Beyond being the first substantial publication on Fourier, this work contains the text of Fourier's seminal paper of 1807 on the propagation of heat, marking the first time it has ever appeared in print. This paper incorporates many of the mathematical creations on which Fourier's fame rests, including derivation of the diffusion equation, the separation of the treatment of surface phenomena from internal phenomena, the use of boundary values and initial conditions, and the development of "Fourier series" and the so-called (...) "Bessel functions."When submitted to the examiners of the Institut de France, the originality of the paper and the surprising nature of some of its mathematical revelations caused great controversy, and it was denied publication both in 1807 and in later years. Fourier had the support, among the examiners, of Laplace and Monge, but Lagrange was adamantly in opposition, so that Fourier's work did not appear in print until 1822, reworked into book form.Fourier's mathematical discoveries are intimately related to his interest in the solution of physical problems and their experimental verification. The mathematical methods he developed in connection with heat diffusion apply to physical situations far beyond the boundaries of this area. Generally, Fourier may be credited with one of the first major extensions of mathematical physics beyond the applications of Newton's laws of motion and universal gravitation.The opening biographical chapter of this book follows Fourier's career up to the submission of the 1807 paper, and the two closing chapters take up his life and work from that point on. Fourier had strong political motivations and spent much of his life in the public service. These chapters trace his political difficulties, both before and after 1807, when he was the prefect of a department of France and was subjected to the dislocations of Napoleon's ups and downs. These chapters also describe aspects of the turbulent but productive development of French science from the Revolution to 1830.The core of the book presents the paper of 1807 in its original French and with the original notation. Grattan-Guinness has divided the paper into sections by the sequence of the problems taken up, and he introduces and, where necessary, closes each section with commentary relevant to Fourier's later work in these areas. The paper itself (cllows the chronology of Fourier's discoveries, and among the topics treated are, in this order: heat diffusion between disjoint bodies and in continuous bodies; the appearance of partial differential equations; the special solution for the lamina; sine and cosine series for an arbitrary function; reflections on the vibrating string problem; solution for the annulus; the full Fourier series for an arbitrary function; reflections on n-body analysis; solution for the sphere; solution for the cylinder; steady-state diffusion in the rectangular prism; time-dependent diffusion in the cube; and Fourier's experimental work. (shrink)

George Boole (1815-1864) is well known to mathematicians for his research and textbooks on the calculus, but his name has spread world-wide for his innovations in symbolic logic and the development and applications made since his day. The utility of "Boolean algebra" in computing has greatly increased curiosity in the nature and extent of his achievements. His work is most accessible in his two books on logic, "A mathematical analysis of logic" (1947) and "An investigation of the laws of thought" (...) (1954). But at various times he wrote manuscript essays, especially after the publication of the second book; several were intended for a non-technical work, "The Philosophy of logic", which he was not able to complete. This volume contains an edited selection which not only relates them to Boole's publications and the historical context of his time, but also describes their strange history of family, followers and scholars have treid to confect an edition. The book will appeal to logicians, mathematicians and philosophers, and those interested in the histories of the corresponding subjects; and also students of the early Victorian Britain in which they were written. (shrink)

The Scottish logician Hugh MacColl is well known for his innovative contributions to modal and nonclassical logics. However, until now little biographical information has been available about his academic and cultural background, his personal and professional situation, and his position in the scientific community of the Victorian era. The present article reports on a number of recent findings.

Introduction Contrary to a widespread assumption the modern history of modal logic did not start with C. I. Lewis’ Survey of Symbolic Logic [Lewis 1918]. His eminent work was preceded by some 20 years by H. MacColl’s fifth article on ‘The Calculus of Equivalent Statements’. This article was read at the London Mathematical Society on 12 November 1896. Some months later it was published in the Society’s Proceedings [MacColl 1896-1897]. During the following years MacColl presented his logic prim...

Introduction Contrary to a widespread assumption the modern history of modal logic did not start with C. I. Lewis’ Survey of Symbolic Logic [Lewis 1918]. His eminent work was preceded by some 20 years by H. MacColl’s fifth article on ‘The Calculus of Equivalent Statements’. This article was read at the London Mathematical Society on 12 November 1896. Some months later it was published in the Society’s Proceedings [MacColl 1896-1897]. During the following years MacColl presented his logic prim...

