This book presents the Riemann Hypothesis, connected problems, and a taste of the body of theory developed towards its solution. It is targeted at the educated non-expert. Almost all the material is accessible to any senior mathematics student, and much is accessible to anyone with some university mathematics.

The appendices include a selection of original papers. This collection is not very large and encompasses only the most important milestones in the evolution of theory connected to the Riemann Hypothesis. The appendices also include some authoritative expository papers. These are the "expert witnesses whose insight into this field is both invaluable and irreplaceable.

The larger part of Yearbook 6 of the Institute Vienna Circle constitutes the proceedings of a symposium on Alfred Tarski and his influence on and interchanges with the Vienna Circle, especially those on and with Rudolf Carnap and Kurt Goedel. It is the first time that this topic has been treated on such a scale and in such depth. Attention is mainly paid to the origins, development and subsequent role of Tarski's definition of truth. Some contributions are primarily historical, others analyze logical aspects of the concept of truth. Contributors include Anita and Saul Feferman, Jan Wolenski, Jan Tarski and Hans Sluga. Several Polish logicians contributed: Gzegorczyk, Wojcicki, Murawski and Rojszczak. The volume presents entirely new biographical material on Tarski, both from his Polish period and on his influential career in the United States: at Harvard, in Princeton, at Hunter, and at the University of California at Berkeley. The high point of the analysis involves Tarski's influence on Carnap's evolution from a narrow syntactical view of language, to the ontologically more sophisticated but more controversial semantical view. Another highlight involves the interchange between Tarski and Goedel on the connection between truth and proof and on the nature of metalanguages. The concluding part of Yearbook 6 includes documentation, book reviews and a summary of current activities of the Institute Vienna Circle. Jan Tarski introduces letters written by his father to Goedel; Paolo Parrini reports on the Vienna Circle's influence in Italy; several reviews cover recent books on logical empiricism, on Goedel, on cosmology, on holistic approaches in Germany, and on Mauthner.

An understanding of developments in Arabic mathematics between the IXth and XVth century is vital to a full appreciation of the history of classical mathematics. This book draws together more than ten studies to highlight one of the major developments in Arabic mathematical thinking, provoked by the double fecondation between arithmetic and the algebra of al-Khwarizmi, which led to the foundation of diverse chapters of mathematics: polynomial algebra, combinatorial analysis, algebraic geometry, algebraic theory of numbers, diophantine analysis and numerical calculus. Thanks to epistemological analysis, and the discovery of hitherto unknown material, the author has brought these chapters into the light, proposes another periodization for classical mathematics, and questions current ideology in writing its history. Since the publication of the French version of these studies and of this book, its main results have been admitted by historians of Arabic mathematics, and integrated into their recent publications. This book is already a vital reference for anyone seeking to understand history of Arabic mathematics, and its contribution to Latin as well as to later mathematics. The English translation will be of particular value to historians and philosophers of mathematics and of science.

In three volumes, a distinguished group of scholars from a variety of disciplines in the natural and social sciences, the humanities and the arts contribute essays in honor of Robert S. Cohen, on the occasion of his 70th birthday. The range of the essays, as well as their originality, and their critical and historical depth, pay tribute to the extraordinary scope of Professor Cohen's intellectual interests, as a scientist-philosopher and a humanist, and also to his engagement in the world of social and political practice. The essays presented in Physics, Philosophy, and the Scientific Community (Volume I of Essays in Honor of Robert S. Cohen) focus on philosophical and historical issues in contemporary physics: on the origins and conceptual foundations of quantum mechanics, on the reception and understanding of Bohr's and Einstein's work, on the emergence of quantum electrodynamics, and on some of the sharp philosophical and scientific issues that arise in current scientific practice (e.g. in superconductivity research). In addition, several essays deal with critical issues within the philosophy of science, both historical and contemporary: e.g. with Cartesian notions of mechanism in the philosophy of biology; with the language and logic of science - e.g. with new insights concerning the issue of a physicalistic' language in the arguments of Neurath, Carnap and Wittgenstein; with the notion of elementary logic'; and with rational and non-rational elements in the history of science. Two original contributions to the history of mathematics and some studies in the comparative sociology of science round off this outstanding collection.

