Major shifts in the field of model theory in the twentieth century have seen the development of new tools, methods, and motivations for mathematicians and philosophers. In this book, John T. Baldwin places the revolution in its historical context from the ancient Greeks to the last century, argues for local rather than global foundations for mathematics, and provides philosophical viewpoints on the importance of modern model theory for both understanding and undertaking mathematical practice. The volume also addresses the impact of model theory on contemporary algebraic geometry, number theory, combinatorics, and differential equations. This comprehensive and detailed book will interest logicians and mathematicians as well as those working on the history and philosophy of mathematics.

Evariste Galois' short life was lived against the turbulent background of the restoration of the Bourbons to the throne of France, the 1830 revolution in Paris and the accession of Louis-Phillipe. This new and scrupulously researched biography of the founder of modern algebra sheds much light on a life led with great intensity and a death met tragically under dark circumstances. Sorting speculation from documented fact, it offers the fullest and most exacting account ever written of Galois' life and work. It took more than seventy years to fully understand the French mathematician's first memoire (published in 1846) which formulated the famous "Galois theory" concerning the solvability of algebraic equations by radicals, from which group theory would follow. Obscurities in his other writings - memoires and numerous fragments of extant papers - persist and his ideas challenge mathematicians to this day. Thus scholars will welcome those chapters devoted specifically to explicating all aspects of Galois' work. A comprehensive bibliography enumerates studies by and also those about the mathematician.

In this volume specialists in mathematics, physics, and linguistics present the first comprehensive analysis of the ideas and influence of Hermann G. Grassmann (1809-1877), the remarkable universalist whose work recast the foundations of these disciplines and shaped the course of their modern development.

The general principles by which the editors and authors of the present edition have been guided were explained in the preface to the first volume of Mathemat ics of the 19th Century, which contains chapters on the history of mathematical logic, algebra, number theory, and probability theory (Nauka, Moscow 1978; En glish translation by Birkhiiuser Verlag, Basel-Boston-Berlin 1992). Circumstances beyond the control of the editors necessitated certain changes in the sequence of historical exposition of individual disciplines. The second volume contains two chapters: history of geometry and history of analytic function theory (including elliptic and Abelian functions); the size of the two chapters naturally entailed di viding them into sections. The history of differential and integral calculus, as well as computational mathematics, which we had planned to include in the second volume, will form part of the third volume. We remind our readers that the appendix of each volume contains a list of the most important literature and an index of names. The names of journals are given in abbreviated form and the volume and year of publication are indicated; if the actual year of publication differs from the nominal year, the latter is given in parentheses. The book History of Mathematics from Ancient Times to the Early Nineteenth Century in Russian], which was published in the years 1970-1972, is cited in abbreviated form as HM (with volume and page number indicated). The first volume of the present series is cited as Bk. 1 (with page numbers)."

..".a story of great mathematicians and their achievements, of practical successes and failures, and of human perfidy and generosity...this is one of the still too rare occasions in which mathematicians are shown as frail, flesh-and-blood creatures...a very worthwhile book." -CHOICE

Discussions of the foundations of mathematics and their history are frequently restricted to logical issues in a narrow sense, or else to traditional problems of analytic philosophy. From Dedekind to GAdel: Essays on the Development of the Foundations of Mathematics illustrates the much greater variety of the actual developments in the foundations during the period covered. The viewpoints that serve this purpose included the foundational ideas of working mathematicians, such as Kronecker, Dedekind, Borel and the early Hilbert, and the development of notions like model and modelling, arbitrary function, completeness, and non-Archimedean structures. The philosophers discussed include not only the household names in logic, but also Husserl, Wittgenstein and Ramsey. Needless to say, such logically-oriented thinkers as Frege, Russell and GAdel are not entirely neglected, either. Audience: Everybody interested in the philosophy and/or history of mathematics will find this book interesting, giving frequently novel insights.

