Richard T. W. Arthur

The main aim of this work is to present a unified treatment of three philosophical problems about time which have come to be regarded as distinct and unrelated. These are: the question of whether time is to be conceived as self-existent or as a relational structure among events; the question of what constitutes the direction of time; and the question of whether temporal becoming may be considered an objective feature of the natural world. ;I contend here that the third of these questions should be answered in the affirmative. This automatically provides us with an answer to the second question, for the direction of time may be construed as the order in which events come to be: that is, in terms of the relation of possible succession. This in turn requires the provision of a foundation for the theory of time in terms of an intrinsic relational structure among possible events. ;In chapter I, I give a preview of these conclusions and lay out my strategy for establishing them in the succeeding chapters. ;In chapter II, I trace the historical development of the modern physical concept of time, and give a detailed discussion of both Leibniz's and Newton's treatments of it. I use this discussion to introduce a number of issues about time: in particular, the opposition between "actualist" and "possibilist" conceptions, the association of time with spacetime and the concept of inertia, and the connection of time homogeneity and the concept of energy. Newton's classical treatment of temporal transiency by means of his fluxional system is also examined in detail, and both his and Leibniz's theories of time are construed as attempts to reassert the continuity of temporal becoming and motion in the face of Descartes' assertions of the atomicity of temporal becoming and the relativity of motion. ;In chapter III, I argue that the absolute-relational controversy may be resolved by taking the essential structure of time to be that of a local chronological ordering intrinsic to possible events. . . . UMI

(Accepted for an edition of Synthese that never saw light of day, and remained on my website. Written 2005.) Noting the status of the Law of Continuity as one of Leibniz’s most cherished axioms, Bertrand Russell charged that his philosophy nevertheless amounted to “a complete denial of the continuous”. Georg Cantor made a similar accusation of inconsistency about Leibniz’s philosophy of the actual infinite. But I argue that neither doctrine is inconsistent when the subtleties of Leibniz’s syncategorematic interpretation are properly taken into account. Leibniz rejects the existence of infinite wholes: an infinite aggregate of actual things forms only a fictitious whole. Analogously, infinitesimals are only fictitious parts, this time of ideal wholes. That is, just as an actual infinity of terms can be understood syncategorematically as more terms than can be assigned a number, without there being any infinite numbers, so too the infinitely small can be given a syncategorematic interpretation by means of the Law of Continuity, without there existing any actual infinitesimals. By examining Leibniz’s justification of infinitesimals in his calculus, I argue that the syncategorematic interpretation is also applicable to series of changes, and thus exonerates Leibniz from Russell’s criticism: on this interpretation all naturally occurring transitions are continuous in that the difference between neighbouring states is smaller than any assignable. This means not that there exists a least difference, but that for any assignable finite difference, there exists a smaller one. Thus there is a true continuous transition, even though the states themselves and all assignable differences between them are actually discrete.

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.

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 these remarks on Ohad Nachtomy’s account of Leibniz’s philosophy of the infinite in his recent book, Living Mirrors, I focus on his suggestion that living creatures be interpreted as exemplifying the second of the three degrees of infinity that Leibniz articulates in 1676, as things which are infinite in their own kind. For the infinity characterizing created substances cannot be the highest degree, which is reserved by Leibniz for the divine substance, while Nachtomy sees the lowest degree as applicable only to “entia rationis such as numbers and relations”. Against this, I argue that the lowest or syncategorematic infinite applies to any multiplicity or magnitude that is greater than any assignable, so that something further can always be added; whereas the second degree applies to the divine attributes or perfections, which are maximal in that nothing further of that kind can be added.

The aim of this paper is to explain the significance of Alfred A. Robb’s philosophy of time stemming from his interpretation of relativity theory; and at the same time, to investigate the reasons for the failure of his philosophical contemporaries to appreciate its significance, with special attention to its reception on Russell’s part. The study of Russell’s reaction to Robb exposes shortcomings in Russell’s own philosophy of time, which has been extremely influential through the years. It also highlights the philosophical differences between Minkowski and Robb, the shared idealistic premises of Russell and McTaggart, and the confluences between the views of Russell, Eddington and McTaggart. The paper closes with an account of how Robb’s interpretation has been vindicated by its adoption in modern physics.

