This is a reproduction of a book published before 1923. This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ++++ The below data was compiled from various identification fields in the bibliographic record of this title. This data is provided as an additional tool in helping to ensure edition identification: ++++ Principia Mathematica, Volume 2; Principia Mathematica; Bertrand Russell Alfred North Whitehead, Bertrand Russell University Press, 1912 Logic, Symbolic and mathematical; Mathematics

This is a reproduction of a book published before 1923. This book may have occasional imperfections such as missing or blurred pages, poor pictures, errant marks, etc. that were either part of the original artifact, or were introduced by the scanning process. We believe this work is culturally important, and despite the imperfections, have elected to bring it back into print as part of our continuing commitment to the preservation of printed works worldwide. We appreciate your understanding of the imperfections in the preservation process, and hope you enjoy this valuable book. ++++ The below data was compiled from various identification fields in the bibliographic record of this title. This data is provided as an additional tool in helping to ensure edition identification: ++++ Principia Mathematica, Volume 2; Principia Mathematica; Bertrand Russell Alfred North Whitehead, Bertrand Russell University Press, 1912 Logic, Symbolic and mathematical; Mathematics

The central contention of this book is that second-order logic has a central role to play in laying the foundations of mathematics. In order to develop the argument fully, the author presents a detailed development of higher-order logic, including a comprehensive discussion of its semantics. Professor Shapiro demonstrates the prevalence of second-order notions in mathematics is practised, and also the extent to which mathematical concepts can be formulated in second-order languages . He shows how first-order languages are insufficient to codify many concepts in contemporary mathematics, and thus that higher-order logic is needed to fully reflect current mathematics. Throughout, the emphasis is on discussing the philosophical and historical issues associated with this subject, and the implications that they have for foundational studies. For the most part, the author assumes little more than a familiarity with logic as might be gained from a beginning graduate course which includes the incompleteness of arithmetic and the Lowenheim-Skolem theorems. All those concerned with the foundations of mathematics will find this a thought-provoking discussion of some of the central issues in this subject.

One of the greatest mathematicians in the world, Michael Atiyah has earned numerous honors, including a Fields Medal, the mathematical equivalent of the Nobel Prize. While the focus of his work has been in the areas of algebraic geometry and topology, he has also participated in research with theoretical physicists. For the first time, these volumes bring together Atiyah's collected papers--both monographs and collaborative works-- including those dealing with mathematical education and current topics of research such as K-theory and gauge theory. The volumes are organized thematically. They will be of great interest to research mathematicians, theoretical physicists, and graduate students in these areas.

Many artists are unaware of the mathematics that bubble beneath their craft, while some consciously use it for inspiration. Our instincts might tell us that these two subjects are incompatible forces with nothing in common, but what if we’re wrong?

Marcus du Sautoy, acclaimed mathematician and Simonyi Professor for the Public Understanding of Science at the University of Oxford, looks to art, music, design and literature to uncover the key mathematical structures that underpin both human creativity and the natural world.

Blueprints takes us from the earliest stone circles to the modernist architecture of Le Corbusier, from Bach’s circular compositions to Radiohead’s disruptive soundscapes, and from Shakespeare’s hidden numerical clues to the Dada artists who embraced randomness. Instead of polar opposites we find a complementary relationship that spans a vast historical and geographic landscape.

Whether we are searching for meaning in an abstract painting or deciphering poetry, there are blueprints everywhere: prime numbers, symmetry, fractals and the weirder worlds of Hamiltonian cycles and hyperbolic geometry. Nature similarly exploits these structures to achieve the wonders of our universe.

In this innovative and delightfully bold exploration of creativity, Marcus explains how we make art, why a creative mindset is vital for discovering new mathematics and how a fundamental connection to the natural world intrinsically links these two subjects.

Beginning with a review of formal languages and their syntax and semantics, Logic, Proof and Computation conducts a computer assisted course in formal reasoning and the relevance of logic to mathematical proof, information processing and philosophy. Topics covered include formal grammars, semantics of formal languages, sequent systems, truth-tables, propositional and first order logic, identity, proof heuristics, regimentation, set theory, databases, automated deduction, proof by induction, Turing machines, undecidability and a computer illustration of the reasoning underpinning Godel's incompleteness proof. LPC is designed as a multidisciplinary reader for students in computing, philosophy and mathematics.

