Ohad Nachtomy

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

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

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.

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

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.

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.

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.

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

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.

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.

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.

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.

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.

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.

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.