Immanent Activity and Equivocity of intensio in Leibniz’s a priori Argument

Introduction

The distinction between the truths of reason and the truths of fact lies at the heart of Leibniz’s system. The former belong to mathematics, geometry, and logic, while the latter pertain to metaphysics, physics, and morality. The methodology Leibniz employs in his new science of dynamics (a science of ‘power and action’) seems to threaten this distinction. Key to Leibniz’s dynamics is the concept of action (actio) and the principle of its conservation. Leibniz advances an a priori demonstration of this principle, bracketing sensible matter, gravity, as well as the physical laws of motion, and considers action in a purely formal sense—as essential to motion, and as arising from the nature of the body in motion. The demonstration proceeds by a strictly deductive reasoning from what Leibniz takes to be the most reasonable combination of abstract phoronomic variables—the product of velocity and space, vs (algebraically equivalent to the product of force and time, v²t). His a priori demonstration, thus, appears to subject physics to the framework of brute geometric necessity. But if this is the case, is Leibniz not undermining the twofold division of truths central to his system?

Such a critical interpretation was pursued by Martial Gueroult in his influential monograph, Leibniz: Dynamique et Métaphysique (1967).¹ In more recent years, Gueroult’s reading has been challenged by Francois Duchesneau (1994, 2019) and Michel Fichant (1998). Duchesneau has argued that Leibniz’s a priori method is guided by his architectonic principles: the principle of equality of the full cause and the entire effect, and the principle of continuity (cf. Garber 2009: 150, n. 61). Fichant, for his part, has pointed out that the conservation of action, though deduced a priori, does not cease to be a contingent truth if the antecedents from which it is deduced are understood as themselves not reducible to the principle of contradiction, while also, however, not being derived from experience.

Besides being tasked with grounding dynamics in metaphysical principles, Leibniz’s a priori argument is also meant to corroborate his metaphysics. While the formal notion of action, grounded in metaphysical principles, renders Leibniz’s dynamics efficacious, the principle of the conservation of action, in turn, is supposed to contribute to the intelligibility of the intrinsic activity of substance.² As Leibniz tells Burchard de Volder, this argument is ‘the gate’ (porta) through which we are to pass to true metaphysics’ (A II, 3, 603/DV 130–31).

The overall aim of this paper is to shed light on the way Leibniz’s a priori argument advances the intelligibility of substantial activity. Although Leibniz derives his argument by eliminating different combinations of nominal variables and arriving at the most reasonable one, the truth which their combination is meant to evince, I argue, surpasses geometric representation and is accessible only to reason. The conservation of action which Leibniz claims to have established by his argument is the action of a body on itself (actio in se ipsum), understood as an immanent action of primitive force, or entelechy, on a body’s primary matter or primitive passive force (these two primitive, active and passive forces being metaphysical constituents of a corporeal substance). The argument thus is primarily concerned with the intrinsic activity of substance, distinct from its derivative, physical force.

A more specific aim of this paper is to offer a fresh analysis of the concept of intensity and its function in Leibniz’s a priori argument. When Leibniz seeks to convince his interlocutors of the truth of the ontological presuppositions on which his argument is based, he draws an important conceptual distinction between intensio and extensio. This distinction, as employed in his correspondence with Denis Papin and, especially, Burchard de Volder, is of strategic importance which has not been fully appreciated in Leibniz scholarship.

Intensity, as Leibniz is keen to underscore to De Volder, can be understood either as velocity, when extension is space, or as force, when extension is time. And so, while in one sense the degree of perfection corresponds to intensity of motion with which action is completed, in another sense it designates the intrinsic degree of primitive force. The precise meaning of Leibniz’s notion of intensio and the details of its metaphysical import call for a careful, close analysis. Several scholars (Rey 2011, 2009; Fichant 1998; Ranea 1994, 1989) have drawn attention to the echoes of the medieval problem of intensification and attenuation of forms (intensio et remissio formarum) in Leibniz’s analysis of action. More specifically, they have in mind the treatment of intensive variation of motion in fourteenth-century physics. In medieval physics, velocity could be understood either in a purely kinematic sense, in terms of the distance traversed by a body in a given time, or as an intrinsic degree of a body’s affection on a par with intensity of accidental forms such as temperature or color. We are invited to understand velocity, in its combination with the extensive element of space, in this latter sense in Leibniz’s analysis of action. However, such an interpretation, focused on the distinction between the two senses of velocity, while correct, is incomplete. It is not only the notion of velocity, in Leibniz, that enjoys an equivocal sense; the very notion of intensity is marked by a subtle ambiguity. When denoting the degree of primitive force or entelechy, it receives, I contend, a much stronger metaphysical inflection, an inflection that is likewise present in scholastic philosophy. It should be understood in the way the notion of degree of perfection was thematized by John Duns Scotus in his theological metaphysics—as a trans-categorial or transcendental term. This Scotistic notion of intensio no longer concerns the variation of physical qualities but denotes an intrinsic degree (gradum intrinsecum) of being, that is, its ontological perfection. That Leibniz was familiar with Scotus’s metaphysical concept of intensive difference is certain; indeed, he employs this notion in texts such as De Ipsa natura (G IV 511/AG 162)³ Insofar as Leibniz places the notion of intensity, employed in this sense, at the heart of his analysis of actio, his a priori argument no longer, strictly speaking, belongs to mathematical physics, but to what he calls a metaphysical mathematics which measures perfections or degrees of reality.

This paper is structured in the following manner. Section 1 gives a brief overview of the key features of Leibniz’s dynamical metaphysics of corporeal substances; sections 2 and 3 discuss the significance of Leibniz’s dynamics, understood in a broad sense of the term (as a general physics of force, rather than in a more restrictive sense of the science of power and action), for his substance metaphysics.⁴ Section 2 focuses on the metaphysical foundations of Leibniz’s dynamic reform inaugurated in his De corporum concursu (written in 1678, unpublished in his lifetime), while section 3 aims to throw into relief the way Leibniz, in assessing the conclusions of his a posteriori argument for the conservation of force, links his foundationalist project of dynamics with his program of rehabilitating substantial forms. Leibniz takes this argument to corroborate his claim that, beyond extension in length, breadth, and depth, we must posit a more fundamental, dynamic element of force, and takes this inference as ‘obliging’ him to reestablish substantial forms. Yet, I argue, insofar as force is understood as constitutive of an individual substance, rather than simply as an extra-geometric, dynamic element of the physical world, only his a priori argument can claim to establish a direct link between his dynamics and his metaphysics of substantial forms. Section 4 offers an interpretation of Leibniz’s a priori argument along these lines; finally, sections 5 and 6 pursue an analysis of the notion of intensio, which Leibniz deploys in his twofold analysis of actio, as an equivocal term denoting the degree of physical activity, on the one hand, and the degree of immanent, metaphysical activity of an individual substance, on the other.

