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. (shrink)

Two of Leibniz's most studied and often quoted works appear in this volume. Published in 1686, the Discourse on Metaphysics consists of the philosopher's explanation of individual perception as an expression of the rest of the universe from a unique perspective. The whole world--the best of all possible worlds, as he famously remarks--is thus contained in each individual substance. The Monadology, written in 1714, offers a concise synopsis of Leibniz's philosophy, establishing the laws of final causes, which underlie God's free (...) choice to create the best possible world--a world that serves as dynamic and perfectly ordered evidence of the wisdom of its creator. Translated by George R. Montgomery. (shrink)

Maimonides’ Latin translation of Moreh Nevukhim | Guide for the Perplexed, was the most influential Jewish work in the last millennia (Di Segni, 2019; Rubio, 2006; Wohlman, 1988, 1995; Kohler, 2017). It marked the beginning of scholasticism, a daughter of Judaism raised by Jewish thinkers, according to historian Heinrich Graetz (Geschichte der Juden, L. 6, Leipzig 1861, p. xii). Printed by Gutenberg's first mechanical press, its influence in the West went as far as the Fifth Lateran Council (1512 — 1517) (...) "where scholars were encouraged to remove the difficulties which seemed to divide the whole of theology and philosophy (Leibniz, Théodicée, 11)." For centuries, the Guide revolutionized the curriculum of school instruction by reintegrating the natural laws of thought in the sphere of faith (the fourth of which became Leibniz’ Principle of sufficient reason). This complete collection of notes expounds the ideas of the Guide and features all the passages selected and rewritten by Leibniz. This first complete annotated bilingual translation of the original manuscripts in Latin serves as an entry point to the faith in conformity with Reason. This complete collection of notes expounds the ideas of the Guide as selected and rewritten by Leibniz, the famous mathematician inventor of computer arithmetic, considered the last universal genius. The first complete bilingual translation in three centuries features on the front cover Rembrandt's Philosopher in meditation, and a recommendation from Leibniz's himself on the back: "Rabbi Maimonides' excellent book, A Guide for the Perplexed, is more philosophical than I imagined and worthy of careful reading." According to Malbim’s translator, Noah Rosenbloom, the book’s epigraph indicates that nineteenth-century Rabbi Leibush ben Yehiel Michel Wisser, also known as Malbim, was familiar with Leibniz’ Theodicy. In Leibniz’ anthology of the Guide, the reader can get a detailed glimpse of Leibniz' impressions of Maimonides. The Foreword by Leibniz' translator Lloyd Strickland suggests that there were sympathies and perhaps even overlaps between the thoughts of both universal luminaries. -/- . (shrink)

I argue that an individual has aspects numerically identical with it and each other that nonetheless qualitatively differ from it and each other. This discernibility of identicals does not violate Leibniz's Law, however, which concerns only individuals and is silent about their aspects. They are not in its domain of quantification. To argue that there are aspects I will appeal to the internal conflicts of conscious beings. I do not mean to imply that aspects are confined to such cases, but (...) the best way to start is to recognize them experientially. We can feel the conflicts within ourselves. In doing so we can feel some of our aspects. I will try to enhance our understanding of the concept of aspect by listing and formalizing some principles for its use. After that I will argue that all sorts of individual things have aspects, not just people who are conflicted. (shrink)

The article is an investigation of a certain form of argument that refers to Leibniz’s Law as its inference ticket (where Leibniz’s Law is understood as the thesis that if x=y.

Since Leibniz first put forward the sufficient reason law in his philosophical work "The Monadology" (1914), the issue of the law of sufficient reason has aroused heated discussions in the field of logic in our country. The question of whether the law of sufficient reason is the basic law of formal logic was particularly heated in the domestic logic circle in 1978-1980. Since then, there has been little discussion, but from the newly compiled formal logic textbooks published one after another, (...) no consensus has been reached. Although they didn't agree on the question of "whether or not", the two sides elaborated their arguments in detail and put forward some new opinions, thus creating new conditions for solving this problem scientifically. Firstly, this paper analyzes the different viewpoints of both sides of the argument. Secondly, it analyzes whether the law of sufficient reason is the basic law of formal logic from whether Leibniz put forward the law of sufficient reason. Finally, on this basis, a new transformation method is put forward. After this transformation, the law of sufficient reason is an important basic law of deductive logic, which can be juxtaposed with the other three logical laws, such as identity, and coordinated with the "three laws" and complement each other. (shrink)

Arguments based on Leibniz's Law seem to show that there is no room for either indefinite or contingent identity. The arguments seem to prove too much, but their conclusion is hard to resist if we want to keep Leibniz's Law. We present a novel approach to this issue, based on an appropriate modification of the notion of logical consequence.

