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Newton's philosophical views are unique and uniquely difficult to categorise. In the course of a long career from the early 1670s until his death in 1727, he articulated profound responses to Cartesian natural philosophy and to the prevailing mechanical philosophy of his day. Newton as Philosopher presents Newton as an original and sophisticated contributor to natural philosophy, one who engaged with the principal ideas of his most important predecessor, Rene; Descartes, and of his most influential critic, G. W. Leibniz. Unlike Descartes and Leibniz, Newton was systematic and philosophical without presenting a philosophical system, but over the course of his life, he developed a novel picture of nature, our place within it, and its relation to the creator. This rich treatment of his philosophical ideas, the first in English for thirty years, will be of wide interest to historians of philosophy, science, and ideas.

A través de una reconstrucción de la evolución de su pensamiento, en este artículo se estudia la utilización de infinitesimales por parte de Newton. Se distingue entre dos concepciones sucesivas de lo que denominó momento. A la primera de estas entidades la caracterizó como un infinitesimal, pero a la segunda (un indivisible generador de magnitudes finitas, que interviene en su método de las primeras y últimas razones) no la consideró como tal. Se entiende así su manifestación de rechazo a los infinitesimales, formulada en una segunda etapa, y se ve que las dudas arrojadas por algunos investigadores sobre la veracidad de tal manifestación se deben a una determinada interpretación de esta última concepción de momento.This paper discusses Newton’s recourse to infinitesimals through the reconstruction of the evolution of his thought. Two successive concepts of what he termed “moment” are told apart. The first of those entities was characterized by him as an infinitesimal, while the second -an indivisible generating finite magnitudes, present in his method of first and last reasons- was not considered such an entity. This move makes understandable his express rejection of infinitesimals in the second stage, and exposes the doubts of some scholars about the sincerity of Newton’s rejection as due to a peculiar interpretation of his last concept of “moment”.

Leibniz's predicate-in-notion principle and some of its alleged consequences, by C. D. Broad.--On Leibniz's metaphysics, by L. Couturat.--Philosophical reflections of Leibniz on law, politics, and the state, by C. J. Friedrich.--The root of contingency, by E. M. Curley.--Monadology, by M. Furth.--Individual substance, by I. Hacking.--Leibniz on plenitude, relations, and the "reign of the law," by J. Hintikka.--Leibniz's theory of the ideality of relations, by H. Ishiguro.--Leibniz and Spinoza on activity, by M. Kneale.--Leibniz and Newton, by A. Koyré.--Plenitude and sufficient reason in Leibniz and Spinoza, by A. O. Lovejoy.--Leibniz on possible worlds, by B. Mates.--Recent work on the philosophy of Leibniz, by B. Russell.--On Leibniz's explication of "necessary truth," by M. D. Wilson.--Bibliography (p. [421]-425).

Broad, C. D. Leibniz's predicate-in-notion principle and some of its alleged consequences.--Couturat, L. On Leibniz's metaphysics.--Friedrich, C. J. Philosophical reflections of Leibniz on law, politics, and the state.--Curley, E. M. The root of contingency. Furth, M. Monadology.--Hacking, I. Individual substance.--Hintikka, J. Leibniz on plenitude, relations, and the "reign of law."--Ishiguro, H. Leibniz's theory of the ideality of relations.--Kneale, M. Leibniz and Spinoza on activity.--Koyré, A. Leibniz and Newton.--Lovejoy, A. O. Plenitude and sufficient reason in Leibniz and Spinoza.--Mates, B. Leibniz on possible worlds.--Russell, B. Recent work on the philosophy of Leibniz.--Wilson, M. D. On Leibniz's explication of "necessary truth.".

Before establishing his mature interpretation of infinitesimals as fictions, Gottfried Leibniz had advocated their existence as actually existing entities in the continuum. In this paper I trace the development of these early attempts, distinguishing three distinct phases in his interpretation of infinitesimals prior to his adopting a fictionalist interpretation: (i) (1669) the continuum consists of assignable points separated by unassignable gaps; (ii) (1670-71) the continuum is composed of an infinity of indivisible points, or parts smaller than any assignable, with no gaps between them; (iii) (1672- 75) a continuous line is composed not of points but of infinitely many infinitesimal lines, each of which is divisible and proportional to a generating motion at an instant (conatus). In 1676, finally, Leibniz ceased to regard infinitesimals as actual, opting instead for an interpretation of them as fictitious entities which may be used as compendia loquendi to abbreviate mathematical reasonings.