Modern mechanics was forged in the seventeenth century from materials inherited from Antiquity and transformed in the period from the Middle Ages through to the sixteenth century. These materials were transmitted through a number of textual traditions and within several disciplines and practices, including ancient and medieval natural philosophy, statics, the theory and design of machines, and mathematics.

This volume deals with a variety of moments in the history of mechanics when conflicts arose within one textual tradition, between different traditions, or between textual traditions and the wider world of practice. Its purpose is to show how the accommodations sometimes made in the course of these conflicts ultimately contributed to the emergence of modern mechanics.

The first part of the volume is concerned with ancient mechanics and its transformations in the Middle Ages; the second part with the reappropriation of ancient mechanics and especially with the reception of the Pseudo-Aristotelian Mechanica in the Renaissance; and the third and final part, with early-modern mechanics in specific social, national, and institutional contexts.

Covering both the history of mathematics and of philosophy, Descartes's Mathematical Thought reconstructs the intellectual career of Descartes most comprehensively and originally in a global perspective including the history of early modern China and Japan. Especially, it shows what the concept of "mathesis universalis" meant before and during the period of Descartes and how it influenced the young Descartes. In fact, it was the most fundamental mathematical discipline during the seventeenth century, and for Descartes a key notion which may have led to his novel mathematics of algebraic analysis.

Leonardo da Pisa, perhaps better known as Fibonacci (ca. 1170 - ca. 1240), selected the most useful parts of Greco-Arabic geometry for the book known as De Practica Geometrie. This translation offers a reconstruction of De Practica Geometrie as the author judges Fibonacci wrote it, thereby correcting inaccuracies found in numerous modern histories. It is a high quality translation with supplemental text to explain text that has been more freely translated. A bibliography of primary and secondary resources follows the translation, completed by an index of names and special words.

This fifth volume of A History of Arabic Sciences and Mathematics is complemented by four preceding volumes which focused on the main chapters of classical mathematics: infinitesimal geometry, theory of conics and its applications, spherical geometry, mathematical astronomy, etc. This book includes seven main works of Ibn al-Haytham (Alhazen) and of two of his predecessors, Thabit ibn Qurra and al-Sijzi: The circle, its transformations and its properties; Analysis and synthesis: the founding of analytical art; A new mathematical discipline: the Knowns; The geometrisation of place; Analysis and synthesis: examples of the geometry of triangles; Axiomatic method and invention: Thabit ibn Qurra; The idea of an Ars Inveniendi: al-Sijzi. 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.

Dreaming Some years ago, in a corridor, in a small space, o? to the side, as in a hidden ravine, they were there. Tey couldn't be anywhere else. Hidden and mysterious, with faces that were dreamy and abstracted, or distracted, or pensive. Caught up in their thoughts, caught up in their space, a space that was distant and that only they could understand. Elusive and yet there, in front of me. Certainly, it was them, the six mathematicians of the Mathematica series by Mimmo Paladino. Tinkers of numbers and shapes. For ten years we have been searching for the mathematicians there. In Venice, the favourite place. An aura of mystery surrounds them, otherwise what kind of mathematicians would they be! Mysterious, dreamy, absent, absorbed are the faces of Paladino's mathemat- V icians. Te voices of ?ve Sardinian shepherds intone the Kyrie,the Libera Me Domine, the Sanctus. Insistent, profound, archaic voices. Te soundtrack of a journey, of a journey towards nothingness. A journey towards Te Wild Blue Yonder ,the last ?lm by Werner Herzog, winner of the international Critic's prize at the Venice f- tival in ????. Te heroes were astronauts, even if by now no one cares about their adventures;astronautswhotakeo?,whotravel,butdon'tknowwheretheyaregoing. And then there are the true heroes, the real gurus of the ?lm, the characters who some years ago burst onto the scene in cinema: the mathematicians.

The manuscript gives a coherent and detailed account of the theory of series in the eighteenth and early nineteenth centuries. It provides in one place an account of many results that are generally to be found - if at all - scattered throughout the historical and textbook literature. It presents the subject from the viewpoint of the mathematicians of the period, and is careful to distinguish earlier conceptions from ones that prevail today.