Arguments based on Leibniz's Law seem to show that there is no room for either indefinite or contingent identity. The arguments seem to prove too much, but their conclusion is hard to resist if we want to keep Leibniz's Law. We present a novel approach to this issue, based on an appropriate modification of the notion of logical consequence.

Stage theorists invoke the idea of counterpart relations to make sense of how objects are able to persist despite their claim that an object is identical with a single instantaneous stage. According to stage theorists, an object persists if and only if it has a later counterpart that bears the appropriate counterpart relation of identity to it. Whilst objects can and do persist, stages cannot and do not. This seems to amount to a refutation of Leibniz’s law. Stage theorists think (...) that the nature of the counterpart relation invoked depends on how it is that the object is referred to. In this paper, I argue that the context sensitivity that they invoke here gives stage theorists a response to the Leibniz’s Law Problem. (shrink)

Moderate composition as identity holds that there is a generalized identity relation, “being the same portion of reality,” of which composition and numerical identity are distinct species. Composition is a genuine kind of identity; but unlike numerical identity, it fails to satisfy Leibniz’s Law. A composite whole and its parts differ with respect to their numerical properties: the whole is one; the parts (collectively) are many. Moderate composition as identity faces the challenge: how, in the absence of Leibniz’s (...) Law, can one characterize what counts as a genuine kind of identity? This paper explores a promising answer: a genuine kind of identity must satisfy a version of Leibniz’s Law restricted to properties that ascribe qualitative character. Strong composition as identity holds that there is only one identity relation, that it satisfies Leibniz’s Law, and that the parts are identical with the whole that they compose. Strong composition as identity faces the challenge of showing that numerical properties do not provide counterexamples to Leibniz’s Law, and doing so in a way that is compatible with the framework of plural logic that is needed to formulate the theory. The most promising way to do this is to hold that plural logic is fundamental at the level of our representations, but not fundamental at the level of being. At the level of being, portions of reality cannot be characterized as either singular or plural. It turns out that the proposed moderate theory and the proposed strong theory are one and the same. In spite of its many attractions, I reject it. The main issue has to do with whether slice-sensitive emergent properties are possible. I argue that they are, making use both of specific examples and general principles of modal plenitude. I do not claim that my arguments are irresistible. But they cannot be evaded as easily as a related argument against strong composition as identity given by Kris McDaniel. I critically examine McDaniel’s argument to pave the way for my own. (shrink)

Leibniz’s Law (or as it sometimes called, ‘the Indiscerniblity of Identicals’) is a widely accepted principle governing the notion of numerical identity. The principle states that if a is identical to b, then any property had by a is also had by b. Leibniz’s Law may seem like a trivial principle, but its apparent consequences are far from trivial. The law has been utilised in a wide range of arguments in metaphysics, many leading to substantive and controversial conclusions. (...) This article discusses the applications of Leibniz’s Law to arguments in metaphysics. It begins by presenting a variety of central arguments in metaphysics which appeal to the law. The article then proceeds to discuss a range of strategies that can be drawn upon in resisting an argument by Leibniz’s Law. These strategies divide into three categories: (i) denying Leibniz’s Law; (ii) denying that the argument in question involves a genuine application of the law; and (iii) denying that the argument’s premises are true. Strategies falling under each of these three categories are discussed in turn. (shrink)

The Philosophical Review, Vol. XCVI, No. 4 (October 1987). Categorically proves that Leibniz's Law (the principle that any instance of _for any x and y, if x=y, then if ...x..., then ..y..._ is true) is not a principle of which is true of natural language objectual quantification.

We draw some metaphysical conclusions about colour and belief from some epistemological commonplaces. It turns out that this requires us to challenge orthodoxy on the causal efficacy of mental properties and to rewrite the standard argument against dualism, but in a way which is good news for functionalists about the mind.

For years philosophers argued for the existence of distinct yet materially coincident things by appealing to modal and temporal properties. For instance, the statue was made on Monday and could not survive being flattened; the lump of clay was made months before and can survive flattening. Such arguments have been thoroughly examined. Kit Fine has proposed a new set of arguments using the same template. I offer a critical evaluation of what I take to be his central lines of reasoning.

The indiscernibility of identicals is incompatible with geach's theory of 'relative' identity, But consistent with the view that x is identical with y iff x is the same a as y, For some count-Noun 'a'. 'x is the same a as y' expresses identity only if x is an a, Otherwise it is merely an equivalence relation.

