Gerade heute, wo sich die Aufmerksamkeit der fuhrenden Philosophen, Logiker und Mathematiker erneut auf die Grundlagen der systematisch-deduktiven Mathematik richtet, ist dieses Buch von zeitnaher und tiefer Bedeutung."

The mathematician and engineer Charles Babbage (1791 1871) is best remembered for his 'calculating machines', which are considered the forerunner of modern computers. Over the course of his life he wrote a number of books based on his scientific investigations, but in this volume, published in 1864, Babbage writes in a more personal vein. He points out at the beginning of the work that it 'does not aspire to the name of autobiography', though the chapters sketch out the contours of his life, beginning with his family, his childhood and formative years studying at Cambridge, and moving through various episodes in his scientific career. However, the work also diverges into his observations on other topics, as indicated by chapter titles such as 'Street Nuisances' and 'Wit'. Babbage's colourful recollections give an intimate portrait of the life of one of Britain's most influential inventors.

Most scholars think of David Hilbert's program as the most demanding and ideologically motivated attempt to provide a foundation for mathematics, and because they see technical obstacles in the way of realizing the program's goals, they regard it as a failure. Against this view, Curtis Franks argues that Hilbert's deepest and most central insight was that mathematical techniques and practices do not need grounding in any philosophical principles. He weaves together an original historical account, philosophical analysis, and his own development of the meta-mathematics of weak systems of arithmetic to show that the true philosophical significance of Hilbert's program is that it makes the autonomy of mathematics evident. The result is a vision of the early history of modern logic that highlights the rich interaction between its conceptual problems and technical development.

Gottlob Frege (1848 1925) was unquestionably one of the most important philosophers of all time. He trained as a mathematician, and his work in philosophy started as an attempt to provide an explanation of the truths of arithmetic, but in the course of this attempt he not only founded modern logic but also had to address fundamental questions in the philosophy of language and philosophical logic. Frege is generally seen (along with Russell and Wittgenstein) as one of the fathers of the analytic method, which dominated philosophy in English-speaking countries for most of the twentieth century. His work is studied today not just for its historical importance but also because many of his ideas are still seen as relevant to current debates in the philosophies of logic, language, mathematics and the mind. The Cambridge Companion to Frege provides a route into this lively area of research.

This book presents a new approach to the epistemology of mathematics by viewing mathematics as a human activity whose knowledge is intimately linked with practice. Charting an exciting new direction in the philosophy of mathematics, Jose Ferreiros uses the crucial idea of a continuum to provide an account of the development of mathematical knowledge that reflects the actual experience of doing math and makes sense of the perceived objectivity of mathematical results. Describing a historically oriented, agent-based philosophy of mathematics, Ferreiros shows how the mathematical tradition evolved from Euclidean geometry to the real numbers and set-theoretic structures. He argues for the need to take into account a whole web of mathematical and other practices that are learned and linked by agents, and whose interplay acts as a constraint. Ferreiros demonstrates how advanced mathematics, far from being a priori, is based on hypotheses, in contrast to elementary math, which has strong cognitive and practical roots and therefore enjoys certainty. Offering a wealth of philosophical and historical insights, Mathematical Knowledge and the Interplay of Practices challenges us to rethink some of our most basic assumptions about mathematics, its objectivity, and its relationship to culture and science.

Offering a collection of fifteen essays that deal with issues at the intersection of phenomenology, logic, and the philosophy of mathematics, this 2005 book is divided into three parts. Part I contains a general essay on Husserl's conception of science and logic, an essay of mathematics and transcendental phenomenology, and an essay on phenomenology and modern pure geometry. Part II is focused on Kurt Godel's interest in phenomenology. It explores Godel's ideas and also some work of Quine, Penelope Maddy and Roger Penrose. Part III deals with elementary, constructive areas of mathematics. These are areas of mathematics that are closer to their origins in simple cognitive activities and in everyday experience. This part of the book contains essays on intuitionism, Hermann Weyl, the notion of constructive proof, Poincare and Frege.

