This volume offers insights into the development of mathematical logic over the last century. Arising from a special session of the history of logic at an American Mathematical Society meeting, the chapters explore technical innovations, the philosophical consequences of work during the period, and the historical and social context in which the logicians worked. The discussions herein will appeal to mathematical logicians and historians of mathematics, as well as philosophers and historians of science.

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 attempts to fill two gaps which exist in the standard textbooks on the History of Mathematics. One is to provide the students with material that could encourage more critical thinking. General textbooks, attempting to cover three thousand or so years of mathematical history, must necessarily oversimplify just about everything, the practice of which can scarcely promote a critical approach to the subject. For this, I think a more narrow but deeper coverage of a few select topics is called for. The second aim is to include the proofs of important results which are typically neglected in the modern history of mathematics curriculum. The most obvious of these is the oft-cited necessity of introducing complex numbers in applying the algebraic solution of cubic equations. This solution, though it is now relegated to courses in the History of Mathematics, was a major occurrence in our history. It was the first substantial piece of mathematics in Europe that was not a mere extension of what the Greeks had done and thus signified the coming of age of European mathematics. The fact that the solution, in the case of three distinct real roots to a cubic, necessarily involved complex numbers both made inevitable the acceptance and study of these numbers and provided a stimulus for the development of numerical approximation methods. Unique features include: * a prefatory essay on the ways in which sources may be unreliable, followed by an annotated bibliography; * a new approach to the historical development of the natural numbers, which was only settled in the 19th century; * construction problems of antiquity, with a proof that the angle cannot be trisected nor the cubeduplicated by ruler and compass alone; * a modern recounting of a Chinese word problem from the 13th century, illustrating the need for consulting multiple sources when the primary source is unavailable; * multiple proofs of the cubic equation, including the proof that the algebraic solution uses complex numbers whenever the cubic equation has three distinct real solutions; * a critical reappraisal of Horner's Method; The final chapter contains lighter material, including a critical look at North Korea's stamps commemorating the 350th birthday of Newton, historically interesting (and hard to find) poems, and humorous song lyrics with mathematical themes. The appendix outlines a few small projects which could serve as replacements for the usual term papers.

Prior to the nineteenth century, algebra meant the study of the solution of polynomial equations. By the twentieth century it came to encompass the study of abstract, axiomatic systems such as groups, rings, and fields. This presentation provides an account of the intellectual lineage behind many of the basic concepts, results, and theories of abstract algebra. The development of abstract algebra was propelled by the need for new tools to address certain classical problems that appeared unsolvable by classical means. A major theme of the approach in this book is to show how abstract algebra has arisen in attempts to solve some of these classical problems, providing a context from which the reader may gain a deeper appreciation of the mathematics involved. Mathematics instructors, algebraists, and historians of science will find the work a valuable reference. The book may also serve as a supplemental text for courses in abstract algebra or the history of mathematics.

This work counters historiographies that search for the origins of modern science within the experimental practices of Europe 's first scientific institutions, such as the Cimento. It proposes that we should look beyond the experimental rhetoric found in published works, to find that the Cimento academicians were participants in a culture of natural philosophical theorising that existed throughout Europe.

In this book, several world experts present (one part of) the mathematical heritage of Kolmogorov. Each chapter treats one of his research themes or a subject invented as a consequence of his discoveries. The authors present his contributions, his methods, the perspectives he opened to us, and the way in which this research has evolved up to now. Coverage also includes examples of recent applications and a presentation of the modern prospects.

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.

Thomas Harriot's "Artis analyticae praxis" is an essential work in the history of algebra. To some extent it is a development work of Viete, who was among the first to use literal symbols to stand for known and unknown quantities. But it was Harriot who took the crucial step of creating an entirely symbolic algebra, so that reasoning could be reduced to a quasi-mechanical manipulation of symbols. Although his algebra was still limited in scope (he insisted. for example, on strict homogeneity, so only terms of the same powers could be added or equated to one another), it is recognizably modern. Although Harriot's book was highly influential in the development of analysis in England before Newton, it has recently become clear that the posthumously published Praxis contains only an incomplete account of Harriot's achievement: his editor substantially rearranged the work before publishing it, and omitted sections that were apparently beyond his comprehension, such as negative and complex roots of equations. The commentary included with the translation attempts to restore the Praxis to the state of Harriot's draft. Basing their work on manuscripts in the British Library, Pentworth House, and Lambeth Palace, the commentary contains some of Harriot's most novel and advanced mathematics, very little of which has been published in the past. It will provide the basis for a reassessment of the development of algebra.