Professor arthur prior has argued that because one thing can become two things we must abandon leibniz's law. ("time, Existence and identity" in 'papers on time and tense', Oxford, 1968). I argue that while it does often happen that one thing becomes two things, We need not say that in such cases one thing is identical with both the things it becomes. Fission need not be seen as preserving identity. Hence prior does not provide us with adequate reasons for abandoning (...) leibniz's law. (shrink)

A standard strategy for defending a claim of non-identity is one which invokes Leibniz’s Law. (1) Fa (2) ~Fb (3) (∀x)(∀y)(x=y ⊃ (∀P)(Px ⊃ Py)) (4) a=b ⊃ (Fa ⊃ Fb) (5) a≠b In Kalderon’s view, this basic strategy underlies both Moore’s Open Question Argument (OQA) as well as (a variant formulation of) Frege’s puzzle (FP). In the former case, the argument runs from the fact that some natural property—call it “F-ness”—has, but goodness lacks, the (2nd order) property of (...) its being an open question whether everything that instantiates it is good to the conclusion that goodness and F-ness are distinct. And in the latter case, the argument runs from the fact that that Hesperus has, but Phosphorus lacks, the property of being believed by the ancient astronomers to be visible in the evening sky to the conclusion that Hesperus and Phosphorus are distinct. Kalderon argues that both the OQA and FP fail because in neither case is there good reason to believe that both (1) and (2) are true. The reason we are tempted to believe that they are true is because we mistake de dicto claims for de re claims. In order for FP to go through, the truth of the following de re claims needs to be established: FP1) Hesperus was believed by the ancient astronomers to be visible in the evening sky. (shrink)

We apply Benacerraf’s distinction between mathematical ontology and mathematical practice to examine contrasting interpretations of infinitesimal mathematics of the seventeenth and eighteenth century, in the work of Bos, Ferraro, Laugwitz, and others. We detect Weierstrass’s ghost behind some of the received historiography on Euler’s infinitesimal mathematics, as when Ferraro proposes to understand Euler in terms of a Weierstrassian notion of limit and Fraser declares classical analysis to be a “primary point of reference for understanding the eighteenth-century theories.” Meanwhile, scholars like (...) Bos and Laugwitz seek to explore Eulerian methodology, practice, and procedures in a way more faithful to Euler’s own. Euler’s use of infinite integers and the associated infinite products are analyzed in the context of his infinite product decomposition for the sine function. Euler’s principle of cancellation is compared to the Leibnizian transcendental law of homogeneity. The Leibnizian law of continuity similarly finds echoes in Euler. We argue that Ferraro’s assumption that Euler worked with a classical notion of quantity is symptomatic of a post-Weierstrassian placement of Euler in the Archimedean track for the development of analysis, as well as a blurring of the distinction between the dual tracks noted by Bos. Interpreting Euler in an Archimedean conceptual framework obscures important aspects of Euler’s work. Such a framework is profitably replaced by a syntactically more versatile modern infinitesimal framework that provides better proxies for his inferential moves. (shrink)

Modern moral and political philosophy is in debt with natural law theory, both in its ancient and mediaeval elaborations. While the very notion of a natural law has proved highly controversial among 20th Century scholars, the last decades have witnessed a renewed interest in it. Indeed, the threats and challenges as result of multiculturalism, plural societies and global changes have generated a renewed attention to natural law theory. Clearly, it offers solid basis as possible framework to a better understanding of (...) human goods without contradictions and partial bias. The purpose of the present volume is to provide an overview of the history of this concept (Cicero, St. Paul, Aquinas, Melanchthon, Montaigne, Descartes, Leibniz, Hume, Burke, Kant, MacIntyre, etc.) as well as a deep understanding of ongoing research, both in Europe and in America. Furthermore, the specificity of these studies will be of particular value to philosophers, law-philosophers, historians, anthropologists, sociologists and theologians, and those concerned on such issues as the relation between law and moral norm, law and practical reason, and the presence of the idea of natural law in several prominent thinkers. It includes a selected bibliography on natural law. The book also provides an excellent introduction to several of the major topics in natural law theory making it useful both as a reference text and as a sourcebook for academics alike. Natural law is a rich, complex, and highly disputed term. Since its first appearances in the history of Western civilization, it has been used both to point to God as the source of the moral order and to assert that there is an objective order of justice in nature that men and their laws ought to respect. In modern times, natural law theory gave birth to what we usually call "human rights." Unlike the meaning of the term, the importance of an ongoing debate on natural law and on the theories related to it is undisputable. This is why I welcome today this new collection of essays edited by Alejandro Nèstor Garcìa Martìnez, Mario Å ilar and Josè M. Torralba. Natural Law: Historical, Systematic and Juridical Approaches includes a wide variety of studies, covering key authors and issues in natural law theory. Younger students will appreciate the clarity of the chapters, and more trained readers the detailed and accurate bibliographical references that each of them offers. The editors's choice to go from a historical approach to contemporary theories, and then to theoretical and more practical issues is also commendable. Students in philosophy and in legal theory will greatly benefit from this book. Fulvio Di Blasi, author of God and the Natural Law: A Rereading of Thomas Aquinas. (shrink)

Neil Feit; Graduate Papers from the Joint Session 1995: On a Famous Counterexample to Leibniz's Law1, Proceedings of the Aristotelian Society, Volume 96, Issue.

