An entertaining history of mathematics as chronicled through fifty short biographies. Mathematics today is the fruit of centuries of brilliant insights by men and women whose personalities and life experiences were often as extraordinary as their mathematical achievements. This entertaining history of mathematics chronicles those achievements through fifty short biographies that bring these great thinkers to life while making their contributions understandable to readers with little math background. Among the fascinating characters profiled are Isaac Newton (1642-1727), the founder of classical physics and infinitesimal calculus--he frequently quarreled with fellow scientists and was obsessed by alchemy and arcane Bible interpretation; Sophie Germain (1776 - 1831), who studied secretly at the Ecole Polytechnique in Paris, using the name of a previously enrolled male student--she is remembered for her work on Fermat's Last Theorem and on elasticity theory; Emmy Noether (1882 - 1935), whom Albert Einstein described as the most important woman in the history of mathematics--she made important contributions to abstract algebra and in physics she clarified the connection between conservation laws and symmetry; and Srinivasa Ramanujan (1887-1920), who came from humble origins in India and had almost no formal training, yet made substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions. The unusual behavior and life circumstances of these and many other intriguing personalities make for fascinating reading and a highly enjoyable introduction to mathematics.

The Twentieth Century has seen a dramatic rise in the use of probability and statistics in almost all fields of research. This has stimulated many new philosophical ideas on probability.
Philosophical Theories of Probability is the first book to present a clear, comprehensive and systematic account of these various theories and to explain how they relate to one another. Gillies also offers a distinctive version of the propensity theory of probability, and the intersubjective interpretation, which develops the subjective theory.

Mancosu provides the first comprehensive account of the relationship between mathematical advances of the seventeenth century and the philosophy of mathematics of the period. Starting with the Renaissance debates on the certainty of mathematics, Mancosu leads the reader through the foundational issues raised by the emergence of these new mathematical techniques. He draws a sophisticated picture of the subtle dependencies between technical development and philosophical reflection in seventeenth century mathematics.

The book treats two approaches to decision theory: (1) the normative, purporting to determine how a 'perfectly rational' actor ought to choose among available alternatives; (2) the descriptive, based on observations of how people actually choose in real life and in laboratory experiments. The mathematical tools used in the normative approach range from elementary algebra to matrix and differential equations. Sections on different levels can be studied independently. Special emphasis is made on 'offshoots' of both theories to cognitive psychology, theoretical biology, and philosophy.

Our much-valued mathematical knowledge rests on two supports: the logic of proof and the axioms from which those proofs begin. Naturalism in Mathematics investigates the status of the latter, the fundamental assumptions of mathematics. These were once held to be self-evident, but progress in work on the foundations of mathematics, especially in set theory, has rendered that comforting notion obsolete. Given that candidates for axiomatic status cannot be proved, what sorts of considerations can be offered for or against them? That is the central question addressed in this book. One answer is that mathematics aims to describe an objective world of mathematical objects, and that axiom candidates should be judged by their truth or falsity in that world. This promising view-realism-is assessed and finally rejected in favour of another-naturalism-which attends less to metaphysical considerations of objective truth and falsity, and more to practical considerations drawn from within mathematics itself. Penelope Maddy defines this naturalism, explains the motivation for it, and shows how it can be helpfully applied in the assessment of candidates for axiomatic status in set theory. Maddy's clear, original treatment of this fundamental issue is informed by current work in both philosophy and mathematics, and will be accessible and enlightening to readers from both disciplines.

Originally published in 1937. This book is a classic work on the philosophy of time, looking at the pshychology, physics and logic of time before investigating the views of Kant, Bergson, Alexander, McTaggart and Dunne. The second half of the book contains more indepth consideration of prediction, the concepts of past and future, and reality.

Originally published in 1980. What is time? How is its structure determined? The enduring controversy about the nature and structure of time has traditionally been a diametrical argument between those who see time as a container into which events are placed, and those for whom time cannot exist without events. This controversy between the absolutist and the relativist theories of time is a central theme of this study. The author's impressive arguments provide grounds for rejecting both these theories, firstly by establishing that 'empty' time is possible, and secondly by showing, through a discussion of the structure of time which involves considering whether time might be cyclical, branching, beginning or non-beginning, that the absolutist theory of time is untenable. This book then advances two new theories, and succeeds in shifting the traditional debate about time to a consideration of time as a theoretical structure and as a theoretical framework.