Jesuit engagement with natural philosophy during the late 16th and early 17th centuries transformed the status of the mathematical disciplines and propelled members of the Order into key areas of controversy in relation to Aristotelianism. Through close investigation of the activities of the Jesuit 'school' of mathematics founded by Christoph Clavius, The Scientific Counter-Revolution examines the Jesuit connections to the rise of experimental natural philosophy and the emergence of the early scientific societies. Arguing for a re-evaluation of the role of Jesuits in shaping early modern science, this book traces the evolution of the Collegio Romano as a hub of knowledge. Starting with an examination of Clavius's Counter-Reformation agenda for mathematics, Michael John Gorman traces the development of a collective Jesuit approach to experimentation and observation under Christopher Grienberger and analyses the Jesuit role in the Galileo Affair and the vacuum debate. Ending with a discussion of the transformation of the Collegio Romano under Athanasius Kircher into a place of curiosity and wonder and the centre of a global information gathering network, this book reveals how the Counter-Reformation goals of the Jesuits contributed to the shaping of modern experimental science.

Why is 7 such a lucky number and 13 so unlucky? Why does a jury traditionally have `12 good men and true', and why are there 24 hours in the day and 60 seconds in a minute? This fascinating new book explores the world of numbers from pin numbers to book titles, and from the sixfold shape of snowflakes to the way our roads, houses and telephone numbers are designated in fact and fiction. Using the numbers themselves as its starting point it investigates everything from the origins and meaning of counting in early civilizations to numbers in proverbs, myths and nursery rhymes and the ancient `science' of numerology. It also focuses on the quirks of odds and evens, primes, on numbers in popular sports - and much, much more. So whether you've ever wondered why Heinz has 57 varieties, why 999 is the UK's emergency phone number but 911 is used in America, why Coco Chanel chose No. 5 for her iconic perfume, or how the title Catch 22 was chosen, then this is the book for you. Dip in anywhere and you'll find that numbers are not just for adding and measuring but can be hugely entertaining and informative whether you're buying a diamond or choosing dinner from the menu.

In 1940 G. H. Hardy published A Mathematician's Apology, a meditation on mathematics by a leading pure mathematician. Eighty-two years later, An Applied Mathematician's Apology is a meditation and also a personal memoir by a philosophically inclined numerical analyst, one who has found great joy in his work but is puzzled by its relationship to the rest of mathematics.

Throughout his career, Keith Hossack has made outstanding contributions to the theory of knowledge, metaphysics and the philosophy of mathematics. This collection of previously unpublished papers begins with a focus on Hossack's conception of the nature of knowledge, his metaphysics of facts and his account of the relations between knowledge, agents and facts. Attention moves to Hossack's philosophy of mind and the nature of consciousness, before turning to the notion of necessity and its interaction with a priori knowledge. Hossack's views on the nature of proof, logical truth, conditionals and generality are discussed in depth. In the final chapters, questions about the identity of mathematical objects and our knowledge of them take centre stage, together with questions about the necessity and generality of mathematical and logical truths. Knowledge, Number and Reality represents some of the most vibrant discussions taking place in analytic philosophy today.

The first critical work to attempt the mammoth undertaking of reading Badiou's Being and Event as part of a sequence has often surprising, occasionally controversial results. Looking back on its publication Badiou declared: "I had inscribed my name in the history of philosophy". Later he was brave enough to admit that this inscription needed correction. The central elements of Badiou's philosophy only make sense when Being and Event is read through the corrective prism of its sequel, Logics of Worlds, published nearly twenty years later. At the same time as presenting the only complete overview of Badiou's philosophical project, this book is also the first to draw out the central component of Badiou's ontology: indifference. Concentrating on its use across the core elements Being and Event-the void, the multiple, the set and the event-Watkin demonstrates that no account of Badiou's ontology is complete unless it accepts that Badiou's philosophy is primarily a presentation of indifferent being. Badiou and Indifferent Being provides a detailed and lively section by section reading of Badiou's foundational work. It is a seminal source text for all Badiou readers.

