During his lifetime, Henri Poincare published three major philosophical books which achieved great success: "La science et l'hypothese" (1902), "La valeur de la science" (1905) and "Science et methode" (1908). After his death in 1913, a fourth volume of his philosophical works was published by his heirs as "Dernieres pensees" (1913). The four books constitute the core of Poincare's philosophic works and were given an ovation by scientific and general public. Around 1919, Gustave Le Bon wrote to Poincare's widow. As the director of the "Bibliotheque de Philosophie Scientifique at Flammarion," he asked her permission to publish a second posthumous volume. "L'Opportunisme scientifique" was intended to be the fifth and final volume of Poincare's philosophical writings. Louis Rougier had elaborated the project, with the collaboration of Gustave Le Bon, and the approval of the philosopher Emile Boutroux and his son Pierre. Because of the reservations of the mathematician's heirs, this book was never published and Dernieres pensees remained his last philosophical book. Nevertheless Poincare's correspondence - which is kept in the Poincare Archives at University Nancy 2 - contains a large amount of documents concerning the project, its justification and the discussions between Louis Rougier and the mathematician's heirs. The aim of this book is to restore this episode, which gives some crucial informations about editorial practices of Poincare and about the posterity of his philosophic thinking."

The chief purpose of the book is to present, in detail, a compilation of proofs of the Cantor-Bernstein Theorem (CBT) published through the years since the 1870's. Over thirty such proofs are surveyed.

The book comprises five parts. In the first part the discussion covers the role of CBT and related notions in the writings of Cantor and Dedekind. New views are presented, especially regarding the general proof of CBT obtained by Cantor, his proof of the Comparability Theorem, the ruptures in the Cantor-Dedekind correspondence and the origin of Dedekind's proof of CBT.

The second part covers the first CBT proofs published (1896-1901). The works of the following mathematicians is considered in detail: Schroder, Bernstein, Bore, Schoenflies and Zermelo. Here a subtheme of the book is launched; it concerns the research project following Bernstein's Division Theorem (BDT).

In its third part the book covers proofs that emerged during the period when the logicist movement was developed (1902-1912). It covers the works of Russell and Whitehead, Jourdain, Harward, Poincare, J. Konig, D. Konig (his results in graph theory), Peano, Zermelo, Korselt. Also Hausdorff's paradox is discussed linking it to BDT.

In the fourth part of the book are discussed the developments of CBT and BDT (including the inequality-BDT) in the hands of the mathematicians of the Polish School of Logic, including Sierpi ski, Banach, Tarski, Lindenbaum, Kuratowski, Sikorski, Knaster, the British Whittaker, and Reichbach.

Finally, in the fifth part, the main discussion concentrates on the attempts to port CBT to intuitionist mathematics (with results by Brouwer, Myhill, van Dalen and Troelstra) and to Category Theory (by Trnkova and Koubek).The second purpose of the book is to develop a methodology for the comparison of proofs. The core idea of this methodology is that a proof can be described by two descriptors, called gestalt and metaphor. It is by comparison of their descriptors that the comparison of proofs is obtained. The process by which proof descriptors are extracted from a proof is named 'proof-processing', and it is conjectured that mathematicians perform proof-processing habitually, in the study of proofs.

Vito Volterra (1860-1940) was one of the most famous representatives of Italian science in his day. Angelo Guerragio and Giovanni Paolini analyze Volterra s most important contributions to mathematics and their applications, as well as his outstanding organizational achievements in scientific policy. Volterra was one of the founding fathers of functional analysis and the author of fundamental contributions in the field of integral equations, elasticity theory and population dynamics (Lotka-Volterra model). He delivered keynote lectures on the occasion of the International Congresses of Mathematicians held in Paris (1900), Rome (1908), Strasbourg (1920) and Bologna (1928). He became involved in the scientific development in united Italy and was appointed senator of the kingdom in 1905. One of his numerous non-mathematical activities was founding the National Research Council (Consiglio Nazionale delle Ricerche, CNR).During the First World War he was active in military research. After the war he took a clear stand against fascism, which was the starting point for his exclusion. In 1926 he resigned as president of the world famous Accademia Nazionale dei Lincei and was later on excluded from the academy. In 1931 he was one of the few university lecturers who denied to swear an oath of allegiance to the fascistic regime. In 1938 he suffered from the impact of the racial laws. The authors draw a comprehensive picture of Vito Volterra, both as a great mathematician and an organizer of science.