1. Leibniz’s Dynamical Ontology

A key thesis of Cartesian ontology is that the nature of corporeal substance consists ‘not in the fact that it is a thing that is hard, or heavy, or colored, or affecting the senses in some other way, but only in the fact that it is a thing extended in length, breadth, and depth’ (AT VIII 42/CSM II 224). Leibniz challenges this thesis. According to him, a metaphysical constituent of a corporeal substance is a twofold principle of motion and rest, or of acting and being affected, which he labels with a Greek neologism το δυναμικον (translatable as ‘dynamic being’, or simply ‘dynamism’⁵). Leibniz conceives of this principle as comprising an active force—entelechy or substantial form— and a passive force, or primary matter. He argues that extension is not a primitive or absolute, but a relative attribute presupposing something that is extended. More specifically, extension signifies a continuous, simultaneous repetition (repetitio continua simultanea) or diffusion of some homogeneous nature (A VI, 5, 264204/AG 251). For example, in milk, there is a continuous repetition or diffusion of whiteness; in gold—a continuous repetition or diffusion of specific weight and yellowness (A VI, 5, 264204/AG 251; G VI 584/AG 261; A II, 3, 546/DV 73). Likewise, in a corporeal substance as such, there is a continuous repetition or diffusion of το δυναμικον (A II, 4, 54/DV 241; cf. A II, 3, 121/AG148; GM VI 235/AG 118). To be more precise, in extended matter there is a diffusion of passive force, consisting of impenetrability and repulsion to motion, which is homogeneous in every part of a body and is proportional to its magnitude (A VI, 5, 264206/AG 252); but it is a combination of active and passive forces that constitutes the substantiality of an extended material body.

Why do we need to posit, over and above extension and its modes—shape, size, and motion—a dynamic element of force, and why is there a need for both an active and a passive kind? If, over and above extension with its modes, there were no active element in matter, we would have to resort to a deus ex machina to account for the activity of corporeal substances, given that extension, as thoroughly indifferent, is devoid of a wherewithal for motion. Extension, ‘or that which is geometrical in bodies, if taken in its bare notion [nude sumatur]’, as Leibniz writes, ‘contains nothing from which action and motion could arise’ (G IV 510/AG 161). Moreover, if matter was devoid of passive force responsible for a body’s resistance to motion, then, Leibniz argues, the laws of motion would be different from the ones we actually observe. For example, upon collision, the smallest body would transfer all its velocity to the largest body at rest and would continue moving with it in the same direction without its speed being diminished the least bit; and so, there would be action without reaction (G VI 240/AG 124).

Further still, force, both active and passive, should be taken, according to Leibniz, in either a primary or a derivative sense. Primary or primitive active and passive forces are immaterial, metaphysical constituents of a corporeal substance, even though they never actually exist without extended matter, while derivative active and passive forces belong to the physical domain of efficient causes and are manifest in such phenomena as collisions, projectile motion, and work. Why such curious multiplication of forces? First, if one is committed to a substance metaphysics, then one presumably subscribes to the idea of identity through change. In addition, Leibniz takes it as an axiom that substances are active, or that actions are of the supposits [actiones sunt suppositorum]’ (GIV 509/AG 160).⁶ Now, the material world in which we observe the play of physical forces is one of constant change or becoming. Since physical forces admit alteration, we must treat them as accidents. But surely, accidents presuppose a substance of which they are accidents. They cannot be modes of an intrinsically passive substance since, according to Leibniz, activity cannot be a modification of something passive. Therefore, active physical forces must be modifications of a principle that is itself active.

Without this principle of persistent activity, Leibniz thinks, no substance would retain its unity and identity through change. This means that activity is not a potential, but a constitutive state of a substance. To be, according to Leibniz, is to be one. So, whatever does not enjoy unity could not be called a being. But if unity can be maintained only through action, then to be is to act. And so, not only is everything that acts a singular substance, but everything that is a singular substance acts without interruption (G IV 509/AG 160).

With this adumbration of Leibniz’s metaphysical commitments in hand, we can now examine the metaphysical foundations of his dynamics and try to understand the role that his a posteriori and a priori arguments for the conservation of force play in his system, before finally looking at the strategic function of the language of intensio Leibniz deploys in defense of the ontological presuppositions of his a priori argument.

2. Physics and Metaphysics in Leibniz’s Dynamic Reform

Leibniz’s dynamics, taken in a broad sense of the term, plays a crucial role in his project of grounding the world of extended substances in a metaphysics of force. Of particular importance, in this respect, is Leibniz’s critique of Descartes’s claim that quantity of motion, measured by the product of mass (or rather ‘size’) and speed, m|v|, is conserved in the universe.⁷ What is conserved in the universe instead, Leibniz argues, is force, which he labels a ‘living force’ (vis viva), measured by the product of mass and the square of velocity, mv² (what we call ‘kinetic energy’, though we write $\frac{1}{2}m{v}^2$ ).

Leibniz, of course, was not the first to argue that the quantity of motion is not always conserved, just as he was not the first to state the conservation of mv². In 1668, the Royal Society of London solicited Christopher Wren, Christiaan Huygens, and John Wallis to submit their accounts of the laws of motion based on the phenomena of impact. Wren and Huygens provided the rules for perfectly elastic collisions (collisions in which kinetic energy is the same before and after the impact), while Wallis offered the rules for inelastic collisions. According to the rules provided by Wren and Huygens, in perfectly elastic collisions the quantity that is always conserved is not the quantity of motion but what we call momentum—the sum of the products of masses and velocities understood as speeds with direction. Wallis, for his part, showed that momentum is conserved in inelastic collisions. Huygens, moreover, stated that the products of masses and the squares of velocities of two colliding bodies are the same before and after the impact.⁸ In August of 1699, Leibniz read and transcribed Huygens’s paper. Though recognizing the significance of the physical problem of impact, he received Huygens’s account with criticism. He does not consider, he says, ‘either Wren or Huygens to have hit the target’ of their investigations (A VI, 2, 159). How come?

