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

While the definition of the ‘genotype’ has undergone dramatic changes in the transition from classical to molecular genetics, the definition of the ‘phenotype’ has remained for a long time within the classical framework. In addition, while the notion of the genotype has received significant attention from philosophers of biology, the notion of the phenotype has not. Recent developments in the technology of measuring gene-expression levels have made it possible to conceive of phenotypic traits in terms of levels of gene expression. (...) We demonstrate that not only has this become possible but it has also become an actual practice. This suggests a significant change in our conception of the phenotype: as in the case of the ‘genotype’, phenotypes can now be conceived in quantitative and measurable terms on a comprehensive molecular level. We discuss in what sense gene expression profiles can be regarded as phenotypic traits and whether these traits are better described as a novel concept of phenotype or as an extension of the classical concept. We argue for an extension of the classical concept and call for an examination of the type of extension involved. (shrink)

In her stimulating article, Catherine Wilson considers the moment of worlds-making in Leibniz’s philosophy. She raises the following question: “How do possible substances give rise to possible worlds?“ and observes that the moment of world-making is as puzzling as it is interesting. In section 2 of her article, Wilson considers two approaches to the question. According to the first, possible individuals logically precede possible worlds and possible worlds are constituted either by combinations of possible individuals or by mechanically checking the (...) compossibility relations among them. This approach assumes that possible individuals are given and complete and that worlds result from their various conjunctions. On this approach, a world is a set of compossible individuals, i.e., a set of complete individuals whose co-existence is possible. (shrink)

This paper argues that God's immanent causation and Spinoza's account of activity as adequate causation (of finite modes) do not always go together in Spinoza's thought. We show that there is good reason to doubt that this is the case in Spinoza's early Short Treatise on God, Man and His Well‐being. In the Short Treatise, Spinoza defends an account of God's immanent causation without fully endorsing the account of activity as adequate causation that he will later introduce in the Ethics (...) (E3def2). We turn to an examination of how God's immanent causation relates to the activity of finite things in the Ethics. We consider two ways to think about the link between God, seen as immanent cause, and the activity of finite things: namely, in terms of entailment and in terms of production. We argue that the productive model is most promising for understanding the way in which the activity of finite things and God's immanent causality are connected in Spinoza's (mature) philosophy. (shrink)

This work presents Leibniz's view of infinity and the central role it plays in his theory of living beings. Nachtomy argues that Leibniz employs three degrees of infinity: absolute infinity, which applies to God; maximum or infinite in kind, which applies to created, living beings; and mathematical infinity.

In notes from 1675-76 Leibniz is using the notion of an infinite number as an illustration of an impossible notion. In the same notes, he is also using this notion in contrast to the possibility of the ‘Ens perfectissumum’ (A.6.3 572; Pk 91; A.6.3 325). I suggest that Leibniz’s concern about the possibility of the notion of ‘the greatest or the most perfect being’ is partly motivated by his observation that similar notions, such as ‘the greatest number’, are impossible. This (...) leads to the question how Leibniz convinced himself that the notion of the greatest number is self-contradictory and that of the greatest being is not. I consider two suggestions, one that stress the difference between beings and numbers and one that stress the difference between two notions of infinity, and conclude that neither of them provides a satisfactory solution to this question. (shrink)

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. (shrink)

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. (shrink)

This paper examines a widely accepted reading of monads as the most fundamental elements of reality. Garber [Leibniz – Body, Substance, Monad, Oxford: Oxford University Press, 2009] argues that simple monads – seen as mind-like atoms without parts and extension – replace the corporeal substance of Leibniz’s middle period. Phemister [Leibniz and the Natural World – Activity, Passivity and Corporeal Substances in Leibniz’s Philosophy, Dordrecht: Springer, 2005] argues that monads figure also at the top as complete corporeal substances. Building on (...) Fichant [‘L’invention métaphysique’ in G.W. Leibniz, Discours de métaphysique suivi de Monadologie et autres textes, edited by Michel Fichant, Paris: Gallimard, 2004], I argue that, for Leibniz, monads function not only as building blocks at the bottom level of composition but also at the top, as grounding the unity and hence the being of complete substances and organic unities. Since organic unities or living beings are seen by Leibniz as natural machines with a nested structure, and since monads are likened to living beings, this would imply that the use of the concept ‘monad’ holds not only at the bottom, and not only at the top, but all over the range between them. (shrink)

