Legendary since his own time as a universal genius, Gottfried Wilhelm Leibniz (1646-1716) contributed significantly to almost every branch of learning. One of the creators of modern mathematics, and probably the most sophisticated logician between the Middle Ages and Frege, as well as a pioneer of ecumenical theology, he also wrote extensively on such diverse subjects as history, geology, and physics. But the part of his work that is most studied today is probably his writings in metaphysics, which have been (...) the focus of particularly lively philosophical discussion in the last twenty years or so. The writings contain one of the great classic systems of modern philosophy, but the system must be pieced together from a vast and miscellaneous array of manuscripts, letters, articles, and books, in a way that makes especially strenuous demands on scholarship. This book presents an in-depth interpretation of three important parts of Leibniz's metaphysics, thoroughly grounded in the texts as well as in philosophical analysis and critique. The three areas discussed are the metaphysical part of Leibniz's philosophy of logic, his essentially theological treatment of the central issues of ontology, and his theory of substance (the famous theory of monads). (shrink)

This paper discusses Leibniz's Trinitarian doctrine in the light of his philosophy, as revealed by a set of virtually unstudied texts. The first part of the paper examines Leibniz's defence of the Trinity against the charge of contradiction as a necessary precondition to the development of his own conception of the Trinity. The second part discusses some of the key features of Leibniz's Trinitarian doctrine, notably his conception of person, the analogy between the human mind and the Trinity, and the (...) problem of Trinitarian relations. (shrink)

In this paper I will discuss certain aspects of Leibniz's theory and practice of 'soft reasoning' as exemplified by his defence of two central mysteries of the Christian revelation: the Trinity and the Incarnation. By theory and practice of 'soft' or 'broad' reasoning, I mean the development of rational strategies which can successefully be applied to the many areas of human understanding which escape strict demonstration, that is, the 'hard' or 'narrow' reasoning typical of mathematical argumentation. These strategies disclose an (...) 'other' reason, i.e. a complementary set of arguments and methods developed by Leibniz in order to deal with crucial issues such as the 'weighting' of probabilities and truths of fact. I will argue that one of the most compelling examples of the importance and fertility of Leibniz's 'other' reason is provided by his solution to the problems posed by the unique epistemological status of theological mysteries. (shrink)

Reminiscing about his early views on the continuum problem in a dialogue penned in 1689,2 Leibniz recalled the period in his youth when he had enthusiastically subscribed to the "New Philosophy", embracing the composition of the continuum out of points and the doctrine that “a slower motion is one interrupted by small intervals of rest.”3 Speaking of himself through the character Lubinianus, he continues: And I indulged other dogmas of this kind, to which people are prone when they are willing (...) to entertain every imagination, and do not notice the infinity lurking everywhere in things. But although when I became a geometer I relinquished these opinions, atoms and the vacuum held out for a long time, like certain relics in my mind rebelling against the idea of infinity; for even though I conceded that every continuum could be divided to infinity in thought, I still did not grasp that in reality there were parts in things exceeding every number, as a consequence of motion in a plenum. That “atoms and the vacuum held out for a long time” among Leibniz’s cherished views is readily confirmed by an examination of his manuscripts. One may find papers containing some measure of commitment to atomism intermittently throughout the period from 1666 to 1676; moreover, if his later memory is to be trusted, he first “gave himself over to” atomism as early as 1661.4 As for his reasons for rejecting atoms, Leibniz’s mature.. (shrink)

I am so in favor of the actual infinite that instead of admitting that Nature abhors it, as is commonly said, I hold that Nature makes frequent use of it everywhere, in order to show more effectively the perfections of its Author. Thus I believe that there is no part of matter which is not, I do not say divisible, but actually divided; and consequently the least particle ought to be considered as a world full of an infinity of different (...) creatures. (shrink)

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. (shrink)