The Companion Encyclopedia is the first comprehensive work to cover all the principal lines and themes of the history and philosophy of mathematics from ancient times up to the twentieth century. In 176 articles contributed by 160 authors of 18 nationalities, the work describes and analyzes the variety of theories, proofs, techniques, and cultural and practical applications of mathematics. The work's aim is to recover our mathematical heritage and show the importance of mathematics today by treating its interactions with the (...) related disciplines of physics, astronomy, engineering and philosophy. It also covers the history of higher education in mathematics and the growth of institutions and organizations connected with the development of the subject. Part 1 deals with mathematics in various ancient and non-Western cultures from antiquity up to medieval and Renaissance times. Part 2 treats developments in all the main areas of mathematics during the medieval and Renaissance periods up to and including the early 17th century. Parts 3-10 are divided into the main branches into which mathematics developed from the early 17th century onwards: calculus and mathematical analysis, logic and foundations, algebras, geometries, mechanics, mathematical physics and engineering, and probability and statistics. Parts 11-13 review the history of mathematics from an international perspective. The teaching of mathematics in higher education is examined in various countries, and mathematics in culture, art and society is covered. The Companion Encyclopedia features annotated bibliographies of both classic and contemporary sources; black and white illustrations, line figures and equations; biographies of major mathematicians and historians and philosophers of mathematics; a chronological table of main events in the developments of mathematics; and a fully integrated index of people, events and topics. (shrink)

Dans la seconde moitié des années 1900, Bertrand Russell et Hugh MacColl échangèrent sans s’entendre sur les questions de l’implication et de l’existence, dans le cadre d’un débat plus général sur la nature de la logique. Il est tentant de voir dans cet échange une opposition entre le moniste logique Russell et le pluraliste MacColl. Dans cet article, j’affirme que cette interprétation est inexacte, et que les deux hommes étaient tous deux monistes, bien qu’ayant des allégeances différentes. La transition du (...) monisme au pluralisme ne s’effectue en réalité qu’à partir du début des années 1910, peu après la mort de MacColl en 1909. Les premiers signes de cette transition sont à trouver particulièrement chez le philosophe américain C. I. Lewis, le mathématicien néerlandais L. E. J. Brouwer, et le logicien polonais Jan Łukasiewicz. Ces auteurs sont des exemples de l’avènement graduel de la métalogique. (shrink)

Dans la seconde moitié des années 1900, Bertrand Russell et Hugh MacColl échangèrent sans s’entendre sur les questions de l’implication et de l’existence, dans le cadre d’un débat plus général sur la nature de la logique. Il est tentant de voir dans cet échange une opposition entre le moniste logique Russell et le pluraliste MacColl. Dans cet article, j’affirme que cette interprétation est inexacte, et que les deux hommes étaient tous deux monistes, bien qu’ayant des allégeances différentes. La transition du (...) monisme au pluralisme ne s’effectue en réalité qu’à partir du début des années 1910, peu après la mort de MacColl en 1909. Les premiers signes de cette transition sont à trouver particulièrement chez le philosophe américain C. I. Lewis, le mathématicien néerlandais L. E. J. Brouwer, et le logicien polonais Jan Łukasiewicz. Ces auteurs sont des exemples de l’avènement graduel de la métalogique. (shrink)

Este artículo presenta un alnplio panorama histórico de las conexiones existentes entre ramas de las matematícas y tipos de lógica durante el periodo 1800-1914. Se observan dos corrientes principales,bastante diferentes entre sí: la lógica algebraica, que hunde sus raíces en la logique yen las algebras de la época revolucionaria francesa y culmina, a través de Boole y De Morgan, en los sistemas de Peirce y de Schröder; y la lógica matematíca, que tiene una fuente de inspiraeión en el analisis matemático (...) de Cauchy y de Weierstrass y culmina, a través de las inieiativas de Peano y de la teoria de conjuntos deCantor, en la obra de Russell. Se extraen algunas conclusiones generales, con referencias relativas a la situaeión posterior a 1914.This article contains a broad historical survey of the connections made between branches of mathematics and types of logic during the period 1800-1914. Two principal streams are noted, rather different from each other: algebraic logic, rooted in French Revolutionary logique and algebras and culminating, via Boole and De Morgan, in the systems of Peirce and Schröder; and mathematical logic, inspired by the mathematical analysis of Cauchy and Weierstrass and culminating, via the initiatives of Peano and the set theory of Cantor, in the work of Russell. Some general conclusions are drawn, with examples given of the state of affairs after 1914. (shrink)