Originally published in 1964. This lively, challenging book, written with enthusiasm, conviction and clarity, sets out to elucidate the shadowy concept of Time. This involves central philosophical issues, which are vigorously discussed. Also relativity theory, in a clear-cut exposition, is made intelligible in a new light. All who are interested in science and its philosophical implications will find this book highly controversial but certainly readable. The author believes philosophy to be important, not only for its professionals, but for everyman. He believes that the fact that this is no longer realised shows that something is wrong with professional philosophy; he also indicates what this is. The book ends, surprisingly but pertinently, with a bold plunge into the questions of telepathy, precognition and psychical research generally. Whilst the phenomena are reasonably admitted, trenchant criticism of their significance confronts parapsychologists.

The Taming of the True poses a broad challenge to the realist views of meaning and truth that have been prominent in recent philosophy. Neil Tennant starts with a careful critical survey of the realism debate, guiding the reader through its complexities; he then presents a sustained defence of the anti-realist view that every truth is knowable in principle, and that grasp of meaning must be able to be made manifest. Sceptical arguments for the indeterminacy or non-factuality of meaning are countered; and the much-maligned notion of analyticity is reinvestigated and rehabilitated. Tennant goes on to show that an effective logical system can be based on his anti-realist view; the logical system that he advocates is justified as a body of analytic truths and inferential principles. Having laid the foundations for global semantic anti-realism, Tennant moves to the world of empirical understanding, and gives an account of the cognitive credentials of natural scientific discourse. He shows that the same canon of constructive and relevant inference suffices both for intuitionistic mathematics and for empirical science. This is an ambitious and contentious book which aims to reform not only theory of meaning, but our deductive practices across a broad range of discourses.

Originally published in 1969. This book is for undergraduates whether specializing in philosophy or not. It assumes no previous knowledge of logic but aims to show how logical notions arise from, or are abstracted from, everyday discourse, whether technical or non-technical. It sets out a knowledge of principles and, while not historical, gives an account of the reasons for which modern systems have emerged from the traditional syllogistic logic, demonstrating how certain central ideas have developed. The text explains the connections between everyday reasoning and formal logic and works up to a brief sketch of systems of propositional calculus and predicate-calculus, using both the axiomatic method and the method of natural deduction. It provides a self-contained introduction but for those who intend to study the subject further it contains many suggestions and a sound basis for more advanced study.

The first European Congress of Mathematics was held in Paris from July 6 to July 10, 1992, at the Sorbonne and Pantheon-Sorbonne universities. It was hoped that the Congress would constitute a symbol of the development of the community of European nations. More than 1,300 persons attended the Congress. The purpose of the Congress was twofold. On the one hand, there was a scientific facet which consisted of forty-nine invited mathematical lectures that were intended to establish the state of the art in the various branches of pure and applied mathematics. This scientific facet also included poster sessions where participants had the opportunity of presenting their work. Furthermore, twenty four specialized meetings were held before and after the Congress. The second facet of the Congress was more original. It consisted of sixteen round tables whose aim was to review the prospects for the interactions of mathe matics, not only with other sciences, but also with society and in particular with education, European policy and industry. In connection with this second goal, the Congress also succeeded in bringing mathematics to a broader public. In addition to the round tables specifically devoted to this question, there was a mini-festival of mathematical films and two mathematical exhibits. Moreover, a Junior Mathematical Congress was organized, in parallel with the Congress, which brought together two hundred high school students."

There are things we routinely say that may strike us as literally false but that we are nonetheless reluctant to give up. This might be something mundane, like the way we talk about the sun setting in the west (it is the earth that moves), or it could be something much deeper, like engaging in talk that is ostensibly about numbers despite believing that numbers do not literally exist. Rather than regard such behaviour as self-defeating, a "fictionalist" is someone who thinks that this kind of discourse is entirely appropriate, even helpful, so long as we treat what is said as a useful fiction, rather than as the sober truth. "Fictionalism" can be broadly understood as a view that uses a notion of pretense or fiction in order to resolve certain puzzles or problems that otherwise do not necessarily have anything to do with literature or fictional creations. Within contemporary analytic philosophy, fictionalism has been on the scene for well over a decade and has matured during that time, growing in popularity. There are now myriad competing views about fictionalism and consequently the discussion has branched out into many more subdisciplines of philosophy. Yet there is widespread disagreement on what philosophical fictionalism actually amounts to and about how precisely it ought to be pursued. This volume aims to guide these discussions, collecting some of the most up-to-date work on fictionalism and tracing the view's development over the past decade. After a detailed discussion in the book's introductory chapter of how philosophers should think of fictionalism and its connection to metaontology more generally, the remaining chapters provide readers with arguments for and against this view from leading scholars in the fields of epistemology, ethics, metaphysics, philosophy of science, philosophy of language, and others.

