2013 Reprint of 1931 edition. Full facsimile of the original edition, not reproduced with Optical Recognition Software. Frank Plumpton Ramsey (1903-1930) was a British mathematician who also made significant and precocious contributions in philosophy and economics before his death at the age of 26. He was a close friend of Ludwig Wittgenstein, and was instrumental in translating Wittgenstein's "Tractatus Logico-Philosophicus" into English, and in persuading Wittgenstein to return to philosophy and to Cambridge. This volume collects Ramsey's most important papers. Contents: The foundations of mathematics.--Mathematical logic.--On a problem of formal logic.--Universals.--Note on the preceding paper.--Facts and propositions.--Truth and probability.--Further considerations.--Last papers.

Aus dem Vorwort von E. Zermelo: "In der Geschichte der Wissenschaften ist es gewiss ein seltener Fall, wenn eine ganze wissenschaftliche Disziplin von grundlegender Bedeutung der schopferischen Tat eines einzelnen zu verdanken ist. Dieser Fall ist verwirklicht in der Schopfung Georg Cantors, der Mengenlehre, einer neuen mathematischen Disziplin, die wahrend eines Zeitraumes von etwa 25 Jahren in einer Reihe von Abhandlungen ein und desselben Forschers in ihren Grundzugen entwickelt, seitdem zum bleibenden Besitze der Wissenschaft geworden ist, so dass alle spateren Forschungen auf diesem Gebiete nur noch als erganzende Ausfuhrungen seiner grundlegenden Gedanken aufzufassen sind. Aber auch abgesehen von dieser ihrer historischen Bedeutung sind die Cantorschen Originalabhandlungen noch fur den heutigen Leser von unmittelbarem Interesse, in ihrer klassischen Einfachheit und Prazision ebenso zur ersten Einfuhrung geeignet und darin noch von keinem neueren Lehrbuch ubertroffen, wie auch fur den Fortgeschrittenen durch die Fulle der zugrunde liegenden Gedanken eine genussreich anregende Lekture.""

In the third volume in the Rutgers Lectures in Philosophy series, distinguished philosopher Robert Stalnaker here offers a defense of an ontology of propositions, and of some logical resources for representing them. He offers an austere formulation of a theory of propositions in a first-order extensional logic, but then uses the commitments of this theory to justify an enrichment to modal logic as an appropriate framework for regimented languages that are constructed to represent any of our scientific and philosophical commitments. His book adopts a self-consciously neo-Quinean methodology, and argues that the theory that is developed helps to motivate and clarify Quine's naturalistic metaphysical picture.

Smart Moves: Developing Mathematical Reasoning with Games and Puzzles is designed to improve your sequential reasoning, explore some mathematics, and have fun along the way. The games and puzzles were created to encourage perseverance and logical thinking. The Mathematical Connections highlight key math concepts. The Game of Racetrack is the perfect introduction to vectors, Tour Puzzles lead to graph theory and Euler paths, and the mathematics behind Magic Squares is revealed.Smart Moves is a very effective way to support mathematical learning and reduce the anxiety that often accompanies the subject.Each chapter is designed to strengthen sequential reasoning, which is necessary for everyday living and problem solving. Whether you are nine or ninety, in the classroom or at home, I invite you to make a smart move and discover how much fun math can be!

From the author of "Zero," comes this "admirable salvo against quantitative bamboozlement by the media and the government" ("The Boston Globe")

In "Zero," Charles Seife presented readers with a thrilling account of the strangest number known to humankind. Now he shows readers how the power of skewed metrics-or "proofiness"- is being used to alter perception in both amusing and dangerous ways. Proofiness is behind such bizarre stories as a mathematical formula for the perfect butt and sprinters who can run faster than the speed of sound. But proofiness also has a dark side: bogus mathematical formulas used to undermine our democracy-subverting our justice system, fixing elections, and swaying public opinion with lies. By doing the real math, Seife elegantly and good-humoredly scrutinizes our growing obsession with metrics while exposing those who misuse them.

This is the first English collection of the work of Albert Lautman, a major figure in philosophy of mathematics and a key influence on Badiou and Deleuze. Albert Lautman (1908-1944) was a French philosopher of mathematics whose work played a crucial role in the history of contemporary French philosophy. His ideas have had an enormous influence on key contemporary thinkers including Gilles Deleuze and Alain Badiou, for whom he is a major touchstone in the development of their own engagements with mathematics. "Mathematics, Ideas and the Physical Real" presents the first English translation of Lautman's published works between 1933 and his death in 1944. Rather than being preoccupied with the relation of mathematics to logic or with the problems of foundation, which have dominated philosophical reflection on mathematics, Lautman undertakes to develop an understanding of the broader structure of mathematics and its evolution. The two powerful ideas that are constants throughout his work, and which have dominated subsequent developments in mathematics, are the concept of mathematical structure and the idea of the essential unity underlying the apparent multiplicity of mathematical disciplines. This collection of his major writings offers readers a much-needed insight into his influence on the development of mathematics and philosophy.