A lively history of the peculiar math of voting Since the very birth of democracy in ancient Greece, the simple act of voting has given rise to mathematical paradoxes that have puzzled some of the greatest philosophers, statesmen, and mathematicians. Numbers Rule traces the epic quest by these thinkers to create a more perfect democracy and adapt to the ever-changing demands that each new generation places on our democratic institutions. In a sweeping narrative that combines history, biography, and mathematics, George Szpiro details the fascinating lives and big ideas of great minds such as Plato, Pliny the Younger, Ramon Llull, Pierre Simon Laplace, Thomas Jefferson, Alexander Hamilton, John von Neumann, and Kenneth Arrow, among many others. Each chapter in this riveting book tells the story of one or more of these visionaries and the problem they sought to overcome, like the Marquis de Condorcet, the eighteenth-century French nobleman who demonstrated that a majority vote in an election might not necessarily result in a clear winner. Szpiro takes readers from ancient Greece and Rome to medieval Europe, from the founding of the American republic and the French Revolution to today's high-stakes elective politics. He explains how mathematical paradoxes and enigmas can crop up in virtually any voting arena, from electing a class president, a pope, or prime minister to the apportionment of seats in Congress. Numbers Rule describes the trials and triumphs of the thinkers down through the ages who have dared the odds in pursuit of a just and equitable democracy.

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.

Karl Menger (1902--1985), a pure mathematician of distinction, also took an active interest in both philosophy and economics. In this memoir, which he was composing at the time of his death, he relates how all these subjects developed and flourished against the Viennese background (itself described in depth and with affection), and did so despite the political developments of the '20s and '30s, which depressed but did not silence him. He continued his work in the United States. The memoir describes his membership of the Vienna Circle (the scientifically minded philosophers that gathered around Moritz Schlick) for whom he was an invaluable intermediary, bringing them into contact with Brouwer's intuitionism, with the work of the Polish logicians, especially that of Tarski, but more generally with rigorous mathematical thinking. Indeed, the other Viennese group described here is the Mathematical Colloquium, which he founded, whose Proceedings (still read) show it to have been a powerhouse of ideas. There are also valuable chapters on philosophy and mathematics in the Poland of the '20s and '30s and the U.S. of the '30s and '40s. The memoir devotes particular attention to Wittgenstein (with whose family Menger was acquainted) and to GAdel, whom he was instrumental in bringing to America. The genesis of Menger's own writings on philosophy is also described and the work abounds in mathematical examples lucidly applied to that subject. This volume (which can now be placed beside the two by Menger already published in the Vienna Circle Collection) gives an unequalled impression of the fruitful interdisciplinarity of the tradition to which he partly belonged and partly created. It testifiesboth to Menger's power to inspire and to the critical eye he always turned on even the philosophers he most approved of. A brief account of his life is given in an Introduction by the Editors (all of whom knew him personally), and his important contribution to the social sciences -- only touched on in the text -- is elucidated by Professor Lionello Punzo.

The calculus has served for three centuries as the principal quantitative language of Western science. In the course of its genesis and evolution some of the most fundamental problems of mathematics were first con fronted and, through the persistent labors of successive generations, finally resolved. Therefore, the historical development of the calculus holds a special interest for anyone who appreciates the value of a historical perspective in teaching, learning, and enjoying mathematics and its ap plications. My goal in writing this book was to present an account of this development that is accessible, not solely to students of the history of mathematics, but to the wider mathematical community for which my exposition is more specifically intended, including those who study, teach, and use calculus. The scope of this account can be delineated partly by comparison with previous works in the same general area. M. E. Baron's The Origins of the Infinitesimal Calculus (1969) provides an informative and reliable treat ment of the precalculus period up to, but not including (in any detail), the time of Newton and Leibniz, just when the interest and pace of the story begin to quicken and intensify. C. B. Boyer's well-known book (1949, 1959 reprint) met well the goals its author set for it, but it was more ap propriately titled in its original edition-The Concepts of the Calculus than in its reprinting."

The story of numbers is a rich, sweeping history that shows how our mathematical achievements contributed to the greatest innovations of civilization. Calvin Clawson, acclaimed author of Conquering Math Phobia, weaves a story of numbers that spans thousands of years. As Clawson so clearly shows, numbers are not only an intrinsic and essential thread in our modern lives, but have always been an integral part of the human psyche - knit into the very fabric of our identity as humans. Clawson travels back through time to the roots of the history of numbers. In exploring early human fascination with numbers, he unearths the clay beads, knotted ropes, and tablets used by our ancestors as counting tools. He then investigates how numeric symbols and concepts developed uniquely and independently in Meso-America, China, and Egypt. As he persuasively argues, the mathematical concepts that arose and flourished in the ancient world enabled the creation of architectural masterpieces as well as the establishment of vast trade networks. Continuing the journey, Clawson brings us to the elegant logic of numbers that soon came to distinguish itself as a discipline and the language of science. From the concepts of infinity contemplated by the Greeks to the complex numbers that are indispensable to scientists on the cutting edge of research today, Clawson breathes life and meaning into the history of great mathematical mysteries and problems. In this spirit of inquiry, he explores, in their times and places, the discovery of numbers that lie outside the province of counting, including irrational numbers, transcendentals, complex numbers, and the enormous transfinite numbers. The personalities and the creative feats surrounding each mathematical invention come alive vividly in Clawson's lucid prose. In this work of breathtaking scope, Clawson guides us through the wonders of numbers and illustrates their monumental impact on civilization.