One of the most puzzling features of Leibniz’s deep metaphysics is the apparent contradiction between his claims that the law of continuity holds everywhere, so that in particular, change is continuous in every monad, and that “changes are not really continuous,” since successive states contradict one another. In this paper I try to show in what sense these claims can be understood as compatible. My analysis depends crucially on Leibniz’s idea that enduring states are “vague,” and abstract away from further changes occurring within them at a higher resolution—consistently with his famous doctrine of "petites perceptions." As Leibniz explains further in a recently transcribed unpublished manuscript, these changes are dense within any actual duration, which is conceived as actually divided by them into states that are syncategorematically infinite in number and unassignably small. The correspondence between these unassignably small intervals between changes and the differentials of his calculus allows processes to be conceived as continuous, despite the discontinuity of the changes that occur in actuality.

In this paper, we endeavour to give a historically accurate presentation of how Leibniz understood his infinitesimals, and how he justified their use. Some authors claim that when Leibniz called them “fictions” in response to the criticisms of the calculus by Rolle and others at the turn of the century, he had in mind a different meaning of “fiction” than in his earlier work, involving a commitment to their existence as non-Archimedean elements of the continuum. Against this, we show that by 1676 Leibniz had already developed an interpretation from which he never wavered, according to which infinitesimals, like infinite wholes, cannot be regarded as existing because their concepts entail contradictions, even though they may be used as if they exist under certain specified conditions—a conception he later characterized as “syncategorematic”. Thus, one cannot infer the existence of infinitesimals from their successful use. By a detailed analysis of Leibniz’s arguments in his De quadratura of 1675–1676, we show that Leibniz had already presented there two strategies for presenting infinitesimalist methods, one in which one uses finite quantities that can be made as small as necessary in order for the error to be smaller than can be assigned, and thus zero; and another “direct” method in which the infinite and infinitely small are introduced by a fiction analogous to imaginary roots in algebra, and to points at infinity in projective geometry. We then show how in his mature papers the latter strategy, now articulated as based on the Law of Continuity, is presented to critics of the calculus as being equally constitutive for the foundations of algebra and geometry and also as being provably rigorous according to the accepted standards in keeping with the Archimedean axiom.

It is well known that Leibniz advocated the actual infinite, but that he did not admit infinite collections or infinite numbers. But his assimilation of this account to the scholastic notion of the syncategorematic infinite has given rise to controversy. A common interpretation is that in mathematics Leibniz’s syncategorematic infinite is identical with the Aristotelian potential infinite, so that it applies only to ideal entities, and is therefore distinct from the actual infinite that applies to the actual world. Against this, I argue in this paper that Leibniz’s actual infinite, understood syncategorematically, applies to any entities that are actually infinite in multitude, whether numbers, actual parts of matter, or monads. It signifies that there are more of them than can be assigned a number, but that there is no infinite number or collection of them, which notion involves a contradiction. Similarly, to say that a magnitude is actually infinitely small in the syncategorematic sense is to say that no matter how small a magnitude one takes, there is a smaller, but there are no actual infinitesimals. In geometry one may calculate with expressions apparently denoting such entities, on the understanding that they are fictions, standing for variable magnitudes that can be made arbitrarily small, so as to produce demonstrations that there is no error in the resulting expressions.

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.

Against Norton's claim that all thought experiments can be reduced to explicit arguments, I defend Brown's position that certain thought experiments yield a priori knowledge. They do this, I argue, not by allowing us to perceive “Platonic universals” (Brown), even though they may contain non-propositional components that are epistemically indispensable, but by helping to identify certain tacit presuppositions or “natural interpretations” (Feyerabend's term) that lead to a contradiction when the phenomenon is described in terms of them, and by suggesting a new natural interpretation in terms of which the phenomenon can be redescribed free of contradiction.