This chapter contains section titled: Introduction General Philosophical Background Substance Substances: Mental and Material Kingdom of Nature: Physics and Laws of Motion Union of Mind and Body: Pre‐established Harmony Kingdom of Nature and of Grace: Grades of Life Human Freedom in the Kingdom of Grace Sufficient Reason and Indiscernibility: Space and Time The Existence of God.

This work is largely the author’s Habilitationsschrift presented at Erlangen. There are three major parts: the first describes ancient and modern forms of scientific reason, the development of which he calls a first and a second Enlightenment; the second explores more deeply certain elements in modern science such as the laws of motion, experimentation, forms of calculation, gravity and force; the third studies rational methodology, including issues of epistemology and artificial languages. The author accepts Kant’s and Nietzsche’s criticism of earlier (...) forms of metaphysics and sees signs of a coming third Enlightenment which will involve criticism of historical and linguistic reason. In parts one and two, considerable space is given to historical and social circumstances as well as to scientific development. Galileo, Newton, Descartes, Leibniz and Kant are among the major figures treated, but many detailed issues are also examined, such as scientific concerns in the late middle ages and renaissance. A forty page bibliography indicates the extent of the author’s range.—R. S. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Hellenistic and Early Modern PhilosophyChristopher S. CelenzaJon Miller and Brad Inwood, editors. Hellenistic and Early Modern Philosophy. New York: Cambridge University Press, 2003. Pp. xii + 330. Cloth, $60.00.There are at least two ways of writing the history of philosophy: the first and most common among those self-identified as "philosophers" treats philosophers of the past as if they were in live dialogue with the present. Only the text (...) is dissected, studied, and analyzed as the interpreter attempts to reconstruct, examine, and occasionally challenge the arguments under consideration. Practitioners of this first way assume that systematic and seemingly internally coherent styles of thought are most worthy of the name "philosophy." These thinkers believe that the term "philosophy," as Richard Rorty pointed out long ago, has an "honorific" sense: philosophy as such stands above other forms of thought, it is regulative in scope, and it serves as something that examines the conclusions of other, necessarily less rigorous fields. The second way is more explicitly contextualist, seeking to understand what a past thinker may have meant in his own time, what outside factors may have shaped his arguments, and how his thought may be part of broader historical currents that are none the less worthy of being remembered and studied now. Practitioners of this way are more likely to believe that ambiguities or seeming deficiencies of clarity that we find in the work of a past thinker are unlikely to be resolved by hypothetical, propositional reconstruction; rather the ambiguities are apparent to us because we are differently situated, so that the study of history is a more powerful tool in helping us satisfactorily access the work of a past thinker than rationalistic reconstruction. (Full disclosure: I am an advocate of the second way.) This volume presents studies that fall into both of the two camps, as Jerome Schneewind points out in his cogent introduction.The volume has a manifest unity: almost all of the articles are concerned in some way with different Hellenistic schools of thought and the way early modern thinkers used, re-imagined, and interpreted them. The studies are often meticulous and all are stimulating. Anthony Long offers a wonderfully synthetic presentation of ancient Stoicism as a coherent system, after which he examines the way Spinoza, Justus Lipsius, and the eighteenth-century Anglican bishop Joseph Butler measure up; unsurprisingly, they don't measure up that well. Margaret Osler seeks to illuminate the way Gassendi appropriated certain aspects of Epicureanism. Philip Mitsis argues that modern scholarship on Locke has not taken Locke's interest in Cicero's De officiis as a tool for teaching morality seriously. Donald Rutherford studies Leibniz's critique of Stoicism, which "centers," he writes, on their "failure to offer an adequate explanation of the nature of moral progress and of the dependence of human happiness on the hope for such progress" (62-63). Catherine Wilson presents a study of the corpuscular aspect of Epicurean natural philosophy among Leibniz and his contemporaries. Jon Miller eloquently and thoughtfully discusses the way Grotius and Spinoza worked with the natural law theory of the Stoics. Among a number of excellent articles, perhaps the most interesting is Stephen Menn's. Menn makes a compelling case that Descartes's intellectual autobiography in the Discourses may be seen as part of a tradition. In it, the writer presents himself as reaching a moment of dissatisfaction with the intellectual customs with which he has come into contact and as needing to strike off into new directions, unencumbered by the vestiges of those customs. While others, most notably Dan Garber, have shown that, as Menn puts it, "none of Descartes's contemporaries seem to have thought in 1637 that they were witnessing a revolution" (142), Menn's study is striking because it shows the depth of the tradition within which Descartes was working. Next, Gail Fine finds that the Cyrenaics, as presented by Sextus Empiricus, seem to have [End Page 207] endorsed "the view that there are truths, beliefs, and knowledge about the subjective" (214). Steven Nadler seeks to show that there is no mysticism in Spinoza's thought, even though it was not uncommon in Spinoza's own day to... (shrink)