Huygens and Wren were committed to a Cartesian ontological worldview, according to which collisions of bodies constitute the source of all phenomena.⁹ The problem, though, is that Descartes’s rules of motion governing collisions, apart from the first rule, patently contradict experience.¹⁰ These two facts explain why the search for the rules of impact preoccupied Huygens and the English virtuosi. Leibniz, however, takes a methodological issue with their reliance on the testimony of sense experience. Motion, according to him, can be treated either by reason or by the senses, and the senses should not prejudge reason, while reason should be permitted to prejudge the senses. When the senses ‘appear to contradict reason’, Leibniz claims, ‘we must conclude that there is something underlying [the appearances] which cannot be sensed except by its effect’ (A VI, 2, 160). For example, according to our sense experience, a large body, such as a house, cannot be dislodged by throwing a pebble at it. According to reason, on the other hand, any object, no matter how large, must be capable of being moved by any object, no matter how small and how slow, given the indifference of matter. Due to this indifference, moreover, a body at rest cannot impart any state to another body—neither motion, nor rest, nor direction. But this truth of reason, as Leibniz points out, contradicts Huygens’s first rule, according to which a hard body in motion, upon hitting an equal, hard body at rest, will be brought to rest, while its velocity will be acquired by the body initially at rest (A VI, 2, 161).¹¹ The shortfall of Wren’s and Huygens’s rules, from the standpoint of demonstrative certainty, is precisely, in Leibniz’s view, the fact that they are derived from observation, and thus based on the senses. As such, these rules are simply descriptions of kinetic phenomena. What we ought to establish, according to Leibniz, are ‘the foundations of motion, such as they are in a pure state of nature [in puro naturae statu]’, based on geometric reasoning and without any reliance on physical demonstrations (A VI, 2, 160).

Guided by the distinction between the two approaches to motion (from reason and from sense experience), Leibniz, at the beginning of the 1670s, embarked on a project of establishing two theories of motion—a purely phoronomic theory abstracted from the contingent aspects of our physical universe and based solely on reason, and a theory of concrete motion aimed at accounting for the empirical phenomena, such as impact, as we actually experience them, based on a speculative hypothesis of an all-permeating subtle fluid, ether. By means of this complementary hypothesis, Leibniz thought to have reconciled abstract laws with natural phenomena.¹²

It is only after his reformist, dynamical turn in 1678, in De corporum concursu, that Leibniz adopts Huygens’s quantity mv². And yet again, Leibniz does not simply follow the Dutch savant, for this quantity, in Leibniz’s system, becomes charged with connotations absent from the former’s thinking. By the late 1670s, Leibniz is no longer satisfied with deducing the rules of motion by abstract geometric reasoning while accounting for the actual phenomena of impact by means of an auxiliary physical hypothesis.¹³ He now seeks to provide a unifying, demonstrative principle of these very phenomena and of the rules of impact established by his contemporaries. The aim, now, is to bring the concrete under the purview of mathematical reasoning and to thereby secure a more satisfactory reconciliation between the abstract and the concrete.¹⁴ He does, to be sure, maintain that neither Huygens nor Wren (nor Wallis nor Edme Mariotte, both of whom, at this time, are in the back of his mind) has fulfilled such a desideratum. Leibniz is, in particular, dissatisfied with the attempts to establish the rules of impact based on the relativist picture of motion, even if by this time he has jettisoned the idea of absolute motion, implicit in his early theory.¹⁵ Motion, understood as the change of position (situs), is indeed, he thinks, relative. If any frame of reference, and thus any hypothesis explaining the relations of motion between objects in a system, is valid, then it is impossible to ascribe veritable motion to a particular body (A VI, 3,104). But while he considers motion, taken formally, as relative, Leibniz comes to believe that it must nonetheless be possible, with respect to cause (ratione causae), to ascribe motion to a body from whose contact the change arises (A VI, 4, 1970/RA 229). To this end, he posits a dynamic element of force grounding the phenomena of motion. As Leibniz insists in the preparatory notes to De corporum concursu, there must be, in motion, ‘that absolute thing [absolutum illud] which I call force or power [vim sive potentiam]’ (DCC 376). Leibniz, we will see, redeploys this argument throughout his career.

Key to Leibniz’s foundationalist project of physics is a principle of equipollence of the full cause and the entire effect, which he announced in 1676. The principle, considered by him as a metaphysical axiom, states that ‘the full cause and the entire effect have the same force [potentia]’ (A VIII, 2, 135). This is to say that the effect must have the capacity to produce a state which is equivalent in magnitude to its cause. For example, a stone attached to a pendulum, upon reaching the lowest point of the swing, must (setting aside external resistance) be able to reach the same height from which it was dropped (A VIII, 2, 136).

It is rather curious, at least in retrospect, that Leibniz initially associates this principle with the conservation of the Cartesian quantity of motion. In the opening of De corporum concursu, Leibniz states: ‘[T]he same force is always preserved in all motion. Force is the quantity of effect, or, what follows thereby, the product of the body multiplied by the quantity of velocity [|v|]’ (A VIII, 3, 530/DCC 71). Leibniz’s project, at its inception, can thus be understood as an attempt to reconcile the principles of conservation put forth by his contemporaries—the conservation of relative velocity and of rectilinear translation of the center of gravity (equivalent to the conservation of momentum)—with the principle of conservation of the scalar quantity of motion.¹⁶ Leibniz, however, soon runs into an impasse that renders such an irenic undertaking barren.¹⁷ Eventually, though, there comes a turning point, marking the moment of his reformatio, which puts an end to Leibniz’s struggles. Leibniz restates the identification of force with the effect in terms of the height to which a body could be lifted and associates it with the square of velocity. Given that, following Galileo, 1) heights are as the squares of velocities, and 2) bodies descending from the same height reach the same velocity regardless of the slope of their descent, the velocity which a body attains upon its descent along the inclined plane, and with which it continues to move horizontally, will be as the square root of the height of the inclined plane. Accordingly, Leibniz claims that the force of that body, allowing it to ascend to its previous height, will be as the square of its velocity (A VIII, 3, 636–37/DCC 152–53). Leibniz has effectively replaced the conservation of mv with that of mv², which is now perfectly compatible with, and is taken by Leibniz as grounding, the conservation of both the rectilinear translation of the center of gravity and relative velocity.¹⁸ It is this quantity, moreover, that must be understood as expressing the force of a body in motion.¹⁹

