Tzuchien Tho

Although Leibniz’s reputation had something to do with the debate about the least action principle within the 18th century, he never stated it as such. He did however address the two conceptual elements in the principle. First, as an interpretation of divine providence in creation, he developed a mathematical theory of natural optimisation in his works on the optics, dynamics, and other physical subjects. Second, in his later dynamical works, he developed a theory of “action” (actio) which is the product of mass, displacement, and speed (a = msv). This second component is also situated within the context of vis viva conservation (mv2). With these two elements in place, and together with the conservation of vis viva, Leibniz seems to have arrived at the doorstep of the least action principle but never entered the open door. This paper argues that while Leibniz did not state a least action principle, his use of the concept of action together with the conservation vis viva implies stationary action, an eventual and more sophisticated interpretation of the least action principle. By reconstructing this argument, we can also show how Leibniz’s defence of a naturalistic, intrinsic teleology developed into a mathematical analysis of action quite distinct from the least action principles of the later Maupertuis and Euler. Instead of an optimisation formed by a sum over the path of motion, Leibniz’s method implies the actualisation of the optimum at each infinitesimal segment of the path. Finality, therefore, is not realised “at the end” but actualised continuously.

The second chapter of this book presents the central argument of the book in a synoptic form. It proposes a first approach to an important interpretive lens for understanding Leibniz’s dynamics project as presenting a theory of causation that is structural in nature. In the preceding introduction, we examined the chronological development of the dynamics and emphasized the growth of the dynamics project from the notion of Leibnizian vis as a structural property, the property of a physical or mechanical system. Now we take a step further. Since the dynamics aimed at developing a science of the cause of motion, the proper object of the dynamics is the nature of causation rather than the mere properties of motion. A structural form of causation is proposed here in order to render explicit the central concept of Leibniz’s dynamics.

This chapter continues with a three-part presentation of the central architectonic components of the dynamics. In this chapter, we examine the status of continuity in the theory of motion developed in Leibniz’s dynamics. The chapter traces some difficulties in the development of continuous motion in order to highlight the productive frictions between Leibniz’s attempts to provide a geometrical and dynamical account of motion. The status of continuity will highlight what is at stake in Leibniz’s focus on vis as the cause of physical phenomenon: the distinction as well as the connection between ideal (geometrical) continuities and dynamical action.

This chapter concludes the three-part presentation of the central architectonic components of the dynamics. In this chapter, we examine the most important (and most explicit) principle of the dynamics. The equipollence of cause and effect, inherited from Scholastic thinkers but creatively reinvented by Leibniz, can be understood as the starting place of the dynamics project. In this chapter, we trace Leibniz’s different interpretations of this principle in order to move from a dynamics based on final causation (teleology) to that of formal cause (via the concept of actio). It is through this development that the very concept of vis matures in Leibniz. By examining this development, this chapter provides a fuller understanding of structural causation in the dynamics.

This paper argues that Cantorian transfinite cardinality is not a necessary assumption for the ontological claims in Badiou’s L’Être et l’Événement (Vol. 1). The necessary structure for Badiou’s mathematical ontology in this work was only the ordinality of sets. The method for reckoning the sizes of sets was only assumed to follow the standard Cantorian measure. In the face of different and compelling forms of measuring non-finite sets (following Benci and Di Nasso, and Mancosu), it is argued that Badiou’s project can indeed accommodate this pluralism of measurement. In turn, this plurality of measurement implies that Badiou’s insistence on the “subtraction of the one”, the move to affirm the unconditioned being of the “inconsistent multiple”, results in the virtuality of the one, a pluralism of counting that further complicates the relationship between the one and the multiple in the post-Cantorian era.

The third chapter of this book begins a three-part presentation of the central architectonic components of the dynamics (continued in the following chapters). In this chapter, we examine the so-called equivalence of hypothesis, inherited from Kepler, Galileo and Huygens, and see how it shaped the concept of force and the method of its measurement. The main aim is to show the theoretical independence of the equivalence of hypotheses and the theory of Leibnzian vis in order to understand how the former principle provides the limits of physical phenomena. It is within these limits that Leibniz vis can be understood as cause.

The sixth and final chapter of this book attempts use the perspective developed in the previous chapters to resolve a seeming contradiction between the doctrine of inherent substantial forces developed in Leibniz’s dynamics and the late doctrine of the autarchy of monads, a mainstay of his late period (post-1695) metaphysics. The immediate problem is how a dynamical theory of corporeal motion can be coherent with the seeming reductive kinematic nature of physical reality implied by monadic perception. This problem is treated first by looking at Leibniz’s theory of space in the analysis situs project and using its insights to understand the nature of relative motion in his correspondences with Clarke. This investigation sheds light on the concept of physical causality in the dynamics that treats the effects of physical phenomenon as a group of internally related empirical variations. Solidifying this concept of structural or dynamical causality by contextualizing it through the methodology of symmetry and invariance that Leibniz inherited from his mentor Huygens, this notion of causality can be shown to be consistent with the autarchy of monads.