This Fourth Edition updates the "Solutions Manual for Econometrics" to match the Sixth Edition of the Econometrics textbook. It adds problems and solutions using latest software versions of Stata and EViews. Special features include empirical examples replicated using EViews, Stata as well as SAS. The book offers rigorous proofs and treatment of difficult econometrics concepts in a simple and clear way, and provides the reader with both applied and theoretical econometrics problems along with their solutions. These should prove useful to students and instructors using this book.

The Third Kurt G-del Symposium, KGC'93, held in Brno, Czech Republic, August1993, is the third in a series of biennial symposia on logic, theoretical computer science, and philosophy of mathematics. The aim of this meeting wasto bring together researchers working in the fields of computational logic and proof theory. While proof theory traditionally is a discipline of mathematical logic, the central activity in computational logic can be foundin computer science. In both disciplines methods were invented which arecrucial to one another. This volume contains the proceedings of the symposium. It contains contributions by 36 authors from 10 different countries. In addition to 10 invited papers there are 26 contributed papers selected from over 50 submissions.

This original and exciting study offers a completely new perspective on the philosophy of mathematics. Most philosophers of mathematics try to show either that the sort of knowledge mathematicians have is similar to the sort of knowledge specialists in the empirical sciences have or that the kind of knowledge mathematicians have, although apparently about objects such as numbers, sets, and so on, isn't really about those sorts of things at all. Jody Azzouni argues that mathematical knowledge is a special kind of knowledge that must be gathered in its own unique way. He analyzes the linguistic pitfalls and misperceptions philosophers in this field are often prone to, and explores the misapplications of epistemic principles from the empirical sciences to the exact sciences. What emerges is a picture of mathematics sensitive both to mathematical practice and to the ontological and epistemological issues that concern philosophers. The book will be of special interest to philosophers of science, mathematics, logic, and language. It should also interest mathematicians themselves.

This is a volume of essays and reviews that delightfully explores mathematics in all its moods - from the light and the witty, and humorous to serious, rational, and cerebral. These beautifully written articles from three great modern mathematicians will provide a source for supplemental reading for almost any math class. Topics include: logic, combinatorics, statistics, economics, artificial intelligence, computer science, and broad applications of mathematics. Readers will also find coverage of history and philosophy, including discussion of the work of Ulam, Kant, and Heidegger, among others.

This book is a sympathetic reconstruction of Henri Poincar's anti-realist philosophy of mathematics. Although Poincar is recognized as the greatest mathematician of the late 19th century, his contribution to the philosophy of mathematics is not highly regarded. Many regard his remarks as idiosyncratic, and based upon a misunderstanding of logic and logicism. This book argues that Poincar's critiques are not based on misunderstanding; rather, they are grounded in a coherent and attractive foundation of neo-Kantian constructivism.

"Creative mathematicians seldom write for outsiders, but when they do, they usually do it well. Jerry King is no exception. His informal, nontechnical book, as its title implies, is organized around what Bertrand Russell called the 'supreme beauty' of mathematics--a beauty 'capable of a stern perfection such as only the greatest art can show.'"
NATURE
In this clear, concise, and superbly written volume, mathematics professor and poet Jerry P. King reveals the beauty that is at the heart of mathematics--and he makes that beauty accessible to all readers. Darting wittily from Euclid to Yeats, from Poincare to Rembrandt, from axioms to symphonies, THE ART OF MATHEMATICS explores the difference between real, rational, and complex numbers; analyzes the intellectual underpinnings of pure and applied mathematics; and reveals the fundamental connection between aesthetics and mathematics. King also sheds light on how mathematicians pursue their research and how our educational system perpetuates the damaging divisions between the "two cultures."

Die preisgekronte Biographie des norwegische Schriftstellers Atle Naess fuhrt den Leser auf eine fesselnde Reise durch die Hohen und Tiefen des Lebens einer der schillerndsten Personlichkeiten der europaischen Wissenschaftsgeschichte - Galileo Galilei. Mit feinsinniger Empathie entwickelt Naess das Portrait eines Mannes, der sich selbst durch die Zwange der romischen Inquisition nicht von seinen wegweisenden Forschungen abbringen liess.

Aus den Rezensionen der norwegischen Ausgabe:

"Mit umfassender Kenntnis und sicherem Erzahlstil hebt Naess die epochemachenden Arbeiten hervor, die die Grundlage der modernen experimentellen Naturwissenschaften bilden. Er packt all die vielen Stationen Galileis] Lebens in ein sehr lesenswertes Buch, das in vielerlei Hinsicht hervorsticht."

