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How do scientific conjectures become laws? Why does proof mean different things in different sciences? Do numbers exist, or were they invented? Why do some laws turn out to be wrong? This book discusses the basis for scientists' claims to knowledge about the world. It looks at science historically, emphasizing not only the achievements of scientists from Galileo onwards, but also their mistakes. The book rejects the claim that all scientific knowledge is provisional, by citing examples from chemistry, biology, and geology. A major feature of the book is its defence of the view that mathematics was invented rather than discovered. While experience has shown that disentangling knowledge from opinion and aspiration is a hard task, this book provides a clear guide to the difficulties. Including many examples and quotations, and with a scope ranging from psychology and evolution to quantum theory and mathematics, this book aims to bring alive issues at the heart of all science.

In the last two decades, philosophers have become increasingly aware of the fact that, in spite of significant differences between the contemporary and romantic contexts, romanticism continues to “persist,” and the questions that the romantics raised remain relevant today. The Relevance of Romanticism: Essays on Early German Romantic Philosophy is the first collection of essays that directly considers the reasons why philosophers have recently become deeply interested in romantic thought. Through historical and systematic reconstructions, the volume offers greater understanding of romanticism as a philosophical movement and deeper insight into the role that romantic thought plays—or can play—in contemporary philosophical debates. Sixteen essays by both established and emerging scholars discussing key romantic themes and concerns highlight the diversity within both romantic thought and its contemporary reception. Part 1 consists of the first published encounter between Manfred Frank and Frederick Beiser, in which the two major scholars discuss their differing interpretations of philosophical romanticism. Part 2 draws significant connections between romantic conceptions of history, sociability, hermeneutics, and education and explores the ways in which these views can illuminate questions in contemporary social-political philosophy and theories of interpretation. Part 3 consists in some of the most innovative takes on romantic aesthetics, which seek to bring romantic thought into dialogue, with, for instance, contemporary analytic aesthetics and theories of cognition. Part 4 offers a rare rigorous engagement with romantic conceptions of science, and demonstrates ways in which the romantic view of nature, experimentation, and mathematics need not be relegated to historical curiosities.

For eight centuries mathematics has been researched and studied at Oxford, and the subject and its teaching have undergone profound changes during that time. This is the story of the intellectual and social life of this community, and of its interactions with the wider world. This highly readable and beautifully illustrated book reveals the richness and influence of Oxford’s mathematical tradition and the fascinating characters that helped to shape it. The story begins with the founding of the University of Oxford and the establishing of the medieval curriculum, in which mathematics had an important role. The Black Death, the advent of printing, the Civil War, and the Newtonian revolution all had a great influence on the development of mathematics at Oxford. So too did many well-known figures: Roger Bacon, Henry Savile, Robert Hooke, Christopher Wren, Edmond Halley, Florence Nightingale, Charles Dodgson (Lewis Carroll), and G. H. Hardy, to name but a few. Later chapters bring us to the 20th century, with some entertaining reminiscences by Sir Michael Atiyah of the thirty years he spent as an Oxford mathematician. In this second edition the story is brought right up to the opening of the new Mathematical Institute in 2013 with a foreword from Marcus du Sautoy and recent developments from Peter M. Neumann.

The history of mathematics is a well-studied and vibrant area of research, with books and scholarly articles published on various aspects of the subject. Yet, the history of combinatorics seems to have been largely overlooked. This book goes some way to redress this and serves two main purposes: it constitutes the first book-length survey of the history of combinatorics, and it assembles, for the first time in a single source, researches on the history of combinatorics that would otherwise be inaccessible to the general reader. Individual chapters have been contributed by sixteen experts. The book opens with an introduction to two thousand years of combinatorics. This is followed by seven chapters on early combinatorics, leading from Indian and Chinese writings on permutations to late-Renaissance publications on the arithmetical triangle. The next seven chapters trace the subsequent story, from Euler’s contributions to such wide-ranging topics as partitions, polyhedra, and latin squares to the 20th-century advances in combinatorial set theory, enumeration, and graph theory. The book concludes with some combinatorial reflections.