A glorious period of Hungarian mathematics started in 1900 when Lipot Fejer discovered the summability of Fourier series.This was followed by the discoveries of his disciples in Fourier analysis and in the theory of analytic functions. At the same time Frederic (Frigyes) Riesz created functional analysis and Alfred Haar gave the first example of wavelets. Later the topics investigated by Hungarian mathematicians broadened considerably, and included topology, operator theory, differential equations, probability, etc. The present volume, the first of two, presents some of the most remarkable results achieved in the twentieth century by Hungarians in analysis, geometry and stochastics.

The book is accessible to anyone with a minimum knowledge of mathematics. It is supplemented with an essay on the history of Hungary in the twentieth century and biographies of those mathematicians who are no longer active. A list of all persons referred to in the chapters concludes the volume. "

This systematic and historical treatment of Russell's contributions to analytic philosophy, from his embrace of analysis in 1898 to his landmark theory of descriptions in 1905, draws important connections between his philosophically motivated conception of analysis and the technical apparatus he devised to facilitate analyses in mathematics

Viewed as a flashpoint of the Scientific Revolution, early modern astronomy witnessed a virtual explosion of ideas about the nature and structure of the world. This study explores these theories in a variety of intellectual settings, challenging our view of modern science as a straightforward successor to Aristotelian natural philosophy. It shows how astronomers dealt with celestial novelties by deploying old ideas in new ways and identifying more subtle notions of cosmic rationality. Beginning with the celestial spheres of Peurbach and ending with the evolutionary implications of the new star Mira Ceti, it surveys a pivotal phase in our understanding of the universe as a place of constant change that confirmed deeper patterns of cosmic order and stability.

To mark the centenary of the 1910 to 1913 publication of the monumental Principia Mathematica by Alfred N. Whitehead and Bertrand Russell, this collection of fifteen new essays by distinguished scholars considers the influence and history of PM over the last hundred years.

In 1915 and 1916 Emmy Noether was asked by Felix Klein and David Hilbert to assist them in understanding issues involved in any attempt to formulate a general theory of relativity, in particular the new ideas of Einstein. She was consulted particularly over the difficult issue of the form a law of conservation of energy could take in the new theory, and she succeeded brilliantly, finding two deep theorems.

But between 1916 and 1950, the theorem was poorly understood and Noether's name disappeared almost entirely. People like Klein and Einstein did little more then mention her name in the various popular or historical accounts they wrote. Worse, earlier attempts which had been eclipsed by Noether's achievements were remembered, and sometimes figure in quick historical accounts of the time.

This book carries a translation of Noether's original paper into English, and then describes the strange history of its reception and the responses to her work. Ultimately the theorems became decisive in a shift from basing fundamental physics on conservations laws to basing it on symmetries, or at the very least, in thoroughly explaining the connection between these two families of ideas. The real significance of this book is that it shows very clearly how long it took before mathematicians and physicists began to recognize the seminal importance of Noether's results. This book is thoroughly researched and provides careful documentation of the textbook literature. Kosmann-Schwarzbach has thus thrown considerable light on this slow dance in which the mathematical tools necessary to study symmetry properties and conservation laws were apparently provided long before the orchestra arrives and the party begins."

This fifth volume of A History of Arabic Sciences and Mathematics is complemented by four preceding volumes which focused on the main chapters of classical mathematics: infinitesimal geometry, theory of conics and its applications, spherical geometry, mathematical astronomy, etc. This book includes seven main works of Ibn al-Haytham (Alhazen) and of two of his predecessors, Thabit ibn Qurra and al-Sijzi: The circle, its transformations and its properties; Analysis and synthesis: the founding of analytical art; A new mathematical discipline: the Knowns; The geometrisation of place; Analysis and synthesis: examples of the geometry of triangles; Axiomatic method and invention: Thabit ibn Qurra; The idea of an Ars Inveniendi: al-Sijzi. Including extensive commentary from one of the world's foremost authorities on the subject, this fundamental text is essential reading for historians and mathematicians at the most advanced levels of research.