Per Anders Madsen, Aftenposten Morgen

"Diese Biographie stellt eine faszinierende kulturhistorische Studie dar und ist daher nicht nur fur Leser mit Interesse an Naturwissenschaft und Wissenschaftsgeschichte geeignet. Sie kann auch hervorragend als Roman gelesen werden."

Atle Abelsen, Teknisk Ukeblad

"

Biographie des ungarischen Mathematikers Janos Bolyai (1802-1860), der etwa gleichzeitig mit dem russischen Mathematiker Nikolai Lobatschewski und unabhangig von ihm die nichteuklidische Revolution eingeleitet hat. Diese erbrachte den Nachweis, dass die euklidische Geometrie keine Denknotwendigkeit ist, wie Kant irrtumlicherweise annahm. Das Verstandnis fur die kuhnen Gedankengange verbreitete sich allerdings erst in der zweiten Halfte des 19. Jahrhunderts durch die Arbeiten von Riemann, Beltrami, Klein und Poincare. Die nichteuklidische Revolution war eine der Grundlagen fur die Entwicklung der Physik im 20. Jahrhundert und fur Einsteins Erkenntnis, dass der uns umgebende reale Raum gekrummt ist. Tibor Weszely schildert das wechselvolle Leben des Offiziers der K.u.K.-Armee, der krank und vereinsamt starb. Bolyai hat sich auch intensiv mit den komplexen Zahlen und mit Zahlentheorie befasst, ebenso auch mit philosophischen und sozialen Fragen ( Allheillehre ) sowie mit Logik und Grammatik.

This volume contains several invited papers as well as a selection of the other contributions. The conference was the first meeting of the Soviet logicians interested in com- puter science with their Western counterparts. The papers report new results and techniques in applications of deductive systems, deductive program synthesis and analysis, computer experiments in logic related fields, theorem proving and logic programming. It provides access to intensive work on computer logic both in the USSR and in Western countries.

An innovative, dramatic graphic novel about the treacherous pursuit of the foundations of mathematics.

This exceptional graphic novel recounts the spiritual odyssey of philosopher Bertrand Russell. In his agonized search for absolute truth, Russell crosses paths with legendary thinkers like Gottlob Frege, David Hilbert, and Kurt Godel, and finds a passionate student in the great Ludwig Wittgenstein. But his most ambitious goal--to establish unshakable logical foundations of mathematics--continues to loom before him. Through love and hate, peace and war, Russell persists in the dogged mission that threatens to claim both his career and his personal happiness, finally driving him to the brink of insanity.