Even the most enthusiastic of maths students probably at one time wondered when exactly it would all prove useful in ‘real life’. Well, maths reaches so far and wide through our world that, love it or hate it, we’re all doing maths almost every minute of every day.

David Darling and Agnijo Banerjee go in search of the perfect labyrinth, journey back to the second century in pursuit of ‘bubble maths’, reveal the weirdest mathematicians in history and transform the bewildering into the beautiful, delighting us once again.

In the social sciences norms are sometimes taken to play a key explanatory role. Yet norms differ from group to group, from society to society, and from species to species. How are norms formed and how do they change? This 'state-of-the-art' collection of essays presents some of the best contemporary research into the dynamic processes underlying the formation, maintenance, metamorphosis and dissolution of norms. The volume combines formal modelling with more traditional analysis, and considers biological and cultural evolution, individual learning, and rational deliberation. In filling a significant gap in the current literature this volume will be of particular interest to economists, political scientists and sociologists, in addition to philosophers of the social sciences.

This book considers the manifold possible approaches, past and present, to our understanding of the natural numbers. They are treated as epistemic objects: mathematical objects that have been subject to epistemological inquiry and attention throughout their history and whose conception has evolved accordingly. Although they are the simplest and most common mathematical objects, as this book reveals, they have a very complex nature whose study illuminates subtle features of the functioning of our thought. Using jointly history, mathematics and philosophy to grasp the essence of numbers, the reader is led through their various interpretations, presenting the ways they have been involved in major theoretical projects from Thales onward. Some pertain primarily to philosophy (as in the works of Plato, Aristotle, Kant, Wittgenstein...), others to general mathematics (Euclid's Elements, Cartesian algebraic geometry, Cantorian infinities, set theory...). Also serving as an introduction to the works and thought of major mathematicians and philosophers, from Plato and Aristotle to Cantor, Dedekind, Frege, Husserl and Weyl, this book will be of interest to a wide variety of readers, from scholars with a general interest in the philosophy or mathematics to philosophers and mathematicians themselves.

This book is an attempt to change our thinking about thinking. Anna Sfard undertakes this task convinced that many long-standing, seemingly irresolvable quandaries regarding human development originate in ambiguities of the existing discourses on thinking. Standing on the shoulders of Vygotsky and Wittgenstein, the author defines thinking as a form of communication. The disappearance of the time-honoured thinking-communicating dichotomy is epitomised by Sfard's term, commognition, which combines communication with cognition. The commognitive tenet implies that verbal communication with its distinctive property of recursive self-reference may be the primary source of humans' unique ability to accumulate the complexity of their action from one generation to another. The explanatory power of the commognitive framework and the manner in which it contributes to our understanding of human development is illustrated through commognitive analysis of mathematical discourse accompanied by vignettes from mathematics classrooms.

The operation of developing a concept is a common procedure in mathematics and in natural science, but has traditionally seemed much less possible to philosophers and, especially, logicians. Meir Buzaglo's innovative study proposes a way of expanding logic to include the stretching of concepts, while modifying the principles which block this possibility. He offers stimulating discussions of the idea of conceptual expansion as a normative process, and of the relation of conceptual expansion to truth, meaning, reference, ontology and paradox, and analyzes the views of Kant, Wittgenstein, Godel, and others, paying especially close attention to Frege. His book will be of interest to a wide range of readers, from philosophers (of logic, mathematics, language, and science) to logicians, mathematicians, linguists, and cognitive scientists.

In this ambitious study, David Corfield attacks the widely held view that it is the nature of mathematical knowledge which has shaped the way in which mathematics is treated philosophically and claims that contingent factors have brought us to the present thematically limited discipline. Illustrating his discussion with a wealth of examples, he sets out a variety of approaches to new thinking about the philosophy of mathematics, ranging from an exploration of whether computers producing mathematical proofs or conjectures are doing real mathematics, to the use of analogy, the prospects for a Bayesian confirmation theory, the notion of a mathematical research programme and the ways in which new concepts are justified. His inspiring book challenges both philosophers and mathematicians to develop the broadest and richest philosophical resources for work in their disciplines and points clearly to the ways in which this can be done.