Leibniz, as is well known, first publicly presented his argument for the conservation of mv² (which is a generalized version of the argument advanced in De corporum concursu) in a paper titled ‘Brevis demonstratio erroris memorabilis Cartesii’ (Acta Eruditorum, March 1686). Now, when Leibniz, in the Discourse on Metaphysics, considers the metaphysical implications of this argument, his foundationalist reform of dynamics and his project, by this time well underway, of rehabilitating substantial forms, enter into a propitious marriage. The notion of force, measured by mv², becomes an index, if an indirect one, of a substantial form.²⁰ If, for Huygens, the quantity mv² is devoid of any metaphysical sense and simply denotes a quantity which remains constant in every frame of reference, for Leibniz it attests to the intrinsic activity of substance. ‘What was a mere number to Huygens’, as Richard Westfall neatly puts it, ‘was invested by Leibniz with cosmic significance’ (Westfall 1971: 284). However, as I argue in the next section, while the a posteriori argument does corroborate Leibniz’s injunction that, over and above extension, one ought to posit a dynamic element evading geometric intuition, we must recognize that the conclusion of the argument falls short of bolstering his attempt to associate the notion of force with the activity constitutive of substantial forms. Only in his a priori argument, I contend, can Leibniz be considered as sealing the marriage of his two complementary projects.

3. Leibniz’s A Posteriori Argument and Its Limits

Leibniz begins his a posteriori demonstration by assuming that a body falling from a particular height acquires, upon hitting the ground, the same force that is necessary to lift it to its previous height. Second, he assumes that the force necessary to raise a body of one pound to the height of four yards is equal to the force necessary to raise a body of four pounds to the height of one yard. From these two assumptions, both of which are shared by Descartes and his supporters, it follows that a body of one pound, dropped from the height of four yards, would acquire the same amount of force upon hitting the ground as a body weighing four pounds dropped from the height of one yard. Further, appealing to Galileo’s law of uniform acceleration, Leibniz states that the speed that the first body acquires upon hitting the ground must be twice the speed acquired by the second body, because distances are directly proportional to the squares of velocities. Accordingly, since the distance that the first body falls is four yards, its speed must be equal to two units (2 squared is 4), and since the distance that the second body falls is one yard, its speed must be equal to one unit (1 squared is 1). But if that is the case, then the quantity of motion in the first case is 1x2, and 4x1 in the second case. This means that the quantity of motion is not preserved. Force, however, is always conserved. So, force cannot be identical to the quantity of motion. Instead, it should be estimated by the quantity of the effect it can produce, such as the height to which it can lift a body (A IV, 4, 2028–29/L 296–97). Its correct measure (which Leibniz, however, does not state explicitly here) is, therefore, mv² rather than m|v|.

In section 18 of the Discourse on Metaphysics, having restated, in section 17, the argument of ‘Brevis demonstratio’, Leibniz explicitly articulates its metaphysical implications:

This consideration of force as distinguished from the quantity of motion is rather important, not only in physics and in mechanics for finding the true laws of nature and the rules of motion, and even for correcting several errors of practice which have slipped into the writings of some capable mathematicians, but also in metaphysics for understanding the principles better. (A IV, 4, 1558–59/AG 51; emphasis added)

By the ‘principles’, Leibniz here primarily has in mind the principle of substantial activity grounding the phenomena of motion. Rehearsing his argument from relativity of motion, Leibniz observes that whenever ‘several bodies change their situation among each other, it is not possible to determine by the sole consideration of these changes to which among them the movement of rest ought to be attributed’ (A IV, 4, 1558–59/AG 51). ‘Yet’, Leibniz insists, ‘the force or the proximate cause of these changes is something more real [quelque chose de plus réel], and there is enough ground [fondement] for attributing it to one body rather than to another’ (A VI, 4, 1559/AG 51). He then directly ties this conclusion to the existence of substantial forms: this force, Leibniz claims, ‘is something different from the magnitude, figure and motion, and we can judge by this that all that is conceived in the bodies does not consist exclusively in extension, as our moderns believe’ and that, therefore, ‘we are again obliged to reestablish some entities or forms which they have banished’ (A VI, 4, 1559/AG 51).

Historically, though, the metaphysical import of Leibniz’s a posteriori argument has often been overlooked. It has routinely been assumed that what is at stake in Leibniz’s critique of Cartesians is simply a correct mathematical measure of the force (assumed to be conserved) of a body in motion. This was, in fact, the concern of the two opposing parties (those who took the side of Leibniz, and those who represented the Cartesian camp) immersed in the so-called vis viva controversy that followed the publication of ‘Brevis demonstratio’.²¹ Leibniz, moreover, has been considered to have illegitimately written off the Cartesian quantity. Early on in the eighteenth century, it was demonstrated, by Roger Boscovich and by Jean le Rond d’Alembert, that both the product of mass and velocity (though not speed), as well as that of mass and the square of velocity, are in fact legitimate measures of different effects of force: the first quantity is the measure of the effect of force acting through time, while the latter is the measure of the effect of force acting through distance.²² Indeed, we can represent both quantities in modern notation: the change with respect to time of the vectorially modified Cartesian quantity of motion is just the change of momentum in the equation of Force, $F=m\frac{dv}{dt}$ ; likewise, the kinetic energy in the equation of work, F×s = ½ mv², is just the Leibnizian concept of living force (although the factor of $\frac{1}{2}$ is absent in Leibniz’s quantity). What, nonetheless, seems to be more significant to us in Leibniz’s critique of Cartesians is not so much the issue of the mathematical measure of force but the metaphysical issue of grounding the phenomena of motion. It is precisely such metaphysical considerations that were jettisoned by both d’Alembert and Boscovich—any talk of forces inherent in a body in motion, according to them, is an obstacle for resolving the controversy.²³

