgang Schwarz.

This book provides the first critical edition of Ibn al-Haytham's On the Shape of the Eclipse with English translation and commentary, which records the first scientific analysis of the camera obscura. On the Shape of the Eclipse includes pioneering research on the conditions of formation of the image, in a time deemed to be committed to aniconism. It also provides an early attempt to merge the two branches of Ancient optics-the theory of light and theory of vision. What perhaps most strongly characterizes this treatise is the close interaction of a geometric analysis of light and experimental reasoning. Ibn al-Haytham conducted his experiments in a systematic way by varying all that could be changed: the shape and size of the aperture, the focal length of the camera obscura, the distance and shape of the celestial bodies. This way, he achieved a thorough understanding. This work represents a decisive step in both the history of optics and the application of the experimental method that was just as efficient in medieval Islam as today.

For a long time, World War I has been shortchanged by the historiography of science. Until recently, World War II was usually considered as the defining event for the formation of the modern relationship between science and society. In this context, the effects of the First World War, by contrast, were often limited to the massive deaths of promising young scientists. By focusing on a few key places (Paris, Cambridge, Rome, Chicago, and others), the present book gathers studies representing a broad spectrum of positions adopted by mathematicians about the conflict, from militant pacifism to military, scientific, or ideological mobilization. The use of mathematics for war is thoroughly examined. This book suggests a new vision of the long-term influence of World War I on mathematics and mathematicians. Continuities and discontinuities in the structure and organization of the mathematical sciences are discussed, as well as their images in various milieux. Topics of research and the values with which they were defended are scrutinized. This book, in particular, proposes a more in-depth evaluation of the issue of modernity and modernization in mathematics. The issue of scientific international relations after the war is revisited by a close look at the situation in a few Allied countries (France, Britain, Italy, and the USA). The historiography has emphasized the place of Germany as the leading mathematical country before WWI and the absurdity of its postwar ostracism by the Allies. The studies presented here help explain how dramatically different prewar situations, prolonged interaction during the war, and new international postwar organizations led to attempts at redrafting models for mathematical developments.

Gerhard Gentzen is best known for his development of the proof systems of natural deduction and sequent calculus, central in many areas of logic and computer science today. Another noteworthy achievement is his resolution of the embarrassing situation created by Goedel's incompleteness results, especially the second one about the unprovability of consistency of elementary arithmetic. After these successes, Gentzen dedicated the rest of his short life to the main problem of Hilbert's proof theory, the question of the consistency of analysis. He was arrested in the summer of 1945 with other professors of the German University of Prague and died soon afterward of starvation in a prison cell. Attempts at locating his lost manuscripts failed at the time, but several decades later, two slim folders of shorthand notes were found. In this volume, Jan von Plato gives an overview of Gentzen's life and scientific achievements, based on detailed archival and systematic studies, and essential for placing the translations of shorthand manuscripts that follow in the right setting. The materials in this book are singular in the way they show the birth and development of Gentzen's central ideas and results, sometimes in a well-developed form, and other times as flashes into the anatomy of the workings of a unique mind.

This is a new annotated edition of Thomas J. Stieltjes' Collected Papers, first published in 1914 (Vol. I) and 1918 (Vol. II) by Noordhoff, Groningen, in French, and now published by Springer-Verlag, originally to mark the occasion of the 100th anniversary of Stieltjes' death (1894). These two volumes will be of great interest to all mathematicians who are anxious to understand the impact of Stieltjes' work on modern mathematics, and in particular on the theory of orthogonal polynomials and continued fractions. In addition to the reproduction of Stieltjes' papers (I-XLVII), Volume I includes about 75 pages of commentaries by contemporary mathematicians on Stieltjes' work. Volume II contains Stieltjes' papers XLVIII-LXXXIV together with English translations of his main paper "Recherches sur les fractions continues" and his short note regarding the Riemann hypothesis. A Bibliography of Stieltjes' papers is included in both volumes for the convenience of the reader.

This is a new annotated edition of Thomas J. Stieltjes' Collected Papers, first published in 1914 (Vol. I) and 1918 (Vol. II) by Noordhoff, Groningen, in French, and now published by Springer-Verlag, originally to mark the occasion of the 100th anniversary of Stieltjes' death (1894). These two volumes will be of great interest to all mathematicians who are anxious to understand the impact of Stieltjes' work on modern mathematics, and in particular on the theory of orthogonal polynomials and continued fractions. In addition to the reproduction of Stieltjes' papers (I-XLVII), Volume I includes about 75 pages of commentaries by contemporary mathematicians on Stieltjes' work. Volume II contains Stieltjes' papers XLVIII-LXXXIV together with English translations of his main paper "Recherches sur les fractions continues" and his short note regarding the Riemann hypothesis. A Bibliography of Stieltjes' papers is included in both volumes for the convenience of the reader.

