Our much-valued mathematical knowledge rests on two supports: the logic of proof and the axioms from which those proofs begin. Naturalism in Mathematics investigates the status of the latter, the fundamental assumptions of mathematics. These were once held to be self-evident, but progress in work on the foundations of mathematics, especially in set theory, has rendered that comforting notion obsolete. Given that candidates for axiomatic status cannot be proved, what sorts of considerations can be offered for or against them? That is the central question addressed in this book. One answer is that mathematics aims to describe an objective world of mathematical objects, and that axiom candidates should be judged by their truth or falsity in that world. This promising view-realism-is assessed and finally rejected in favour of another-naturalism-which attends less to metaphysical considerations of objective truth and falsity, and more to practical considerations drawn from within mathematics itself. Penelope Maddy defines this naturalism, explains the motivation for it, and shows how it can be helpfully applied in the assessment of candidates for axiomatic status in set theory. Maddy's clear, original treatment of this fundamental issue is informed by current work in both philosophy and mathematics, and will be accessible and enlightening to readers from both disciplines.

Originally published in 1980. What is time? How is its structure determined? The enduring controversy about the nature and structure of time has traditionally been a diametrical argument between those who see time as a container into which events are placed, and those for whom time cannot exist without events. This controversy between the absolutist and the relativist theories of time is a central theme of this study. The author's impressive arguments provide grounds for rejecting both these theories, firstly by establishing that 'empty' time is possible, and secondly by showing, through a discussion of the structure of time which involves considering whether time might be cyclical, branching, beginning or non-beginning, that the absolutist theory of time is untenable. This book then advances two new theories, and succeeds in shifting the traditional debate about time to a consideration of time as a theoretical structure and as a theoretical framework.

Originally published in 1964. This lively, challenging book, written with enthusiasm, conviction and clarity, sets out to elucidate the shadowy concept of Time. This involves central philosophical issues, which are vigorously discussed. Also relativity theory, in a clear-cut exposition, is made intelligible in a new light. All who are interested in science and its philosophical implications will find this book highly controversial but certainly readable. The author believes philosophy to be important, not only for its professionals, but for everyman. He believes that the fact that this is no longer realised shows that something is wrong with professional philosophy; he also indicates what this is. The book ends, surprisingly but pertinently, with a bold plunge into the questions of telepathy, precognition and psychical research generally. Whilst the phenomena are reasonably admitted, trenchant criticism of their significance confronts parapsychologists.

The Taming of the True poses a broad challenge to the realist views of meaning and truth that have been prominent in recent philosophy. Neil Tennant starts with a careful critical survey of the realism debate, guiding the reader through its complexities; he then presents a sustained defence of the anti-realist view that every truth is knowable in principle, and that grasp of meaning must be able to be made manifest. Sceptical arguments for the indeterminacy or non-factuality of meaning are countered; and the much-maligned notion of analyticity is reinvestigated and rehabilitated. Tennant goes on to show that an effective logical system can be based on his anti-realist view; the logical system that he advocates is justified as a body of analytic truths and inferential principles. Having laid the foundations for global semantic anti-realism, Tennant moves to the world of empirical understanding, and gives an account of the cognitive credentials of natural scientific discourse. He shows that the same canon of constructive and relevant inference suffices both for intuitionistic mathematics and for empirical science. This is an ambitious and contentious book which aims to reform not only theory of meaning, but our deductive practices across a broad range of discourses.

Originally published in 1937. This book is a classic work on the philosophy of time, looking at the pshychology, physics and logic of time before investigating the views of Kant, Bergson, Alexander, McTaggart and Dunne. The second half of the book contains more indepth consideration of prediction, the concepts of past and future, and reality.