It is important to note that Leibniz, however, considered the product of mass and velocity (speed with direction), not only that of mass and velocity squared, to be conserved. To be sure, in the public polemic that ensued after the publication of ‘Brevis demonstratio’, Leibniz does make it sound as if he believes that there is a single conservation principle (that of mv²). Yet, as we saw, once Leibniz replaced the conservation of the quantity of motion with that of mv², he considered the constancy of the rectilinear translation of the center of gravity, as well as the conservation of relative velocity, to follow as a matter of fact. And the reason Leibniz felt justified in positing the conservation of mv² in the first place was that he regarded it as evincing his metaphysical principle of equipollence. In two of his unpublished essays by the same title, Essay de Dynamique (one written in 1692, and the other around 1700), Leibniz, likewise, affirms the conservation of a modified Cartesian quantity, mv, which he calls quantity of progress, as well as the conservation of relative velocity. When, in his public polemic, Leibniz is arguing for the invalidity of the conservation of the Cartesian quantity, he therefore should be understood as having in mind the scalar quantity, m|v| (see Iltis 1971: 22). As Idan Shimony points out in an important paper, ‘Leibniz had already argued, half a century before d’Alembert proposed his solution, that when investigated quantitatively, indeed both forces turn out to be true and legitimate’ (Shimony 2010: 53). While Leibniz does privilege the conservation of mv² over the conservation of mv, he has no mathematical or physical reasons to do so. Insofar as he prefers the conservation of the former, his reasons are metaphysical.²⁴

These metaphysical reasons explicitly come to the fore in the (later) Essay de Dynamique, in which Leibniz states the equations for the conservation of three different quantities: the quantity of relative velocity (in elastic collisions), given by the formula v – y = z – x (whereby the left-hand side represents relative velocity with which the bodies approach each other before the impact, and the right-hand side represents relative velocity with which the bodies depart after the impact); the quantity of progress, given by the formula av + by = ax + bx (assuming that no external force intervenes); and the quantity of absolute force, given by the formula avv + byy = axx + bzz, in both elastic and inelastic collisions.²⁵ He observes that any of these equations can be derived from the other two, and so, mathematically, there is nothing special about any of the quantities. Leibniz notes, however, that the last equation ‘has this excellent feature, that all the variations of signs which can only come from the different direction of the speeds y, x, z […] cease, because all the letters which express these speeds are here raised to the square’ (it is irrelevant whether y is negative or positive, whether the body is coming from the right or from the left; GM VI 227–28/Leibniz 1896: 667–68). This is why this equation ‘yields something absolute [quelque chose d’absolu] independent of the respective speeds or the progress from a certain side’ (GM VI 228/Leibniz 1896: 668). This conclusion, Leibniz says, ‘satisfies at the same time the rigor of the mathematicians and the wish of the philosophers, the experiments and reasons drawn from different principles’ (GM VI 228/Leibniz 1896: 668).

And yet, in what sense does this equation yield something absolute? Gueroult’s classical study of Leibniz’s dynamics, I think, allows us to appreciate the significance of this metaphysical import. A claim that a metaphysical element is brought into relief by a banal operation of squaring the variable of velocity might sound odd. However, Gueroult points out, ‘if one wishes, which is something that Leibniz without a doubt demands, to think of the things which correspond to symbols, one will perceive that the expression “square of velocity”, when related to a purely geometrical intuition of magnitude and velocity, is unintelligible’ (Gueroult 1967: 47). Velocity squared, as Gueroult observes, ‘is neither a velocity actually determinable in intuition, nor a magnitude of such a velocity’ (1967: 47). While the expression mv² is, of course, a strictly mathematical expression and can even be illustrated by geometrical constructions, ‘that which it expresses, [viz., the ability to do work], has in itself nothing pertaining to the immediately represented in the cartesian intuition of space’ (Gueroult 1967: 47). This is why, despite the formal equivalence of the formula of work and Leibniz’s principle of vis viva, Leibniz’s quantity is inflated with ontological connotations that are foreign to, and indeed explicitly barred from, the mechanical world picture.

Hence the metaphysical, anti-Cartesian implications of Leibniz’s dynamical quantity. Descartes’s ontology construes the reality of the world in terms of passive matter (extension in length, breadth, and depth) and its modes (shape, size, and motion). This ontological assumption is fundamental to Descartes’s physics. But if Descartes’s thesis of conservation of the quantity of motion proves to be false, then his ontology must be fundamentally revised. Leibniz, therefore, considers himself poised to thematize extension and its modes as epiphenomena presupposed by a ‘more real’, deeper ontological element of force measured by mv².²⁶

Still, how justified is Leibniz in professing himself, as a corollary to his a posteriori argument, obliged to reestablish substantial forms—the bearers of force? While Leibniz might be in a position, by appealing to his dynamical quantity, to corroborate the claim that over and above extension there must be an extra-geometric element, force, he does not quite appear to be positioned to substantialize this element, that is, to thematize it as a substantial form. What Leibniz’s a posteriori argument establishes is a measure of living force. In Leibniz’s system, living force, we shall recall, is a derivative force. As such, it enjoys only a caducous, accidental reality, one that presupposes an invariable, primitive active force with which he comes to identify the notion of substantial form. Without this primitive activity associated with substantial form, no corporeal substance would retain its unity and identity through change. The activity of primitive force is not, in contrast to the violent action of living force, an actualization of an ability to complete a particular effect, but an always-already-actual, interminable state.

Leibniz, in 1686, has not yet worked out the distinction between the primitive and derivative forces. However, as things stand, the issue is that Leibniz’s a posteriori argument, with all its metaphysical implications, does not prevent us from thinking of an underlying reality of phenomena in terms of a continuous becoming of living force without a foundation in a subject retaining its identity through change. Insofar as the role of his dynamics in grounding extended phenomena in a metaphysics of force as substance metaphysics is concerned, only his a priori argument, I contend, can claim to do the required corroborative work.

