Michael Potter presents a comprehensive new philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. Potter offers a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates. He discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set theory. Potter offers a strikingly simple version of the most widely accepted response to the paradoxes, which classifies sets by means of a hierarchy of levels. What makes the book unique is that it interweaves a careful presentation of the technical material with a penetrating philosophical critique. Potter does not merely expound the theory dogmatically but at every stage discusses in detail the reasons that can be offered for believing it to be true. Set Theory and its Philosophy is a key text for philosophy, mathematical logic, and computer science.

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.

David Corfield provides a variety of innovative approaches to research in the philosophy of mathematics. His study ranges 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 program; and the ways in which new concepts are justified. This highly original book will challenge philosophers as well as mathematicians to develop the broadest and most complete philosophical resources for research in their disciplines.

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.

Philosophy of Science Today offers a state-of-the-art guide to this fast-developing area. 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.

Marcus Giaquinto tells the story of one of the great intellectual adventures of the modern era -- the attempt to find firm foundations for mathematics. From the late nineteenth century to the present day, this project has stimulated some of the most original and influential work in logic and philosophy.

This book studies the important issue of the possibility of conceptual change--a possibility traditionally denied by logicians--from the perspective of philosophy of mathematics. The author also looks at aspects of language, and his conclusions have implications for a theory of concepts, truth and thought. The book will appeal to readers in the philosophy of mathematics, logic, and the philosophy of mind and language.

Charles Chihara gives a thorough critical exposition of modal realism, the philosophical doctrine that there exist many possible worlds of which the actual world--the universe in which we live--is just one. The striking success of possible-worlds semantics in modal logic has made this ontological doctrine attractive. Modal realists maintain that philosophers must accept the existence of possible worlds if they wish to have the benefit of using possible-worlds semantics to assess modal arguments and explain modal principles. Chihara challenges this claim, and argues instead for modality without worlds; he offers a new account of the role of interpretations or structures of the formal languages of logic.

How should we reason in science? Jan Sprenger and Stephan Hartmann offer a refreshing take on classical topics in philosophy of science, using a single key concept to explain and to elucidate manifold aspects of scientific reasoning. They present good arguments and good inferences as being characterized by their effect on our rational degrees of belief. Refuting the view that there is no place for subjective attitudes in 'objective science', Sprenger and Hartmann explain the value of convincing evidence in terms of a cycle of variations on the theme of representing rational degrees of belief by means of subjective probabilities (and changing them by Bayesian conditionalization). In doing so, they integrate Bayesian inference-the leading theory of rationality in social science-with the practice of 21st century science. Bayesian Philosophy of Science thereby shows how modeling such attitudes improves our understanding of causes, explanations, confirming evidence, and scientific models in general. It combines a scientifically minded and mathematically sophisticated approach with conceptual analysis and attention to methodological problems of modern science, especially in statistical inference, and is therefore a valuable resource for philosophers and scientific practitioners.

In this book, Balaguer demonstrates that there are no good arguments for or against mathematical platonism. He does this by establishing that both platonism and anti-platonism are defensible views. Introducing a form of platonism ("full-blooded platonism") that solves all problems traditionally associated with the view, he proceeds to defend anti-platonism (in particular, mathematical fictionalism) against various attacks, most notably the Quine-Putnam indispensability attack. He concludes by arguing that it is not simply that we do not currently have any good argument for or against platonism, but that we could never have such an argument and, indeed, that there is no fact of the matter as to whether platonism is correct.

The initial volume of a comprehensive edition of Gödel's works, this book makes available for the first time in a single source all his publications from 1929 to 1936. The volume begins with an informative overview of Gödel's life and work and features facing English translations for all German originals, extensive explanatory and historical notes, and a complete biography. Volume 2 will contain the remainder of Gödel's published work, and subsequent volumes will include unpublished manuscripts, lectures, correspondence and extracts from the notebooks.

Naturalism in Mathematics investigates how the most fundamental assumptions of mathematics can be justified. One prevalent philosophical approach to the problem--realism--is examined and rejected in favour of another approach--naturalism--which attends more closely to practical considerations drawn from within mathematics itself. Penelope Maddy defines naturalism, explains the motivation for it, and shows how it can be successfully applied in set theory. Her clear, original discussion is informed by current work in both philosophy and mathematics, and will be accessible and enlightening to readers from both disciplines.

Bayesian ideas have recently been applied across such diverse fields as philosophy, statistics, economics, psychology, artificial intelligence, and legal theory. Fundamentals of Bayesian Epistemology examines epistemologists' use of Bayesian probability mathematics to represent degrees of belief. Michael G. Titelbaum provides an accessible introduction to the key concepts and principles of the Bayesian formalism, enabling the reader both to follow epistemological debates and to see broader implications Volume 1 begins by motivating the use of degrees of belief in epistemology. It then introduces, explains, and applies the five core Bayesian normative rules: Kolmogorov's three probability axioms, the Ratio Formula for conditional degrees of belief, and Conditionalization for updating attitudes over time. Finally, it discusses further normative rules (such as the Principal Principle, or indifference principles) that have been proposed to supplement or replace the core five. Volume 2 gives arguments for the five core rules introduced in Volume 1, then considers challenges to Bayesian epistemology. It begins by detailing Bayesianism's successful applications to confirmation and decision theory. Then it describes three types of arguments for Bayesian rules, based on representation theorems, Dutch Books, and accuracy measures. Finally, it takes on objections to the Bayesian approach and alternative formalisms, including the statistical approaches of frequentism and likelihoodism.