This volume brings together a collection of essays on the history and philosophy of probability and statistics by one of the eminent scholars in these subjects. Written over the last fifteen years, they fall into three broad categories. The first deals with the use of symmetry arguments in inductive probability, in particular, their use in deriving rules of succession (Carnap's 'continuum of inductive methods'). The second group deals with four outstanding individuals who made lasting contributions to probability and statistics in very different ways: Frank Ramsey, R. A. Fisher, Alan Turing, and Abraham de Moivre. The last group of essays deals with the problem of 'predicting the unpredictable' - making predictions when the range of possible outcomes is unknown in advance. The essays weave together the history and philosophy of these subjects and document the fascination that they have exercised for more than three centuries.

This book is a collection of fifteen essays that deal with issues at the intersection of phenomenology, logic, and the philosophy of mathematics. The book is divided into three parts. Part I, Reason, Science, and Mathematics contains a general essay on Husserl's conception of science and logic, an essay of mathematics and transcendental phenomenology, and an essay of phenomenology and modern pure geometry. Part II is focused on Kurt Godel's interest in phenomenology. It explores Godel's ideas and also some work of Quine, Penelope Maddy and Roger Penrose. Part III deals with elementary, constructive areas of mathematics. These are areas of mathematics that are closer to their origins in simple cognitive activities and in everyday experience. This part of the book contains essays on intuitionism, Hermann Weyl, the notion of constructive proof, Poincave and Frege.

This volume brings together a collection of essays on the history and philosophy of probability and statistics by one of the eminent scholars in these subjects. Written over the last fifteen years, they fall into three broad categories. The first deals with the use of symmetry arguments in inductive probability, in particular, their use in deriving rules of succession (Carnap's 'continuum of inductive methods'). The second group deals with four outstanding individuals who made lasting contributions to probability and statistics in very different ways: Frank Ramsey, R. A. Fisher, Alan Turing, and Abraham de Moivre. The last group of essays deals with the problem of 'predicting the unpredictable' - making predictions when the range of possible outcomes is unknown in advance. The essays weave together the history and philosophy of these subjects and document the fascination that they have exercised for more than three centuries.

Belief revision is a topic of much interest in theoretical computer science and logic, and it forms a central problem in research into artificial intelligence. In simple terms: how do you update a database of knowledge in the light of new information? What if the new information is in conflict with something that was previously held to be true? An intelligent system should be able to accommodate all such cases. This book contains a collection of research articles on belief revision that are completely up to date and an introductory chapter that presents a survey of current research in the area and the fundamentals of the theory. Thus this volume will be useful as a textbook on belief revision.

Philosophical considerations, which are often ignored or treated casually, are given careful consideration in this introduction. Thomas Forster places the notion of inductively defined sets (recursive datatypes) at the center of his exposition resulting in an original analysis of well established topics. The presentation illustrates difficult points and includes many exercises. Little previous knowledge of logic is required and only a knowledge of standard undergraduate mathematics is assumed.

New to the Fourth Edition Reorganised and revised chapter seven and thirteen New exercises and examples Expanded, updated references Further historical material on figures besides Galois: Omar Khayyam, Vandermonde, Ruffini, and Abel A new final chapter discussing other directions in which Galois Theory has developed: the inverse Galois problem, differential Galois theory, and a (very) brief introduction to p-adic Galois representations.

}There are two kinds of people: those who can do mathematics, and then theres the rest of us.Math is boring.Females have no facility for mathematics (and really dont need it, anyway).For many people who do not like math, these myths ring true.Calvin Clawson, the celebrated author of Mathematical Mysteries , has a unique talent for opening the door for the uninitiated to the splendors of mathematics. A writer in love with his subject, Clawson offers readers the perfect antidote to the phobias and misconceptions surrounding mathematics in MATHEMATICAL SORCERY . Contending that the power and beauty of mathematics are gifts in which we all can partake, he shows that the field of mathematics holds a bounty of wonder that can be reaped by any one of us in the hopes of discovering new truths.In this captivating quest for pure knowledge, Clawson takes us on a journey to the amazing discoveries of our ancient ancestors. He divulges the wisdom of the Ancient Greeks, Sumerians, Babylonians, and Egyptians, whose stunning revelations still have deep meaning to us today. The secrets of the constellations, the enigma of the golden mean, and the brilliance of a proof are just some of the breakthroughs he explores with unbridled delight.Enabling us to appreciate the achievements of Newton and other intellectual giants, Clawson inspires us through his eloquence and zeal to actually do mathematics, urging us to leap to the next level. He helps us intuitively comprehend and follow the very building blocks that too long have been a mystery to most of us, including infinity, functions, and the limit. As he elegantly states: Mathematics is pursued not only for the sheer joy of the pursuit, as with the Ancient Greeks, but for the truths it reveals about our universe. Through MATHEMATICAL SORCERY , we taste the fruit of knowledge that has eluded us until now. }