The aim of this paper is to analyze Leibniz and Newton’s conception of space, and to point out where their agreements and disagreements lie with respect to its mode of existence. I shall offer a definite characterization of Leibniz and Newton’s conceptions of space. I will show that, according to their own concepts of substance, both Newtonian and Leibnizian spaces are not substantiva!. The reason of that consists in the fact that space is not capable of action. Moreover, there is a sense in which space is relational, because their parts are individuated only by means of their mutual relations.

In this paper I challenge the usual interpretations of Newton's and Leibniz's views on the nature of space and the relativity of motion. Newton's ‘relative space’ is not a reference frame; and Leibniz did not regard space as defined with respect to actual enduring bodies. Newton did not subscribe to the relativity of intertial motions; whereas Leibniz believed no body to be at rest, and Newton's absolute motion to be a useful fiction. A more accurate rendering of the opposition between them, I argue, leads to a wholly different understanding of Leibniz's theory of space, one which is not susceptible to the objections Newton had raised against Descartes regarding the representation of motion. This in turn suggests a new approach for contemporary theory of space, one which neither hypostatizes space nor tries to reduce it to relations among actual things. * This work was generously supported by the National Endowment for the Humanities, with a Fellowship for College Teachers and Independent Scholars (FB-26897-89), and also by a sabbatical leave from my institution, Middlebury College. Iam very grateful to various members of faculty of York University for their appreciative reception of an earlier one-week-old version of this paper. ‘Relative Space in Newton and Leibniz’, read to the Department of Philosophy there in January 1990, and to Robert Rynasiewicz for criticisms of an extract read at the 1991 History of Science meeting.

Sir Isaac Newton (1642-1727) left a voluminous legacy of writings. Despite his influence on the early modern period, his correspondence, manuscripts, and publications in natural philosophy remain scattered throughout many disparate editions. In this volume, Newton's principal philosophical writings are for the first time collected in a single place. They include excerpts from the Principia and the Opticks, his famous correspondence with Boyle and with Bentley, and his equally significant correspondence with Leibniz, which is often ignored in favor of Leibniz's later debate with Samuel Clarke. Newton's exchanges with Leibniz place their different understandings of natural philosophy in sharp relief. The volume also includes 'De Gravitatione', offered here in a corrected translation, which is crucial for understanding Newton's relation to his great predecessor Descartes. In a historical and philosophical introduction, Andrew Janiak examines Newton's philosophical positions and his relations to canonical figures in early modern philosophy.

In this paper I attempt to trace the development of Gottfried Leibniz’s early thought on the status of the actually infinitely small in relation to the continuum. I argue that before he arrived at his mature interpretation of infinitesimals as fictions, he had advocated their existence as actually existing entities in the continuum. From among his early attempts on the continuum problem I distinguish four distinct phases in his interpretation of infinitesimals: (i) (1669) the continuum consists of assignable points separated by unassignable gaps; (ii) (1670-71) the continuum is composed of an infinity of indivisible points, or parts smaller than any assignable, with no gaps between them; (iii) (1672-75) a continuous line is composed not of points but of infinitely many infinitesimal lines, each of which is divisible and proportional to a generating motion at an instant (conatus); (iv) (1676 onward) infinitesimals are fictitious entities, which may be used as compendia loquendi to abbreviate mathematical reasonings; they are justifiable in terms of finite quantities taken as arbitrarily small, in such a way that the resulting error is smaller than any pre-assigned margin. Thus according to this analysis Leibniz arrived at his interpretation of infinitesimals as fictions already in 1676, and not in the 1700's in response to the controversy between Nieuwentijt and Varignon, as is often believed.

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 (SIA), as propounded by John Bell. Despite many parallels between SIA and Leibniz’s approach —the non-punctiform nature of infinitesimals, their acting as parts of the continuum, the dependence on variables (as opposed to the static quantities of both Standard and Non-standard Analysis), the resolution of curves into infinitesided polygons, and the finessing of a commitment to the existence of infinitesimals— I find some salient differences, especially with regard to higher-order infinitesimals. These differences are illustrated by a consideration of how each approach might be applied to Newton’s Proposition 6 of the Principia, and the derivation from it of the v2/r law for the centripetal force on a body orbiting around a centre of force. It is found that while Leibniz’s syncategorematic approach is adequate to ground a Leibnizian version of the v2/r law for the “solicitation” ddr experienced by the orbiting body, there is no corresponding possibility for a derivation of the law by nilsquare infinitesimals; and while SIA can allow for second order differentials if nilcube infinitesimals are assumed, difficulties remain concerning the compatibility of nilcube infinitesimals with the principles of SIA, and in any case render the type of infinitesimal analysis adopted dependent on its applicability to the problem at hand.