The contributions gathered here demonstrate how categorical ontology can provide a basis for linking three important basic sciences: mathematics, physics, and philosophy. Category theory is a new formal ontology that shifts the main focus from objects to processes. The book approaches formal ontology in the original sense put forward by the philosopher Edmund Husserl, namely as a science that deals with entities that can be exemplified in all spheres and domains of reality. It is a dynamic, processual, and non-substantial ontology in which all entities can be treated as transformations, and in which objects are merely the sources and aims of these transformations. Thus, in a rather surprising way, when employed as a formal ontology, category theory can unite seemingly disparate disciplines in contemporary science and the humanities, such as physics, mathematics and philosophy, but also computer and complex systems science.

Features Provides a uniquely historical perspective on the mathematical underpinnings of a comprehensive list of games Suitable for a broad audience of differing mathematical levels. Anyone with a passion for games, game theory, and mathematics will enjoy this book, whether they be students, academics, or game enthusiasts Covers a wide selection of topics at a level that can be appreciated on a historical, recreational, and mathematical level.

This handbook features essays written by both literary scholars and mathematicians that examine multiple facets of the connections between literature and mathematics. These connections range from mathematics and poetic meter to mathematics and modernism to mathematics as literature. Some chapters focus on a single author, such as mathematics and Ezra Pound, Gertrude Stein, or Charles Dickens, while others consider a mathematical topic common to two or more authors, such as squaring the circle, chaos theory, Newton's calculus, or stochastic processes. With appeal for scholars and students in literature, mathematics, cultural history, and history of mathematics, this important volume aims to introduce the range, fertility, and complexity of the connections between mathematics, literature, and literary theory. Chapter 1 is available open access under a Creative Commons Attribution 4.0 International License via [link.springer.com|http://link.springer.com/].

This monograph examines the private annotations that Ludwig Wittgenstein made to his copy of G.H. Hardy's classic textbook, A Course of Pure Mathematics. Complete with actual images of the annotations, it gives readers a more complete picture of Wittgenstein's remarks on irrational numbers, which have only been published in an excerpted form and, as a result, have often been unjustly criticized. The authors first establish the context behind the annotations and discuss the historical role of Hardy's textbook. They then go on to outline Wittgenstein's non-extensionalist point of view on real numbers, assessing his manuscripts and published remarks and discussing attitudes in play in the philosophy of mathematics since Dedekind. Next, coverage focuses on the annotations themselves. The discussion encompasses irrational numbers, the law of excluded middle in mathematics and the notion of an "improper picture," the continuum of real numbers, and Wittgenstein's attitude toward functions and limits.

Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. This volume, the second publication in the Lecture Notes in Logic series, is the proceedings of the Association for Symbolic Logic meeting held in Helsinki, Finland, in July 1990. It contains eighteen papers by leading researchers, covering all fields of mathematical logic from the philosophy of mathematics, through model theory, proof theory, recursion theory, and set theory, to the connections of logic to computer science. The articles published here are still widely cited and continue to provide ideas for ongoing research projects.

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.

There is a growing awareness among researchers in the humanities and social sciences of the rhetorical force of mathematical discourse-whether in regard to gerrymandering, facial recognition technologies, or racial biases in algorithmic automation. This book proposes a novel way to engage with and understand mathematics via a theoretical framework that highlights how math transforms the social-material world. In this study, G. Mitchell Reyes applies contemporary rhetorical analysis to mathematical discourse, calling into question the commonly held view that math equals truth. Examining mathematics in historical context, Reyes traces its development from Plato's teaching about abstract numbers to Euclidian geometry and the emergence of calculus and infinitesimals, imaginary numbers, and algorithms. This history reveals that mathematical innovation has always relied on rhetorical practices of making meaning, such as analogy, metaphor, and invention. Far from expressing truth hidden deep in reality, mathematics is dynamic and evolving, shaping reality and our experience of it. By bringing mathematics back down to the material-social world, Reyes makes it possible for scholars of the rhetoric and sociology of science, technology, and math to collaborate with mathematicians themselves in order to better understand our material world and public culture.

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.

Originally published in 1938, this book focuses on the area of elliptic and hyperelliptic integrals and allied theory. The text was a posthumous publication by William Westropp Roberts (1850-1935), who held the position of Vice-Provost at Trinity College, Dublin from 1927 until shortly before his death. This book will be of value to anyone with an interest in the history of mathematics.