4. Leibniz’s A Priori Argument and Its Metaphysical Implications

Leibniz developed the a priori argument following polemical exchanges with Cartesian critics of his new measure of force (the argument, however, was never made public by him). This argument concerns action abstracted ‘from sensible matter [materia sensibili]’ and is derived ‘from contemplation of space and time alone’, without considering gravity, Galileo’s law of uniform acceleration, or any ‘other hypotheses posterior in nature’ (GM VI 291–92/AG 110–11). The notion of action here should be taken in a purely formal sense, that is, as essential to motion (GM VI 346), or as flowing intrinsically (per se) from the nature of the body in motion (A II, 3, 594). It is opposed to what Leibniz, in a Peripatetic lingo, calls a violent action, viz., an exercise of force against external impediments (lifting an object to a certain height, stretching a string, etc.), with which he dealt in the a posteriori argument. There is a clear advantage to this methodological move of abstraction. Gravity (for Leibniz and his Cartesian critics, not a force but a product of ethereal vortices), and thus also the law of constant acceleration, are contingent features of the world.²⁷ While God could have created a world in which falling bodies did not accelerate at a uniform rate, he could not have created a world in which the principle of contradiction, from which Leibniz’s demonstration seems to draw its a priori characteristic, did not hold. If Leibniz could establish the conservation of force and show that it is measured by mv², not by mv, without appealing to those contingent truths, then surely he could not but win over the most intransigent Cartesians?²⁸ Moreover, he would be able to satisfy the critics, such as Abbé de Catelan, who reproached him for not taking into account the variable of time in his a posteriori argument.

But there is, I argue, more to this methodological approach. Despite appearances, Leibniz is not simply giving a different argument for the conservation of living force. As will become clear in our analysis of his polemic with Papin and De Volder, Leibniz, by bracketing the contingent aspects of the world, should be understood as having conceptually reduced the activity of force to the factor of actio as denoting an intrinsic, immanent activity of substance. Consequently, the quantity v², on my reading, gains not merely an extra-geometric significance, as it did in the a posteriori argument, but an extra-phenomenal sense, becoming an index of the activity of primitive, metaphysical, and no longer of the derivative, physical, force. From the way Leibniz sets up the argument, it may seem that, having bracketed the factors of external resistance, he is simply considering the motion of a body in an inertial frame. However, to read Leibniz in such a deflationary way would be to miss the point. In fact, as we shall see, if this were all there is to it, the argument would not make sense from the point of view of classical mechanics (which Leibniz endorses).²⁹ In the rest of this section, I look at the details of his a priori argument and focus on its metaphysical implications.

In measuring formal action, we are to consider it under its two constituent components—the effect and the velocity with which that effect is produced. Given a purely formal notion of action, we will, naturally, deal with a purely formal effect, as opposed to a real effect which is an effect of a violent action (A II, 3, 594/DV 119). A formal effect is simply a transfer of a body through a certain distance. The difference between the two kinds of effect is that while the latter consumes the force, the former conserves it (the formal effect remains the same at each moment of a continuous uniform motion). Since the same formal effect can be produced either slower or quicker, in measuring action we should, furthermore, consider the promptitude with which that effect is brought about. Indeed, Leibniz takes it as an axiom that completing the same distance in a shorter time requires more action (GM VI 349), or that ‘it is more to achieve the same [effect] in a shorter time [majus sit efficere idem minore tempore]’ (GM VI 353). Given these assumptions, the argument goes as follows:

1. An action producing a double [effect] in a double interval of time is twice the action producing a single [effect] in a single interval of time. For example, the action completing two leagues in two hours is twice the action completing one league in one hour, for the first action formally contains the second, or repeats it exactly, given that it traverses one league in one hour twice.

2. An action producing a simple [effect] in a single interval of time is twice the action producing a single [effect] in a double interval of time. For example, the action completing one league in one hour is twice the action completing one league in two hours. For […] that which brings about [praestat] the same [effect] more promptly accomplishes more. And I assume that the actions bringing about the same effect are in the ratio of speeds or in a reciprocal ratio of times. Thus, the action which completes a distance with a double speed has a double value of that which traverses it with a simple speed, or, which amounts to the same thing, the latter is virtually contained twice in the former. Whence follows a conclusion, namely:

3. An action producing a double [effect] in a double interval of time is four times the action producing the same simple [effect] in a double interval of time. For example, the action traversing two leagues in two hours is four times the action traversing one league in two hours. (A II, 3, 549/DV 79)³⁰

And so, as if by serendipity, we get the same quantitative expression as in the a posteriori argument!³¹

Leibniz’s reasoning is based on a simple procedure of algebraic substitution. First, consider the fact that distance, s, is given by the product of velocity and time, tv. Second, assume, with Leibniz, that quantity of action, a, is calculated as the product of velocity and formal effect (distance traversed), sv. Substituting s in vs with tv, we get tvv. If we denote force as p, then a = pt, whereby p = v² (given that a = v²t), or forces are like squares of velocity.³² If one suspects some kind of sneaky subterfuge here, Leibniz will reassure us that he has followed a rigorous procedure and derived his conclusion by eliminating different combinations of phoronomic variables until he was left with the most reasonable one, which so happens to be vs. If, for example, we consider action as the product of velocity and time, vt, then, Leibniz argues, we will end up with an absurdity. Since velocity is directly proportional to space and inversely proportional to time $(v=\frac{s}{t})$ , then actions will be as the distances traversed, and the velocity with which those distances were traversed will be irrelevant. If, on the other hand, we consider actions to be directly proportional to velocities and inversely proportional to times $(a=\frac{v}{t})$ , then this will also result in absurdity, since those actions will be greater that, when their velocities are equal, last a shorter time. Finally, if we consider actions to be directly proportional to spaces and inversely proportional to times $(a=\frac{s}{t})$ , then action will simply be equal to velocity, and now the time in which an action is completed will be irrelevant. In violation of Leibniz’s axiom (majus sit efficere idem minore tempore), a short action will be equal to an action which, with the same velocity as the first, is completed in a longer interval of time (A III, 7, 864–65). The most reasonable combination, therefore, is the product of velocity and distance or, equivalently, the product of the square of velocity and time.

