Leibniz on Infinite Numbers,Infinite Wholes,and Composite Substances

Leibniz claims that nature is actually infinite but rejects infinite number. Are his mathematical commitments out of step with his metaphysical ones? It is widely accepted that Leibniz has a viable response to this problem: there can be infinitely many created substances, but no infinite number of them. But there is a second problem that has not been satisfactorily resolved. It has been suggested that Leibniz's argument against the world soul relies on his rejection of infinite number, and, as such, Leibniz cannot assert that any body has a soul without also accepting infinite number, since any body has infinitely many parts. Previous attempts to address this concern have misunderstood the character of Leibniz's rejection of infinite number. I argue that Leibniz draws an important distinction between ‘wholes’ – collections of parts that can be thought of as a single thing – and ‘fictional wholes’ – collections of parts that cannot be thought of as a single thing, which allows us to make sense of his rejection of infinite number in a way that does not conflict either with his view that the world is actually infinite or that the bodies of substances have infinitely many parts.     

Situating Time in the Leibnizian Hierarchy of Beings

Leibniz's widely influential account of time provides a significant puzzle for those seeking to locate this account within his hierarchical ontology. Leibniz follows his scholastic predecessors in supposing that there are different grades of being, with substances being the most real and all other things possessing their reality via their relationships to substance. Following this picture, Leibniz suggests that phenomenal bodies only possess the being that they derive from the substances (i.e., monads) that ground them. Some would argue that time likewise only possesses its being based on the bodies that it relates. Contrary to this suggestion (i.e., that time is twice removed from substances), I will argue that time is derived directly from rational souls. Thus, I will argue that time is on an ontological par with the phenomenal world of bodies.     

A Tale of Two Thinkers,One Meeting,and Three Degrees of Infinity: Leibniz and Spinoza (1675–8)

The article presents Leibniz's preoccupation (in 1675–6) with the difference between the notion of infinite number, which he regards as impossible, and that of the infinite being, which he regards as possible. I call this issue ‘Leibniz's Problem’ and examine Spinoza's solution to a similar problem that arises in the context of his philosophy. ‘Spinoza's solution’ is expounded in his letter on the infinite (Ep.12), which Leibniz read and annotated in April 1676. The gist of Spinoza's solution is to distinguish between three kinds of infinity and, in particular, between one that applies to substance, and one that applies to numbers, seen as auxiliaries of the imagination. The rest of the paper examines the extent to which Spinoza's solution solves Leibniz's problem. The main thesis I advance is that, when Spinoza and Leibniz say that the divine substance is infinite, in most contexts it is to be understood in non-numerical and non-quantitative terms. Instead, for Spinoza and Leibniz, a substance is said to be infinite in a qualitative sense stressing that it is complete, perfect and indivisible. I argue that this approach solves one strand of Leibniz's problem and leaves another unsolved.     

Leibniz's Two Realms Revisited

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.     

Leibniz on Memory and Consciousness

In this article, I develop a higher-order interpretation of Leibniz's theory of consciousness according to which memory is constitutive of consciousness. I offer an account of Leibniz's theory of memory on which his theory of consciousness may be based, and I then show that Leibniz could have developed a coherent higher-order account. However, it is not clear whether Leibniz held (or should have held) such an account of consciousness; I sketch an alternative that has at least as many advantages as the higher-order theory. This analysis provides an important antecedent to the contemporary discussions of higher-order theories of consciousness.     

Disinterested Love: Understanding Leibniz's Reconciliation of Self- and Other-Regarding Motives

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.     

Leibniz on the Metaphysics of Color

Drawing on remarks scattered through his writings, I argue that Leibniz has a highly distinctive and interesting theory of color. The central feature of the theory is the way in which it combines a nuanced subjectivism about color with a reductive approach of a sort usually associated with objectivist theories of color. After reconstructing Leibniz's theory and calling attention to some of its most notable attractions, I turn to the apparent incompatibility of its subjective and reductive components. I argue that this apparent tension vanishes in light of his rejection of a widely accepted doctrine concerning the nature of bodies and their geometrical qualities.     

Socio-Cultural Learning as a 'Transcendental Fact': Habermas's Postmetaphysical Perspective

In the Discourse on Metaphysics Leibniz writes, 'Our perceptions are always true, it is our judgments that come from ourselves that deceive us' (section 14). Leroy Loemker in his 'Leibniz's Doctrine of Ideas' criticizes this account of error. His main worry can be presented in the form of the following syllogistic argument, which he derives from Leibniz's doctrine of ideas: (a) There cannot be a false perception; (b) All judgments are perceptions; and therefore (c) There cannot be a false judgment. However, in this paper I argue that Leibniz can distinguish between a perception and a judgment in their semantic aspect. The sense in which a perception turns out to be false (or true) for Leibniz is different from the sense in which a judgment turns out to be false (or true). This is because, while a perception, understood in Leibniz's terms, lacks truth-conditions, a judgment, in being representational, has such conditions. Thus while a perception remains true always, a judgment can be false. Pointing to the equivocal use of 'false' in (a) and (c) above, I conclude that (c) does not follow from (a) and (b).     

