A state-of-the-art guide to the fast-developing area of the philosophy of science. An eminent international team of authors covers a wide range of topics at the intersection of philosophy and the sciences, including causation, realism, methodology, epistemology and the philosophical foundations of physics, biology and psychology. An expanded version of a special issue of the journal "BJPS", this collection should be valuable to advanced students as well as to scholars.

Essentials of Mathematical Thinking addresses the growing need to better comprehend mathematics today. Increasingly, our world is driven by mathematics in all aspects of life. The book is an excellent introduction to the world of mathematics for students not majoring in mathematical studies. The author has written this book in an enticing, rich manner that will engage students and introduce new paradigms of thought. Careful readers will develop critical thinking skills which will help them compete in today's world. The book explains: What goes behind a Google search algorithm How to calculate the odds in a lottery The value of Big Data How the nefarious Ponzi scheme operates Instructors will treasure the book for its ability to make the field of mathematics more accessible and alluring with relevant topics and helpful graphics. The author also encourages readers to see the beauty of mathematics and how it relates to their lives in meaningful ways.

Dieses zweibAndige Werk handelt von Mathematik und ihrer Geschichte. Die sorgfAltige Analyse dessen, was die Alten bewiesen - meist sehr viel mehr, als sie ahnten -, fA1/4hrt zu einem besseren VerstAndnis der Geschichte und zu einer guten Motivation und einem ebenfalls besseren VerstAndnis heutiger Mathematik.
Die Themen des ersten Bandes reichen von der Konstruktion der reellen Zahlen mittels dedekindscher Schnitte bis hin zum Fundamentalsatz der Algebra. Dazwischen werden die BA1/4cher V bis X der euklidischen Elemente abgehandelt, wobei insbesondere die eudoxische Proportionenlehre (Buch V) eine zentrale Rolle spielt. Sie bietet einen eleganten Zugang zu den Logarithmen, so dass auch Neper ausfA1/4hrlich zu Wort kommt. Weitere Themen sind die natA1/4rlichen Zahlen und das Induktionsprinzip; die Entdeckung der LAsungsformeln der Gleichungen dritten und vierten Grades; Polynomringe in beliebig vielen Unbestimmten; symmetrische Polynome und der Satz von Waring.

Die innerdeutsche Grenze verlief nicht nur zwischen zwei Staaten, sondern spiegelte sich sogar in den Grundlagenwissenschaften wie der Mathematik wider. Aus personlicher Sicht zeigt der Autor den subjektiven Umgang mit Erzeugung, Bewertung und Propagierung wissenschaftlicher Resultate in den zwei unterschiedlichen Gesellschaftssystemen. Auf unterhaltsame Art werden Innensichten aus Forschungsinstitutionen, der Wissenschaftsforderung und die verschiedenen Einstellungen zur Zweckbestimmung reiner und angewandter Forschung dargelegt."

McCarthy develops a theory of Radical Interpertation - The project of characterizing from scratch the language and attituteds of an agent or population - and applies the theory to the problems of indeterminacy of interperation first descrided in the writings of Quine. The major theme in McCarthy's study is that a relatively modest set of interpertive principles, properly applied, can serve to resolve the major indeterminacies of interperation. Its most substantive contribution is in proposing a solution to problems of indeterminacy that remain unsloved in the literature.

Contemporary thinking on philosophy and the social sciences has primarily focused on the centrality of language in understanding societies and individuals; important developments which have been under-utilised by researchers in mathematics education. In this revised and extended edition this book reaches out to contemporary work in these broader fields, adding new material on how progression in mathematical learning might be variously understood. A new concluding chapter considers how teachers experience the new demands they face.