The present work is the first ever English translation of the original text of Thomas Harriota (TM)s Artis Analyticae Praxis, first published in 1631 in Latin. Thomas Harriota (TM)s Praxis is an essential work in the history of algebra. Even though Harriota (TM)s contemporary, Viete, was among the first to use literal symbols to stand for known and unknown quantities, it was Harriott who took the crucial step of creating an entirely symbolic algebra. This allowed reasoning to be reduced to a quasi-mechanical manipulation of symbols. Although Harriota (TM)s algebra was still limited in scope (he insisted, for example, on strict homogeneity, so only terms of the same powers could be added or equated to one another), it is recognizably modern.

While Harriota (TM)s book was highly influential in the development of analysis in England before Newton, it has recently become clear that the posthumously published Praxis contains only an incomplete account of Harriota (TM)s achievement: his editor substantially rearranged the work before publishing it, and omitted sections that were apparently beyond comprehension, such as negative and complex roots of equations.

The commentary included with this translation relates the contents of the Praxis to the corresponding pages in his manuscript papers, which enables much of Harriot's most novel and advanced mathematics to be explored. This publication will become an important contribution to the history of mathematics, and it will provide the basis for a reassessment of the development of algebra.

The Italian mathematician Mario Pieri (1860-1913) played an integral part in the research groups of Corrado Segre and Giuseppe Peano, and thus had a significant, yet somewhat underappreciated impact on several branches of mathematics, particularly on the development of algebraic geometry and the foundations of mathematics in the years around the turn of the 20th century. This book is the first in a series of three volumes that are dedicated to countering that neglect and comprehensively examining Pieria (TM)s life, mathematical work and influence in such diverse fields as mathematical logic, algebraic geometry, number theory, inversive geometry, vector analysis, and differential geometry.

The Legacy of Mario Pieri in Geometry and Arithmetic introduces readers to Pieria (TM)s career and his studies in foundations, from both historical and modern viewpoints, placing his life and research in context and tracing his influence on his contemporaries as well as more recent mathematicians. The text also provides a glimpse of the Italian academic world of Pieri's time, and its relationship with the developing international mathematics community. Included in this volume are the first English translations, along with analyses, of two of his most important axiomatizationsa "his postulates for arithmetic, which Peano judged superior to his own; and his foundation of elementary geometry on the basis of point and sphere, which Alfred Tarski used as a basis for his own system.

Combining an engaging exposition, little-known historical information, exhaustive references and an excellent index, this text will be of interest to graduate students, researchers and historians with a general knowledgeof logic and advanced mathematics, and it requires no specialized experience in mathematical logic or the foundations of geometry.

Written by a distinguished specialist in functional analysis, this book presents a comprehensive treatment of the history of Banach spaces and (abstract bounded) linear operators. Banach space theory is presented as a part of a broad mathematics context, using tools from such areas as set theory, topology, algebra, combinatorics, probability theory, logic, etc. Equal emphasis is given to both spaces and operators. The book may serve as a reference for researchers and as an introduction for graduate students who want to learn Banach space theory with some historical flavor.

One of the paradoxes of the physical sciences is that as our knowledge has progressed, more and more diverse physical phenomena can be explained in terms of fewer underlying laws, or principles. In Hidden Unity, eminent physicist John Taylor puts many of these findings into historical perspective and documents how progress is made when unexpected, hidden unities are uncovered between apparently unrelated physical phenomena. Taylor cites examples from the ancient Greeks to the present day, such as the unity of celestial and terrestrial dynamics (17th century), the unity of heat within the rest of dynamics (18th century), the unity of electricity, magnetism, and light (19th century), the unity of space and time and the unification of nuclear forces with electromagnetism (20th century). Without relying on mathematical detail, Taylor's emphasis is on fundamental physics, like particle physics and cosmology. Balancing what is understood with the unestablished theories and still unanswered questions, Taylor takes readers on a fascinating ongoing journey. John C. Taylor is Professor Emeritus of Mathematical Physics at the University of Cambridge. A student of Nobel laureate Abdus Salam, Taylor's research career has spanned the era of developments in elementary particle physics since the 1950s. He taught theoretical physics at Imperial College, London, and at the Universities of Oxford and Cambridge, and he has lectured worldwide. He is a Fellow of the Royal Society and a Fellow of the Institute of Physics.