Responding to widespread interest within cultural studies and social inquiry, this book addresses the question 'what is a mathematical concept?' using a variety of vanguard theories in the humanities and posthumanities. Tapping historical, philosophical, sociological and psychological perspectives, each chapter explores the question of how mathematics comes to matter. Of interest to scholars across the usual disciplinary divides, this book tracks mathematics as a cultural activity, drawing connections with empirical practice. Unlike other books in this area, it is highly interdisciplinary, devoted to exploring the ontology of mathematics as it plays out in different contexts. This book will appeal to scholars who are interested in particular mathematical habits - creative diagramming, structural mappings, material agency, interdisciplinary coverings - that shed light on both mathematics and other disciplines. Chapters are also relevant to social sciences and humanities scholars, as each offers philosophical insight into mathematics and how we might live mathematically.

The infinite No other question has ever moved so profoundly the spirit of man; no other idea has so fruitfully stimulated his intellect; yet no other concept stands in greater need of clarification than that of the infinite. David Hilbert This interdisciplinary study of infinity explores the concept through the prism of mathematics and then offers more expansive investigations in areas beyond mathematical boundaries to reflect the broader, deeper implications of infinity for human intellectual thought. More than a dozen world renowned researchers in the fields of mathematics, physics, cosmology, philosophy, and theology offer a rich intellectual exchange among various current viewpoints, rather than displaying a static picture of accepted views on infinity. The book starts with a historical examination of the transformation of infinity from a philosophical and theological study to one dominated by mathematics. It then offers technical discussions on the understanding of mathematical infinity. Following this, the book considers the perspectives of physics and cosmology: Can infinity be found in the real universe? Finally, the book returns to questions of philosophical and theological aspects of infinity."

This book is an original-the first-ever treatment of the mathematics of Luck. Setting out from the principle that luck can be measured by the gap between reasonable expectation and eventual realization, the book develops step-by-step a mathematical theory that accommodates the entire range of our pre-systematic understanding of the way in which luck functions in human affairs. In so moving from explanatory exposition to mathematical treatment, the book provides a clear and accessible account of the way in which luck assessment enters into the calculations of rational decision theory.

Alain Badiou has claimed that Quentin Meillassoux's book After Finitude (Bloomsbury, 2008) "opened up a new path in the history of philosophy." And so, whether you agree or disagree with the speculative realism movement, it has to be addressed. Lacanian Realism does just that. This book reconstructs Lacanian dogma from the ground up: first, by unearthing a new reading of the Lacanian category of the real; second, by demonstrating the political and cultural ingenuity of Lacan's concept of the real, and by positioning this against the more reductive analyses of the concept by Slavoj Zizek, Alain Badiou, Saul Newman, Todd May, Joan Copjec, Jacques Ranciere, and others, and; third, by arguing that the subject exists intimately within the real. Lacanian Realism is an imaginative and timely exploration of the relationship between Lacanian psychoanalysis and contemporary continental philosophy.

The chapters in this timely volume aim to answer the growing interest in Arthur Schopenhauer's logic, mathematics, and philosophy of language by comprehensively exploring his work on mathematical evidence, logic diagrams, and problems of semantics. Thus, this work addresses the lack of research on these subjects in the context of Schopenhauer's oeuvre by exposing their links to modern research areas, such as the "proof without words" movement, analytic philosophy and diagrammatic reasoning, demonstrating its continued relevance to current discourse on logic. Beginning with Schopenhauer's philosophy of language, the chapters examine the individual aspects of his semantics, semiotics, translation theory, language criticism, and communication theory. Additionally, Schopenhauer's anticipation of modern contextualism is analyzed. The second section then addresses his logic, examining proof theory, metalogic, system of natural deduction, conversion theory, logical geometry, and the history of logic. Special focus is given to the role of the Euler diagrams used frequently in his lectures and their significance to broader context of his logic. In the final section, chapters discuss Schopenhauer's philosophy of mathematics while synthesizing all topics from the previous sections, emphasizing the relationship between intuition and concept. Aimed at a variety of academics, including researchers of Schopenhauer, philosophers, historians, logicians, mathematicians, and linguists, this title serves as a unique and vital resource for those interested in expanding their knowledge of Schopenhauer's work as it relates to modern mathematical and logical study.