This challenging book argues that a new way of speaking of mathematics and describing it emerged at the end of the sixteenth century. Leading mathematicians like Hariot, Stevin, Galileo, and Cavalieri began referring to their field in terms drawn from the exploration accounts of Columbus and Magellan. As enterprising explorers in search of treasures of knowledge, these mathematicians described themselves as sailing the treacherous seas of mathematics, facing shipwreck on the shoals of paradox, and seeking shelter and refuge on the shores of geometrical demonstrations. Mathematics, formerly praised for its logic, clarity, and inescapable truths, was for them a hazardous voyage in inhospitable geometrical lands.
Significantly, many of the same practitioners who promoted the vision of mathematics as heroic exploration also played central roles in developing the most important mathematical innovation of the period--the infinitesimal methods. This was no coincidence: the heroic tales of exploration and discovery helped shape a new form of mathematical practice, complete with new questions, new acceptable answers, and new standards of evidence. It was this new vision of mathematics as a grand adventure that allowed for the development of the new techniques that led to the Newtonian calculus.
In demonstrating this, the book moves from real voyages to imaginary ones, from the coasts of the Canadian Arctic to the tropical forests of Guyana, and from the inner structure of matter to the intricacies of the mathematical continuum. Throughout, a common rhetoric and imagery of exploration and discovery run like a thread through these diverse elements and bind them together.

Very Short Introductions: Brilliant, Sharp, Inspiring Kurt Goedel first published his celebrated theorem, showing that no axiomatization can determine the whole truth and nothing but the truth concerning arithmetic, nearly a century ago. The theorem challenged prevalent presuppositions about the nature of mathematics and was consequently of considerable mathematical interest, while also raising various deep philosophical questions. Goedel's Theorem has since established itself as a landmark intellectual achievement, having a profound impact on today's mathematical ideas. Goedel and his theorem have attracted something of a cult following, though his theorem is often misunderstood. This Very Short Introduction places the theorem in its intellectual and historical context, and explains the key concepts as well as common misunderstandings of what it actually states. A. W. Moore provides a clear statement of the theorem, presenting two proofs, each of which has something distinctive to teach about its content. Moore also discusses the most important philosophical implications of the theorem. In particular, Moore addresses the famous question of whether the theorem shows the human mind to have mathematical powers beyond those of any possible computer ABOUT THE SERIES: The Very Short Introductions series from Oxford University Press contains hundreds of titles in almost every subject area. These pocket-sized books are the perfect way to get ahead in a new subject quickly. Our expert authors combine facts, analysis, perspective, new ideas, and enthusiasm to make interesting and challenging topics highly readable.

Jeffrey Barrett presents the most comprehensive study yet of a problem that has puzzled physicists and philosophers since the 1930s. Quantum mechanics is in one sense the most successful physical theory ever, accurately predicting the behaviour of the basic constituents of matter. But it has an apparent ambiguity or inconsistency at its heart; Barrett gives a careful, clear, and challenging evaluation of attempts to deal with this problem.

Hartry Field presents a selection of thirteen of his most important essays on a set of related topics at the foundations of philosophy; one essay is previously unpublished, and eight are accompanied by substantial new postscripts. Five of the essays are primarily about truth, meaning, and propositional attitudes, five are primarily about semantic indeterminacy and other kinds of 'factual defectiveness' in our discourse, and three are primarily about issues concerning objectivity, especially in mathematics and in epistemology. This influential work by a key figure in contemporary philosophy will reward the attention of any philosopher interested in language, epistemology, or mathematics.

In The Non-Local Universe, Nadeau and Kafatos offer a revolutionary look at the breathtaking implications of non-locality. They argue that since every particle in the universe has been "entangled" with other particles, physical reality on the most basic level is an undivided wholeness. In addition to demonstrating that physical processes are vastly interdependent and interactive, they also show that more complex systems in both physics and biology display emergent properties and/or behaviours that cannot be explained in terms of the sum of the parts. One of the most startling implications of nonlocality in human, terms, claim the authors, is that there is no longer any basis for believing in the stark division between mind and world that has preoccupied much of Western thought since the seventeenth century.

Robert Hanna presents a fresh view of the Kantian and analytic traditions that have dominated continental European and Anglo-American philosophy over the last two centuries, and of the relation between them. The rise of analytic philosophy decisively marked the end of the hundred-year dominance of Kant's philosophy in Europe. But Hanna shows that the analytic tradition also emerged from Kant's philosophy in the sense that its members were able to define and legitimate their ideas only by means of an intensive, extended engagement with, and a partial or complete rejection of, the Critical Philosophy. Hanna puts forward a new 'cognitive-semantic' interpretation of transcendental idealism, and a vigorous defence of Kant's theory of analytic and synthetic necessary truth. These will make Kant and the Foundations of Analytic Philosophy compelling reading not just for specialists in the history of philosophy, but for all who are interested in these fundamental philosophical issues.