This article aims to interpret Leibniz’s dynamics project through a theory of the causation of corporeal motion. It presents an interpretation of the dynamics that characterizes physical causation as the structural organization of phenomena. The measure of living force by mv2 must then be understood as an organizational property of motion conceptually distinct from the geometrical or otherwise quantitative magnitudes exchanged in mechanical phenomena. To defend this view, we examine one of the most important theoretical discrepancies of Leibniz’s dynamics with classical mechanics, the measure of vis viva as mv2 rather than ½ mv2. This “error”, resulting from the limits of Leibniz’s methodology, reveals the systematic role of this quantity mv2 in the dynamics. In examining the evolution of the quantity mv2 in the refinement of the force concept from potentia to actio, I argue that Leibniz’s systematic limitations help clarify dynamical causality as neither strictly metaphysical nor mechanical but a distinct level of reality to which Leibniz dedicates the “dynamica” as “nova scientia”.

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.

This book presents a systematic reconstruction of Leibniz’s dynamics project (c. 1676-1700) that contributes to a more comprehensive understanding of the concepts of physical causality in Leibniz’s work and 17th century physics. It argues that Leibniz’s theory of forces privileges the causal relationship between structural organization and physical phenomena instead of body-to-body mechanical causation. The mature conception of Leibnizian force is not the power of one body to cause motion in another, but a kind of structural causation related to the configuration of integral systems of bodies in physical evolution. By treating the immanent philosophy of Leibniz’s dynamics, this book makes explicit the systematic aims and inherent limits of Leibniz's physical project, in addition to providing an alternative vision of the scientific understanding of the physical world in the late 17th and early 18th century.

This article aims to treat the question of the reality of Leibniz’s infinitesimals from the perspective of their application in his account of corporeal motion. Rather than beginning with logical foundations or mathematical methodology, I analyze Leibniz’s use of an allegedly “instantiated” infinitesimal magnitude in his treatment of dead force in the Specimen Dynamicum. In this analysis I critique the interpretive strategy that uses the Leibnizian distinction, drawn from the often cited 1706 letter to De Volder, between actual and ideal for understanding the meaning of Leibniz’s infinitesimal fictionalism. In particular, I demonstrate the ambiguity that results from sticking too closely with the idea that ideal mathematical terms merely “represent” concrete or actual things. In turn I suggest that, rather than something that had to be prudentially separated from the realm of actual things, the mathematics of infi nitesimals was part of how Leibniz conceived of the distinction between the actual and ideal within the Specimen Dynamicum.

Alain Badiou’s reception in the English-speaking world has centred on his project of a “mathematical ontology” undertaken in Being and Event. Its reception has raised serious concerns about how mathematics could be relevant to concrete situations. Caution must be taken in applying mathematics to concrete situationsand, without making explicit the equivocal senses of “consistency” as it operates in Badiou’s thought, this caution cannot be precisely applied. By examining Being and Event as well as looking backwards at his first philosophical work, The Concept of Model, some key distinctions on the meaning of “consistency” will be clarified.

In this article, I address two different kinds of equivocations in reading Leibniz’s fictional infinite and infinitesimal. These equivocations form the background of a reductive reading of infinite and infinitesimal fictions either as ultimately finite or as something whose status can be taken together with any other mathematical object as such. The first equivocation is the association of a foundation of infinitesimals with their ontological status. I analyze this equivocation by criticizing the logicist influence on 20th century Anglophone reception of the syncategorematical infinite and infinitesimal. The second equivocation is the association of the rigor of mathematical demonstration with the problem of the admissibility of infinite or infinitesimal terms. I analyze this by looking at Leibniz’s constructive method and apagogic argument style in his quadrature method. In treating these equivocations, I critique some assumptions that underlie the reductive reading of Leibniz’s fictionalism concerning infinite and infinitesimals. In turn, I suggest that these infinitesimal “fictions” pointed to a problematic within Leibniz’s work that was conceived and reconsidered in Leibniz’s work from a range of different contexts and methods.

A central controversy in the reception of Leibniz’s philosophy, not only during his lifetime, but also in the immediately posthumous period and more recently, concerns the role that substantial forms play in Leibniz’s ontology. Interpreters like Garber argue that the Leibnizian defense of the quasi-Scholastic substantial forms in the 1680’s-1690’s demonstrate an ontology of corporeal substance irreducible to an idealist ontology. On the other hand interpreters like Adams argue that corporeal substances reduce to a fully idealist ontology and that this period in Leibniz’s work only demonstrate a modification of idealism. In this paper I argue that without clarifying the ambiguous status of what constitutes “ontology” for Leibniz, the stakes of this longstanding debate are unclear and the anti-idealist position appears to be a self-defeating one. By turning to a thorough reading of Leibniz’s transition from the middle to the late years and noting key turns in its historical reception, I argue that the anti-phenomenalist position becomes meaningful in light of an idealist ontology rather than in spite of it. My aim is not to defend either idealism or anti-idealism but rather to reconfi gure the nature of the controversy concerning substantial forms by outlining the limits of current debates over Leibniz’s ontology.