The first European Congress of Mathematics was held in Paris from July 6 to July 10, 1992, at the Sorbonne and Pantheon-Sorbonne universities. It was hoped that the Congress would constitute a symbol of the development of the community of European nations. More than 1,300 persons attended the Congress. The purpose of the Congress was twofold. On the one hand, there was a scientific facet which consisted of forty-nine invited mathematical lectures that were intended to establish the state of the art in the various branches of pure and applied mathematics. This scientific facet also included poster sessions where participants had the opportunity of presenting their work. Furthermore, twenty four specialized meetings were held before and after the Congress. The second facet of the Congress was more original. It consisted of sixteen round tables whose aim was to review the prospects for the interactions of mathe matics, not only with other sciences, but also with society and in particular with education, European policy and industry. In connection with this second goal, the Congress also succeeded in bringing mathematics to a broader public. In addition to the round tables specifically devoted to this question, there was a mini-festival of mathematical films and two mathematical exhibits. Moreover, a Junior Mathematical Congress was organized, in parallel with the Congress, which brought together two hundred high school students."

The Third Kurt G-del Symposium, KGC'93, held in Brno, Czech Republic, August1993, is the third in a series of biennial symposia on logic, theoretical computer science, and philosophy of mathematics. The aim of this meeting wasto bring together researchers working in the fields of computational logic and proof theory. While proof theory traditionally is a discipline of mathematical logic, the central activity in computational logic can be foundin computer science. In both disciplines methods were invented which arecrucial to one another. This volume contains the proceedings of the symposium. It contains contributions by 36 authors from 10 different countries. In addition to 10 invited papers there are 26 contributed papers selected from over 50 submissions.

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.

There are things we routinely say that may strike us as literally false but that we are nonetheless reluctant to give up. This might be something mundane, like the way we talk about the sun setting in the west (it is the earth that moves), or it could be something much deeper, like engaging in talk that is ostensibly about numbers despite believing that numbers do not literally exist. Rather than regard such behaviour as self-defeating, a "fictionalist" is someone who thinks that this kind of discourse is entirely appropriate, even helpful, so long as we treat what is said as a useful fiction, rather than as the sober truth. "Fictionalism" can be broadly understood as a view that uses a notion of pretense or fiction in order to resolve certain puzzles or problems that otherwise do not necessarily have anything to do with literature or fictional creations. Within contemporary analytic philosophy, fictionalism has been on the scene for well over a decade and has matured during that time, growing in popularity. There are now myriad competing views about fictionalism and consequently the discussion has branched out into many more subdisciplines of philosophy. Yet there is widespread disagreement on what philosophical fictionalism actually amounts to and about how precisely it ought to be pursued. This volume aims to guide these discussions, collecting some of the most up-to-date work on fictionalism and tracing the view's development over the past decade. After a detailed discussion in the book's introductory chapter of how philosophers should think of fictionalism and its connection to metaontology more generally, the remaining chapters provide readers with arguments for and against this view from leading scholars in the fields of epistemology, ethics, metaphysics, philosophy of science, philosophy of language, and others.

This is a volume of essays and reviews that delightfully explores mathematics in all its moods - from the light and the witty, and humorous to serious, rational, and cerebral. These beautifully written articles from three great modern mathematicians will provide a source for supplemental reading for almost any math class. Topics include: logic, combinatorics, statistics, economics, artificial intelligence, computer science, and broad applications of mathematics. Readers will also find coverage of history and philosophy, including discussion of the work of Ulam, Kant, and Heidegger, among others.

Originally published in 1969. This book is for undergraduates whether specializing in philosophy or not. It assumes no previous knowledge of logic but aims to show how logical notions arise from, or are abstracted from, everyday discourse, whether technical or non-technical. It sets out a knowledge of principles and, while not historical, gives an account of the reasons for which modern systems have emerged from the traditional syllogistic logic, demonstrating how certain central ideas have developed. The text explains the connections between everyday reasoning and formal logic and works up to a brief sketch of systems of propositional calculus and predicate-calculus, using both the axiomatic method and the method of natural deduction. It provides a self-contained introduction but for those who intend to study the subject further it contains many suggestions and a sound basis for more advanced study.

This book deals with a crucial period in the formation of twentieth-century analytic philosophy. It discusses the tradition of British Idealism, and the rejection of that tradition by Bertrand Russell and G. E. Moore at the beginning of this century. It goes on to examine the very influential work of Russell in the period up to the First World War, and addresses the question of what we can learn about the nature of analytic philosophy through a close examination of its origins.