And yet, despite Leibniz’s claim to have grounded his argument in ‘contemplation of space and time alone’, the product of velocity and distance (a measure of action in the second premise of the syllogism), and the axiom on which it relies, presupposes a metaphysical principle of an intrinsic activity of a substance. In the absence of external impediments, a substance’s causal efficacy will certainly, to Leibniz’s mind, be greater if the effect is brought about quicker. However, this axiom does not carry even an iota of geometric self-evidence. Moreover, from the standpoint of classical mechanics, to say that a body in motion, encountering no external resistance, exerts action, is a confused way of speaking in the first place. How can we talk of action when there is no resistance to be overcome? In an unimpeded uniform motion, a body simply persists in its state of motion without exercising the slightest amount of force. As Papin writes to Leibniz,

I take this axiom, omne agens agendo repatitur, as incontestable. Thus, supposing that a body is moving without encountering anything on which it acts and from which it can also receive alteration, I say that a body does not act but that it simply persists in the state in which it is. (A III, 7, 173)

Indeed, assuming an inertial framework, a body in motion does not need to exert effort to stay in the state of motion any more than a body at rest does to stay in the state of rest. As Papin writes to Leibniz, since a body, no matter how fast it moves, ‘does so without any effort’, to say that such a body ‘exercises force’ is to speak improperly; a body in motion persists in its motion ‘with the same ease [facilité] as a body at rest persists in its rest, when nothing impacts [it]’ (A III, 7, 191). This is why Papin refuses to entertain Leibniz’s axiom and cannot accept his measure of action by the product of velocity and distance. Based on Papin’s axiom, action ought to be measured by the quantity of resistance a body overcomes instead. It often happens, after all, that we overcome more resistance in traversing a certain distance slowly than in traversing it fast (A III, 7, 851). Furthermore, even if, for the sake of argument, we considered velocity of a body as a key factor in measuring the quantity of action, how could we possibly say that one body exerts more action than another given the relativity of motion—something Papin takes, and expects Leibniz to, for granted?³³

If we look at Leibniz’s retorts to these objections, it becomes immediately clear that the a priori argument, despite its analytical form, concerns action as a metaphysical constituent of a substance rather than as a purely physical factor. Although, Leibniz admits, the axiom omne agens agendo repatitur applies to agents which produce an effect outside themselves, we can also, he argues, apply it to an agent in which there is only a change of place with no external resistance if we consider it as acting on itself. And so, Leibniz tells Papin, if an agent ‘only acts on itself [sur soy même], it is also itself that is being affected [qui souffre]’ (A III, 7, 182). As Leibniz, similarly, writes to De Volder three years later, ‘in the free or formal action of a moving body, if the latter is conceived of as acting on itself [agens in se ipsum], we can conceive analogically of a sort of real effect, which is not a change of place (which I consider only as something modal), but the very fact that a moving body with a given velocity is proceeding to a following moment’ (A II, 3, 596/DV 121–23). Even when a body moves unimpeded and thus exercises no force against an external resistance, we can therefore speak of an internal, immanent exercise of force, a force that a body continuously exercises on itself as it persists in uniform motion. At the same time, as it acts on itself, we can conceive of it as being acted upon by, or ‘suffering’ from, its own mass due to ‘the inertia or resistance of this mass’ (A III, 8, 27).

By inertia, Leibniz here does not mean the Cartesian or the Newtonian concept, but what he terms a natural inertia, a passive force intrinsic to a body by means of which that body resists change (A III, 7, 863). Papin, of course, refuses to accept Leibniz’s claim that a body’s mass naturally resists motion, or that a moving body ‘suffers from its own mass’ in continuously trying to overcome its resistance, since he holds that matter is thoroughly indifferent to motion and rest (A III, 8, 59–60). He reminds Leibniz that a body, rather than intrinsically resisting motion, will continue to stay in motion until an external force brings it to rest: ‘[F]ar from it being the case that a body resists this motion or transport by its natural inertia […], on the contrary, it can never be reduced to rest except by some force which is opposed to it’ (A III, 7, 879).

Now, Leibniz does not think that a mechanical notion of inertia is inadequate for the analysis of motion.³⁴ However, Leibniz cannot accept a metaphysical claim that a body is indifferent to motion and rest. For, if it were indifferent, then, according to Leibniz, this would, as we saw, result in a situation whereby the lightest object will drag along with it the heaviest body at rest upon bumping into it, without the slightest change in speed (a result that Leibniz, in his youth, embraced). Such a scenario would entail the possibility of an action without any reaction. Papin’s axiom, omne agens agendo repatitur, must hold. However, it can only hold if a body enjoys an intrinsic passive force that resists change! Likewise, even if Leibniz (again, rehearsing the argument from relative velocity) recognizes that, at the level of phenomena, motion is relative, he insists that it is necessary to presuppose an element of force to account for the very possibility of motion. ‘Let it be the case’, Leibniz tells Papin, that motion is relative; nonetheless, ‘there is truly always motion in [a particular body] as there is in other bodies to which you attribute it, each contributing their own to the total change, otherwise there will be no motion in the world’ (A III, 7, 224). So, if, Leibniz claims, Papin attributes ‘a true motion to some body in a sense of nominatione vera intrinseca’, then he (Leibniz) himself is justified in attributing to it ‘a true action or change’ (A III, 7, 224).

And so, when Leibniz says that, in continuing its motion, a body is acting on itself and that, at the same time, it resists its own motion due to natural inertia, he has in mind the action and the resistance of primitive forces which together constitute a twofold principle of acting and being acted upon—το δυναμικον. ‘When the body is in motion, and resists rest’, Leibniz writes in what is perhaps his most philosophically candid letter to Papin, ‘I hold that it thereby has a force or entelechy which makes it tend to continue the motion. Whence it follows that the mass continuously resists the entelechy, and that, therefore, there is action and reaction in the body itself’ (A III, 8, 68–69; emphases added).

Leibniz, in other words, has isolated, in his analysis, the activity of the primitive active force which is exercised on the complementary primitive passive force. He conceives of this formal exercise of primitive force as grounding the violent action of physical, derivative force, even if, in the physical world, there is never an action without external resistance (A II, 3, 594/DV 119).³⁵ In this way, he could be taken as also having established the measure and conservation of living force. Leibniz is, indeed, quite explicit that he thinks of his a priori argument as fulfilling such a foundationalist aim. As he writes to Papin:

I judge this measure of force by the formal actions to be more profound and more a priori, whereby each thing must be measured in its source; and the source of this force [puissance] capable of producing the actions of the second kind [i.e., violent actions], is the faculty of producing the formal action of the first kind. (A III, 7, 891)

Leibniz’s approach, in the a priori argument, resembles the aprioristic method employed in his early, abstract theory of motion. But now, despite the nominal features of the variables at play, the truth which their most reasonable combination is meant to evince surpasses representation and is accessible only to reason. By eliminating different combinations of his phoronomic variables and choosing the most reasonable one, Leibniz, whilst satisfying the demands of analytical reasoning, brings into relief a metaphysical element which nonetheless evades geometric intuition. Leibniz, we saw, had successfully indexed an extra-geometric, dynamic element of force in his a posteriori argument. Since, however, the domain of sensible matter and empirical laws is now bracketed, the factor v² receives an extra-phenomenal, metaphysical inflection designating the activity of primitive force rather than the capacity of a derivative, living force. Leibniz’s two complementary projects—the project of dynamics and the program of rehabilitating substantial forms—are here combined in a more direct manner. Here, unlike in his a posteriori argument, it is no longer the concrete physical phenomena that are brought under the purview of demonstrative reason, but the immanent substantial activity independent of the contingent features of the world.