As is well known, one of Leibniz’s seminal insights in his work on series concerned sums of differences. If from a given series A one forms a difference series B whose terms are the differences of the successive terms of A, the sum of the terms in the B series is simply the difference between the last and first terms of the original series: “the sum of the differences is the difference between the first term and the last” (A vii.3, (...) p. 95). This insight, so far as I now, has never been named; I shall call it the Difference Principle. Suitably generalized, it becomes the basis for the fundamental theorem of the calculus: the sum (integral) of the differentials equals the difference of the sums (the definite integral evaluated between last and first terms), ∫ Bdx = [A]fi. As is well known, the Difference Principle has its origin in the problem set him by Huygens in September 1672 to find the sum of the reciprocal triangular numbers. This date is, incidentally, confirmed by Leibniz himself in Summa fractionum a figuratis, per aequationes (A vii.3, p. 365 ), as well as by the first piece in A vii.3, De summa numerorum triangulorum recipricorum (p. 3), although in his Origo inventionis trianguli harmonici of the Winter of 1675-76 he misremembers it as “Anno 1673” (p. 712). Leibniz explains the algorithm in a letter to Meissner 21 years later: “If one wants to add, for example, the first five.. (shrink)

In a recent note in this review (Leibniz e gli Zenonisti, n. 3, 2001, pp. 15-22) Paolo Rossi stresses the importance of a philosophical sect that he claims has been unjustly ignored in accounts of the history of modern philosophy, the Jesuit philosophers of Louvain and Spain of the late sixteenth and early seventeenth century known as the Zenonists. The occasion for his complaint is Massimo Mugnai’s admirable new introduction to Leibniz’s thought (Introduzione alla filosofia di Leibniz, Torino, Einaudi, 2001), (...) which in all other respects than its failure to mention the Zenonists, Rossi compliments and commends: justly, for in my opinion it is the best introduction to Leibniz yet written. (shrink)

Newton and Leibniz had profound disagreements concerning metaphysics and the relationship of mathematics to natural philosophy, as well as deeply opposed attitudes towards analysis. Nevertheless, or so I shall argue, despite these deeply held and distracting differences in their background assumptions and metaphysical views, there was a considerable consilience in their positions on the status of infinitesimals. In this paper I compare the foundation Newton provides in his Method Of First and Ultimate Ratios (sketched at some time between 1671 and (...) 1684, and published in the Principia of 1687) with that provided independently by Leibniz in his unpublished manuscript De quadratura arithmetica (1675-6) as well as in later writings. I argue that both appeal to a version of the Archimedean Axiom to underwrite their use of infinitesimal techniques, which must be interpreted as a shorthand for rigorously finitist methods. (shrink)

In this paper we argue for the robustness of Leibniz's commitment to the reality (but not substantiality) of body. We claim that a number of his most important metaphysical doctrines — among them, psychophysical parallelism, the harmony between efficient and final causes, the connection of all things, and the argument for the plurality of substances stemming from his solution to the continuum problem— make no sense if he is interpreted as giving an eliminative reduction of bodies to perceptions.

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. (shrink)

Gottfried Leibniz is well known for his claim to have “rehabilitated” the substantial forms of scholastic philosophy, forging a reconciliation of the New Philosophy of Descartes, Mersenne and Gassendi with Aristotelian metaphysics (in his so-called Discourse on Metaphysics, 1686). Much less celebrated is the fact that fifty years earlier (in his Hypomnemata Physica, 1636) the Bratislavan physician and natural philosopher Daniel Sennert had already argued for the indispensability to atomism of (suitably re-interpreted) Aristotelian forms, in explicit opposition to the rejection (...) of substantial forms by his fellow atomist Sébastien Basson.1.. (shrink)

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. (shrink)