This book offers a detailed history of parametric statistical inference. Covering the period between James Bernoulli and R.A. Fisher, it examines: binomial statistical inference; statistical inference by inverse probability; the central limit theorem and linear minimum variance estimation by Laplace and Gauss; error theory, skew distributions, correlation, sampling distributions; and the Fisherian Revolution. Lively biographical sketches of many of the main characters are featured throughout, including Laplace, Gauss, Edgeworth, Fisher, and Karl Pearson. Also examined are the roles played by DeMoivre, James Bernoulli, and Lagrange.

China's most sophisticated system of computational astronomy was created for a Mongol emperor who could neither read nor write Chinese, to celebrate victory over China after forty years of devastating war. This book explains how and why, and reconstructs the observatory and the science that made it possible.

For two thousand years, a fundamental ritual of government was the emperor's "granting the seasons" to his people at the New Year by issuing an almanac containing an accurate lunisolar calendar. The high point of this tradition was the "Season-granting system" (Shou-shih li, 1280). Its treatise records detailed instructions for computing eclipses of the sun and moon and motions of the planets, based on a rich archive of observations, some ancient and some new.

Sivin, the West's leading scholar of the Chinese sciences, not only recreates the project's cultural, political, bureaucratic, and personal dimensions, but translates the extensive treatise and explains every procedure in minimally technical language. The book contains many tables, illustrations, and aids to reference. It is clearly written for anyone who wants to understand the fundamental role of science in Chinese history. There is no comparable study of state science in any other early civilization.

Creativity plays an important role in all human activities, from the visual arts to cinema and theatre, and in particular in science and mathematics .

This volume, published only in English in the series "Mathematics and Culture," stresses the strong links between mathematics, culture and creativity in architecture, contemporary art, geometry, computer graphics, literature, theatre and cinema. So this book is designed not only for mathematicians but for all the people who have an interest in the various aspects of culture, both scientific and literary, with a special emphasis on the visual aspects.

We live in a space, we get about in it. We also quantify it, we think of it as having dimensions. Ever since Euclid's ancient geometry, we have thought of bodies occupying parts of this space (including our own bodies), the space of our practical orientations (our 'moving abouts'), as having three dimensions. Bodies have volume specified by measures of length, breadth and height. But how do we know that the space we live in has just these three dimensions? It is theoreti cally possible that some spaces might exist that are not correctly described by Euclidean geometry. After all, there are the non Euclidian geometries, descriptions of spaces not conforming to the axioms and theorems of Euclid's geometry. As one might expect, there is a history of philosophers' attempts to 'prove' that space is three-dimensional. The present volume surveys these attempts from Aristotle, through Leibniz and Kant, to more recent philosophy. As you will learn, the historical theories are rife with terminology, with language, already tainted by the as sumed, but by no means obvious, clarity of terms like 'dimension', 'line', 'point' and others. Prior to that language there are actions, ways of getting around in the world, building things, being interested in things, in the more specific case of dimensionality, cutting things. It is to these actions that we must eventually appeal if we are to understand how science is grounded."

two main (interacting) ways. They constitute that with which exploration into problems or questions is carried out. But they also constitute that which is exchanged between scholars or, in other terms, that which is shaped by one (or by some) for use by others. In these various dimensions, texts obviously depend on the means and technologies available for producing, reproducing, using and organizing writings. In this regard, the contribution of a history of text is essential in helping us approach the various historical contexts from which our sources originate. However, there is more to it. While shaping texts as texts, the practitioners of the sciences may create new textual resources that intimately relate to the research carried on. One may think, for instance, of the process of introduction of formulas in mathematical texts. This aspect opens up a wholerangeofextremelyinterestingquestionstowhichwewillreturnatalaterpoint.But practitioners of the sciences also rely on texts produced by themselves or others, which they bring into play in various ways. More generally, they make use of textual resources of every kind that is available to them, reshaping them, restricting, or enlarging them. Among these, one can think of ways of naming, syntax of statements or grammatical analysis, literary techniques, modes of shaping texts or parts of text, genres of text and so on.Inthissense, thepractitionersdependon, anddrawon, the"textualcultures"available to the social and professional groups to which they belong.