This book demonstrates how a radical version of physicalism ('No-Self Physicalism') can offer an internally coherent and comprehensive philosophical worldview. It first argues that a coherent physicalist should explicitly treat a cognitive subject merely as a physical thing and should not vaguely assume an amorphous or even soul-like subject or self. This approach forces the physicalist to re-examine traditional core philosophical notions such as truth, analyticity, modality, apriority because our traditional understandings of them appear to be predicated on a cognitive subject that is not literally just a physical thing. In turn, working on the assumption that a cognitive subject is itself completely physical, namely a neural network-based robot programmed by evolution (hence the term 'No-Self'), the book proposes physicalistic theories on conceptual representation, truth, analyticity, modality, the nature of mathematics, epistemic justification, knowledge, apriority and intuition, as well as a physicalistic ontology. These are meant to show that this No-Self Physicalism, perhaps the most minimalistic and radical version of physicalism proposed to date, can accommodate many aspects that have traditionally interested philosophers. Given its refreshingly radical approach and painstakingly developed content, the book is of interest to anyone who is seeking a coherent philosophical worldview in this age of science.

Computation is revolutionizing our world, even the inner world of the 'pure' mathematician. Mathematical methods - especially the notion of proof - that have their roots in classical antiquity have seen a radical transformation since the 1970s, as successive advances have challenged the priority of reason over computation. Like many revolutions, this one comes from within. Computation, calculation, algorithms - all have played an important role in mathematical progress from the beginning - but behind the scenes, their contribution was obscured in the enduring mathematical literature. To understand the future of mathematics, this fascinating book returns to its past, tracing the hidden history that follows the thread of computation. Along the way it invites us to reconsider the dialog between mathematics and the natural sciences, as well as the relationship between mathematics and computer science. It also sheds new light on philosophical concepts, such as the notions of analytic and synthetic judgment. Finally, it brings us to the brink of the new age, in which machine intelligence offers new ways of solving mathematical problems previously inaccessible. This book is the 2007 winner of the Grand Prix de Philosophie de l'Academie Francaise.

This volume contains fourteen papers that were presented at the 2016 Annual Meeting of the Canadian Society for History and Philosophy of Mathematics/La Societe Canadienne d'Histoire et de Philosophie des Mathematiques, held at the University of Calgary in Alberta, Canada. In addition to showcasing rigorously reviewed modern scholarship on an interesting variety of topics in the history and philosophy of mathematics, this meeting also honored the life and work of the logician and philosopher of mathematics Aldo Antonelli (1962-2015). The first four papers in this book are part of that remembrance and have a philosophical focus. Included in these are a discussion of Bolzano's objections to Kant's philosophy of mathematics and an examination of the influence of rhetorical and poetic aesthetics on the development of symbols in the 16th and 17th Centuries. The remaining papers deal with the history of mathematics and cover such subjects as Early schemes for polar ordinates in the work of L'Hopital, based on lessons given to him by Bernoulli A method devised by Euler for determining if a number is a sum of two squares Playfair's Axiom and what it reveals about the history of 19th-Century mathematics education The modern library classification system for mathematical subjects An exploration of various examples of sundials throughout Paris Written by leading scholars in the field, these papers are accessible to not only mathematicians and students of the history and philosophy of mathematics, but also anyone with a general interest in mathematics.

To what extent are the subjects of our thoughts and talk real? This is the question of realism. In this book, Justin Clarke-Doane explores arguments for and against moral realism and mathematical realism, how they interact, and what they can tell us about areas of philosophical interest more generally. He argues that, contrary to widespread belief, our mathematical beliefs have no better claim to being self-evident or provable than our moral beliefs. Nor do our mathematical beliefs have better claim to being empirically justified than our moral beliefs. It is also incorrect that reflection on the genealogy of our moral beliefs establishes a lack of parity between the cases. In general, if one is a moral antirealist on the basis of epistemological considerations, then one ought to be a mathematical antirealist as well. And, yet, Clarke-Doane shows that moral realism and mathematical realism do not stand or fall together - and for a surprising reason. Moral questions, insofar as they are practical, are objective in a sense that mathematical questions are not, and the sense in which they are objective can only be explained by assuming practical anti-realism. One upshot of the discussion is that the concepts of realism and objectivity, which are widely identified, are actually in tension. Another is that the objective questions in the neighborhood of factual areas like logic, modality, grounding, and nature are practical questions too. Practical philosophy should, therefore, take center stage.

