This volume contains the official record of the Congress of Mathematicians held in Edinburgh from 14 to 21 August 1958.

Today the notion of the algorithm is familiar not only to mathematicians. It forms a conceptual base for information processing; the existence of a corresponding algorithm makes automatic information processing possible. The theory of algorithms (together with mathematical logic ) forms the the oretical basis for modern computer science (see [Sem Us 86]; this article is called "Mathematical Logic in Computer Science and Computing Practice" and in its title mathematical logic is understood in a broad sense including the theory of algorithms). However, not everyone realizes that the word "algorithm" includes a transformed toponym Khorezm. Algorithms were named after a great sci entist of medieval East, is al-Khwarizmi (where al-Khwarizmi means "from Khorezm"). He lived between c. 783 and 850 B.C. and the year 1983 was chosen to celebrate his 1200th birthday. A short biography of al-Khwarizmi compiled in the tenth century starts as follows: "al-Khwarizmi. His name is Muhammad ibn Musa, he is from Khoresm" (cited according to [Bul Rozen Ah 83, p.8]).

The Arabic contribution is fundamental to the history of science, mathematics and technology, but until now no single publication has offered an up-to-date synthesis of knowledge in this area. In three fully-illustrated volumes the Encyclopedia of the History of Arabic Science documents the history and philosophy of Arabic science from the earliest times to the present day. The set as a whole covers seven centuries. Thirty chapters, written by an international team of specialists from Europe, America, the Middle East and Russia cover such areas as astronomy, mathematics, music, engineering, nautical science and scientific institutions.

The Dictionary of Modern American Philosophers includes both academic and non-academic philosophers, anda large number of female and minority thinkers whose work has been neglected. It includes those intellectualsinvolved in the development of psychology, pedagogy, sociology, anthropology, education, theology, politicalscience, and several other fields, before these disciplines came to be considered distinct from philosophy in thelate nineteenth century.Each entry contains a short biography of the writer, an exposition and analysis of his or her doctrines and ideas, abibliography of writings, and suggestions for further reading. While all the major post-Civil War philosophers arepresent, the most valuable feature of this dictionary is its coverage of a huge range of less well-known writers,including hundreds of presently obscure thinkers. In many cases, the Dictionary of Modern AmericanPhilosophers offers the first scholarly treatment of the life and work of certain writers. This book will be anindispensable reference work for scholars working on almost any aspect of modern American thought.

The book presents the history of ICMI trough a prosopographical approach. In other words, it pays a lot of attention to the actors of the International movement. The portraits of the members of the ICMI Central Committees (1908-1936) and ICMI Executive Committees (1952-2008), and other eminent figures in ICMI history, who have passed away in the first 100 years of its life, are the guiding thread of the volume. Each portrait includes: · Biographical information · An outline of the various contributions made by the individual in question to the study of problems pertaining to mathematics teaching/education · Primary bibliography · Secondary with particular attention to the publications concerning the teaching of mathematics · Images: photos, book frontispieces, relevant manuscripts The authors of the portraits (30 altogether) are researchers in the history of mathematics, mathematics, and mathematics education. The focus on the officer’s role within ICMI and on his/her contributions to mathematics education, make the portraits different from usual biographies. In particular, since most officers were active mathematicians, the portraits shed light on aspects of their lesser-known activity. Connecting chapters place the action of these figures in the historical context and in the different phases of ICMI history.

Peu aprt!s la mort d'E. W. Beth, j'ai pu organiser a Paris, a l'Institut Henri Poincare, un Colloque a sa memoire. Je suis heureux qu'aujourd'hui les Actes de ce Colloque soient publies en un volume qui materialise l'hom mage que ses amis, collegues et eleves ont voulu lui rendre. Ce Colloque international qui groupait des specialistes de nationalites diverses appartenant a de nombreuses disciplines, a fait ressortir la ri chesse et la portee de l'oeuvre d'E. W. Beth et Ie rayonnement de sa personnalite. Que Madame Beth veuille bien trouver ici l'hommage que no us adres sons tous a la memoire de son mario C'est grace aux soins de la Maison d'edition D. Reidel que ce volume a pu etre realise sous une forme digne de celui que no us voulons honorer. Paris, mai 1967 J.-L. DES TOUCHES v TABLE OF CONTENTS Preface, by Jean-Louis Destouches v G.H. VON WRIGHT / Memorial Address 1 J.-L. DES TOUCHES / Allocution 2 A. HEYTING / Remarques sur la theorie intuitionniste des espaces lineaires 4 K.L. DE BOUVERE / Some Remarks about Synonymity and the Theorem of Beth 10 J.J.F. NIELAND / Beth's Tableau-Method 19 MARCEL GUILLAUME / Quelques remarques sur les "Tableaux de Beth" 39 ROLAND FRAisSE / Une hypothese sur I'extension des relations finies et sa verification dans certaines classes particulieres (deuxieme partie) 46 J.J.A.