Such Silver Currents is the first biography of a mathematical genius and his literary wife, their wide circle of well-known intellectual and artistic friends, and through them of the age in which they lived. William Clifford is now recognised not only for his innovative and lasting mathematics, but also for his philosophy, which embraced the fundamentals of scientific thought, the nature of the physical universe, Darwinian theory, the nature of consciousness, personal morality and law, and the whole mystery of being. Clifford algebra is seen as the basis for Dirac's theory of the electron, fundamental to modern physics, and Clifford also anticipated Einstein's idea that space is curved. The book includes a personal reflection on William Clifford's mathematics by the Nobel Prize winner Sir Roger Penrose O.M. The year after his election to the Royal Society, Clifford married Lucy Lane, the journalist and novelist. During their four years of marriage they held Sunday salons attended by many well-known scientific, literary and artistic personalities. Following William's early death, Lucy became a close friend and confidante of Henry James. Her wide circle of friends included Rudyard Kipling, Thomas Hardy, George Eliot, Leslie Stephen, Thomas Huxley, Sir Frederick Macmillan and Oliver Wendell Holmes Jr.

This volume commemorates the life, work, and foundational views of Kurt Godel (1906 1978), most famous for his hallmark works on the completeness of first-order logic, the incompleteness of number theory, and the consistency with the other widely accepted axioms of set theory of the axiom of choice and of the generalized continuum hypothesis. It explores current research, advances, and ideas for future directions not only in the foundations of mathematics and logic, but also in the fields of computer science, artificial intelligence, physics, cosmology, philosophy, theology, and the history of science. The discussion is supplemented by personal reflections from several scholars who knew Godel personally, providing some interesting insights into his life. By putting his ideas and life's work into the context of current thinking and perceptions, this book will extend the impact of Godel's fundamental work in mathematics, logic, philosophy, and other disciplines for future generations of researchers."

The famous and prolific nineteenth-century mathematician, engineer and inventor Charles Babbage (1791 1871) was an early pioneer of computing. He planned several calculating machines, but none was built in his lifetime. On his death his youngest son, Henry P. Babbage, was charged with the task of completing an unfinished volume of papers on the machines, which was finally published in 1889 and is reissued here. The papers, by a variety of authors, were collected from journals including The Philosophical Magazine, The Edinburgh Review and Scientific Memoirs. They relate to the construction and potential application of Charles Babbage's calculating engines, notably the Difference Engine and the more complex Analytical Engine, which was to be programmed using punched cards. The book also includes correspondence with members of scientific societies, as well as proceedings, catalogues and drawings. Included is a complete catalogue of the drawings of the Analytical Engine.

This is a charming collection of essays on life and science, by one of the leading mathematicians of our day. Vladimir Igorevich Arnold is renowned for his achievements in mathematics, and nearly as famous for his informal teaching style, and for the clarity and accessibility of his writing. The chapter headings convey Arnold's humor and restless imagination. A few examples: My first recollections; The combinatorics of Plutarch; The topology of surfaces according to Alexander of Macedon; Catching a pike in Cambridge. Yesterday and Long Ago offers a rare opportunity to appreciate the life and work of one of the world's outstanding living mathematicians.

Paul Erdos was an amazing and prolific mathematician whose life as a world-wandering numerical nomad was legendary. He published almost 1500 scholarly papers before his death in 1996, and he probably thought more about math problems than anyone in history. Like a traveling salesman offering his thoughts as wares, Erdos would show up on the doorstep of one mathematician or another and announce, "My brain is open." After working through a problem, he'd move on to the next place, the next solution. Hoffman's book, like Sylvia Nasar's biography of John Nash, A Beautiful Mind, reveals a genius's life that transcended the merely quirky. But Erdos's brand of madness was joyful, unlike Nash's despairing schizophrenia. Erdos never tried to dilute his obsessive passion for numbers with ordinary emotional interactions, thus avoiding hurting the people around him, as Nash did. Oliver Sacks writes of Erdos: "A mathematical genius of the first order, Paul Erdos was totally obsessed with his subject--he thought and wrote mathematics for nineteen hours a day until the day he died. He traveled constantly, living out of a plastic bag, and had no interest in food, sex, companionship, art--all that is usually indispensable to a human life."The Man Who Loved Only Numbers is easy to love, despite his strangeness. It's hard not to have affection for someone who referred to children as "epsilons," from the Greek letter used to represent small quantities in mathematics; a man whose epitaph for himself read, "Finally I am becoming stupider no more"; and whose only really necessary tool to do his work was a quiet and open mind. Hoffman, who followed and spoke with Erdos over the last 10 years of his life, introduces us to an undeniably odd, yet pure and joyful, man who loved numbers more than he loved God--whom he referred to as SF, for Supreme Fascist. He was often misunderstood, and he certainly annoyed people sometimes, but Paul Erdos is no doubt missed. --Therese Littleton