Während Leibniz' Vorstellung eines vollständigen Begriffs viel Beachtung fand, blieb die Frage seiner Begründung im Verstand Gottes eher unbeachtet. In diesem Aufsatz versuche ich auf diese Frage einzugehen, indem ich den Zeitraum (ungefähr 1672-1679), in dem Leibniz die Vorstellung eines vollständigen Begriffs als eine explizite Definition eines Individuums entwickelte, näher untersuche. Meine Darstellung über die Begründung des individuellen Begriffs im Verstand Gottes beinhaltet drei Thesen: (1) Leibniz sieht einen inneren Zusammenhang zwischen der Bildung einfacher Begriffe zu zusammengesetzten Begriffen und der (...) Individuation von Begriffen, so daß die Zusammengesetztheit der Begriffe zu ihrer Individualität und Einzigartigkeit beiträgt. (2) Ein individueller Begriff sollte nicht als eine Menge von Prädikaten betrachtet werden, sondern vielmehr als eine einzigartige Struktur von Prädikaten, in der die Reihenfolge der Prädikate eine essentielle Rolle spielt. (3) Ein individueller Begriff sollte nicht nur mit einer einzigartigen Struktur in Verbindung gebracht werden, sondern auch mit einer kombinatorischen Regel, die eine solche einzigartige Struktur von Prädikaten im Verstand Gottes erzeugt. (shrink)

This work develops a Leibnizian approach to possibility by explicating the notions of possibility in general, in chapter 1; possible individuals in chapter 2; possible worlds in chapter 3; and actualization in chapter 4. ;A Leibnizian notion of possibility is characterized against the traditional view of an intelligible realm of thoughts in God's mind. It is understood in terms of self-consistent thoughts and is developed by explicating the notions of thought and of possibility in terms of the combinatorial structure common (...) to both, that is, the construction of complex structures by progressive combinations of simple elements. According to this view, God mentally combines his simple forms, and complex concepts arise in his mind. Such complex and self-consistent concepts are deemed possible. ;Against this general view of possibility, a possible individual is identified with the combinatorial rule that produces a unique and maximally consistent structure of predicates. Such a rule is termed the "production rule" of the individual. The "production rule" unifies a series of predicates and constitutes the individual's logical subject and its principle of individuation. ;Given this view of a possible individual, the notion of a possible world is defined as a set of compossible individuals. It is argued that the compossiblity relation result from the consideration of two individuals in the same logical space. It is suggested that the inner structure of each individual entails its possible connections with other individuals, and that God's consideration of the individuals splits them into compossible sets or possible worlds. ;Finally, it is suggested that actualization involves endowing the "production rule" in God's mind with power of action. The union of a divine prescription of action with power of action yields an active and self-sufficient agent. Upon creation, the agent's activity generates the sequence of predicates prescribed to it by its production rule. ;The explication of the above notions accounts for the conceptual dependence of possible individuals and for the causal independence of created individuals--both of which are indispensable for Leibniz's notion of preestablished harmony. (shrink)

This paper develops some important observations from a recent article by Maria Rosa Antognazza published in The Leibniz Review 2015 under the title “The Hypercategorematic Infinite”, from which I take up the characterization of God, the most perfect Being, as infinite in a hypercategorematic sense, i.e., as a being beyond any determination. By contrast, creatures are determinate beings, and are thus limited and particular expressions of the divine essence. But since Leibniz takes both God and creatures to be infinite, creatures (...) are simultaneously infinite and limited. This leads to seeing creatures as infinite in kind, in distinction from the absolute and hypercategorematic infinity of God. I present three lines of argument to substantiate this point: (1) seeing creatures as entailing a particular sequence of perfections and imperfections; (2) seeing creatures under the rubric of an intermediate degree of infinity and perfection that Leibniz, in 1676, calls “maximum in kind”; and (3) observing that primitive force, a defining feature of created substance, may be seen as infinite in a metaphysical sense. This leads to viewing Leibniz’s use of infinity within a Neoplatonic framework of descending degrees of Being: from the hypercategorematic infinite, identified with the most perfect Being; to the intermediate degree of maximum in kind, identified with creatures; to the lowest degree of entia rationis (or beings of reason), identified with mathematical and abstract entities. (shrink)

In his Pensées, Blaise Pascal gives vivid voice to both the wonder and anxiety that many early modern thinkers felt towards infinity. Contemplating our place between the infinite expanse of space and the infinite divisibility of matter, Pascal writes.