"Creative mathematicians seldom write for outsiders, but when they do, they usually do it well. Jerry King is no exception. His informal, nontechnical book, as its title implies, is organized around what Bertrand Russell called the 'supreme beauty' of mathematics--a beauty 'capable of a stern perfection such as only the greatest art can show.'"
NATURE
In this clear, concise, and superbly written volume, mathematics professor and poet Jerry P. King reveals the beauty that is at the heart of mathematics--and he makes that beauty accessible to all readers. Darting wittily from Euclid to Yeats, from Poincare to Rembrandt, from axioms to symphonies, THE ART OF MATHEMATICS explores the difference between real, rational, and complex numbers; analyzes the intellectual underpinnings of pure and applied mathematics; and reveals the fundamental connection between aesthetics and mathematics. King also sheds light on how mathematicians pursue their research and how our educational system perpetuates the damaging divisions between the "two cultures."

When engaged in mathematics, most people tend to think of themselves as scientists investigating the features of real mathematical things, and the wildly successful application of mathematics in the physical sciences reinforces this picture of mathematics as an objective study. For philosophers, however, this realism about mathematics raises serious questions: What are mathematical things? Where are they? How do we know about them? Penelope Maddy delineates and defends a novel version of mathematical realism that answers the traditional questions and refocuses philosophical attention on the pressing foundational issues of contemporary mathematics.

Chihara here develops a mathematical system in which there are no existence assertions but only assertions of the constructibility of certain sorts of things. He utilizes this system in the analysis of the nature of mathematics, and discusses many recent works in the philosophy of mathematics from the viewpoint of the constructibility theory developed. This innovative analysis will appeal to mathematicians and philosophers of logic, mathematics, and science.

This volume contains several invited papers as well as a selection of the other contributions. The conference was the first meeting of the Soviet logicians interested in com- puter science with their Western counterparts. The papers report new results and techniques in applications of deductive systems, deductive program synthesis and analysis, computer experiments in logic related fields, theorem proving and logic programming. It provides access to intensive work on computer logic both in the USSR and in Western countries.

Die preisgekronte Biographie des norwegische Schriftstellers Atle Naess fuhrt den Leser auf eine fesselnde Reise durch die Hohen und Tiefen des Lebens einer der schillerndsten Personlichkeiten der europaischen Wissenschaftsgeschichte - Galileo Galilei. Mit feinsinniger Empathie entwickelt Naess das Portrait eines Mannes, der sich selbst durch die Zwange der romischen Inquisition nicht von seinen wegweisenden Forschungen abbringen liess.

Aus den Rezensionen der norwegischen Ausgabe:

"Mit umfassender Kenntnis und sicherem Erzahlstil hebt Naess die epochemachenden Arbeiten hervor, die die Grundlage der modernen experimentellen Naturwissenschaften bilden. Er packt all die vielen Stationen Galileis] Lebens in ein sehr lesenswertes Buch, das in vielerlei Hinsicht hervorsticht."

Per Anders Madsen, Aftenposten Morgen

"Diese Biographie stellt eine faszinierende kulturhistorische Studie dar und ist daher nicht nur fur Leser mit Interesse an Naturwissenschaft und Wissenschaftsgeschichte geeignet. Sie kann auch hervorragend als Roman gelesen werden."

Atle Abelsen, Teknisk Ukeblad

"

M and N, the last two books of the Metaphysics, are Aristotle's only sustained venture into the philosophy of mathematics. In them, he criticizes Plato's theories and suggests alternatives of his own. This commentary concentrates on the continuing philosophical interest of these books rather than on scholarly controversies, and will provide a clear introduction for students, including those without Greek, to an unjustly neglected part of Aristotle's work. This paperback edition replaces the outstandingly successful hardback. 'Dr Annas's translation is clear, readable, and accurate...an enjoyable volume, stimulating both as intellectual history and as philosophical argument.' Times Literary Supplement

Hermann Grassmann, Gymnasiallehrer in Stettin und bekannt als Begrunder der n-dimensionalen Vektoralgebra, erwarb sich auch in der Physik und der Sprachforschung bleibende Verdienste. Gestutzt auf die Dialektik Schleiermachers entwickelte er in seinem Hauptwerk, der Ausdehnungslehre, mit philosophischer Methode eine vollig neue mathematische Disziplin. Zunachst von der Fachwelt abgelehnt, wurde sein Werk Jahrzehnte spater als wegweisend gefeiert. Die Biographie geht dem komplexen Geflecht innerer und ausserer Einflusse nach, innerhalb derer Grassmann sein Schopfertum entfaltete."