We seek means of distinguishing logical knowledge from other kinds of knowledge, especially mathematics. The attempt is restricted to classical two-valued logic and assumes that the basic notion in logic is the proposition. First, we explain the distinction between the parts and the moments of a whole, and theories of ?sortal terms?, two theories that will feature prominently. Second, we propose that logic comprises four ?momental sectors?: the propositional and the functional calculi, the calculus of asserted propositions, and rules for (...) (in)valid deduction, inference or substitution. Third, we elaborate on two neglected features of logic: the various modes of negating some part(s) of a proposition R, not only its ?external? negation not-R; and the assertion of R in the pair of propositions ?it is (un)true that R? belonging to the neglected logic of asserted propositions, which is usually left unstated. We also address the overlooked task of testing the asserted truth-value of R. Fourth, we locate logic among other foundational studies: set theory and other theories of collections, metamathematics, axiomatisation, definitions, model theory, and abstract and operator algebras. Fifth, we test this characterisation in two important contexts: the formulation of some logical paradoxes, especially the propositional ones; and indirect proof-methods, especially that by contradiction. The outcomes differ for asserted propositions from those for unasserted ones. Finally, we reflect upon self-referring self-reference, and on the relationships between logical and mathematical knowledge. A subject index is appended. (shrink)

Russell argued against the Great War, but he also wanted to drop atomic bombs on the Soviet Union after World War II, and later he advocated nuclear disarmament. How could a great logician accommodate such inconsistencies? How, as a private citizen, did he make such a world-wide impact in his late years?

In 1803 Louis Poinsot published a textbook on statics, in which he made clear that the subject dealt not only with forces but also with 'couples' (his word), pairs of coplanar non-collinear forces equal in magnitude and direction but opposite in sense. His innovation was not understood or even welcomed by some contemporary mathematicians. Later he adapted his theory to put forward a new relationship between rectilinear and rotational motion in dynamics; its reception was more positive, although not always appreciative (...) of the generality. After summarising the creation of these two theories and noting their respective receptions, this paper considers his advocacy of spatial and geometrical thinking in mechanics and the fact that, despite its importance, historians of statics who cover his period usually ignore his theory of couples. (shrink)

Este artículo presenta un alnplio panorama histórico de las conexiones existentes entre ramas de las matematícas y tipos de lógica durante el periodo 1800-1914. Se observan dos corrientes principales,bastante diferentes entre sí: la lógica algebraica, que hunde sus raíces en la logique yen las algebras de la época revolucionaria francesa y culmina, a través de Boole y De Morgan, en los sistemas de Peirce y de Schröder; y la lógica matematíca, que tiene una fuente de inspiraeión en el analisis matemático (...) de Cauchy y de Weierstrass y culmina, a través de las inieiativas de Peano y de la teoria de conjuntos deCantor, en la obra de Russell. Se extraen algunas conclusiones generales, con referencias relativas a la situaeión posterior a 1914.This article contains a broad historical survey of the connections made between branches of mathematics and types of logic during the period 1800-1914. Two principal streams are noted, rather different from each other: algebraic logic, rooted in French Revolutionary logique and algebras and culminating, via Boole and De Morgan, in the systems of Peirce and Schröder; and mathematical logic, inspired by the mathematical analysis of Cauchy and Weierstrass and culminating, via the initiatives of Peano and the set theory of Cantor, in the work of Russell. Some general conclusions are drawn, with examples given of the state of affairs after 1914. (shrink)

Popper emphasised both the problem-solving nature of human knowledge, and the need to criticise a scientific theory as strongly as possible. These aims seem to contradict each other, in that the former stresses the problems that motivate scientific theories while the one ignores the character of the problems that led to the formation of the theories against which the criticism is directed. A resolution is proposed in which problems as such are taken as prime in the search for knowledge, and (...) subject to discussion. This approach is then applied to the problem of induction. Popper set great stake to his solution of it, but others doubted its legitimacy, in ways that are clarified by changing the form of the induction problem itself. That change draws upon logic, which is the subject of another application: namely, in contrast to Popper’s adhesion to classical logic as the only welcome form (because of the maximal strength of criticism that it dispenses), can other logics be used without abandoning his philosophy of criticism? (shrink)

It is with profound sadness that the editors note the passing on 12 December 2014 of Professor Ivor Grattan-Guinness. Born 23 June 1941, Ivor was the founding editor of History and Philosophy of Lo...