The work of Professor Eduard Cech had a si~ificant influence on the development of algebraic and general topology and differential geometry. This book, which appears on the occasion of the centenary of Cech's birth, contains some of his most important papers and traces the subsequent trends emerging from his ideas. The body of the book consists of four chapters devoted to algebraic topology, Cech-Stone compactification, dimension theory and differential geometry. Each of these includes a selection of Cech's papers, a brief summary of some results which followed from his work or constituted solutions to the problems he posed, and several selected papers by various authors concerning the areas of study he initiated. The book also contains a concise biography borrowed with minor changes from the book Topological papers of E. tech, a list of Cech's publications and a very brief note on his activity in the didactics of mathematics. The editors wish to express their sincere gratitude to all who contributed to the completion and publication of this book.

This study discusses the history of the central limit theorem and related probabilistic limit theorems from about 1810 through 1950. In this context the book also describes the historical development of analytical probability theory and its tools, such as characteristic functions or moments. The central limit theorem was originally deduced by Laplace as a statement about approximations for the distributions of sums of independent random variables within the framework of classical probability, which focused upon specific problems and applications.

Making this theorem an autonomous mathematical object was very important for the development of modern probability theory.

- Following on from the 2000 edition of Jan De Witt's Elementa Curvarum Linearum, Liber Primus, this book provides the accompanying translation of the second volume of Elementa Curvarum Linearum (Foundations of Curved Lines). One of the first books to be published on Analytic Geometry, it was originally written in Latin by the Dutch statesman and mathematician Jan de Witt, soon after Descartes' invention of the subject.

- Born in 1625, Jan de Witt served with distinction as Grand Pensionary of Holland for much of his adult life. In mathematics, he is best known for his work in actuarial mathematics as well as extensive contributions to analytic geometry.

- Elementa Curvarum Linearum, Liber Secondus moves forward from the construction of the familiar conic sections covered in the Liber Primus, with a discussion of problems connected with their classification; given an equation, it covers how one can recover the standard form, and additionally how one can find the equation's geometric properties.

- This volume, begun by Albert Grootendorst (1924-2004) and completed after his death by Jan Aarts, Reinie Erne and Miente Bakker, is supplemented by:

- annotation explaining finer points of the translation;

- extensive commentary on the mathematics These features make the work of Jan de Witt broadly accessible to today's mathematicians."

This volume develops the major themes of time series analysis from its formal beginnings in the early part of the 20th century to the present day through the research of six distinguished British statisticians, all of whose work is characterised by the British traits of pragmatism and the desire to solve practical problems of importance.

A new translation makes this classic and important text more generally accessible. The text is placed in its contemporary context, but also related to the interests of practising mathematicians today. This book will be of interest to mathematical historians, researchers, and numerical analysts.

The implicit function theorem is part ofthe bedrock of mathematical analysis and geometry. Finding its genesis in eighteenth century studies of real analytic functions and mechanics, the implicit and inverse function theorems have now blossomed into powerful tools in the theories of partial differential equations, differential geometry, and geometric analysis.

There are many different forms of the implicit function theorem, including (i) the classical formulation for"Ck"functions, (ii) formulations in other function spaces, (iii) formulations for non-smooth function, and (iv) formulations for functions with degenerate Jacobian. Particularly powerful implicit function theorems, such as the Nash Moser theorem, have been developed for specific applications (e.g., the imbedding of Riemannian manifolds). All of these topics, and many more, are treated in the present uncorrected reprint of this classicmonograph.