Mental Manipulations and the Problem of Causal Exclusion

Leibniz has long faced a challenge about the coherence of the distinction between necessary and contingent truths in his philosophy. In this paper, I propose and examine a new way to save genuine contingency within a Leibnizian framework. I conclude that it succeeds in formally solving the problem, but at unbearable cost. I present Leibniz's challenge by considering God's choice of the best possible world (Sect. 2). God necessarily exists and necessarily chooses to actualize the best possible world. The actual world therefore could not be different, for if it were different it would be a distinct and inferior world and hence would not be created. In Section 3, I defend Leibniz from this challenge. I argue that, while it is necessary for God to choose to create the best possible world, it is not necessary for any world to be the best possible. This is because the criterion for judging perfection can itself be contingent. Different criteria will judge different worlds as the best. Thus it is necessary for God to create the best, but not necessary which is the best. Distinguishing between possible worlds in Leibniz's sense and in the modern sense allows a fuller exposition of this position. There are worries that can arise with the claim that the criterion of perfection is contingent. I consider two of the most pressing (Sect. 4). The first argues that the criterion is in God's understanding and hence is necessary; the second alleges that a contingent criterion of perfection violates Leibniz's cherished Principle of Sufficient Reason. These worries are well grounded, and examining them reveals a deep incompatibility between this solution and Leibniz's metaphysical views. I conclude that there is a real solution available, but that it is unacceptable to Leibniz or a Leibnizian. The search for a genuine solution that is genuinely Leibnizian goes on.     

MALEBRANCHE AND LEIBNIZ ON THE BEST OF ALL POSSIBLE WORLDS

In this article I explore Leibniz's claim in the Theodicy that on the essential points Malebranche's theodicy “reduces to” his own view. This judgment may seem to be warranted given that both thinkers emphasize that evils are justified by the fact that they follow from the simple and uniform laws that govern that world which is worthy of divine creation. However, I argue that Leibniz's theodicy differs in several crucial respects from Malebranche's. I begin with a qualified endorsement of Charles Larmore's recent claim that remarks in Malebranche's correspondence with Leibniz indicate that their theodicies rely on incompatible conceptions of the moral rationality of divine action. I also attempt to go beyond Larmore's discussion in highlighting further differences concerning the sort of freedom involved in the divine act of creation. My conclusion is that these differing conceptions of divine morality and divine freedom reveal that in contrast to the case of Leibniz, Malebranche's theodicy not only does not require that God create anything at all, but also is compatible with the result that the world he decides to create is not uniquely the best possible.     

Leibniz's non-tensed theory of time

Leibniz's philosophy of time, often seen as a precursor to current forms of relationalism and causal theories of time, has rightly earned the admiration of his more recent counterparts in the philosophy of science. In this article, I examine Leibniz's philosophy of time from a new perspective: the role that tense and non-tensed temporal properties/relations play in it. Specifically, I argue that Leibniz's philosophy of time is best (and non-anachronistically) construed as a non-tensed theory of time, one that dispenses with tensed temporal properties such as past, present, and future. In arguing for this thesis, I focus on the three facets of Leibniz's philosophy most relevant for evaluating his commitment to a B-theory of time: (1) the nature of change, (2) the reality of the future, and (3) the truth-conditions for tensed temporal statements. Despite prima facie evidence to the contrary, I show that a close examination of Leibniz's views on these topics provides compelling evidence for interpreting his philosophy of time as a B-theory of time.     

German Idealism and the Concept of Punishment

This paper examines the Leibnizian background to Kant's critique of the ontological argument. I present Kant's claim that existence is not a real predicate, already formulated in his pre-critical essay of 1673, as a generalization of Leibniz's reasoning regarding the existence of created things. The first section studies Leibniz's equivocations on the notion of existence and shows that he employs two distinct notions of existence – one for God and another for created substances. The second section examines Kant's position in his early paper of 1763. My claim is that Kant's view of existence in 1763, namely that it is not a predicate, is strongly related to the logical notion of possibility, formulated by Leibniz and accepted by Kant.     

Leibniz and the Square: A Deontic Logic for the Vir Bonus

Seventeenth century philosopher Gottfried Leibniz's contributions to metaphysics, mathematics, and logic are well known. Lesser known is his ‘invention’ of deontic logic, and that his invention derives from the alethic logic of the Aristotelian square of opposition. In this paper, I show how Leibniz developed this ‘logic of duties’, which designates actions as ‘possible, necessary, impossible, and omissible’ for a ‘vir bonus’ (good person). I show that for Leibniz, deontic logic can determine whether a given action, e.g. as permitted, is therefore obligatory or prohibited (impossible). Secondly, since the deontic modes are derived from what is possible, necessary, etc., for a good person to do, and that ‘right and obligation’ are the ‘moral qualities’ of a good person, we can see how Leibniz derives deontic logic from these moral qualities. Finally, I show how Leibniz grounds a central deontic concept, namely obligation, in the human capacity for freedom.     