In this reassessment of Descartes' debt to his mentor Isaac Beeckman, I argue that they share the same basic conception of motion: the force of a body's motion—understood as the force of persisting in that motion, shorn of any connotations of internal cause—is conserved through God's direct action, is proportional to the speed and magnitude of the body, and is gained or lost only through collisions. I contend that this constitutes a fully coherent ontology of motion, original with Beeckman and consistent with his atomism, which, notwithstanding Descartes' own profoundly original contributions to the theory of motion, is basic to all Descartes' further work in natural philosophy.

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).

This volume contains essays that examine infinity in early modern philosophy. The essays not only consider the ways that key figures viewed the concept. They also detail how these different beliefs about infinity influenced major philosophical systems throughout the era. These domains include mathematics, metaphysics, epistemology, ethics, science, and theology. Coverage begins with an introduction that outlines the overall importance of infinity to early modern philosophy. It then moves from a general background of infinity up through Kant. Readers will learn about the place of infinity in the writings of key early modern thinkers. The contributors profile the work of Descartes, Spinoza, Leibniz, and Kant. Debates over infinity significantly influenced philosophical discussion regarding the human condition and the extent and limits of human knowledge. Questions about the infinity of space, for instance, helped lead to the introduction of a heliocentric solar system as well as the discovery of calculus. This volume offers readers an insightful look into all this and more. It provides a broad perspective that will help advance the present state of knowledge on this important but often overlooked topic.

It is commonly held that there is no place for the 'now’ in physics, and also that the passing of time is something subjective, having to do with the way reality is experienced but not with the way reality is. Indeed, the majority of modern theoretical physicists and philosophers of physics contend that the passing of time is incompatible with modern physical theory, and excluded in a fundamental description of physical reality. This book provides a forceful rebuttal of such claims. In successive chapters the author explains the historical precedents of the modern opposition to time flow, giving careful expositions of matters relevant to becoming in classical physics, the special and general theories of relativity, and quantum theory, without presupposing prior expertise in these subjects. Analysing the arguments of thinkers ranging from Aristotle, Russell, and Bergson to the proponents of quantum gravity, he contends that the passage of time, understood as a local becoming of events out of those in their past at varying rates, is not only compatible with the theories of modern physics, but implicit in them.

Despite his fame as a philosopher, Leibniz was a diplomat by profession, and seldom managed to engage in sustained philosophical activity for any length of time. One exception to this, though, is the period towards the end of his stay in Paris and a little afterwards, when he launched a concerted attack on most of the profoundest problems in metaphysics, tackling them with a penetration and persistence that is remarkable by any standards. The resulting series of “meditations”, to use his own descriptive term, penned between December of 1675 and his arrival at the court in Hanover in December of 1676, has been available for the greater part of this century only to a handful of Leibniz scholars. An edition was published in Kazan in 1913 by Ivan Jagodinsky, and it is from this source that Loemker drew to give a tantalizingly brief selection of the ‘Paris Notes’ in his well known volume of English translations. But Jagodinsky’s edition is as unreliable as it is rare, and it was not until recently that Leibniz scholars were presented with an authoritative and wholly reliable Latin edition by the German Academy of Science at Berlin, volume 3 of Series VI of Gottfried Wilhelm Leibniz: Sämtliche Schriften und Briefe. In this volume most of the notes on metaphysics are collected together in Section F under the title De Summa Rerum, where they consitute articles 57 through 87. This fine edition, however, is of limited use for those having no Latin or German, and even for those who do, it is scarcely either convenient or affordable.

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.

At the turn of the century Bertrand Russell advocated an absolutist theory of space and time, and scornfully rejected Leibniz’s relational theory in his Critical Exposition of the Philosophy of Leibniz. But by the time of the second edition, he had proposed highly influential relational theories of space and time that had much in common with Leibniz’s own views. Ironically, he never acknowledges this. In trying to get to the bottom of this enigma, I looked further at contemporary texts by Russell, and also those he might have relied on, especially that of Robert Latta. I found that, like Latta’s, Russell’s interpretation of Leibniz was heavily conditioned by his immersion in neo-Hegelian and neo-Kantian philosophy prior to 1898, and that the doctrine of internal relations he attributes to Leibniz was more nearly the view of Lotze.