In this book, David Stump traces alternative conceptions of the a priori in the philosophy of science and defends a unique position in the current debates over conceptual change and the constitutive elements in science. Stump emphasizes the unique epistemological status of the constitutive elements of scientific theories, constitutive elements being the necessary preconditions that must be assumed in order to conduct a particular scientific inquiry. These constitutive elements, such as logic, mathematics, and even some fundamental laws of nature, were once taken to be a priori knowledge but can change, thus leading to a dynamic or relative a priori. Stump critically examines developments in thinking about constitutive elements in science as a priori knowledge, from Kant's fixed and absolute a priori to Quine's holistic empiricism. By examining the relationship between conceptual change and the epistemological status of constitutive elements in science, Stump puts forward an argument that scientific revolutions can be explained and relativism can be avoided without resorting to universals or absolutes.

The world around us is saturated with numbers. They are a fundamental pillar of our modern society, and accepted and used with hardly a second thought. But how did this state of affairs come to be? In this book, Leo Corry tells the story behind the idea of number from the early days of the Pythagoreans, up until the turn of the twentieth century. He presents an overview of how numbers were handled and conceived in classical Greek mathematics, in the mathematics of Islam, in European mathematics of the middle ages and the Renaissance, during the scientific revolution, all the way through to the mathematics of the 18th to the early 20th century. Focusing on both foundational debates and practical use numbers, and showing how the story of numbers is intimately linked to that of the idea of equation, this book provides a valuable insight to numbers for undergraduate students, teachers, engineers, professional mathematicians, and anyone with an interest in the history of mathematics.

First published in 1982, this reissue contains a critical exposition of the views of Frege, Dedekind and Peano on the foundations of arithmetic. The last quarter of the 19th century witnessed a remarkable growth of interest in the foundations of arithmetic. This work analyses both the reasons for this growth of interest within both mathematics and philosophy and the ways in which this study of the foundations of arithmetic led to new insights in philosophy and striking advances in logic. This historical-critical study provides an excellent introduction to the problems of the philosophy of mathematics - problems which have wide implications for philosophy as a whole. This reissue will appeal to students of both mathematics and philosophy who wish to improve their knowledge of logic.

The volume is the first collection of essays that focuses on Gottlob Frege's Basic Laws of Arithmetic (1893/1903), highlighting both the technical and the philosophical richness of Frege's magnum opus. It brings together twenty-two renowned Frege scholars whose contributions discuss a wide range of topics arising from both volumes of Basic Laws of Arithmetic. The original chapters in this volume make vivid the importance and originality of Frege's masterpiece, not just for Frege scholars but for the study of the history of logic, mathematics, and philosophy.

If mathematics is the purest form of knowledge, the perfect foundation of all the hard sciences, and a uniquely precise discipline, then how can the human brain, an imperfect and imprecise organ, process mathematical ideas? Is mathematics made up of eternal, universal truths? Or, as some have claimed, could mathematics simply be a human invention, a kind of tool or metaphor? These questions are among the greatest enigmas of science and epistemology, discussed at length by mathematicians, physicians, and philosophers. But, curiously enough, neuroscientists have been absent in the debate, even though it is precisely the field of neuroscience-which studies the brain's mechanisms for thinking and reasoning-that ought to be at the very center of these discussions. How our Emotions and Bodies are Vital for Abstract Thought explores the unique mechanisms of cooperation between the body, emotions, and the cortex, based on fundamental physical principles. It is these mechanisms that help us to overcome the limitations of our physiology and allow our imperfect, human brains to make transcendent mathematical discoveries. This book is written for anyone who is interested in the nature of abstract thought, including mathematicians, physicists, computer scientists, psychologists, and psychiatrists.