THE DEFENCE OF THE MYSTERIES OF THE TRINITY AND THE INCARNATION: AN EXAMPLE OF LEIBNIZ'S 'OTHER' REASON

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.¹ 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.     

Leibniz,Pufendorf, and the Possibility of Moral Self-Governance

In his On the Duties of Man and Citizen, seventeenth century natural law theorist Samuel Pufendorf argues that the source of obligation lies in ‘the command of a superior’. This so-called ‘voluntarist’ position was famously criticized by the ‘rationalist’ Gottfried Leibniz. However, I wish to highlight several neglected aspects of the debate. Leibniz implicitly proposes a solution to a central moral problem: how one can be obligated voluntarily. His answer reflects a sort of motivational internalism, whereby the ideas of justice provide some motive cause of action, and virtue provides the rest. In this way, the agent acts voluntarily by making the principles of justice the principles of her action. Secondly, I show how this argument for the principles depends implicitly on his ‘science of right’, established in his earliest writings on jurisprudence. These principles are constituent of the nature of rational substance. It then becomes clear that Leibniz had long developed a foundation for self-governance, similar to Kantian autonomy, consisting in the agent's internal moral power to act (jus) and moral necessity to act (obligation). These points are exposed through a close reading of Leibniz's criticisms of Pufendorf on the end, object and efficient cause of natural law.     

Leibniz's Twofold Gap Between Moral Knowledge and Motivation

Moral rationalists and sentimentalists traditionally disagree on at least two counts, namely regarding the source of moral knowledge or moral judgements and regarding the source of moral motivation. I will argue that even though Leibniz's moral epistemology is very much in line with that of mainstream moral rationalists, his account of moral motivation is better characterized as sentimentalist. Just like Hume, Leibniz denies that there is a necessary connection between knowing that something is right and the motivation to act accordingly. Instead, he believes that certain affections are necessary for moral motivation. On my interpretation, then, Leibniz is an externalist about judgements and motivation: he is committed to a gap between the judgement that something is morally right and the motivation to act accordingly. As a matter of fact, I will argue that there are two gaps. The first and less controversial one has to do with the fact that Leibniz reconciles his psychological egoism with ethical altruism through his account of love. The second gap between moral judgements and motivation is a more fundamental one: Leibniz denies that there are any necessary connections between beliefs and motivation, or even more generally, between perceptions and appetitions.     

Brandom's Leibniz

I discuss an objection by Margaret Wilson against Robert Brandom's interpretation of Leibniz's account of perceptual distinctness. According to Brandom, Leibniz holds that (i) the relative distinctness of a perception is a function of its inferentially articulated content and (ii) apperception, or awareness, is explicable in terms of degrees of perceptual distinctness. Wilson alleges that Brandom confuses ‘external deducibility’ from a perceptual state of a monad to the existence of properties in the world, with ‘internally accessible content’ for the monad in that state. Drawing on Leibniz, I develop a response to Wilson on Brandom's behalf.     

Leibniz's Concept of Substance and his Reception of John Calvin's Doctrine of the Eucharist

Leibniz saw the question of the eucharist as a crucial stumbling block to the agreement between Lutherans and Calvinists. Mandated together with Daniel Ernst Jablonsky to prepare working documents for the negotiations between Hanover and Brandenburg in 1697, Leibniz carefully read through the Calvinist Confessions of faith and the works of Calvin in their 1671 edition. He made an extensive collection of excerpts from the Confessions of faith and from Calvin's Institutes all intended to show that Calvinists admitted the substantial presence of Christ's body in the eucharist. (This collection of excerpts is analysed here for the first time and compared with another little-known document, the Unvorgreiffliches Bedencken). L. had argued previously in 1691/92 that, contrary to the assertions of Pellisson-Fontanier, his own conception of substance and of Christ's presence in the eucharist was completely different from Calvin's. However, by 1697, it was clear to Leibniz that Calvin's concept of substance, which was broadly speaking Aristotelian, was never defined clearly by the reformer, and could be made to coincide with Leibniz's own notion of substance as force rather than substance in its dimensional sense. At the same time L. dissociated Ubiquitarianism (doctrine characteristic of late sixteenth century Lutheranism, which defended the dimensional presence of Christ's body in heaven and in the eucharist, by arguing that Christ in his divine nature could cause his physical body to be present in several places at the same time) from Lutheranism. He also drove a wedge between the doctrines of Zwingli and Calvin. L. thus attempted to find religious union on a common ontology and he might well have succeeded if it were not for complex political circumstances, which ultimately caused the failure of the negotiations.     

Monadic Interaction

Leibniz has almost universally been represented as denying that created substances, including human minds and the souls of animals, can causally interact either with one another or with bodies. Yet he frequently claims that such substances are capable of interacting in the special sense of what he calls ‘ideal’ interaction. In order to reconcile these claims with their favored interpretation, proponents of the traditional reading often suppose that ideal action is not in fact a genuine form of causation but instead a merely apparent influence which serves to ‘save the appearances.’ I argue that this traditional reading distorts Leibniz's thought and that he actually considers ideal action a genuine (though non-standard) form of causation.