This book grew out of a graduate student paper [261] in which I set down some criticisms of J. R. Lucas' attempt to refute mechanism by means of G6del's theorem. I had made several such abortive attempts myself and had become familiar with their pitfalls, and especially with the double edged nature of incompleteness arguments. My original idea was to model the refutation of mechanism on the almost universally accepted G6delian refutation of Hilbert's formalism, but I kept getting stuck on questions of mathematical philosophy which I found myself having to beg. A thorough study of the foundational works of Hilbert and Bernays finally convinced me that I had all too naively and uncritically bought this refutation of formalism. I did indeed discover points of surprisingly close contact between formalism and mechanism, but also that it was possible to under mine certain strong arguments against these positions precisely by invok ing G6del's and related work. I also began to realize that the Church Turing thesis itself is the principal bastion protecting mechanism, and that G6del's work was perhaps the best thing that ever happened to both mechanism and formalism. I pushed these lines of argument in my dis sertation with the patient help of my readers, Raymond Nelson and Howard Stein. I would especially like to thank the latter for many valuable criticisms of my dissertation as well as some helpful suggestions for reor ganizing it in the direction of the present book.

Mathematics education research has blossomed into many different areas which we can see in the programmes of the ICME conferences as well as in the various survey articles in the Handbooks. However, all of these lines of research are trying to grapple with a common problem, the complexity of the process of learning mathematics. Although our knowledge of the process is more extensive and deeper despite the fragmented nature of research in this area, there is still a need to overcome this fragmentation and to see learning as one process with different aspects. To overcome this fragmentation, this book identifies six themes: (1) mathematics, culture and society, (2) the structure of mathematics and its influence on the learning process, (3) mathematics learning as a cognitive process, (4) mathematics learning as a social process, (5) affective conditions of the mathematics learning process, (6) new technologies and mathematics learning. This book is addressed to all researchers in mathematic education. It gives an orientation and overview by addressing some carefully chosen questions on what is going on and what are the main results and questions what are important books or papers if further information is needed.

This Biographical Dictionary provides detailed accounts of the lives, works, influence and reception of thinkers from all the major philosophical schools and traditions of the twentieth-century. This unique volume covers the lives and careers of thinkers from all areas of philosophy - from analytic philosophy to Zen and from formal logic to aesthetics. All the major figures of philosophy, such as Nietzsche, Wittgenstein and Russell are examined and analysed. The scope of the work is not merely restricted to the major figures in western philosophy but also covers in depth a significant number of thinkers from the near and far east and from the non-European Hispanic-language communities. The Biographical Dictionary also includes a number of general entries dealing with important schools of philosophy, such as the Vienna Circle, or currents of thought, such as vitalism. These allow the reader to set the individual biographies in the context of the philosophical history of the period. With entries written by over 100 leading philosophy scholars, the Biographical Dictionary is the most comprehensive survey of twentieth-century thinkers to date. Structure The book is structured alphabetically by philosopher. Each entry is identically structured for ease of access and covers: * nationality * dates and places of birth and death * philosophical style or school * areas of interest * higher education * significant influences * main appointments * main publications * secondary literature * account of intellectual development and main ideas * critical reception and impact At the end of the book a glossary gives accounts of the schools, movements and traditions to which these philosophers belonged, and thorough indexes enable the reader to access the information in several ways: * by nationality * by major areas of contribution to philosophy e.g. aesthetics * by major influences on the thinker concerned e.g. Plato, Kant, Wittgenstein

Language and Philosophical Problems investigates problems about mind, meaning and mathematics rooted in preconceptions of language. It deals in particular with problems which are connected with our tendency to be misled by certain prevailing views and preconceptions about language. Philosophical claims made by theorists of meaning are scrutinized and shown to be connected with common views about the nature of certain mathematical notions and methods. Drawing in particular on Wittgenstein's ideas, Sren Stenlund demonstrates a strategy for tracing out and resolving conceptual and philosophical problems. By a critical examination of examples from different areas of philosophy, he shows that many problems arise through the transgression of the limits of the use of technical concepts and formal methods. Many prima facie different kinds of problems are shown to have common roots, and should thus be dealt and resolved together. Such an approach is usually prevented by the influence of traditional philosophical terminology and classification. The results of this investigation make it clear that the received ways of subdividing the subject matter of philosophy often conceal the roots of the problem.

Newton/Descartes. Einstein/Gödel. The seventeenth century had its scientific and philosophical geniuses. Why shouldn't ours have them as well? Kurt Gödel was indisputably one of the greatest thinkers of our time, and in this first extended treatment of his life and work, Hao Wang, who was in close contact with Gödel in his last years, brings out the full subtlety of Gödel's ideas and their connection with grand themes in the history of mathematics and philosophy. The subjects he covers include the completeness of elementary logic, the limits of formalization, the problem of evidence, the concept of set, the philosophy of mathematics, time, and relativity theory, metaphysics and religion, as well as general ideas on philosophy as a worldview. Wang, whose reflections on his colleague also serve to clarify his own philosophical thoughts, distinguishes his ideas from those of Gödel's and on points of agreement develops Gödel's views further. The book provides a generous array of information on and interpretation of the two main phases of Gödel's career - the years between 1924 and 1939 at the University of Vienna, which were marked by intense mathematical creativity, and the period from 1940 to his death in 1978, during which he was affiliated with the Institute for Advanced Studies in Princeton, a time in which Gödel's interests steadily shifted from questions of logic to metaphysics. And it also examines Gödel's relations with the Vienna Circle, his philosophical differences with Carnap and Wittgenstein, the intimate and mutually fruitful friendship with Einstein, and the periodic bouts of depression for which Gödel was hospitalized a number of times over the course of his life. A Bradford Book.

Die beiden ursprünglich 1992 veröffentlichten Bände liegen nun in zusammengefaßter Paperback-Form vor. Reality Rules beleuchten die Syntax und die Semantik der Sprache, in der mathematische Modellierungsregeln niedergelegt werden. Eine Vielzahl von Beispielen zeigt praktische Anwendungen auf; auch ein Lösungsband zur Unterstützung des Selbststudiums ist erhältlich.

In this book, Livingston develops the political implications of formal results obtained over the course of the twentieth century in set theory, metalogic, and computational theory. He argues that the results achieved by thinkers such as Cantor, Russell, Godel, Turing, and Cohen, even when they suggest inherent paradoxes and limitations to the structuring capacities of language or symbolic thought, have far-reaching implications for understanding the nature of political communities and their development and transformation. Alain Badiou's analysis of logical-mathematical structures forms the backbone of his comprehensive and provocative theory of ontology, politics, and the possibilities of radical change. Through interpretive readings of Badiou's work as well as the texts of Giorgio Agamben, Jacques Lacan, Jacques Derrida, Gilles Deleuze, and Ludwig Wittgenstein, Livingston develops a formally based taxonomy of critical positions on the nature and structure of political communities. These readings, along with readings of Parmenides and Plato, show how the formal results can transfigure two interrelated and ancient problems of the One and the Many: the problem of the relationship of a Form or Idea to the many of its participants, and the problem of the relationship of a social whole to its many constituents.

These proceedings comprise the invited lectures and an edited sampling of few other contributions to the 4th Teaching Modern Physics Conference held in Badajoz (Spain) on July 1992, devoted to THERMODYNAMICS AND STATISTICAL PHYSICS: CRITICAL PHENOMENA, PHASE TRANSITIONS, NONLINEAR EVOLUTION, FRACTALS, COMPLEXITY,… COMPUTER SIMULATIONS forms the core of the contents.