The Legacy of Freudenthal pays homage to Freudenthal and his work on mathematics, its history and education. Almost all authors were his scholars or co-workers. They testify to what they learned from him. Freudenthal himself contributes posthumously. His didactical phenomenology of the concept of force is both provocative and revealing in its originality, compared with what is usually found in physics instruction. Freudenthal is portrayed as a universal human being by Josette Adda. He made considerable contributions to mathematics itself, e.g. on homotopy theory and Lie groups in geometry. The exposition of Freudenthal's mathematical life and work is on Van Est's account. Henk Bos discusses his historical work. The essay review of the 8th edition of Hilbert's Grundlagen der Geometrie serves as a vehicle of thought. The main part of the book, however, concerns Freudenthal's work on mathematics education. Christine Keitel reviews his final book Revisiting Mathematics Education (1991). Fred Goffree describes Freudenthal's Working on Mathematics Education' both from an historical as well as a theoretical perspective. Adrian Treffers analyses Freudenthal's influence on the development of realistic mathematics education at primary level in the Netherlands, especially his influence on the Wiskobas-project of the former IOWO. Freudenthal once predicted the disappearance of mathematics as an individual subject in education sometime around the year 2000, because it would by then have merged with integrated thematic contexts. Jan de Lange anticipates this future development and shows that Freudenthal's prediction will not come true after all. Reflective interludes unveil how he might haveinfluenced those developments. Freudenthal contributed a wealth of ideas and conceptual tools to the development of mathematics education -- on contexts, didactical phenomenology, guided reinvention, mathematisation, the constitution of mental objects, the development of reflective thinking, levels in learning processes, the development of a mathematical attitude and so on -- but he did not design very much concrete material. Leen Streefland deals with the question of design from a theoretical point of view, while applying Freudenthal's ideas on changing perspective and shifting. For teachers, researchers, mathematics educators, mathematicians, educationalists, psychologists and policy makers.

How can we be sure that Pythagoras's theorem is really true? Why is the 'angle in a semicircle' always 90 degrees? And how can tangents help determine the speed of a bullet? David Acheson takes the reader on a highly illustrated tour through the history of geometry, from ancient Greece to the present day. He emphasizes throughout elegant deduction and practical applications, and argues that geometry can offer the quickest route to the whole spirit of mathematics at its best. Along the way, we encounter the quirky and the unexpected, meet the great personalities involved, and uncover some of the loveliest surprises in mathematics.

In his monumental 1687 work, Philosophiae Naturalis Principia Mathematica, known familiarly as the Principia, Isaac Newton laid out in mathematical terms the principles of time, force, and motion that have guided the development of modern physical science. Even after more than three centuries and the revolutions of Einsteinian relativity and quantum mechanics, Newtonian physics continues to account for many of the phenomena of the observed world, and Newtonian celestial dynamics is used to determine the orbits of our space vehicles. This authoritative, modern translation by I. Bernard Cohen and Anne Whitman, the first in more than 285 years, is based on the 1726 edition, the final revised version approved by Newton; it includes extracts from the earlier editions, corrects errors found in earlier versions, and replaces archaic English with contemporary prose and up-to-date mathematical forms. Newton's principles describe acceleration, deceleration, and inertial movement; fluid dynamics; and the motions of the earth, moon, planets, and comets. A great work in itself, the Principia also revolutionized the methods of scientific investigation. It set forth the fundamental three laws of motion and the law of universal gravity, the physical principles that account for the Copernican system of the world as emended by Kepler, thus effectively ending controversy concerning the Copernican planetary system. The illuminating Guide to Newton's Principia by I. Bernard Cohen makes this preeminent work truly accessible for today's scientists, scholars, and students. Designed with collectors in mind, this deluxe edition has faux leather binding covered with a beautiful dustjacket.