In this article we examine the “phenotype” concept in light of recent technological advances in Genome-Wide Association Studies . By observing the technology and its presuppositions, we put forward the thesis that at least in this case genotype and phenotype are effectively coidentifled one by means of the other. We suggest that the coidentiflcation of genotype-phenotype couples in expression-based GWAS also indicates a conceptual dependence, which we call “co-deñnition.” We note that viewing these terms as codeflned runs against possible expectations, (...) viz., that genotypes and phenotypes could ultimately be expressed independently from one another. In addition, the co-definition of genotypes and phenotypes in this context emphasizes the correlative character of both genotypes and phenotypes in GWAS. (shrink)

In her stimulating article, Catherine Wilson considers the moment of worlds-making in Leibniz’s philosophy. She raises the following question: “How do possible substances give rise to possible worlds?“ and observes that the moment of world-making is as puzzling as it is interesting. In section 2 of her article, Wilson considers two approaches to the question. According to the first, possible individuals logically precede possible worlds and possible worlds are constituted either by combinations of possible individuals or by mechanically checking the (...) compossibility relations among them. This approach assumes that possible individuals are given and complete and that worlds result from their various conjunctions. On this approach, a world is a set of compossible individuals, i.e., a set of complete individuals whose co-existence is possible. (shrink)

In her stimulating article, Catherine Wilson considers the moment of worlds-making in Leibniz’s philosophy. She raises the following question: “How do possible substances give rise to possible worlds?“ and observes that the moment of world-making is as puzzling as it is interesting. In section 2 of her article, Wilson considers two approaches to the question. According to the first, possible individuals logically precede possible worlds and possible worlds are constituted either by combinations of possible individuals or by mechanically checking the (...) compossibility relations among them. This approach assumes that possible individuals are given and complete and that worlds result from their various conjunctions. On this approach, a world is a set of compossible individuals, i.e., a set of complete individuals whose co-existence is possible. (shrink)

Italo Calvino’s Invisible Cities presents a wide array of possible cities—cities whose composition turns on a productive ambiguity of their being described or invented by Marco Polo in his conversations with Kublai Khan. Implicit in this book is also a theory about how all possible cities are composed. The method turns on decomposing a city down to its basic elements and recomposing it in different ways through the imagination. I argue that there is a close affinity between Calvino’s theory of (...) fictional cities and Leibniz’s theory of possible worlds. The main similarity is that both theo­ries are combinatorial—they suppose that possibilities are produced by combination and variation of basic elements. The paper presents Leibniz’s theory of possibility in its metaphysical context and explores the similarity with Calvino’s cities in their literary context. I suggest that there is a rather strong relation between the theory of literary fiction implicit in Invisible Cities and Leibniz’s theory of possibility, in that both define the possible in terms of the conceivable. Indeed, Leibniz often refers to literary examples to substantiate his position, and I argue that this reveals an essential feature of his theory. (shrink)

This chapter seeks to highlight some of the main threads that Leibniz used in developing his views on infinity in his early years in Paris. In particular, I will be focusing on Leibniz’s encounters with Descartes, Galileo, and Spinoza. Through these encounters, some of the most significant features of Leibniz’s view of infinity will begin to emerge. Leibniz’s response to Descartes reveals his positive attitude to infinity. He rejects Descartes’s view that, since we are finite, we cannot comprehend the infinite (...) and therefore should refrain from studying it. Likewise, Leibniz rejects Descartes’s view that the term ‘infinite’ should be reserved to God alone, as well as Descartes’s distinction between the infinite and the indefinite. Leibniz’s encounter with Galileo brings out his rejection of infinite number in response to Galileo’s paradox. This, in turn, leads him to face another formidable challenge, viz., to defend the claim that an infinite being is possible, while an infinite number is not. Leibniz’s encounter with Spinoza, I suggest, highlights the way he is approaching this problem by distinguishing between quantitative and non- quantitative senses of infinity. The strategy of employing different senses of infinity in different contexts will remain central in Leibniz’s approach to infinity for the rest of his career. (shrink)

We argue in this paper that Leibniz’s characterization of a substance as “un être” in his correspondence with Arnauld stresses the per se unity of substance rather than oneness in number. We employ two central lines of reasoning. The first is a response to Mogens Lærke’s claim that one can mark the difference between Spinoza and Leibniz by observing that, while Spinoza’s notion of substance is essentially non-numerical, Leibniz’s view of substance is numerical. We argue that Leibniz, like Spinoza, qualifies (...) the substance as “one” primarily in a non-numerical sense, where non-numerical means per se unity or qualitative uniqueness. The second line of reasoning suggests that the term “one” should be understood as a-unity-presupposed-by-multiplicity in two senses: a) externally, in the sense of being presupposed by higher complex structures, such as aggregates, and, b) internally, in the sense of having itself a complex structure. We develop an analogy along these lines between the role the notion of a fundamental unity plays in Leibniz’s view of numbers and his view of substance. In other words, we suggest that looking at the role units play in Leibniz’s view of mathematics can shed some light on the role they play in his metaphysics. (shrink)