New light is shed on Leibniz’s commitment to the metaphysical priority of the intensional interpretation of logic by considering the arithmetical and graphical representations of syllogistic inference that Leibniz studied. Crucial to understanding this connection is the idea that concepts can be intensionally represented in terms of properties of geometric extension, though significantly not the simple geometric property of part-whole inclusion. I go on to provide an explanation for how Leibniz could maintain the metaphysical priority of the intensional interpretation while (...) holding that logically the intensional and the extensional stand in strictly inverse relation to each other. (shrink)

THE CLOSE CONNECTION BETWEEN mathematics and philosophy has long been recognized by practitioners of both disciplines. The apparent timelessness of mathematical truth, the exactness and objective nature of its concepts, its applicability to the phenomena of the empirical world—explicating such facts presents philosophy with some of its subtlest problems. We shall discuss some of the attempts made by philosophers and mathematicians to explain the nature of mathematics. We begin with a brief presentation of the views of four major classical philosophers: (...) Plato, Aristotle, Leibniz, and Kant. We conclude with a more detailed discussion of the three “schools” of mathematical philosophy which have emerged in the twentieth century: Logicism, Formalism, and Intuitionism. (shrink)

In this illuminating, highly engaging book, Jonathan Bennett acquaints us with the ideas of six great thinkers of the early modern period: Descartes, Spinoza, Leibniz, Locke, Berkeley, and Hume. For newcomers to the early modern scene, this lucidly written work is an excellent introduction. For those already familiar with the time period, this book offers insight into the great philosophers, treating them as colleagues, antagonists, students, and teachers.

For most early Medieval and Scholastic philosophers working in the Aristotelian tradition, knowledge of any specific subject is knowledge of its causes and principles. Knowledge of individuals was no exception. As Jorge Gracia has written "To know individuality [for early Medieval and Scholastic philosophers] is to be able to determine the causes and principles that are responsible for it."1 The achievement of such ability is also known as the problem of individuation. This paper will be concerned with the solution to (...) the problem suggested by Leibniz’s writings and how it relates to the contemporary metaphysical debate. In the first section I introduce the problem of individuation along with the solution Leibniz proposed during the latter part of his life. The second section analyzes Leibniz’s solution in a contemporary perspective. I argue that, unlike during the Medieval and early Modern periods, today the epistemic side of the problem of individuation plays a major role in the debate. In this light, Leibniz’s proposal that humans cannot grasp what the individuality of an individual consists in seems problematic. I show, however, that Leibniz’s proposal can stand on its feet also nowadays, provided we are willing to give up the pretenses that there is a definitive count of individuals and that re-identifying individuals across time and space is part of the problem of individuation. (shrink)

The main purpose of the article is to get clear what Leibniz's concerns about relations were. His: I do not believe that you will admit an accident that is in two subjects at the same time. My judgement about relations is that paternity in David is one thing, sonship in Solomon another, but that the relation common to both is a merely mental thing whose basis is the modifications of the individuals is best seen as akin to: Father is true (...) of David. Son is true of Solomon. But Being a father of is not true of any individual. Leibniz, like modern nominalist Nelson Goodman, could not allow the ordered pair . To establish this I must argue against Hidé Ishiguro's claim that Leibniz should have straightforwardly constructed a logic of relations, and Jaakko Hintikka's claim that Leibniz could have allowed the use of relational predicates in such forms as (Ex) Rax and (Ey) Ryb. I must also argue that what they say about the windowlessness doctrine (especially the as if formulation) is beside the point. (shrink)

As Newton realized, his absolute space was a ‘conspiracy of nature’ in the sense that his laws dictated that nobody could discover who, among all possible observers (in various states of motion relative to one another), was at rest in absolute space. So absolute space was an unverifiable element of his theory.

: In this paper, I investigate Leibniz's conception of final causation. I focus especially on the role that Leibnizian final causes play in intentional action, and I argue that for Leibniz, final causes are a species of efficient causation. It is the intentional nature of final causation that distinguishes it from mechanical efficient causation. I conclude by highlighting some of the implications of Leibniz's conception of final causation for his views on human freedom, and on the unconscious activity of substances.