An exploration of mathematical style through 99 different proofs of the same theorem This book offers a multifaceted perspective on mathematics by demonstrating 99 different proofs of the same theorem. Each chapter solves an otherwise unremarkable equation in distinct historical, formal, and imaginative styles that range from Medieval, Topological, and Doggerel to Chromatic, Electrostatic, and Psychedelic. With a rare blend of humor and scholarly aplomb, Philip Ording weaves these variations into an accessible and wide-ranging narrative on the nature and practice of mathematics. Inspired by the experiments of the Paris-based writing group known as the Oulipo-whose members included Raymond Queneau, Italo Calvino, and Marcel Duchamp-Ording explores new ways to examine the aesthetic possibilities of mathematical activity. 99 Variations on a Proof is a mathematical take on Queneau's Exercises in Style, a collection of 99 retellings of the same story, and it draws unexpected connections to everything from mysticism and technology to architecture and sign language. Through diagrams, found material, and other imagery, Ording illustrates the flexibility and creative potential of mathematics despite its reputation for precision and rigor. Readers will gain not only a bird's-eye view of the discipline and its major branches but also new insights into its historical, philosophical, and cultural nuances. Readers, no matter their level of expertise, will discover in these proofs and accompanying commentary surprising new aspects of the mathematical landscape.

This open access book explores commentaries on an influential text of pre-Copernican astronomy in Europe. It features essays that take a close look at key intellectuals and how they engaged with the main ideas of this qualitative introduction to geocentric cosmology. Johannes de Sacrobosco compiled his Tractatus de sphaera during the thirteenth century in the frame of his teaching activities at the then recently founded University of Paris. It soon became a mandatory text all over Europe. As a result, a tradition of commentaries to the text was soon established and flourished until the second half of the 17th century. Here, readers will find an informative overview of these commentaries complete with a rich context. The essays explore the educational and social backgrounds of the writers. They also detail how their careers developed after the publication of their commentaries, the institutions and patrons they were affiliated with, what their agenda was, and whether and how they actually accomplished it. The editor of this collection considers these scientific commentaries as genuine scientific works. The contributors investigate them here not only in reference to the work on which it comments but also, and especially, as independent scientific contributions that are socially, institutionally, and intellectually contextualized around their authors.

This revealing work examines an approach from ancient astronomy to what was then a particularly important question, namely that of understanding the relationship between the position in the ecliptic and the time it takes for a fixed-length of the ecliptic beginning at that point to rise above the eastern horizon. Schemes known as "rising time schemes" were used to give lengths of the celestial equator corresponding to each of the twelve zodiacal signs which make up the ecliptic. This book investigates the earliest known examples of these schemes which come from Babylonia and date to the mid to late first millennium BC. Making an important contribution to our knowledge of astronomy in the ancient world, this volume includes editions and translations of all of the known Babylonian rising time texts, including several texts that are identified for the first time. Through a close examination of the preserved texts it has been possible to reconstruct the complete Babylonian rising time scheme. This reconstruction is unprecedented in its completeness, and it is also now possible to situate the scheme within a genre of Babylonian astronomy known as schematic astronomy which presents theoretical descriptions of the astronomical phenomena. The unique discoveries and fresh explorations in this book will be of interest to historians of ancient astronomy, scholars of Babylonian history and those investigating the origins of scientific thought.

Luis Antonio Santalo (Spain 1911 - Argentina 2001) contributed to several branches of Geometry, his laying of the mathematical foundations of Stereology and its applications perhaps being his most outstanding achievement. A considerable power of abstraction, a brilliant geometric intuition and an outstanding gift as a disseminator of science were among his virtues. The present volume contains a selection of his best papers. Part I consists of a short biography and some photographs together with a complete list of his publications, classified into research papers, books, and articles on education and the popularization of mathematics, as well as a comprehensive analysis of his contribution to science. Part II, the main part of the book, includes selected papers, arranged into five sections according to the nature of their contents: Differential Geometry, Integral Geometry, Convex Geometry, Affine Geometry, and Statistics and Stereology. Each section is preceded by a commentary by a renowned specialist: Teufel, Langevin, Schneider, Leichtweiss, and Cruz-Orive, respectively. Finally, Part III emphasizes the influence of his work. It contains commentaries by several specialists regarding modern results based on, or closely related to, those of Santalo, some book reviews written by Santalo, as well as some reviews of his books. As a curious addendum, a ranking of his articles, given by Santalo himself, is included.

In this book the classical Greek construction problems are explored in a didactical, enquiry based fashion using Interactive Geometry Software (IGS). The book traces the history of these problems, stating them in modern terminology. By focusing on constructions and the use of IGS the reader is confronted with the same problems that ancient mathematicians once faced. The reader can step into the footsteps of Euclid, Viete and Cusanus amongst others and then by experimenting and discovering geometric relationships far exceed their accomplishments. Exploring these problems with the neusis-method lets him discover a class of interesting curves. By experimenting he will gain a deeper understanding of how mathematics is created. More than 100 exercises guide him through methods which were developed to try and solve the problems. The exercises are at the level of undergraduate students and only require knowledge of elementary Euclidean geometry and pre-calculus algebra. It is especially well-suited for those students who are thinking of becoming a mathematics teacher and for mathematics teachers.