Nonstandard Analysis enhances mathematical reasoning by introducing new ways of expression and deduction. Distinguishing between standard and nonstandard mathematical objects, its inventor, the eminent mathematician Abraham Robinson, settled in 1961 the centuries-old problem of how to use infinitesimals correctly in analysis. Having also worked as an engineer, he saw not only that his method greatly simplified mathematically proving and teaching, but also served as a powerful tool in modelling, analyzing and solving problems in the applied sciences, among others by effective rescaling and by infinitesimal discretizations.

This book reflects the progress made in the forty years since the appearance of Robinson s revolutionary book Nonstandard Analysis: in the foundations of mathematics and logic, number theory, statistics and probability, in ordinary, partial and stochastic differential equations and in education. The contributions are clear and essentially self-contained."

This book presents a historical overview of number theory. It examines texts that span some thirty-six centuries of arithmetical work, from an Old Babylonian tablet to Legendre's Essai sur la Theorie des Nombres, written in 1798. Coverage employs a historical approach in the analysis of problems and evolving methods of number theory and their significance within mathematics. The book also takes the reader into the workshops of four major authors of modern number theory: Fermat, Euler, Lagrange and Legendre and presents a detailed and critical examination of their work.

Written by one of the most important historians of statistics of the 20th century, Anders Hald. This book can be viewed as a follow-up to his two most recent books, although this current text is much more streamlined and contains new analysis of many ideas and developments. And unlike his other books, which were encyclopedic by nature, this book can be used for a course on the topic, the only prerequisites being a basic course in probability and statistics. The book is divided into five main sections: 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, 1912-1935. Throughout each of the chapters, the author provides lively biographical sketches of many of the main characters, including Laplace, Gauss, Edgeworth, Fisher, and Karl Pearson. He also examines the roles played by DeMoivre, James Bernoulli, and Lagrange, and he provides an accessible exposition of the work of R.A. Fisher.

Euler was not only by far the most productive mathematician in the history of mankind, but also one of the greatest scholars of all time. He attained, like only a few scholars, a degree of popularity and fame which may well be compared with that of Galilei, Newton, or Einstein.

Moreover he was a cosmopolitan in the truest sense of the word; he lived during his first twenty years in Basel, was active altogether for more than thirty years in Petersburg and for a quarter of a century in Berlin.

Leonhard Euler's unusually rich life and broadly diversified activity in the immediate vicinity of important personalities which have made history, may well justify an exposition.

This book is based in part on unpublished sources and comes right out of the current research on Euler. It is entirely free of formulae as it has been written for a broad audience with interests in the history of culture and science.

Intellectual History and the Identity of John Dee In April 1995, at Birkbeck College, University of London, an interdisciplinary colloquium was held so that scholars from diverse fields and areas of expertise could 1 exchange views on the life and work of John Dee. Working in a variety of fields - intellectual history, history of navigation, history of medicine, history of science, history of mathematics, bibliography and manuscript studies - we had all been drawn to Dee by particular aspects of his work, and participating in the colloquium was to c- front other narratives about Dee's career: an experience which was both bewildering and instructive. Perhaps more than any other intellectual figure of the English Renaissance Dee has been fragmented and dispersed across numerous disciplines, and the various attempts to re-integrate his multiplied image by reference to a particular world-view or philosophical outlook have failed to bring him into focus. This volume records the diversity of scholarly approaches to John Dee which have emerged since the synthetic accounts of I. R. F. Calder, Frances Yates and Peter French. If these approaches have not succeeded in resolving the problematic multiplicity of Dee's activities, they will at least deepen our understanding of specific and local areas of his intellectual life, and render them more historiographically legible.

This book is at once an analytical study of one of the most important mathematical texts of antiquity, the Mathematical Collection of the fourth-century AD mathematician Pappus of Alexandria, and also an examination of the work's wider cultural setting. This is one of very few books to deal extensively with the mathematics of Late Antiquity. It sees Pappus' text as part of a wider context and relates it to other contemporary cultural practices and opens new avenues to research into the public understanding of mathematics and mathematical disciplines in antiquity.