This book uses Ludwig Wittgenstein's philosophical methodology to solve a problem that has perplexed thinkers for thousands of years: 'how come (abstract) mathematics applies so wonderfully well to the (concrete, physical) world?' The book is distinctive in several ways. First, it gives the reader a route into understanding important features of Wittgenstein's writings and lectures by using his methodology to tackle this long-standing and seemingly intractable philosophical problem. More than this, though, it offers an outline of important (sometimes little-known) aspects of the development of mathematical thought through the ages, and an engagement of Wittgenstein's philosophy with this and with contemporary philosophy of mathematics on its own terms. A clear overview of all this in the context of Wittgenstein's philosophy of mathematics is interesting in its own right; it is also just what is needed to solve the problem of mathematics and world.

This collection examines the uses of quantification in climate science, higher education, and health. Numbers are both controlling and fragile. They drive public policy, figuring into everything from college rankings to vaccine efficacy rates. At the same time, they are frequent objects of obfuscation, manipulation, or outright denial. This timely collection by a diverse group of humanists and social scientists challenges undue reverence or skepticism toward quantification and offers new ideas about how to harmonize quantitative with qualitative forms of knowledge. Limits of the Numerical focuses on quantification in several contexts: climate change; university teaching and research; and health, medicine, and well-being more broadly. This volume shows the many ways that qualitative and quantitative approaches can productively interact-how the limits of the numerical can be overcome through equitable partnerships with historical, institutional, and philosophical analysis. The authors show that we can use numbers to hold the powerful to account, but only when those numbers are themselves democratically accountable.

The idea that mathematics is reducible to logic has a long history, but it was Frege who gave logicism an articulation and defense that transformed it into a distinctive philosophical thesis with a profound influence on the development of philosophy in the twentieth century. This volume of classic, revised and newly written essays by William Demopoulos examines logicism's principal legacy for philosophy: its elaboration of notions of analysis and reconstruction. The essays reflect on the deployment of these ideas by the principal figures in the history of the subject - Frege, Russell, Ramsey and Carnap - and in doing so illuminate current concerns about the nature of mathematical and theoretical knowledge. Issues addressed include the nature of arithmetical knowledge in the light of Frege's theorem; the status of realism about the theoretical entities of physics; and the proper interpretation of empirical theories that postulate abstract structural constraints.

This volume is a collection of essays in honour of Professor Mohammad Ardeshir. It examines topics which, in one way or another, are connected to the various aspects of his multidisciplinary research interests. Based on this criterion, the book is divided into three general categories. The first category includes papers on non-classical logics, including intuitionistic logic, constructive logic, basic logic, and substructural logic. The second category is made up of papers discussing issues in the contemporary philosophy of mathematics and logic. The third category contains papers on Avicenna's logic and philosophy. Mohammad Ardeshir is a full professor of mathematical logic at the Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran, where he has taught generations of students for around a quarter century. Mohammad Ardeshir is known in the first place for his prominent works in basic logic and constructive mathematics. His areas of interest are however much broader and include topics in intuitionistic philosophy of mathematics and Arabic philosophy of logic and mathematics. In addition to numerous research articles in leading international journals, Ardeshir is the author of a highly praised Persian textbook in mathematical logic. Partly through his writings and translations, the school of mathematical intuitionism was introduced to the Iranian academic community.

This book introduces students and researchers to the philosophical issues at play in the growing field of formal (or Bayesian) epistemology. It focuses not on how to do particular calculations but instead on the philosophical foundations at the convergence of belief and mathematical representation. Its central questions are: What is the nature of quantifying belief? What is the source of its norms? How is it reasonable to represent belief numerically? Accessible to those without any mathematical background, this book will become a much used classic in the field.

This book analyses the straw man fallacy and its deployment in philosophical reasoning. While commonly invoked in both academic dialogue and public discourse, it has not until now received the attention it deserves as a rhetorical device. Scott Aikin and John Casey propose that straw manning essentially consists in expressing distorted representations of one’s critical interlocutor. To this end, the straw man comprises three dialectical forms, and not only the one that is usually suggested: the straw man, the weak man and the hollow man. Moreover, they demonstrate that straw manning is unique among fallacies as it has no particular logical form in itself, because it is an instance of inappropriate meta-argument, or argument about arguments. They discuss the importance of the onlooking audience to the successful deployment of the straw man, reasoning that the existence of an audience complicates the dialectical boundaries of argument. Providing a lively, provocative and thorough analysis of the straw man fallacy, this book will appeal to postgraduates and researchers alike, working in a range of fields including fallacies, rhetoric, argumentation theory and informal logic.