Originally published in 2002, "The Implicit Function Theorem"is an accessible and thorough treatment of implicit and inverse function theorems and their applications. It will be of interest to mathematicians, graduate/advanced undergraduate students, and to those who apply mathematics. The book unifies disparate ideas that have played an important role in modern mathematics. It serves to document and placein context a substantial body of mathematical ideas. "

This clearly written and enlightening textbook provides a concise, introductory guide to the key mathematical concepts and techniques used by computer scientists. Topics and features: ideal for self-study, offering many pedagogical features such as chapter-opening key topics, chapter introductions and summaries, review questions, and a glossary; places our current state of knowledge within the context of the contributions made by early civilizations, such as the ancient Babylonians, Egyptians and Greeks; examines the building blocks of mathematics, including sets, relations and functions; presents an introduction to logic, formal methods and software engineering; explains the fundamentals of number theory, and its application in cryptography; describes the basics of coding theory, language theory, and graph theory; discusses the concept of computability and decideability; includes concise coverage of calculus, probability and statistics, matrices, complex numbers and quaternions.

Moritz Pasch (1843-1930) is justly celebrated as a key figure in the history of axiomatic geometry. Less well known are his contributions to other areas of foundational research. This volume features English translations of 14 papers Pasch published in the decade 1917-1926. In them, Pasch argues that geometry and, more surprisingly, number theory are branches of empirical science; he provides axioms for the combinatorial reasoning essential to Hilbert's program of consistency proofs; he explores "implicit definition" (a generalization of definition by abstraction) and indicates how this technique yields an "empiricist" reconstruction of set theory; he argues that we cannot fully understand the logical structure of mathematics without clearly distinguishing between decidable and undecidable properties; he offers a rare glimpse into the mind of a master of axiomatics, surveying in detail the thought experiments he employed as he struggled to identify fundamental mathematical principles; and much more. This volume will: Give English speakers access to an important body of work from a turbulent and pivotal period in the history of mathematics, help us look beyond the familiar triad of formalism, intuitionism, and logicism, show how deeply we can see with the help of a guide determined to present fundamental mathematical ideas in ways that match our human capacities, will be of interest to graduate students and researchers in logic and the foundations of mathematics.

The editors of the present series had originally intended to publish an integrated work on the history of mathematics in the nineteenth century, passing systemati cally from one discipline to another in some natural order. Circumstances beyond their control, mainly difficulties in choosing authors, led to the abandonment of this plan by the time the second volume appeared. Instead of a unified mono graph we now present to the reader a series of books intended to encompass all the mathematics of the nineteenth century, but not in the order of the accepted classification of the component disciplines. In contrast to the first two books of The Mathematics of the Nineteenth Century, which were divided into chapters, this third volume consists of four parts, more in keeping with the nature of the publication. 1 We recall that the first book contained essays on the history of mathemati 2 cal logic, algebra, number theory, and probability, while the second covered the history of geometry and analytic function theory. In the present third volume the reader will find: 1. An essay on the development of Chebyshev's theory of approximation of functions, later called "constructive function theory" by S. N. Bernshtein. This highly original essay is due to the late N. I. Akhiezer (1901-1980), the author of fundamental discoveries in this area. Akhiezer's text will no doubt attract attention not only from historians of mathematics, but also from many specialists in constructive function theory."

The significance of foundational debate in mathematics that took place in the 1920s seems to have been recognized only in circles of mathematicians and philosophers. A period in the history of mathematics when mathematics and philosophy, usually so far away from each other, seemed to meet. The foundational debate is presented with all its brilliant contributions and its shortcomings, its new ideas and its misunderstandings.

Philanthropies funded by the Rockefeller family have been prominent in the social history of the twentieth century for their involvement in medicine and applied science. This book provides the first detailed study of their relatively brief but nonetheless influential foray into the field of mathematics. The careers of a generation of pathbreakers in modern mathematics, such as S.Banach, B.L.van der Waerden and Andre Weil, were decisively affected by their becoming fellows of the Rockefeller-funded International Education Board in the 1920s. To help promote cooperation between physics and mathematics Rockefeller funds supported the erection of the new Mathematical Institute in Gottingen between 1926 and 1929, while the rise of probability and mathematical statistics owes much to the creation of the Institut Henri Poincare in Paris by American philanthropy at about the same time. This account draws upon the documented evaluation processes behind these personal and institutional involvements of philanthropies. It not only sheds light on important events in the history of mathematics and physics of the 20th century but also analyzes the comparative developments of mathematics in Europe and the United States. Several of the documents are given in their entirety as significant witnesses to the gradual shift of the centre of world mathematics to the USA. This shift was strengthened by the Nazi purge of German and European mathematics after 1933 to which the Rockefeller Foundation reacted with emergency programs that subsequently contributed to the American war effort. The general historical and political background of the events discussed in this book is the mixture of competition and cooperation between the various European countries and the USA after World War I, and the consequences of the Nazi dictatorship after 1933. Ideological positions of both the philanthropists and mathematicians mattered heavily in that process. Cultural bias in the selection of fellows and of disciplines supported, and the economic predominance of American philanthropy, led among other things to a restriction of the programs to Europe and America, to an uneven consideration of European candidates, and to preferences for Americans. Political self-isolation of the Soviet Union contributed to an increasing alienation of that important mathematical culture from Western mathematics. By focussing on a number of national cultures the investigation aims to represent a step toward a true inter-cultural comparison in mathematics."