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). (shrink)

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. (shrink)

Leibniz is not commonly numbered amongst canonical writers on toleration. One obvious reason is that, unlike Locke, he wrote no treatise specifically devoted to that doctrine. Another is the enormous amount of energy which he famously devoted to ecclesiastical reunification. Promoting the reunification of Christian churches is an objective quite different from promoting the toleration of different religious faiths – so different, in fact, that they are sometimes even construed as mutually exclusive. Ecclesiastical reunification aims to find agreement at least (...) on the most important doctrines. Religious toleration involves accepting and respecting disagreement even on the most important doctrines. If one regards these two projects as alternative rather than complementary strategies for dealing with religious diversity, Leibniz might more readily be characterised as an ecumenist rather than a tolerationist. Such appears to be the conclusion, at any rate, implicit in the balance of critical literature on these topics: whereas a steady stream of studies has discussed Leibniz’s project of ecclesiastical reunification, only a meager trickle has been devoted to his views on toleration; and some of these studies go so far as to conclude that Leibniz had no doctrine of toleration at all. This paper will attempt to show, on the contrary, that a robust, many-layered, and unusually inclusive doctrine of toleration can be gleaned from Leibniz’s writings. This doctrine operated at least at three different levels: philosophical, theological, and pragmatic. As this implies, the doctrine of religious toleration, rather than being in collision with Leibniz’s core project of ecclesiastical reunification, was an essential element of that very project: it should be seen, in other words, not as a competing goal, but as a necessary first step toward ecclesiastical reunification. Yet at the same time it will also emerge that Leibniz’s doctrine of toleration was no mere function of this intra-Christian project of reunification: rather it was a still more fundamental principle which extended beyond the ecumenical project as well. It had the resources for accepting and respecting irreducible religious diversity and disagreement on points of importance not only between Christian communities, but also between Christians and non-Christians, including, in principle, even the atheists. These points will emerge from a discussion of statements on the philosophical, theological, and pragmatic justifications of religious toleration scattered throughout Leibniz’s sprawling corpus of writings. (shrink)

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. (shrink)

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)

Leibniz’s mature philosophical understanding of the laws of nature emerges rather suddenly in the late 1670’s to early 1680’s and is signaled by his embrace of three central theses.1 The first, what I’ll call the thesis of Contingency, suggests that the laws of nature are not only contingent, but, in some sense, paradigmatically contingent; they are supposed to provide insight into the very nature of contingency as Leibniz comes to understand it. The second, what I’ll call the thesis of (...) Providence, suggests that the laws of nature provide a basis for a new argument from design by showing how reflection on God’s ends can be positively useful in the practice of natural philosophy. The third, what I’ll call thesis of .. (shrink)

Leibniz speaks, in a variety of contexts, of there being two realms—a "kingdom of power or efficient causes" and "a kingdom of wisdom or final causes." This essay explores an often overlooked application of Leibniz's famous "two realms doctrine." The first part turns to Leibniz's work in optics for the roots of his view that nature can be seen as being governed by two complete sets of equipotent laws, with one set corresponding to the efficient causal order of the world, (...) and the other to its teleological order. The second part offers an account of how this picture of lawful over-determination is to be reconciled with Leibniz's mature metaphysics. The third addresses a line of objection proposed by David Hirschmann to the effect that Leibniz's doctrine undermines his stated commitment to an efficient, broadly mechanical account of the natural world. Finally, the fourth part suggests that Leibniz's thinking about the harmony of final and efficient causes in connection with corporeal nature may help to shed light on his understanding of the teleological unfolding of monads as well. (shrink)

Leibniz’s Méditation sur la notion commune de la justice is his most important writing on justice as “wise charity” and “universal benevolence” ; we now observe the 300th anniversary of its composition, and a reproduction of part of Leibniz’s manuscript appears in the Appendix to this article. But Leibniz’s essay might with equal justice be called, “Meditation on the Common Notion of Platonism”—for the Méditation opens with a nearly-verbatim paraphrase of Euthyphro 9e-10e, moves on to reduce Hobbes (...) to Thrasymachus in Republic Book I, and ends with Platonic “ascent,” in the manner of Phaedrus and Symposium, from mere negative forbearance from harm to justice as wise love, caritas, Roman-law honeste vivere, and benevolent “aid to others”—as in Augustine’s Platonizing De Civitate Dei XIX). (shrink)