PYTHAGORAS (fl. 500 B.C.E.), the first man to call himself a philosopher, was both a brilliant mathematician and spiritual teacher. This anthology is the largest collection of Pythagorean writings ever to appear in the English language. It contains the four ancient biographies of Pythagoras and over twenty-five Pythagorean and Neopythagorean writings from the classical and Hellenistic periods. The Pythagorean ethical and political tractates are especially interesting, for they are based on the premise that the universal principles of Harmony, Proportion, and Justice govern the physical cosmos, and these writings show how individuals and societies alike attain their peak of excellence when informed by these same principles. Indexed, illustrated, with appendices and an extensive bibliography, this work also contains an introductory essay by David Fideler.

The Equation of Knowledge: From Bayes' Rule to a Unified Philosophy of Science introduces readers to the Bayesian approach to science: teasing out the link between probability and knowledge. The author strives to make this book accessible to a very broad audience, suitable for professionals, students, and academics, as well as the enthusiastic amateur scientist/mathematician. This book also shows how Bayesianism sheds new light on nearly all areas of knowledge, from philosophy to mathematics, science and engineering, but also law, politics and everyday decision-making. Bayesian thinking is an important topic for research, which has seen dramatic progress in the recent years, and has a significant role to play in the understanding and development of AI and Machine Learning, among many other things. This book seeks to act as a tool for proselytising the benefits and limits of Bayesianism to a wider public. Features Presents the Bayesian approach as a unifying scientific method for a wide range of topics Suitable for a broad audience, including professionals, students, and academics Provides a more accessible, philosophical introduction to the subject that is offered elsewhere

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."

Our knowledge of mathematics has structured much of what we think we know about ourselves as individuals and communities, shaping our psychologies, sociologies, and economies. In pursuit of a more predictable and more controllable cosmos, we have extended mathematical insights and methods to more and more aspects of the world. Today those powers are greater than ever, as computation is applied to virtually every aspect of human activity. Yet, in the process, are we losing sight of the human? When we apply mathematics so broadly, what do we gain and what do we lose, and at what risk to humanity? These are the questions that David and Ricardo L. Nirenberg ask in Uncountable, a provocative account of how numerical relations became the cornerstone of human claims to knowledge, truth, and certainty. There is a limit to these number-based claims, they argue, which they set out to explore. The Nirenbergs, father and son, bring together their backgrounds in math, history, literature, religion, and philosophy, interweaving scientific experiments with readings of poems, setting crises in mathematics alongside world wars, and putting medieval Muslim and Buddhist philosophers in conversation with Einstein, Schroedinger, and other giants of modern physics. The result is a powerful lesson in what counts as knowledge and its deepest implications for how we live our lives.

This collection presents significant contributions from an international network project on mathematical cultures, including essays from leading scholars in the history and philosophy of mathematics and mathematics education. Mathematics has universal standards of validity. Nevertheless, there are local styles in mathematical research and teaching, and great variation in the place of mathematics in the larger cultures that mathematical practitioners belong to. The reflections on mathematical cultures collected in this book are of interest to mathematicians, philosophers, historians, sociologists, cognitive scientists and mathematics educators.

This edited work presents contemporary mathematical practice in the foundational mathematical theories, in particular set theory and the univalent foundations. It shares the work of significant scholars across the disciplines of mathematics, philosophy and computer science. Readers will discover systematic thought on criteria for a suitable foundation in mathematics and philosophical reflections around the mathematical perspectives. The volume is divided into three sections, the first two of which focus on the two most prominent candidate theories for a foundation of mathematics. Readers may trace current research in set theory, which has widely been assumed to serve as a framework for foundational issues, as well as new material elaborating on the univalent foundations, considering an approach based on homotopy type theory (HoTT). The third section then builds on this and is centred on philosophical questions connected to the foundations of mathematics. Here, the authors contribute to discussions on foundational criteria with more general thoughts on the foundations of mathematics which are not connected to particular theories. This book shares the work of some of the most important scholars in the fields of set theory (S. Friedman), non-classical logic (G. Priest) and the philosophy of mathematics (P. Maddy). The reader will become aware of the advantages of each theory and objections to it as a foundation, following the latest and best work across the disciplines and it is therefore a valuable read for anyone working on the foundations of mathematics or in the philosophy of mathematics.