Kurt Gödel (1906 - 1978) was the most outstanding logician of the twentieth century, famous for his hallmark works on the completeness of logic, the incompleteness of number theory, and the consistency of the axiom of choice and the continuum hypothesis. He is also noted for his work on constructivity, the decision problem, and the foundations of computability theory, as well as for the strong individuality of his writings on the philosophy of mathematics. He is less well known for his discovery of unusual cosmological models for Einstein's equations, in theory permitting time travel into the past. The Collected Works is a landmark resource that draws together a lifetime of creative thought and accomplishment. The first two volumes were devoted to Gödel's publications in full (both in original and translation), and the third volume featured a wide selection of unpublished articles and lecture texts found in Gödel's Nachlass. These long-awaited final two volumes contain Gödel's correspondence of logical, philosophical, and scientific interest. Volume IV covers A to G, with H to Z in volume V; in addition, Volume V contains a full inventory of Gödel's Nachlass. All volumes include introductory notes that provide extensive explanatory and historical commentary on each body of work, English translations of material originally written in German (some transcribed from the Gabelsberger shorthand), and a complete bibliography of all works cited. Kurt Gödel: Collected Works is designed to be useful and accessible to as wide an audience as possible without sacrificing scientific or historical accuracy. The only comprehensive edition of Gödel's work available, it will be an essential part of the working library of professionals and students in logic, mathematics, philosophy, history of science, and computer science and all others who wish to be acquainted with one of the great minds of the twentieth century.

The aim of the first volume of the present Handbook of Philosophical Logic is essentially two-fold: First of all, the chapters in this volume should provide a concise overview of the main parts of classical logic. Second, these chapters are intended to present all the relevant background material necessary for the understanding of the contributions which are to follow in the next three volumes. We have thought it to be of importance that the connections between classical logic and its 'extensions' (covered in Volume 11) as well as its most important 'alternatives' (covered in Volume Ill) be brought out clearly from the start. The first chapter presents a clear and detailed picture of the range of what is generally taken to be the standard logical framework, namely, predicate (or first-order quantificational) logic. On the one hand, this chapter surveys both propositionai logic and first-order predicate logic and, on the other hand, presents the main metalogical results obtained for them. Chapter 1. 1 also contains a discussion of the limits of first-order logic, i. e. it presents an answer to the question: Why has predicate logic played such a formidable role in the formalization of mathematics and in the many areas of philo sophical and linguistic applications? Chapter 1. 1 is prerequisite for just about all the other chapters in the entire Handbook, while the other chapters in Volume I provide more detailed discussions of material developed or hinted at in the first chapter.

In this collection of essays written over a period of twenty years, Solomon Feferman explains advanced results in modern logic and employs them to cast light on significant problems in the foundations of mathematics. Most troubling among these is the revolutionary way in which Georg Cantor elaborated the nature of the infinite, and in doing so helped transform the face of twentieth-century mathematics. Feferman details the development of Cantorian concepts and the foundational difficulties they engendered. He argues that the freedom provided by Cantorian set theory was purchased at a heavy philosophical price, namely adherence to a form of mathematical platonism that is difficult to support. Beginning with a previously unpublished lecture for a general audience, Deciding the Undecidable, Feferman examines the famous list of twenty-three mathematical problems posed by David Hilbert, concentrating on three problems that have most to do with logic. Other chapters are devoted to the work and thought of Kurt Gödel, whose stunning results in the 1930s on the incompleteness of formal systems and the consistency of Cantors continuum hypothesis have been of utmost importance to all subsequent work in logic. Though Gödel has been identified as the leading defender of set-theoretical platonism, surprisingly even he at one point regarded it as unacceptable. In his concluding chapters, Feferman uses tools from the special part of logic called proof theory to explain how the vast part--if not all--of scientifically applicable mathematics can be justified on the basis of purely arithmetical principles. At least to that extent, the question raised in two of the essays of the volume, Is Cantor Necessary?, is answered with a resounding no. This volume of important and influential work by one of the leading figures in logic and the foundations of mathematics is essential reading for anyone interested in these subjects.