Now widely available in English for the first time, this is Carlo Rovelli's first book: the thrilling story of a little-known man who created one of the greatest intellectual revolutions Over two thousand years ago, one man changed the way we see the world. Since the dawn of civilization, humans had believed in the heavens above and the Earth below. Then, on the Ionian coast, a Greek philosopher named Anaximander set in motion a revolution. He not only conceived that the Earth floats in space, but also that animals evolve, that storms and earthquakes are natural, not supernatural, that the world can be mapped and, above all, that progress is made by the endless search for knowledge. Carlo Rovelli's first book, now widely available in English, tells the origin story of scientific thinking: our rebellious ability to reimagine the world, again and again.

How our understanding of calculus has evolved over more than three centuries, how this has shaped the way it is taught in the classroom, and why calculus pedagogy needs to change Calculus Reordered takes readers on a remarkable journey through hundreds of years to tell the story of how calculus evolved into the subject we know today. David Bressoud explains why calculus is credited to seventeenth-century figures Isaac Newton and Gottfried Leibniz, and how its current structure is based on developments that arose in the nineteenth century. Bressoud argues that a pedagogy informed by the historical development of calculus represents a sounder way for students to learn this fascinating area of mathematics. Delving into calculus's birth in the Hellenistic Eastern Mediterranean-particularly in Syracuse, Sicily and Alexandria, Egypt-as well as India and the Islamic Middle East, Bressoud considers how calculus developed in response to essential questions emerging from engineering and astronomy. He looks at how Newton and Leibniz built their work on a flurry of activity that occurred throughout Europe, and how Italian philosophers such as Galileo Galilei played a particularly important role. In describing calculus's evolution, Bressoud reveals problems with the standard ordering of its curriculum: limits, differentiation, integration, and series. He contends that the historical order-integration as accumulation, then differentiation as ratios of change, series as sequences of partial sums, and finally limits as they arise from the algebra of inequalities-makes more sense in the classroom environment. Exploring the motivations behind calculus's discovery, Calculus Reordered highlights how this essential tool of mathematics came to be.

WILEY-INTERSCIENCE PAPERBACK SERIES

The Wiley-Interscience Paperback Series consists of selected books that have been made more accessible to consumers in an effort to increase global appeal and general circulation. With these new unabridged softcover volumes, Wiley hopes to extend the lives of these works by making them available to future generations of statisticians, mathematicians, and scientists.

From the Reviews of History of Probability and Statistics and Their Applications before 1750

"This is a marvelous book . . . Anyone with the slightest interest in the history of statistics, or in understanding how modern ideas have developed, will find this an invaluable resource."
–Short Book Reviews of ISI

A few years ago, in the Wren Library of Trinity College, Cambridge, I came across a remarkable but then little-known album of pencil and watercolour portraits. The artist of most (perhaps all) was Thomas Charles Wageman. Created during 1829-1852, these portraits are of pupils of the famous mat- matical tutor William Hopkins. Though I knew much about several of the subjects, the names of others were then unknown to me. I was prompted to discover more about them all, and gradually this interest evolved into the present book. The project has expanded naturally to describe the Cambridge educational milieu of the time, the work of William Hopkins, and the later achievements of his pupils and their contemporaries. As I have taught applied mathematics in a British university for forty years, during a time of rapid change, the struggles to implement and to resist reform in mid-nineteenth-century Cambridge struck a chord of recognition. So, too, did debates about academic standards of honours degrees. And my own experiences, as a graduate of a Scottish university who proceeded to C- bridge for postgraduate work, gave me a particular interest in those Scots and Irish students who did much the same more than a hundred years earlier. As a mathematician, I sometimes felt frustrated at having to suppress virtually all of the ? ne mathematics associated with this period: but to have included such technical material would have made this a very different book.