La communication qui suit porte sur le concept de relation tel que le définit Leibniz dans sa correspondance avec Arnauld. La première partie présente trois des présupposés impliqués dans ce concept, à savoir 1) qu'il y a des relations entre des individus possibles, 2) que ces relations sont nécessaires à la notion de mondes possibles et 3) qu'elles sont également nécessaires pour compléter l'individuation des individus possibles. Dans la deuxième partie, on verra que le premier présupposé semble entrer en conflit (...) avec le nominalisme de Leibniz et son refus d'attribuer une réalité aux relations. Un problème de circularité entrave la résolution de ce conflit: d'une part, Leibniz considère en effet que les relations interindividuelles présupposent des individus, mais d'autre part, le troisième présupposé indique que de telles relations sont nécessaires à l'individuation même des individus possibles. Dans la troisième partie, on proposera une solution à ce problème de circularité au moyen du concept d'espace logique de possibilités. Dans cet espace logique, la place qu'occupe un individu est à la fois 1) constitutive de son concept et 2) une notion relationnelle. Grâce au concept de relation que propose Leibniz, les concepts d'individus et l'espace logique en viennent ainsi à se constituer mutuellement, ce qui ouvre une avenue hors de la circularité. (shrink)

This volume explores the intersection between early modern philosophy and the life sciences by presenting the contributions of important but often neglected figures such as Cudworth, Grew, Glisson, Hieronymus Fabricius, Stahl, Gallego, Hartsoeker, and More, as well as familiar figures such as Descartes, Spinoza, Leibniz, Malebranche, and Kant.

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

We argue in this paper that Leibniz’s characterization of a substance as “un être” in his correspondence with Arnauld stresses the per se unity of substance rather than oneness in number. We employ two central lines of reasoning. The first is a response to Mogens Lærke’s claim that one can mark the difference between Spinoza and Leibniz by observing that, while Spinoza’s notion of substance is essentially non-numerical, Leibniz’s view of substance is numerical. We argue that Leibniz, like Spinoza, qualifies (...) the substance as “one” primarily in a non-numerical sense, where non-numerical means per se unity or qualitative uniqueness. The second line of reasoning suggests that the term “one” should be understood as a-unity-presupposed-by-multiplicity in two senses: a) externally, in the sense of being presupposed by higher complex structures, such as aggregates, and, b) internally, in the sense of having itself a complex structure. We develop an analogy along these lines between the role the notion of a fundamental unity plays in Leibniz’s view of numbers and his view of substance. In other words, we suggest that looking at the role units play in Leibniz’s view of mathematics can shed some light on the role they play in his metaphysics. (shrink)

Acouple of years ago I gave a talk on Leibniz’s approach to human freedom. I tried to apply some current philosophical distinctions in order to resolve the tension between Leibniz’s doctrine of complete concept, which entails every truth about an individual, and Leibniz’s insistence that such an individual—whose identity and individuality are defined by its complete concept—acts freely.

Acouple of years ago I gave a talk on Leibniz’s approach to human freedom. I tried to apply some current philosophical distinctions in order to resolve the tension between Leibniz’s doctrine of complete concept, which entails every truth about an individual, and Leibniz’s insistence that such an individual—whose identity and individuality are defined by its complete concept—acts freely.

Ohad Nachtomy restates the main points of “Plenitude and Compossibility” with admirable fidelity and economy. His proposed revisions, based on the distinction between incomplete and complete substances and on the mind-relativity of relations, are intriguing additions to his earlier paper in Studia Leibnitiana and deserve careful consideration. Some brief remarks on the context of the problem, will, I hope, help to set the stage for the assessment of our various views.

Ohad Nachtomy restates the main points of “Plenitude and Compossibility” with admirable fidelity and economy. His proposed revisions, based on the distinction between incomplete and complete substances and on the mind-relativity of relations, are intriguing additions to his earlier paper in Studia Leibnitiana and deserve careful consideration. Some brief remarks on the context of the problem, will, I hope, help to set the stage for the assessment of our various views.