Richard T. W. Arthur

Reductio arguments are notoriously inconclusive, a fact which no doubt contributes to their great fecundity. For once a contradiction has been proved, it is open to interpretation which premise should be given up. Indeed, it is often a matter of great creativity to identify what can be consistently given up. A case in point is a traditional paradox of the infinite provided by Galileo Galilei in his Two New Sciences, which has since come to be known as Galileo’s Paradox. It concerns the set of all numbers, N: since to every number there is a corresponding square, there are as many squares as numbers. But since there are non-squares between the squares, “all numbers, comprising the squares and the non-squares, are greater than the squares alone”, i.e. there must be fewer numbers in the set of all squares S than in N. Thus N is both equal to and greater than S. This is a contradiction, so one of the premises must be given up: the question is, which one?

In contrast with some recent theories of infinitesimals as non-Archimedean entities, Leibniz’s mature interpretation was fully in accord with the Archimedean Axiom: infinitesimals are fictions, whose treatment as entities incomparably smaller than finite quantities is justifiable wholly in terms of variable finite quantities that can be taken as small as desired, i.e. syncategorematically. In this paper I explain this syncategorematic interpretation, and how Leibniz used it to justify the calculus. I then compare it with the approach of Smooth Infinitesimal Analysis, as propounded by John Bell. I find some salient differences, especially with regard to higher-order infinitesimals. I illustrate these differences by a consideration of how each approach might be applied to propositions of Newton’s Principia concerning the derivation of force laws for bodies orbiting in a circle and an ellipse. “If the Leibnizian calculus needs a rehabilitation because of too severe treatment by historians in the past half century, as Robinson suggests (1966, 250), I feel that the legitimate grounds for such a rehabilitation are to be found in the Leibnizian theory itself.”—(Bos 1974–1975, 82–83).

Few philosophers have left a legacy like that of Gottfried Wilhelm Leibniz. He has been credited not only with inventing the differential calculus, but also with anticipating the basic ideas of modern logic, information science, and fractal geometry. He made important contributions to such diverse fields as jurisprudence, geology and etymology, while sketching designs for calculating machines, wind pumps, and submarines. But the common presentation of his philosophy as a kind of unworldly idealism is at odds with all this bustling practical activity. In this book Richard. T. W. Arthur offers a fresh reading of Leibniz’s philosophy, clearly situating it in its scientific, political and theological contexts. He argues that Leibniz aimed to provide an improved foundation for the mechanical philosophy based on a new kind of universal language. His contributions to natural philosophy are an integral part of this programme, which his metaphysics, dynamics and organic philosophy were designed to support. Rather than denying that substances really exist in space and time, as the idealist reading proposes, Leibniz sought to provide a deeper understanding of substance and body, and a correct understanding of space as an order of situations and time as an order of successive things. This lively and approachable book will appeal to students of philosophy, as well as anyone seeking a stimulating introduction to Leibniz's thought and its continuing relevance

This is the first volume compiling English translations of Leibniz's journal articles on natural philosophy, presenting a selection of 26 articles, only three of which have appeared before in English translation. It also includes in full Leibniz's public controversies with De Catelan, Papin, and Hartsoeker. The articles include work in optics, on the fracture strength of materials, and on motion in a resisting medium, and Leibniz's pioneering applications of his calculus to these issues by construing them as mini-max and inverse tangent problems. Other topics covered by the articles include: criticisms of the Cartesian estimate of motive force and Leibniz's proposal of a different way of estimating force to replace it; a proposed theory of celestial motions and gravitation, and derivation of the inverse square law; challenge problems concerning the isochronous curve and the catenary; a sample of work on gaming theory; and Leibniz's critique of atomism.

This book gathers together for the first time an important body of texts written between 1672 and 1686 by the great German philosopher and polymath Gottfried Leibniz. These writings, most of them previously untranslated, represent Leibniz's sustained attempt on a problem whose solution was crucial to the development of his thought, that of the composition of the continuum. The volume begins with excerpts from Leibniz's Paris writings, in which he tackles such problems as whether the infinite division of matter entails perfect points, whether matter and space can be regarded as true wholes, whether motion is truly continuous, and the nature of body and substance. Comprising the second section is Pacidius Philalethi, Leibniz's brilliant dialogue of late 1676 on the problem of the continuity of motion. In the selections of the final section, from his Hanover writings of 1677-1686, Leibniz abandons his earlier transcreationism and atomism in favor of the theory of corporeal substance, where the reality of body and motion is founded in substantial form or force.Leibniz's texts (one in French, the rest in Latin) are presented with facing-page English translations, together with an introduction, notes, appendixes containing related excerpts from earlier works by Leibniz and his predecessors, and a valuable glossary detailing important terms and their translations.

In lively and readable prose, Arthur presents a new approach to the study of logic, one that seeks to integrate methods of argument analysis developed in modern “informal logic” with natural deduction techniques. The dry bones of logic are given flesh by unusual attention to the history of the subject, from Pythagoras, the Stoics, and Indian Buddhist logic, through Lewis Carroll, Venn, and Boole, to Russell, Frege, and Monty Python. A previous edition of this book appeared under the title _Natural Deduction_. This new edition adds clarifications of the notions of explanation, validity and formal validity, a more detailed discussion of derivation strategies, and another rule of inference, Reiteration.

Mario Bunge wrote his classic Causality and Modern Science more than 60 years ago, and a third revised edition was published by Dover in 1979. With its impressive scope and historical perspective it was a long way ahead of its time. But many of its insights still have not been sufficiently appreciated by physicists and philosophers alike. These include Bunge’s distinction between causation and other types of determination, his critique of the still-dominant Humean accounts of causality as leaving out the productive aspect of determination, his critique of the conflation of determination with predictability, his insistence on the fictional character of isolated causal chains, and his demonstration that “causal connectability” depends only on the principle of retarded action, not causation. It will be argued that there is a degree of agreement between his views on determinism and causality and those of Leibniz, and a comparison between the two thinkers’ views is used to throw further light on these matters.

Leibniz never tired of stressing the fundamental importance of the concept of continuity for philosophy, nor was he shy of attributing major importance to his own struggle through “the labyrinth of the continuum” for the subsequent development of his whole system of thought. Unfortunately, however, his own thought on the subject is something of a labyrinth itself, and from a modern point of view many of his pronouncements are apt to seem blatantly contradictory.Certain quotations seem to commit him unambiguously to atomism. Thus to de Voider he writes: “Matter is not continuous, but discrete…. The same holds for changes, which are not truly continuous.” (To de Voider, 11th October 1705: G.II.279).2.

In lively and readable prose, Arthur presents a new approach to the study of logic, one that seeks to integrate methods of argument analysis developed in modern “informal logic” with natural deduction techniques. The dry bones of logic are given flesh by unusual attention to the history of the subject, from Pythagoras, the Stoics, and Indian Buddhist logic, through Lewis Carroll, Venn, and Boole, to Russell, Frege, and Monty Python. A previous edition of this book appeared under the title _Natural Deduction_. This new edition adds clarifications of the notions of explanation, validity and formal validity, a more detailed discussion of derivation strategies, and another rule of inference, Reiteration.