Joseph Liouville was the most important French mathematician in the gen- eration between Galois and Hermite. This is reflected in the fact that even today all mathematicians know at least one of the more than six theorems named after him and regularly study Liouville's Journal, as the Journal de Mathematiques pures et appliquees is usually nicknamed after its creator. However, few mathematicians are aware of the astonishing variety of Liou- ville's contributions to almost all areas of pure and applied mathematics. The reason is that these contributions have not been studied in their histor- ical context. In the Dictionary of Scientific Biography 1973, Taton [1973] gave a rather sad but also true picture of the Liouville studies carried out up to that date: The few articles devoted to Liouville contain little biographical data. Thus the principal stages of his life must be reconstructed on the ba- sis of original documentation. There is no exhausti ve list of Liou ville's works, which are dispersed in some 400 publications ...His work as a whole has been treated in only two original studies of limited scope those of G. Chrystal and G. Loria. Since this was written, the situation has improved somewhat through the publications of Peiffer, Edwards, Neuenschwander, and myself. Moreover, C. Houzel and I have planned on publishing Liouville's collected works.

The mathematics of ancient Egypt was fundamentally different from our math today. Contrary to what people might think, it wasn't a primitive forerunner of modern mathematics. In fact, it can't be understood using our current computational methods. "Count Like an Egyptian" provides a fun, hands-on introduction to the intuitive and often-surprising art of ancient Egyptian math. David Reimer guides you step-by-step through addition, subtraction, multiplication, and more. He even shows you how fractions and decimals may have been calculated--they technically didn't exist in the land of the pharaohs. You'll be counting like an Egyptian in no time, and along the way you'll learn firsthand how mathematics is an expression of the culture that uses it, and why there's more to math than rote memorization and bewildering abstraction.

Reimer takes you on a lively and entertaining tour of the ancient Egyptian world, providing rich historical details and amusing anecdotes as he presents a host of mathematical problems drawn from different eras of the Egyptian past. Each of these problems is like a tantalizing puzzle, often with a beautiful and elegant solution. As you solve them, you'll be immersed in many facets of Egyptian life, from hieroglyphs and pyramid building to agriculture, religion, and even bread baking and beer brewing.

Fully illustrated in color throughout, "Count Like an Egyptian" also teaches you some Babylonian computation--the precursor to our modern system--and compares ancient Egyptian mathematics to today's math, letting you decide for yourself which is better.

From the reviews: "This collection of essays surveys the most important of Fisher's papers in various areas of statistics. ... ... the monograph will be a useful source of reference to most of Fisher's major papers; it will certainly provide background material for much vigorous discussion." #"Australian Journal of Statistics"#1

Diese Arbeit enthiilt zwei grof3ere Fallstudien zur Beziehung zwischen theo- retischer Mathematik und Anwendungen im 19. Jahrhundert. Sie ist das Ergebnis eines mathematikhistorischen Forschungsprojekts am Mathemati- schen Fachbereich der Universitiit-Gesamthochschule Wuppertal und wurde dort als Habilitationsschrift vorgelegt. Ohne das wohlwollende Interesse von Herrn H. Scheid und den Kollegen der Abteilung fUr Didaktik der Mathema- tik ware das nicht moglich gewesen: Inhaltlich verdankt sie - direkt oder indirekt - vielen Beteiligten et- was. So wurde mein Interesse an den kristallographischen Symmetriekon- zepten, dem Thema der ersten Fallstudie, durch Anregungen und Hinweise von Herrn E. Brieskorn geweckt. Sowohl von seiner Seite als auch von Herrn J. J. Burckhardt stammen uberdies viele wert volle Hinweise zum Manuskript von Kapitel I. Herrn C. J. Scriba mochte ich fur seine die gesamte Arbeit betreffenden priizisen Anmerkungen danken und Herrn W. Borho ebenso fUr seine ubergreifenden Kommentare und Vorschlage. Beziiglich der in Kapitel II behandelten projektiven Methoden in der Baustatik des 19. Jahrhunderts gilt mein besonderer Dank den Herren K. -E. Kurrer und T. Hiinseroth fUr ihre zum Teil sehr detaillierten Anmerkungen aus dem Blickwinkel der Geschichte der Bauwissenschaften. Schliefilich geht mein Dank an alle nicht namentlich Erwiihnten, die in Gesprachen, technisch oder auch anderweitig zur Fertig- stellung dieser Arbeit beigetragen haben. Fur die vorliegende Publikation habe ich einen Anhang mit einer Skizze von in unserem Zusammenhang besonders wichtig erscheinenden Aspekten der Theorie der kristallographischen Raumgruppen hinzugefUgt. Ich hoffe, daB er zum Verstiindnis des mathematischen Hintergrunds der historischen Arbeiten des ersten Kapitels beitragt.