Cognitive mathematics provides insights into how mathematics works inside the brain and how it is interconnected with other faculties through so-called blending and other associative processes. This handbook is the first large collection of various aspects of cognitive mathematics to be amassed into a single title, covering decades of connection between mathematics and other figurative processes as they manifest themselves in language, art, and even algorithms. It will be of use to anyone working in math cognition and education, with each section of the handbook edited by an international leader in that field.

Die Mathematik im mittelalterlichen Islam hatte gro en Einfluss auf die allgemeine Entwicklung des Faches. Der Autor beschreibt diese Periode der Geschichte der Mathematik und bezieht sich dabei auf die arabischsprachigen Quellen. Zu den behandelten Themen geh ren Dezimalrechnen, Geometrie, ebene und sph rische Trigonometrie, Algebra sowie die Approximation von Wurzeln von Gleichungen. Das Buch wendet sich an Mathematikhistoriker und -studenten, aber auch an alle Interessierten mit Mathematikkenntnissen der weiterf hrenden Schule.

Archytas of Tarentum is one of the three most important philosophers in the Pythagorean tradition, a prominent mathematician, who gave the first solution to the famous problem of doubling the cube, an important music theorist, and the leader of a powerful Greek city-state. He is famous for sending a trireme to rescue Plato from the clutches of the tyrant of Syracuse, Dionysius II, in 361 BC. This 2005 study was the first extensive enquiry into Archytas' work in any language. It contains original texts, English translations and a commentary for all the fragments of his writings and for all testimonia concerning his life and work. In addition there are introductory essays on Archytas' life and writings, his philosophy, and the question of authenticity. Carl A. Huffman presents an interpretation of Archytas' significance both for the Pythagorean tradition and also for fourth-century Greek thought, including the philosophies of Plato and Aristotle.

A dynamic exploration of infinity In Infinity and the Mind, Rudy Rucker leads an excursion to that stretch of the universe he calls the "Mindscape," where he explores infinity in all its forms: potential and actual, mathematical and physical, theological and mundane. Using cartoons, puzzles, and quotations to enliven his text, Rucker acquaints us with staggeringly advanced levels of infinity, delves into the depths beneath daily awareness, and explains Kurt Goedel's belief in the possibility of robot consciousness. In the realm of infinity, mathematics, science, and logic merge with the fantastic. By closely examining the paradoxes that arise, we gain profound insights into the human mind, its powers, and its limitations. This Princeton Science Library edition includes a new preface by the author.

Philosophy of mathematics is moving in a new direction: away from a foundationalism in terms of formal logic and traditional ontology, and towards a broader range of approaches that are united by a focus on mathematical practice. The scientific research network PhiMSAMP (Philosophy of Mathematics: Sociological Aspects and Mathematical Practice) consisted of researchers from a variety of backgrounds and fields, brought together by their common interest in the shift of philosophy of mathematics towards mathematical practice. Hosted by the Rheinische Friedrich- Wilhelms-Universitat Bonn and funded by the Deutsche Forschungsgemeinschaft (DFG) from 2006-2010, the network organized and contributed to a number of workshops and conferences on the topic of mathematical practice. The refereed contributions in this volume represent the research results of the network and consists of contributions of the network members as well as selected paper versions of presentations at the network's mid-term conference, "Is mathematics special?" (PhiMSAMP-3) held in Vienna 2008.

"The Virtue of Heresy - Confessions of a Dissident Astronomer" is a narrative account of the 30-year struggle by the author to put the "physical" back into "physics". With sporadic assistance from a fictional alter-ego character named Haquar, the author traces the history of astronomy and physics to the point of their confluence with meta-mathematics. From there on, the fundamental hypotheses of cosmology, and indeed of physical science generally, became increasingly detached from observed reality and more like psychedelic mind games than works of empirical science. Hilton Ratcliffe guilelessly confronts these issues head-on, spicing the tale with humour and fascinating anecdotes of his association with some of the finest scientific minds of our era. His passion for true science and child-like awe at the wonders of the Universe are infused in every line. A classic.