The controversy between the wave theory and the emission theory of light early in the nineteenth century has been a subject of numerous studies. Yet many is sues remain unclear, in particular, the reasons for rejecting Young's theory of light. It appears that further progress in the field requires a better grasp of the overall situation in optics and related subjects at the time and a more thorough study of every factor suggested to be of importance for the dispute. This book is intended to be a step in this direction. It examines the impact of the concept of interference of light on the development of the early nineteenth century optics in general, and the theory of light, in particular. This is not a his tory of the wave theory of light, nor is it a history of the debate on the nature of light in general: it covers only that part of the controversy which involved the concept of interference. Although the book deals with a number of scientists, scientific institutions, and journals, its main character is a scientific concept, the principle of interference. While discussing the reasons for accepting or rejecting this concept I have primarily focused on scientific factors, although in some cases the human factor is examined as well. The book is a revised Ph. D. dissertation (University of Minnesota, 1984) writ ten under Alan E. Shapiro."

To attempt to compile a relatively complete bibliography of the theory of functions of a real variable with the requisite bibliographical data, to enumer ate the names of the mathematicians who have studied this subject, exhibit their fundamental results, and also include the most essential biographical data about them, to conduct an inventory of the concepts and methods that have been and continue to be applied in the theory of functions of a real variable ... in short, to carry out anyone of these projects with appropriate completeness would require a separate book involving a corresponding amount of work. For that reason the word essays occurs in the title of the present work, allowing some freedom in the selection of material. In justification of this selection, it is reasonable to try to characterize to some degree the subject to whose history these essays are devoted. The truth of the matter is that this is a hopeless enterprise if one requires such a characterization to be exhaustively complete and concise. No living subject can be given a final definition without provoking some objections, usually serious ones. But if we make no such claims, a characterization is possible; and if the first essay of the present book appears unconvincing to anyone, the reason is the personal fault of the author, and not the objective necessity of the attempt."

This collection of historical research studies covers the evolution of technology as knowledge, the emergence of an autonomous engineering science in the Industrial Age, the idea of scientific managment of production and operation systems, and the interaction between mathematical models and technological concepts.

The book is published with the support of the UNESCO Venice Office - Regional Office for Science & Technology in Europe as an activity of the Project: The evolution of events, concepts and models in engineering systems.

The present work represents a unique undertaking in scientific publishing to honor Nick Metropolis, who passed away in October, 1999. Nick was the last survivor of the Manhattan Project that began during World War II in Los Alamos and later became the Los Alamos National Laboratory. In this volume, some of the leading scientists and humanists of our time have contributed essays related to their respective disciplines exploring various aspects of future developments in science, technology, and society. Speculations on the future developments of science and society, philosophy, national security, nuclear power, pure and applied mathematics, physics and biology, particle physics, computing, information science, among many others, are included. Contributors include: H. Agnew * R. Ashenhurst * K. Baclawski * G. Baker * N. Balazs * J.A. Freed * R. Hamming * M. Hawrylycz * O. Judd * D. Kleitman * M. Krieger * N. Krikorian * P. Lax * J.D. Louck * T. Puck * M. Raju * R. Richtmyer * J. Schwartz * R. Sokolowski * E. Teller * M. Waterman