This story is at the same time a historical novel and an accessible explication of some of the biggest ideas of mathematics and modern philosophy. With rich characterizations and expressive, atmospheric artwork, the book spins the pursuit of these ideas into a highly satisfying tale. Probing and ingeniously layered, the book throws light on Russell's inner struggles while setting them in the context of the timeless questions he spent his life trying to answer. At its heart, "Logicomix "is a story about the conflict between an ideal rationality and the unchanging, flawed fabric of reality.Apostolos Doxiadis studied mathematics at Columbia University. His international bestseller "Uncle Petros and Goldbach's Conjecture" spearheaded the impressive entrance of mathematics into the world of storytelling. Apart from his work in fiction, Apostolos has also worked in film and theater and is an internationally recognized expert on the relationship of mathematics to narrative. Christos H. Papadimitriou is C . Lester Hogan professor of computer science at the University of California, Berkeley. He was won numerous international awards for his pioneering work in computational complexity and algorithmic game theory. Christos is the author of the novel "Turing: A Novel about Computation." Alecos Papadatos worked for over twenty years in film animation in France and Greece. In 1997, he became a cartoonist for the major Athens daily "To Vima." He lives in Athens with his wife, Annie Di Donna, and their two children. Annie Di Donna studied graphic arts and painting in France and has worked as animator on many productions, among them "Babar" and "Tintin." Since 1991, she has been running an animation studio with her husband, Alecos Papadatos. This innovative graphic novel is based on the early life of the brilliant philosopher Bertrand Russell. Russell and his impassioned pursuit of truth. Haunted by family secrets and unable to quell his youthful curiosity, Russell became obsessed with a Promethean goal: to establish the logical foundation of all mathematics. In his agonized search for absolute truth, Russell crosses paths with legendary thinkers like Gottlob Frege, David Hilbert, and Kurt Godel, and finds a passionate student in the great Ludwig Wittgenstein. But the object of his defining quest continues to loom before him. Through love and hate, peace and war, Russell persists in the dogged mission that threatens to claim both his career and his personal happiness, finally driving him to the brink of insanity. "Logicomix" is at the same time a historical novel and an accessible explication to some of the biggest ideas of mathematics and modern philosophy. With rich characterizations and expressive, atmospheric artwork, the book spins the pursuit of these ideas into a captivating tale. Probing and ingeniously layered, the book throws light on Russell's inner struggles while setting them in the context of the timeless questions he spent his life trying to answer. At its heart, "Logicomix" is a story about the conflict between an ideal rationality and the unchanging, flawed fabric of reality. "At the heart of Logicomix stands Sir Bertrand Russell, a man determined to find a way of arriving at absolutely right answers. It's a tale within a tale, as the two authors and two graphic artists ardently pursue their own search for truth and appear as characters in the book. As one of them assures us, this won't be 'your typical, usual comic book.' Their quest takes shape and revolves around a lecture given by Russell at an unnamed American university in 1939, a lecture that is really, as he himself tells us, the story of his life and of his pursuit of real logical truth. With Proustian ambition and exhilarating artwork, "Logicomix"'s search for truth encounters head-on the horrors of the Second World War and the agonizing question of whether war can ever be the right choice. Russell himself had to confront that question personally: he endured six months in jail for his pacifism. Russell was determined to find the perfect logical method for solving all problems and attempted to remold human nature in his experimental school at Beacon Hill. Despite repeated failures, Russell never stopped being 'a sad little boy desperately seeking ways out of the deadly vortex of uncertainty.' The book is a visual banquet chronicling Russell's lifelong pursuit of 'certainty in total rationality.' As Logic and Mathematics, the last bastions of certainty, fail him, and as Reason proves not absolute, Russell is forced to face the fact that there is no Royal Road to Truth. Authors Dosiadis and Papadimitriou perfectly echo Russell's passion, with a sincere, easily grasped text amplified with breathtaking visual richness, making this the most satisfying graphic novel of 2009, a titanic artistic achievement of more than 300 pages, all of it pure reading joy."--Nick DiMartino, "Shelf Awareness" "This is an extraordinary graphic novel, wildly ambitious in daring to put into words and drawings the life and thought of one of the great philosophers of the last century, Bertrand Russell. The book is a rare intellectual and artistic achievement, which will, I am sure, lead its readers to explore realms of knowledge they thought were forbidden to them."--Howard Zinn "This magnificent book is about ideas, passions, madness, and the fierce struggle between well-defined principle and the larger good. It follows the great mathematicians--Russell, Whitehead, Frege Cantor, Hilbert--as they agonized to make the foundations of mathematics exact, consistent, and complete. And we see the band of artists and researchers--and the all-seeking dog Manga--creating, and participating in, this glorious narrative."--Barry Mazur, Gerhard Gade University Professor at Harvard University, and author of "Imagining Numbers (Particularly the Square Root of Minus Fifteen)" "The lives of ideas (and those who think them) can be as dramatic and unpredicteable as any superhero fantasy. "Logicomix" is witty, engaging, stylish, visually stunning, and full of surprising sound effects, a masterpiece in a genre for which there is as yet no name."--Michael Harris, professor of mathematics at Universite Paris 7 and member of the Institut Universitaire de France

From cells in our bodies to measuring the universe, big numbers are everywhere We all know that numbers go on forever, that you could spend your life counting and never reach the end of the line, so there can't be such a thing as a 'biggest number'. Or can there? To find out, David Darling and Agnijo Banerjee embark on an epic quest, revealing the answers to questions like: are there more grains of sand on Earth or stars in the universe? Is there enough paper on Earth to write out the digits of a googolplex? And what is a googolplex? Then things get serious. Enter the strange realm between the finite and the infinite, and float through a universe where the rules we cling to no longer apply. Encounter the highest number computable and infinite kinds of infinity. At every turn, a cast of wild and wonderful characters threatens the status quo with their ideas, and each time the numbers get larger.

First published in 1982, this reissue contains a critical exposition of the views of Frege, Dedekind and Peano on the foundations of arithmetic. The last quarter of the 19th century witnessed a remarkable growth of interest in the foundations of arithmetic. This work analyses both the reasons for this growth of interest within both mathematics and philosophy and the ways in which this study of the foundations of arithmetic led to new insights in philosophy and striking advances in logic. This historical-critical study provides an excellent introduction to the problems of the philosophy of mathematics - problems which have wide implications for philosophy as a whole. This reissue will appeal to students of both mathematics and philosophy who wish to improve their knowledge of logic.