This volume contains ten papers that have been collected by the Canadian Society for History and Philosophy of Mathematics/Societe canadienne d'histoire et de philosophie des mathematiques. It showcases rigorously-reviewed contemporary scholarship on an interesting variety of topics in the history and philosophy of mathematics from the seventeenth century to the modern era. The volume begins with an exposition of the life and work of Professor Boleslaw Sobocinski. It then moves on to cover a collection of topics about twentieth-century philosophy of mathematics, including Fred Sommers's creation of Traditional Formal Logic and Alexander Grothendieck's work as a starting point for discussing analogies between commutative algebra and algebraic geometry. Continuing the focus on the philosophy of mathematics, the next selections discuss the mathematization of biology and address the study of numerical cognition. The volume then moves to discussing various aspects of mathematics education, including Charles Davies's early book on the teaching of mathematics and the use of Gaussian Lemniscates in the classroom. A collection of papers on the history of mathematics in the nineteenth century closes out the volume, presenting a discussion of Gauss's "Allgemeine Theorie des Erdmagnetismus" and a comparison of the geometric works of Desargues and La Hire. Written by leading scholars in the field, these papers are accessible not only to mathematicians and students of the history and philosophy of mathematics, but also to anyone with a general interest in mathematics.

Visual thinking -- visual imagination or perception of diagrams and symbol arrays, and mental operations on them -- is omnipresent in mathematics. Is this visual thinking merely a psychological aid, facilitating grasp of what is gathered by other means? Or does it also have epistemological functions, as a means of discovery, understanding, and even proof? By examining the many kinds of visual representation in mathematics and the diverse ways in which they are used, Marcus Giaquinto argues that visual thinking in mathematics is rarely just a superfluous aid; it usually has epistemological value, often as a means of discovery. Drawing from philosophical work on the nature of concepts and from empirical studies of visual perception, mental imagery, and numerical cognition, Giaquinto explores a major source of our grasp of mathematics, using examples from basic geometry, arithmetic, algebra, and real analysis. He shows how we can discern abstract general truths by means of specific images, how synthetic a priori knowledge is possible, and how visual means can help us grasp abstract structures.
Visual Thinking in Mathematics reopens the investigation of earlier thinkers from Plato to Kant into the nature and epistemology of an individual's basic mathematical beliefs and abilities, in the new light shed by the maturing cognitive sciences. Clear and concise throughout, it will appeal to scholars and students of philosophy, mathematics, and psychology, as well as anyone with an interest in mathematical thinking.

This volume presents interviews that have been conducted from the 1980s to the present with important scholars of social choice and welfare theory. Starting with a brief history of social choice and welfare theory written by the book editors, it features 15 conversations with four Nobel Laureates and other key scholars in the discipline. The volume is divided into two parts. The first part presents four conversations with the founding fathers of modern social choice and welfare theory: Kenneth Arrow, John Harsanyi, Paul Samuelson, and Amartya Sen. The second part includes conversations with scholars who made important contributions to the discipline from the early 1970s onwards. This book will appeal to anyone interested in the history of economics, and the history of social choice and welfare theory in particular.

This book consists of eleven new essays that provide new insights into classical and contemporary issues surrounding free will and human agency. They investigate topics such as the nature of practical knowledge and its role in intentional action; mental content and explanations of action; recent arguments for libertarianism; the situationist challenge to free will; freedom and a theory of narrative configuration; the moral responsibility of the psychopath; and free will and the indeterminism of quantum mechanics. Also tackling some historical precursors of contemporary debates, taken together these essays demonstrate the need for an approach that recognizes the multifaceted nature of free will. This book provides essential reading for anyone interested in the current scholarship on free will.