Leibniz’s Méditation sur la notion commune de la justice is his most important writing on justice as “wise charity” and “universal benevolence” ; we now observe the 300th anniversary of its composition, and a reproduction of part of Leibniz’s manuscript appears in the Appendix to this article. But Leibniz’s essay might with equal justice be called, “Meditation on the Common Notion of Platonism”—for the Méditation opens with a nearly-verbatim paraphrase of Euthyphro 9e-10e, moves on to reduce Hobbes (...) to Thrasymachus in Republic Book I, and ends with Platonic “ascent,” in the manner of Phaedrus and Symposium, from mere negative forbearance from harm to justice as wise love, caritas, Roman-law honeste vivere, and benevolent “aid to others”—as in Augustine’s Platonizing De Civitate Dei XIX). (shrink)

entry for the Stanford Encyclopedia of Philosophy (SEP) This entry will attempt to provide a broad overview of the central themes of Leibniz’s philosophy of physics, as well as an introduction to some of the principal arguments and argumentative strategies he used to defend his positions. It tentatively includes sections entitled, The Historical Development of Leibniz’s Physics, Leibniz on Matter, Leibniz’s Dynamics, Leibniz on the Laws of Motion, Leibniz on Space and Time. A bibliography arranged by topic (...) is also included. A working draft is available here (comments, as always welcome): draft. (shrink)

I have shown (to my satisfaction) that Leibniz's final attempt at a generalized syllogistico-propositional calculus in the Generales Inquisitiones was pretty successful. The calculus includes the truth-table semantics for the propositional calculus. It contains an unorthodox view of conjunction. It offers a plethora of very important logical principles. These deserve to be called a set of fundamentals of logical form. Aside from some imprecisions and redundancies the system is a good systematization of propositional logic, its semantics, and a correct account (...) of general syllogistics. For 1686 it was quite an accomplishment. It is a pity that Leibniz himself did not fully appreciate what he had achieved. It does seem to me that this was due in part, as the Kneales urge (Note 4), to his having kept the focus of his attention on traditional syllogistics. It is a great pity that he did not polish GI 195–200 for publication. The publication of GI 195, 198, and 200 would have most likely promoted further research. MAJR- Humanities, Social Sciences and Law. (shrink)

While he was in the employ of the Elector of Mainz, between 1668 and 1671, Leibniz produced a series of important studies in natural law. One of these, dated between 1670 and 1671, is especially noteworthy since it contains Leibniz's earliest sustained attempt to develop an account of justice. Central to this account is the notion of what Leibniz would later come to call `disinterested love', a notion that remained essentially unchanged in Leibniz's work from this period to the end (...) of his life. Through his notion of disinterested love, Leibniz sought to resolve the supposed conflict between self- and other-regarding motives. For a variety of reasons, many commentators have failed to understand the basis of Leibniz's proposed resolution. My purpose in the present paper is to clarify the terms in which Leibniz effected this resolution, as well as to point out important developments in his later thought concerning the relation between pleasure, good, and happiness. (shrink)