Shapiro argues that both realist and anti-realist accounts of mathematics are problematic. To resolve this dilemma, he articulates a 'structuralist' approach, arguing that the subject matter of a mathematical theory is not a fixed domain of numbers that exist independent of each other, but rather is the natural structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle.

Stewart Shapiro presents a distinctive original view of the foundations of mathematics, arguing that second-order logic has a central role to play in laying these foundations. He gives an accessible account of second-order and higher-order logic, paying special attention to philosophical and historical issues. Foundations without Foundationalism is a key contribution both to philosophy of mathematics and to mathematical logic.

'In this excellent treatise Shapiro defends the use of second-order languages and logic as frameworks for mathematics. His coverage of the wide range of logical and philosophical . . . is thorough, clear, and persuasive.' Michael D. Resnik, History and Philosophy of Logic

Numbers and other mathematical objects are exceptional in having no locations in space or time or relations of cause and effect. This makes it difficult to account for the possibility of the knowledge of such objects, leading many philosophers to embrace nominalism, the doctrine that there are no abstract entities, and to embark on ambitious projects for interpreting mathematics so as to preserve the subject while eliminating its objects. A Subject With No Object cuts through a host of technicalities that have obscured previous discussions of these projects, and presents clear, concise accounts, with minimal prerequisites, of a dozen strategies for nominalistic interpretation of mathematics, thus equipping the reader to evaluate each and to compare different ones. The authors also offer critical discussion, rare in the literature, of the aims and claims of nominalistic interpretation, suggesting that it is significant in a very different way from that usually assumed.

Mathematics as a Science of Patterns is the definitive exposition of Michael Resnik's distinctive view of the nature of mathematics. He calls mathematics a science on the grounds that it has a factual subject-matter and that mathematical knowledge is on a par with other scientific knowledge; in calling it a science of patterns he expresses his commitment to a structuralist philosophy of mathematics. He links this to a defence of realism about the metaphysics of mathematics - the view that mathematics is about things that really exist.

'interesting, well written . . . [a] timely and important addition to contemporary philosophy of mathematics.' British Journal for the Philosophy of Science

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 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.

THE INTERNATIONAL BESTSELLER 'An entertaining tour that will change how you see the world' Sean Carroll, author of Something Deeply Hidden Is there a secret formula for improving your life? For making something a viral hit? For deciding how long to stick with your current job, Netflix series, or even relationship? This book is all about the equations that make our world go round. Ten of them, in fact. They are integral to everything from investment banking to betting companies and social media giants. And they can help you to increase your chance of success, guard against financial loss, live more healthily and see through scaremongering. They are known only by mathematicians - until now. With wit and clarity, mathematician David Sumpter shows that it isn't the technical details which make these formulas so successful. It is the way they allow mathematicians to view problems from a different angle - a way of seeing the world that anyone can learn. Empowering and illuminating, The Ten Equations that Rule the World shows how maths really can change your life.

The Philosophy of Mathematics Today gives a panorama of the best current work in this lively field, through twenty essays specially written for this collection by leading figures. The topics include indeterminacy, logical consequence, mathematical methodology, abstraction, and both Hilbert's and Frege's foundational programmes. The collection will be an important source for research in the philosophy of mathematics for years to come.

Contributors Paul Benacerraf, George Boolos, John P. Burgess, Charles S. Chihara, Michael Detlefsen, Michael Dummett, Hartry Field, Kit Fine, Bob Hale, Richard G. Heck, Jnr., Geoffrey Hellman, Penelope Maddy, Karl-Georg Niebergall, Charles D. Parsons, Michael D. Resnik, Matthias Schirn, Stewart Shapiro, Peter Simons, W.W. Tait, Crispin Wright.

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 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 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.