In this book, Johnny Ball tells one of the most important stories in world history - the story of mathematics.

By introducing us to the major characters and leading us through many historical twists and turns, Johnny slowly unravels the tale of how humanity built up a knowledge and understanding of shapes, numbers and patterns from ancient times, a story that leads directly to the technological wonderland we live in today. As Galileo said, 'Everything in the universe is written in the language of mathematics', and Wonders Beyond Numbers is your guide to this language.

Mathematics is only one part of this rich and varied tale; we meet many fascinating personalities along the way, such as a mathematician who everyone has heard of but who may not have existed; a Greek philosopher who made so many mistakes that many wanted his books destroyed; a mathematical artist who built the largest masonry dome on earth, which builders had previously declared impossible; a world-renowned painter who discovered mathematics and decided he could no longer stand the sight of a brush; and a philosopher who lost his head, but only after he had died.

Enriched with tales of colourful personalities and remarkable discoveries, there is also plenty of mathematics for keen readers to get stuck into. Written in Johnny Ball's characteristically light-hearted and engaging style, this book is packed with historical insight and mathematical marvels; join Johnny and uncover the wonders found beyond the numbers.

I am very pleased that my books about David Hilbert, published in 1970, and Richard Courant, published in 1976, are now being issued by Springer Verlag in a single volume. I have always felt that they belonged together, Courant being, as I have written, the natural and necessary sequel to Hilbert the rest of the story. To make the two volumes more compatible when published as one, we have combined and brought up to date the indexes of names and dates. U nfortu nately we have had to omit Hermann Weyl's article on "David Hilbert and his mathematical work," but the interested reader can always find it in the hard back edition of Hilbert and in Weyl's collected papers. At the request of a number of readers we have included a listing of all of Hilbert's famous Paris problems. It was, of course, inevitable that we would give the resulting joint volume the title Hilbert-Courant."

STANISLAW MARCIN ULAM, or Stan as his friends called him, was one of those great creative mathematicians whose interests ranged not only over all fields of mathematics, but over the physical and biological sciences as well. Like his good friend "Johnny" von Neumann, and unlike so many of his peers, Ulam is unclassifiable as a pure or applied mathematician. He never ceased to find as much beauty and excitement in the applications of mathematics as in working in those rarefied regions where there is a total un concern with practical problems. In his Adventures of a Mathematician Ulam recalls playing on an oriental carpet when he was four. The curious patterns fascinated him. When his father smiled, Ulam remembers thinking: "He smiles because he thinks I am childish, but I know these are curious patterns. I know something my father does not know." The incident goes to the heart of Ulam's genius. He could see quickly, in flashes of brilliant insight, curious patterns that other mathematicians could not see. "I am the type that likes to start new things rather than improve or elaborate," he wrote. "I cannot claim that I know much of the technical material of mathematics."

In a broad sense Design Science is the grammar of a language of images rather than of words. Modern communication techniques enable us to transmit and reconstitute images without the need of knowing a specific verbal sequential language such as the Morse code or Hungarian. International traffic signs use international image symbols which are not specific to any particular verbal language. An image language differs from a verbal one in that the latter uses a linear string of symbols, whereas the former is multidimensional. Architectural renderings commonly show projections onto three mutually perpendicular planes, or consist of cross sections at differ ent altitudes representing a stack of floor plans. Such renderings make it difficult to imagine buildings containing ramps and other features which disguise the separation between floors; consequently, they limit the creativity of the architect. Analogously, we tend to analyze natural structures as if nature had used similar stacked renderings, rather than, for instance, a system of packed spheres, with the result that we fail to perceive the system of organization determining the form of such structures.