This paper focuses on Leibniz’s conception of modality and its application to the issue of natural laws. The core of Leibniz’s investigation of the modality of natural laws lays in the distinction between necessary, geometrical laws on the one hand, and contingent, physical laws of nature on the other. For Leibniz, the contingency of physical laws entailed the assumption of the existence of an additional form of causality beyond mechanical or efficient ones. While geometrical truths, being necessary, do (...) not require the use of the principle of sufficient reason, physical laws are not strictly determined by geometry and therefore are logically distinct from geometrical laws. As a consequence, the set of laws that regulate the physical laws could have been created otherwise by God. However, in addition to this, the contingency of natural laws does not consist only in the fact that God has chosen them over other possible ones. On the contrary, Leibniz understood the status of natural laws as arising from the action internal to physical substances. Hence the actuality of physical laws results from a causal power that is inherent to substances rather than being the mere consequence of the way God arranged the relations between physical objects. Focusing on three instances of Leibniz’s treatment of contingency in physics, this paper argues that, in order to account for the contingency of physical laws, Leibniz maintained that final causes, in addition to efficient and mechanical ones, must operate in physical processes and operations. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Leibniz’s Political and Moral Philosophy in the Novissima Sinica, 1699–1999Patrick RileyThe Preface to Leibniz’s Novissima Sinica 1 contains an important but highly compressed and abbreviated quintessence of his theory of justice or jurisprudence universelle—a version so compressed and abbreviated that one must have a broader and fuller understanding of this universal jurisprudence before one can entirely appreciate what Leibniz has to say about Christian charity, Platonism, and (...) geometry in the Novissima Sinica itself. Above all, one wants to know why Leibniz, in the key paragraph of the Preface to Novissima Sinica, should describe the Chinese emperor as just and charitable and as a (more or less) “Platonic” geometer who is as wise as he is charitable. And why should Leibniz contrast this “wisely charitable” Chinese ruler with the Pontius Pilate who irresponsibly asked, “What is truth?” and then permitted the judicial murder of the very Christ who did most to make charity and justice coextensive by saying, “A new law I give unto you, that ye love one another”? 2 The wise course, therefore, is to throw enough light on Leibnizian universal jurisprudence in general to make his moral-political utterances in Novissima Sinica intelligible in particular.In 1693, four years before the publication of Novissima Sinica, Leibniz revealed the outlines of his jurisprudence universelle in the Codex Iuris Gentium:a good man is one who loves everybody, so far as reason permits. Justice, then, which is the virtue which regulates that affection which [End Page 217] the Greeks call philanthropy, will be most conveniently defined... as the charity of the wise man, that is, charity which follows the dictates of wisdom... Charity is a universal benevolence, and benevolence the habit of loving or of willing the good. Love then signifies rejoicing in the happiness of another..., the happiness of those whose happiness pleases us turns into our own happiness, since things which please us are desired for their own sakes. 3Slightly later, in La véritable piété, Leibniz indicated what this view of justice entails:those who... reduce justice to [mere] rigor, and who fail altogether to understand that one cannot be just without being benevolent... in a word, not only those who look for their profit, pleasure, and glory in the misery of others, but also those who are not at all anxious to procure the common good and to lift out of misery those who are in their care, and generally those who show themselves to be without enlightenment and without charity, boast in vain of piety which they do not know at all, whatever appearance they create. 4The central idea of Leibniz’s “universal jurisprudence,” which aims to find quasi-geometrical, eternal moral verities equally valid for all rational beings, human or divine, is that justice is “the charity of the wise” (caritas sapientis), 5 that it is not mere conformity to sovereign-ordained “positive” law given ex plenitudo potestatis (in the manner of Hobbes); nor is it mere “refraining from harm” or even “rendering what is due” (the neminem laedere and suum cuique tribuere of Roman law). 6 The equal stress on “charity” and on “wisdom” suggests that Leibniz’s practical thought is a kind of fusing of Platonism—in which the wise know the eternal truths such as absolute goodness (Phaedo 75d), which the gods themselves also know and love (Euthyphro 93-10e) and which therefore deserve to rule (Republic 443d-e) 7 —and of Pauline Christianity, whose key moral idea is that charity or love is the first of the virtues (“though I speak [End Page 218] with the tongues of men and of angels, and have not charity, I am become as sounding brass or a tinkling cymbal”). 8 Historically, there is nothing remarkable in trying to fuse Platonism and Christianity. Augustine’s thought (particularly the early De Libero Arbitrio) is just such a fusion. 9 But Leibniz was the last of the great Christian Platonists, and left the world just as Hume, Rousseau, and Kant were about to transform and “secularize” it.I. If one decomposes caritas sapientis into its parts, charity and wisdom, the provenance of both elements is clear enough... (shrink)

I have developed an informational interpretation of Leibniz’s metaphysics and dynamics, but in this paper I will concentrate on his theory of time. According to my interpretation, each monad is an incorporeal automaton programed by God, and likewise each organized group of monads is a cellular automaton governed by a single dominant monad. The activities of these produce phenomena, which must be “coded appearances” of these activities; God determines this coding. A crucially important point here is that we have (...) to distinguish the phenomena for a monad from its states. Both are a kind of representation: a state represents the whole world of monads, and phenomena for a monad “result” from the activities of monads. But the coding for each must be different; R for the first, Ph for the second, where W is a state of the monadic world. The reason for this is that no monadic state is in space and time, but phenomena occur in space and time. Now, the basis of the phenomenal time must be in the timeless realm of monads. This basis is the order of state-transition of each monads. All the changes of these states are given at once by God, and these do not presuppose time. The coded appearances of this order occur in time, and its metric must depend on God’s coding for phenomena. For humans, in particular, this metric time is derived from spatial distance via the laws of dynamics. Thus there may well be an interrelation between spatial and temporal metric. This means that the Leibnizian frame allows relativistic metric of space-time. I will show this after outlining Leibniz’s scenario. (shrink)