More than three centuries after its creation, calculus remains a dazzling intellectual achievement and the gateway to higher mathematics. This book charts its growth and development by sampling from the work of some of its foremost practitioners, beginning with Isaac Newton and Gottfried Wilhelm Leibniz in the late seventeenth century and continuing to Henri Lebesgue at the dawn of the twentieth. Now with a new preface by the author, this book documents the evolution of calculus from a powerful but logically chaotic subject into one whose foundations are thorough, rigorous, and unflinching-a story of genius triumphing over some of the toughest, subtlest problems imaginable. In touring The Calculus Gallery, we can see how it all came to be.

The problem of approximating a given quantity is one of the oldest challenges faced by mathematicians. Its increasing importance in contemporary mathematics has created an entirely new area known as Approximation Theory. The modern theory was initially developed along two divergent schools of thought: the Eastern or Russian group, employing almost exclusively algebraic methods, was headed by Chebyshev together with his coterie at the Saint Petersburg Mathematical School, while the Western mathematicians, adopting a more analytical approach, included Weierstrass, Hilbert, Klein, and others.

This work traces the history of approximation theory from Leonhard Euler's cartographic investigations at the end of the 18th century to the early 20th century contributions of Sergei Bernstein in defining a new branch of function theory. One of the key strengths of this book is the narrative itself. The author combines a mathematical analysis of the subject with an engaging discussion of the differing philosophical underpinnings in approach as demonstrated by the various mathematicians. This exciting exposition integrates history, philosophy, and mathematics. While demonstrating excellent technical control of the underlying mathematics, the work is focused on essential results for the development of the theory.

The exposition begins with a history of the forerunners of modern approximation theory, i.e., Euler, Laplace, and Fourier. The treatment then shifts to Chebyshev, his overall philosophy of mathematics, and the Saint Petersburg Mathematical School, stressing in particular the roles played by Zolotarev and the Markov brothers. A philosophical dialectic then unfolds, contrastingEast vs. West, detailing the work of Weierstrass as well as that of the Goettingen school led by Hilbert and Klein. The final chapter emphasizes the important work of the Russian Jewish mathematician Sergei Bernstein, whose constructive proof of the Weierstrass theorem and extension of Chebyshev's work serve to unify East and West in their approaches to approximation theory.

Appendices containing biographical data on numerous eminent mathematicians, explanations of Russian nomenclature and academic degrees, and an excellent index round out the presentation.

This book contains several contributions on the most outstanding events in the development of twentieth century mathematics, representing a wide variety of specialities in which Russian and Soviet mathematicians played a considerable role. The articles are written in an informal style, from mathematical philosophy to the description of the development of ideas, personal memories and give a unique account of personal meetings with famous representatives of twentieth century mathematics who exerted great influence in its development.

This book will be of great interest to mathematicians, who will enjoy seeing their own specialities described with some historical perspective. Historians will read it with the same motive, and perhaps also to select topics for future investigation.

This book provides a way to understand a momentous development in human intellectual history: the phenomenon of deductive argument in classical Greek mathematics. The argument rests on a close description of the practices of Greek mathematics, principally the use of lettered diagrams and the regulated, formulaic use of language.

This book contains around 80 articles on major writings in mathematics published between 1640 and 1940. All aspects of mathematics are covered: pure and applied, probability and statistics, foundations and philosophy. Sometimes two writings from the same period and the same subject are taken together. The biography of the author(s) is recorded, and the circumstances of the preparation of the writing are given. When the writing is of some lengths an analytical table of its contents is supplied. The contents of the writing is reviewed, and its impact described, at least for the immediate decades. Each article ends with a bibliography of primary and secondary items.
.First book of its kind
.Covers the period 1640-1940 of massive development in mathematics
.Describes many of the main writings of mathematics
.Articles written by specialists in their field

Heavenly Mathematics traces the rich history of spherical trigonometry, revealing how the cultures of classical Greece, medieval Islam, and the modern West used this forgotten art to chart the heavens and the Earth. Once at the heart of astronomy and ocean-going navigation for two millennia, the discipline was also a mainstay of mathematics education for centuries and taught widely until the 1950s. Glen Van Brummelen explores this exquisite branch of mathematics and its role in ancient astronomy, geography, and cartography; Islamic religious rituals; celestial navigation; polyhedra; stereographic projection; and more. He conveys the sheer beauty of spherical trigonometry, providing readers with a new appreciation of its elegant proofs and often surprising conclusions. Heavenly Mathematics is illustrated throughout with stunning historical images and informative drawings and diagrams. This unique compendium also features easy-to-use appendixes as well as exercises that originally appeared in textbooks from the eighteenth to the early twentieth centuries.