Mathieu Marion offers a careful, historically informed study of Wittgenstein's philosophy of mathematics. This area of his work has frequently been undervalued by Wittgenstein specialists and philosophers of mathematics alike; but the surprising fact that he wrote more on this subject than any other indicates its centrality in his thought. Marion traces the development of Wittgenstein's thinking from the 1920s through to the 1950s, in the context of the mathematical and philosophical work of the times, to make coherent sense of ideas that have too often been misunderstood because they have been presented in a disjointed and incomplete way. He shows that study of Wittgenstein's writings on mathematics is essential to a proper understanding of his philosophy, and also that it can do much to illuminate current debates about the foundations of mathematics.

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.

The book treats two approaches to decision theory: (1) the normative, purporting to determine how a 'perfectly rational' actor ought to choose among available alternatives; (2) the descriptive, based on observations of how people actually choose in real life and in laboratory experiments. The mathematical tools used in the normative approach range from elementary algebra to matrix and differential equations. Sections on different levels can be studied independently. Special emphasis is made on 'offshoots' of both theories to cognitive psychology, theoretical biology, and philosophy.

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.

Proposing social constructivism as a novel philosophy of mathematics, this book is inspired by current work in sociology of knowledge and social studies of science. It extends the ideas of social constructivism to the philosophy of mathematics, developing a whole set of new notions. The outcome is a powerful critique of traditional absolutist conceptions of mathematics, as well as of the field of philosophy of mathematics itself. Proposed are a reconceptualization of the philosophy of mathematics and a new set of adequacy criteria.

The book offers novel analyses of the important but under-recognized contributions of Wittgenstein and Lakatos to the philosophy of mathematics. Building on their ideas, it develops a theory of mathematical knowledge and its relation to the social context. It offers an original theory of mathematical knowledge based on the concept of conversation, and develops the rhetoric of mathematics to account for proof in mathematics. Another novel feature is the account of the social construction of subjective knowledge, which relates the learning of mathematics to philosophy of mathematics via the development of the individual mathematician. It concludes by considering the values of mathematics and its social responsibility.

Mathematics as a Science of Patterns expounds a system of ideas about the nature of mathematics which Michael Resnik has been elaborating for a number of years. In calling mathematics a science he implies 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.

Leibniz's dispute with Newton over the physico-mathematical theories expounded in the Principia Mathematica (1687) have long been identified as a crucial episode in the history of science. Dr Bertoloni Meli examines several hitherto unpublished manuscripts in Leibniz's own hand illustrating his first reading of and reaction to Newton's Principia. Six of the most important manuscripts are here edited for the first time. Contrary to Leibniz's own claims, this new evidence shows that he had studied Newton's masterpiece before publishing An Essay on the Causes of Celestial Motions. This article, representing his response to Newton, is included here in English translation. "Bertoloni's book provides a very detailed and deep analysis of Leibniz's calculus and dynamics by focusing on a consistent and important set of previously unknown manuscripts..." Niccolo Guicciardini, Universita de Bologna "...Equivalence and Priority is a major contribution to our understanding of the development of mathematical physics in the late seventeenth and early eighteenth century." Daniel Garber, University of Chigago

Chinese Remainder Theorem, CRT, is one of the jewels of mathematics. It is a perfect combination of beauty and utility or, in the words of Horace, omne tulit punctum qui miscuit utile dulci. Known already for ages, CRT continues to present itself in new contexts and open vistas for new types of applications. So far, its usefulness has been obvious within the realm of "three C's". Computing was its original field of application, and continues to be important as regards various aspects of algorithmics and modular computations. Theory of codes and cryptography are two more recent fields of application.This book tells about CRT, its background and philosophy, history, generalizations and, most importantly, its applications. The book is self-contained. This means that no factual knowledge is assumed on the part of the reader. We even provide brief tutorials on relevant subjects, algebra and information theory. However, some mathematical maturity is surely a prerequisite, as our presentation is at an advanced undergraduate or beginning graduate level. We have tried to make the exposition innovative, many of the individual results being new. We will return to this matter, as well as to the interdependence of the various parts of the book, at the end of the Introduction.A special course about CRT can be based on the book. The individual chapters are largely independent and, consequently, the book can be used as supplementary material for courses in algorithmics, coding theory, cryptography or theory of computing. Of course, the book is also a reference for matters dealing with CRT.

This book will present a collection of mathematical problems - lighthearted in nature - intended to entertain the general readership. Problems will be selected largely for the unusual and unexpected solutions to which they lend themselves. Some interesting contents included: All in all, the book is meant to entertain the general readership and to convince them about the power and beauty of mathematics.

Chemists, working with only mortars and pestles, could not get very far unless they had mathematical models to explain what was happening "inside" of their elements of experience -- an example of what could be termed mathematical learning.
This volume contains the proceedings of Work Group 4: Theories of Mathematics, a subgroup of the Seventh International Congress on Mathematical Education held at Universite Laval in Quebec. Bringing together multiple perspectives on mathematical thinking, this volume presents elaborations on principles reflecting the progress made in the field over the past 20 years and represents starting points for understanding mathematical learning today. This volume will be of importance to educational researchers, math educators, graduate students of mathematical learning, and anyone interested in the enterprise of improving mathematical learning worldwide.