"The Ideal and the Real should prove valuable to two particular sets of readers: (i) those with an interest in Kant and little or no background in the philosophy of mathematics, or (ii) those with an interest in the philosophy of mathematics and little or no background in Kant...The book contains much that is suggestive which should promote further discussion...(and) offers more than a simple examination of Kant's philosophy of mathematics. Of particular interest is his suggestion that Newton's thought experiments have been changed and idealized by commentators." R.R.Wojtowicz, (Canadian Philosophical Review) This book argues that Kant's theory of space, time and mathematics has contemporary significance principally because of its roots in the ideas of construction and schematism. These concepts are analysed in the light of the central Kantian distinction between the ideal and the empirically real. A reassessment of Newton's arguments for absolute space is followed by an examination of Leibniz's theory of space, time and continuity. The metaphysical frameworks of these theories are presented as essential precursors of Kant's critical programme. The ideas of construction and schematism illuminate all aspects of Kant's philosophy of mathematics, and have important implications for understanding both the task and the achievement of the critical philosophy. Through an analysis of these concepts, the role of intuition, and in particular the argument from incongruent counterparts, is given added significance. "While he intends The Ideal and the Real as a limited commentary on space, time, and mathematical construction, it also brings the reader into contact with a whole series of problems treated by Kant in the First Critique and the Prolegomena...While the discussion of Newton displays a sensitivity to the complexity of Newton's position, Winterbourne's own exposition develops clearly...(and) advances with such sensitivity both to primary and secondary sources that one could hardly find a better summary of the issues surrounding the Leibniz-Clarke controversy...The discussion of incongruent counterparts provides the most interesting part of the monograph...Winterbourne avoids technical jargon and obscure explanation in an admirable way...(and) gives us one of the best treatments of the Schematism available. Kantian scholars would do well to take note of Winterbourne's conclusions." John Treloar, (The Modern Schoolman) "One of the main strengths of Winterbourne's book is his treatment of Kant's philosophy of mathematics...and (it) offers an interesting overview of the ideas of Leibniz and Newton..." Grant West (Isis)

Archimedes was the greatest scientist of antiquity and one of the greatest of all time. This book is Volume I of the first authoritative translation of his works into English. It is also the first publication of a major ancient Greek mathematician to include a critical edition of the diagrams and the first translation into English of Eutocius' ancient commentary on Archimedes. Furthermore, it is the first work to offer recent evidence based on the Archimedes Palimpsest, the major source for Archimedes, lost between 1915 and 1998. A commentary on the translated text studies the cognitive practice assumed in writing and reading the work, and it is Reviel Netz's aim to recover the original function of the text as an act of communication. Particular attention is paid to the aesthetic dimension of Archimedes' writings. Taken as a whole, the commentary offers a groundbreaking approach to the study of mathematical texts.

The Discourse on the Method is a philosophical and mathematical treatise published by Ren Descartes in 1637. Its full name is Discourse on the Method of Rightly Conducting the Reason, and Searching for Truth in the Sciences. The Discourse on Method is best known as the source of the famous quotation "Je pense, donc je suis," "I think, therefore I am."This is one of the most influential works in the history of modern science

Reprint. Paperback. 387 pp. Diophantus of Alexandria, sometimes called "the father of algebra," was an Alexandrian mathematician and the author of a series of books called Arithmetica. These texts deal with solving algebraic equations, many of which are now lost. In studying Arithmetica, Pierre de Fermat concluded that a certain equation considered by Diophantus had no solutions, and noted without elaboration that he had found "a truly marvelous proof of this proposition," now referred to as Fermat's Last Theorem. This led to tremendous advances in number theory, and the study of diophantine equations ("diophantine geometry") and of diophantine approximations remain important areas of mathematical research. Diophantus was the first Greek mathematician who recognized fractions as numbers; thus he allowed positive rational numbers for the coefficients and solutions. In modern use, diophantine equations are usually algebraic equations with integer coefficients, for which integer solutions are sought. Diophantus also made advances in mathematical notation. Heath's work is one of the standard books in the field.

"Philosophy of Mathematics: An Introduction" provides a critical analysis of the major philosophical issues and viewpoints in the concepts and methods of mathematics - from antiquity to the modern era.
Offers beginning readers a critical appraisal of philosophical viewpoints throughout historyGives a separate chapter to predicativism, which is often (but wrongly) treated as if it were a part of logicismProvides readers with a non-partisan discussion until the final chapter, which gives the author's personal opinion on where the truth liesDesigned to be accessible to both undergraduates and graduate students, and at the same time to be of interest to professionals

A survey of the major figures and mathematical movements of the 19th century, this is a thorough examination of every significant foundation stone of today's modern mathematics. Providing clear and concise articles on the fundamental definition of numbers through to quantics and infinite series, as well as exposition on the relationships between theorems, this volume, which was first published in 1896, cements itself as an essential reference work, a solid jumping-off point for all students of mathematics, and a fascinating glimpse at the once-cutting edge that now is taken for granted in an ever-changing scientific field. New York lawyer and mathematician DAVID EUGENE SMITH (1860-1944) authored a number of books while a professor of mathematics at Columbia University, including The Teaching of Elementary Mathematics (1900), A History of Japanese Mathematics (1914), and The Sumario Compendioso of Brother Juan Diez (1921).