Larry Jorgensen aims to show that "Leibniz offers a fully natural theory of mind", recommending Leibniz to our contemporary discussion of naturalism. Readers of Leibniz will, however, hesitate to call him a naturalist. After all, he considered natural laws to be subordinated rules below general divine laws and rejected explaining the soul's action by bodily motion. Jorgensen does have a point, though, when he refers to Leibniz's frequent pleas for natural explanations and his continuity principle.Jorgensen's project unfolds in four parts. (...) The first offers a discussion of Leibniz's concept of nature and a clear presentation of Leibniz's principles, above all the... (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Leibniz’s Universal Jurisprudence: Justice as the Charity of the Wise by Patrick RileySusanna GoodinPatrick Riley. Leibniz’s Universal Jurisprudence: Justice as the Charity of the Wise. Cambridge: Harvard University Press, 1996. Pp. xiii + 338. Cloth, $39.95.Leibniz’s political views are often downplayed, if not simply ignored, by philosophers focusing on his metaphysical accounts of substance and force. That Leibniz himself does not view these two areas (...) as distinct is forcefully argued in Patrick Riley’s detailed account of Leibniz’s theory of justice. Under Riley’s presentation, Leibniz’s definition of justice as the “charity of the wise” depends crucially on both his monadology and his theodicy. And his political maneuverings, particularly with France, are dominated by this unique definition of justice.Riley begins his argument by detailing Leibniz’s objections to Descartes’s voluntarism and Hobbes’s mechanistic view of human nature and the emphasis on power as the foundation of moral truths. The central Leibnizian theme to which Riley returns repeatedly is perfection. A perfect God does not act on whim or power but on reason. Monads are created with the freedom to strive toward perfection. Love, or charity, becomes perfect by becoming universal. Justice is not, as Hobbes claimed, that which is determined to be good for the whole by an absolute power, but a love or benevolence for the whole, striving through reason for the best. Wisdom is the heart of justice, not power. And for Leibniz, wisdom is inseparable from charity. God wills to actualize the best of all possible worlds out of love and wisdom, thus, wise charity.But Leibniz’s view of justice as wise charity is not without problems, most significantly the problem of freedom. In his first chapter, “Foundations,” Riley skillfully raises this problem in the context not only of Leibniz’s political stance on justice but also in the context of his metaphysics and theodicy. It is in the second chapter, “Monadology and Justice,” that Riley does the bulk of the work relating the metaphysics with the moral-political: “... on the knowledge of substance, and in consequence of the soul, depends the knowledge of virtue and of justice” (Leibniz, as quoted in Riley, 51). Political philosophy cannot be done correctly without first acquiring an adequate and proper account of substance, something the materialist Hobbes had failed to do. But because Leibniz has so closely linked his politics with his metaphysics, the perhaps insurmountable problem of freedom in his metaphysics hounds his politics. “If there are no autonomous persons with nonillusory ‘wills’, there cannot be ‘general benevolence’—for benevolence is just [End Page 470] ‘good willing’, and Leibniz invariably equates benevolentia with caritas sapientis. Without a certain kind of will, then, wise charity cannot serve as the heart of a justice which is more than sovereign-ordained law” (61). To prevent Hobbes from ruling the day, Leibniz must, as Riley correctly notes, devise a metaphysics which allows for freedom for both God and man.And here arises the one troubling aspect of Riley’s work. He addresses the problem of freedom in Leibniz’s metaphysics, but there is no discussion of the recent work done by such scholars as Robert Adams, whose book Leibniz: Determinist, Theist, Idealist makes significant contributions on exactly this topic. Regarding Leibniz’s serpentine defense of freedom, Riley does little more than bring up the problem and comment that Leibniz had a problem which, ultimately, he seemed unable to resolve. In fact, Riley shows little appreciation for the work done by mainstream scholars in the area of Leibniz’s metaphysics. If Riley’s work centered primarily on Leibniz’s political writings this might not matter, but given that the thrust of the first part of the book is to argue that Leibniz’s political theory is not merely compatible with his metaphysics but arises out of it, so that an understanding of Leibniz’s metaphysics is essential to grasping the logic of his political views, the absence of current scholarship on the metaphysics is troubling. References in footnotes confirm that Riley had read Adams’s book, as well as The Cambridge Companion to Leibniz, which contains David Blumenfeld’s excellent... (shrink)

Many historians of the calculus deny significant continuity between infinitesimal calculus of the seventeenth century and twentieth century developments such as Robinson’s theory. Robinson’s hyperreals, while providing a consistent theory of infinitesimals, require the resources of modern logic; thus many commentators are comfortable denying a historical continuity. A notable exception is Robinson himself, whose identification with the Leibnizian tradition inspired Lakatos, Laugwitz, and others to consider the history of the infinitesimal in a more favorable light. Inspite of his Leibnizian sympathies, (...) Robinson regards Berkeley’s criticisms of the infinitesimal calculus as aptly demonstrating the inconsistency of reasoning with historical infinitesimal magnitudes. We argue that Robinson, among others, overestimates the force of Berkeley’s criticisms, by underestimating the mathematical and philosophical resources available to Leibniz. Leibniz’s infinitesimals are fictions, not logical fictions, as Ishiguro proposed, but rather pure fictions, like imaginaries, which are not eliminable by some syncategorematic paraphrase. We argue that Leibniz’s defense of infinitesimals is more firmly grounded than Berkeley’s criticism thereof. We show, moreover, that Leibniz’s system for differential calculus was free of logical fallacies. Our argument strengthens the conception of modern infinitesimals as a development of Leibniz’s strategy of relating inassignable to assignable quantities by means of his transcendental law of homogeneity. (shrink)