This collection of new essays offers a "state-of-the-art" conspectus of major trends in the philosophy of logic and philosophy of mathematics. A distinguished group of philosophers addresses issues at the center of contemporary debate: semantic and set-theoretic paradoxes, the set/class distinction, foundations of set theory, mathematical intuition and many others. The volume includes Hilary Putnam's 1995 Alfred Tarski lectures published here for the first time. The essays are presented to honor the work of Charles Parsons.

How can we identify events due to intelligent causes and distinguish them from events due to undirected natural causes? If we lack a causal theory how can we determine whether an intelligent cause acted? This book presents a reliable method for detecting intelligent causes: the design inference. The design inference uncovers intelligent causes by isolating the key trademark of intelligent causes: specified events of small probability. Design inferences can be found in a range of scientific pursuits from forensic science to research into the origins of life to the search for extraterrestrial intelligence. This challenging and provocative book will be read with particular interest by philosophers of science and religion, other philosophers concerned with epistemology and logic, probability and complexity theorists, and statisticians.

This book, written by one of philosophy's preeminent logicians, argues that many of the basic assumptions common to logic, philosophy of mathematics and metaphysics are in need of change. Jaakko Hintikka proposes a new basic first-order logic and uses it to explore the foundations of mathematics. This new logic enables logicians to express on the first-order level such concepts as equicardinality, infinity, and truth in the same language. Hintikka's new logic is highly original and will prove appealing to logicians, philosophers of mathematics, and mathematicians concerned with the foundations of the discipline.

Information is a central topic in computer science, cognitive science, and philosophy. In spite of its importance in the "information age," there is no consensus on what information is, what makes it possible, and what it means for one medium to carry information about another. Drawing on ideas from mathematics, computer science, and philosophy, this book addresses the definition and place of information in society. The authors, observing that information flow is possible only within a connected distribution system, provide a mathematically rigorous, philosophically sound foundation for a science of information. They illustrate their theory by applying it to a wide range of phenomena, from file transfer to DNA, from quantum mechanics to speech act theory.

The genesis of the digital idea and why it transformed civilization A few short decades ago, we were informed by the smooth signals of analog television and radio; we communicated using our analog telephones; and we even computed with analog computers. Today our world is digital, built with zeros and ones. Why did this revolution occur? The Discrete Charm of the Machine explains, in an engaging and accessible manner, the varied physical and logical reasons behind this radical transformation. The spark of individual genius shines through this story of innovation: the stored program of Jacquard’s loom; Charles Babbage’s logical branching; Alan Turing’s brilliant abstraction of the discrete machine; Harry Nyquist’s foundation for digital signal processing; Claude Shannon’s breakthrough insights into the meaning of information and bandwidth; and Richard Feynman’s prescient proposals for nanotechnology and quantum computing. Ken Steiglitz follows the progression of these ideas in the building of our digital world, from the internet and artificial intelligence to the edge of the unknown. Are questions like the famous traveling salesman problem truly beyond the reach of ordinary digital computers? Can quantum computers transcend these barriers? Does a mysterious magical power reside in the analog mechanisms of the brain? Steiglitz concludes by confronting the moral and aesthetic questions raised by the development of artificial intelligence and autonomous robots. The Discrete Charm of the Machine examines why our information technology, the lifeblood of our civilization, became digital, and challenges us to think about where its future trajectory may lead.