The life of Vito Volterra, one of the finest scientists and mathematicians Italy ever produced, spans the period from the unification of the Italian peninsula in 1860 to the onset of the Second World War--an era of unparalleled progress and unprecedented turmoil in the history of Europe. Born into an Italian Jewish family in the year of the liberation of Italy's Jewish ghettos, Volterra was barely in his twenties when he made his name as a mathematician and took his place as a leading light in Italy's modern scientific renaissance. By his early forties, he was a world-renowned mathematician, a sought-after figure in European intellectual and social circles, the undisputed head of Italy's mathematics and physics school--and still living with his mother, who decided the time was ripe to arrange his marriage. When Italy entered World War I in 1915, the fifty-five-year-old Volterra served with distinction and verve as a lieutenant and did not put on civilian clothes again until the Armistice of 1918. This book, based in part on unpublished personal letters and interviews, traces the extraordinary life and times of one of Europe's foremost scientists and mathematicians, from his teenage struggles to avoid the stifling life of a ""respectable"" bank clerk in Florence, to his seminal mathematical work--which today influences fields as diverse as economics, physics, and ecology--and from his spirited support of Italy's scientific and democratic institutions during his years as an Italian Senator, to his steadfast defiance of the Fascists and Mussolini. In recounting the life of this outstanding scientist, European Jewish intellectual, committed Italian patriot, and devoted if frequently distracted family man, The Volterra Chronicles depicts a remarkable individual in a prodigious age and takes the reader on a vivid and splendidly detailed historical journey.

This volume is a collection of papers on philosophy of mathematics which deal with a series of questions quite different from those which occupied the minds of the proponents of the three classic schools: logicism, formalism, and intuitionism. The questions of the volume are not to do with justification in the traditional sense, but with a variety of other topics. Some are concerned with discovery and the growth of mathematics. How does the semantics of mathematics change as the subject develops? What heuristics are involved in mathematical discovery, and do such heuristics constitute a logic of mathematical discovery? What new problems have been introduced by the development of mathematics since the 1930s? Other questions are concerned with the applications of mathematics both to physics and to the new field of computer science. Then there is the new question of whether the axiomatic method is really so essential to mathematics as is often supposed, and the question, which goes back to Wittgenstein, of the sense in which mathematical proofs are compelling. Taking these questions together they give part of an emerging agenda which is likely to carry philosophy of mathematics forward into the twenty first century.

Gregory Chaitin, one of the world's foremost mathematicians, leads us on a spellbinding journey, illuminating the process by which he arrived at his groundbreaking theory.
Chaitin's revolutionary discovery, the Omega number, is an exquisitely complex representation of unknowability in mathematics. His investigations shed light on what we can ultimately know about the universe and the very nature of life. In an infectious and enthusiastic narrative, Chaitin delineates the specific intellectual and intuitive steps he took toward the discovery. He takes us to the very frontiers of scientific thinking, and helps us to appreciate the art--and the sheer beauty--in the science of math.

A book about both material's and space's non-local geometric relation to fiber groups of principle fiber bundles, it subsequently reveals the fundamental structure of life's intent and our perception.

In his most ambitious book yet, Clifford Pickover bridges the gulf between logic, spirit, science, and religion. While exploring the concept of omniscience, Pickover explains the kinds of relationships limited beings can have with an all-knowing God. Pickover's thought exercises, controversial experiments, and practical analogies help us transcend our ordinary lives while challenging us to better understand our place in the cosmos and our dreams of a supernatural God. Through an inventive blend of science, history, philosophy, science fiction, and mind-stretching brainteasers, Pickover unfolds the paradoxes of God like no other writer. He provides glimpses into the infinite, allowing us to think big, and to have daring, limitless dreams.

In this brief treatise, Carus traces the roots of his belief in the philosophical basis for mathematics and analyzes that basis after a historical overview of Euclid and his successors. He then examines his base argument and proceeds to a study of different geometrical systems, all pulled together in his epilogue, which examines matter, mathematics, and, ultimately, the nature of God.

This important book by a major American philosopher brings together eleven essays treating problems in logic and the philosophy of mathematics. A common point of view, that mathematical thought is central to our thought in general, underlies the essays. In his introduction, Parsons articulates that point of view and relates it to past and recent discussions of the foundations of mathematics.Mathematics in Philosophy is divided into three parts. Ontology the question of the nature and extent of existence assumptions in mathematics is the subject of Part One and recurs elsewhere. Part Two consists of essays on two important historical figures, Kant and Frege, and one contemporary, W. V. Quine. Part Three contains essays on the three interrelated notions of set, class, and truth."