Karl Menger, one of the founders of dimension theory, is among the most original mathematicians and thinkers of the twentieth century. He was a member of the Vienna Circle and the founder of its mathematical equivalent, the Viennese Mathematical Colloquium. Both during his early years in Vienna and, after his emigration, in the United States, Karl Menger made significant contributions to a wide variety of mathematical fields, and greatly influenced many of his colleagues. These two volumes contain Menger's major mathematical papers, based on his own selection from his extensive writings. They deal with topics as diverse as topology, geometry, analysis and algebra, and also include material on economics, sociology, logic and philosophy. The Selecta Mathematica is a monument to the diversity and originality of Menger's ideas.

Joseph W. Dauben, a leading authority on the history of mathematics in Europe, China, and North America, has played a pivotal role in promoting international scholarship over the last forty years. This Festschrift volume, showcasing recent historical research by leading experts on three continents, offers a global perspective on important themes in this field.

Up to now there have been scarcely any publications on Leibniz dedicated to investigating the interrelations between philosophy and mathematics in his thought. In part this is due to the previously restricted textual basis of editions such as those produced by Gerhardt. Through recent volumes of the scientific letters and mathematical papers series of the Academy Edition scholars have obtained a much richer textual basis on which to conduct their studies - material which allows readers to see interconnections between his philosophical and mathematical ideas which have not previously been manifested. The present book draws extensively from this recently published material. The contributors are among the best in their fields. Their commissioned papers cover thematically salient aspects of the various ways in which philosophy and mathematics informed each other in Leibniz's thought.

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 existing literature 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, Lalouvere, Guldin, Barrow, Mengoli, Wallis, Leibniz, Newton.

Walter Gautschi has written extensively on topics ranging from special functions, quadrature and orthogonal polynomials to difference and differential equations, software implementations, and the history of mathematics. He is world renowned for his pioneering work in numerical analysis and constructive orthogonal polynomials, including a definitive textbook in the former, and a monograph in the latter area. This three-volume set, Walter Gautschi: Selected Works with Commentaries, is a compilation of Gautschi's most influential papers and includes commentaries by leading experts. The work begins with a detailed biographical section and ends with a section commemorating Walter's prematurely deceased twin brother. This title will appeal to graduate students and researchers in numerical analysis, as well as to historians of science. Selected Works with Commentaries, Vol. 1 Numerical Conditioning Special Functions Interpolation and Approximation Selected Works with Commentaries, Vol. 2 Orthogonal Polynomials on the Real Line Orthogonal Polynomials on the Semicircle Chebyshev Quadrature Kronrod and Other Quadratures Gauss-type Quadrature Selected Works with Commentaries, Vol. 3 Linear Difference Equations Ordinary Differential Equations Software History and Biography Miscellanea Works of Werner Gautschi

Walter Gautschi has written extensively on topics ranging from special functions, quadrature and orthogonal polynomials to difference and differential equations, software implementations, and the history of mathematics. He is world renowned for his pioneering work in numerical analysis and constructive orthogonal polynomials, including a definitive textbook in the former, and a monograph in the latter area. This three-volume set, Walter Gautschi: Selected Works with Commentaries, is a compilation of Gautschi's most influential papers and includes commentaries by leading experts. The work begins with a detailed biographical section and ends with a section commemorating Walter's prematurely deceased twin brother. This title will appeal to graduate students and researchers in numerical analysis, as well as to historians of science. Selected Works with Commentaries, Vol. 1 Numerical Conditioning Special Functions Interpolation and Approximation Selected Works with Commentaries, Vol. 2 Orthogonal Polynomials on the Real Line Orthogonal Polynomials on the Semicircle Chebyshev Quadrature Kronrod and Other Quadratures Gauss-type Quadrature Selected Works with Commentaries, Vol. 3 Linear Difference Equations Ordinary Differential Equations Software History and Biography Miscellanea Works of Werner Gautschi

Lazare Carnot was the unique example in the history of science of someone who inadvertently owed the scientific recognition he eventually achieved to earlier political prominence. He and his son Sadi produced work that derived from their training as engineering and went largely unnoticed by physicists for a generation or more, even though their respective work introduced concepts that proved fundamental when taken up later by other hands. There was, moreover, a filial as well as substantive relation between the work of father and son. Sadi applied to the functioning of heat engines the analysis that his father had developed in his study of the operation of ordinary machines. Specifically, Sadi's idea of a reversible process originated in the use his father made of geometric motions in the analysis of machines in general. This unique book shows how the two Carnots influenced each other in their work in the fields of mechanics and thermodynamics and how future generations of scientists have further benefited from their work.