The question regarding the validity of the argument, of course, jumps to the fore. Previously, Leibniz took the conclusions of his (a posteriori) argument for the conservation of mv² as indirectly supporting his metaphysics of substantial forms. But now, in contrast, the substance’s constant, immanent activity, thematized under the formal notion of actio, is presupposed at the outset of the argument. If the a priori argument is meant to demonstrate the conservation of this activity, then it is clearly circular. It is, however, more accurate to interpret his a priori argument as rendering the constancy of immanent activity of primitive force intelligible according to the principles of analytical rigor. This activity, as we pointed out, is implicitly entailed in his axiom, majus sit efficere idem minore tempore, on which the second premise of the argument is based. If we accept it, then, Leibniz thinks, we can articulate the intensity of this immanent activity according to the exigencies of geometric intelligibility in abstract, nominal terms of distance and time, and thereby give it a mathematical expression. This expression, following the most reasonable combination of phoronomic variables, involves the square of velocity. Furthermore, given that this mathematical expression is equivalent to the quantity denoting living force, the latter can now also enjoy an a priori foundation. And so, just as the a priori argument contributes to the intelligibility of substantial activity, it also renders the physical, dynamical notion of action more efficacious.³⁶

The a priori argument, then, no longer (or not only) belongs to mathematical physics in the strict sense of the term, but to what Leibniz calls ‘a certain metaphysical mathematics’ which measures ‘perfections or degrees of reality’ (A II, 3, 656/DV 181). It is to the significance of this curious tool of metaphysical mathematics, and to the notion of intensio at the heart of it, that I now turn. The language of intensity, as we will see in the next section, is marked by an equivocity corresponding to the ambiguity of Leibniz’s notion of action understood as, on the one hand, physical activity (which the a priori argument is meant to render efficacious), and as immanent activity of substance (which the argument is meant to make more intelligible), on the other. Leibniz’s mobilization of the language of intensity, as I show in section 6, plays a decisive, strategic function in his attempt to persuade De Volder, an open-minded Cartesian, of the soundness of his a priori argument.

5. Equivocity of Intensio

Leibniz thematizes the distinction between the two elements of action—formal effect and velocity—as, respectively, its extension (extensio) and its intensity (intensio). The quantity of formal effect—the distance traversed—is, we are told, the diffusion or extension of action, while the quantity of velocity with which it is completed is its intensity (GM VI 355). Leibniz’s use of this conceptual distinction is marked by a metaphysical inflection, for it does not concern the properties of phoronomic variables but a magnitude of ontological perfection of action. As he puts it to Papin,

the perfection or the degree of reality in things, and particularly in motion, can be measured according to two reasons, that is, by extension, which is here the magnitude of the space traversed, and by intensity, which is here the promptitude or the speed of the change or motion. (A III, 8, 129; emphasis added)

The metaphysical import of the language of intensiones and extensiones is even more salient in his correspondence with De Volder. The ultimate reason as to why, Leibniz writes to De Volder, he is invoking the intensive estimation (aestimatio intensiva) of action, is

so that the sources of the most beautiful things may be known more closely [intimius], and that it may be recognized that the principles of nature are no less metaphysical than mathematical; or rather, that the causes of things are hidden in a certain metaphysical mathematics which measures perfections or degrees of reality. (A II, 3, 656/DV 181; emphasis added)

Leibniz’s a priori argument decidedly takes us outside the purview of mathematical physics. But what, exactly, are we to understand by metaphysical mathematics and the perfections or degrees of reality it measures? Alberto Ranea, in his analysis of the Leibniz–Papin correspondence, has argued that by calling the space traversed the extension of action, and velocity its intensity, understood as a perfection of reality, Leibniz ‘takes us back to the Physics of the c a l c u l a t i o n e s in the Fourteenth Century’ (Ranea 1989: 56; cf. Fichant 1998: 227; Rey 2009: 50; 2011: 255). Ranea has in mind the so-called Oxford Calculators, who were the first in the Western tradition to attempt a rigorous quantification of motion, and their successors in the continent (such as Marsilius of Inghen, ca. 1340–1396). The most prominent among the calculators were Thomas Bradwardine (ca.1300–1349), William of Heytesbury (ca. 1313–1372/1373), and Richard Swineshead (or Suiseth, fl. 1340–1354).³⁷ A central problem with which the Calculators were concerned was that of relating the change in velocity of a body in motion to the change in the relation of the motive (i.e., active) force and the force of resistance that are jointly causing motion (V ∝ F/R).³⁸ The Calculators were working against the backdrop of another shared problem, viz., the problem of intensification and attenuation of forms (intensio et remissio formarum). The problem had to do with the manner of increase and decrease in the strength of qualities, be they virtues, such as charity, or physical qualities, such as light, color, or heat.³⁹ Local motion was treated on a par with other physical qualities, and the Calculators’ aim was to compare the intensification (acceleration) and attenuation (deceleration) of motion to the relevant changes in the ratio of the motive force to the force of resistance. The ontological theory that enabled them to quantify these changes was the so-called ‘additive’ theory, according to which a quality is made more intense by the addition of an intensive part, a degree (gradus). On this picture, a physical quality was understood as having a continuum of homogeneous, infinitely divisible degree-parts, just as a line consists of a continuum of infinitely divisible extensive parts.⁴⁰

The most important, for our analysis, aspect of medieval physics, to which Leibniz’s distinction of intensio and extensio alludes, is the fact that, in it, velocity, as intensity of motion, lends itself to a metaphysical sense which is quite distinct from a purely kinematic meaning. As Ranea notes, velocity, in medieval physics, has two meanings: either a quotient of space and time, or an intensity of motion understood as an accidens intrinsecum of the moving body (Ranea 1989: 57).⁴¹ In this latter sense, velocity ‘becomes a metaphysical or “quasi-physical” sign of the inner perfection of motion, a magnitude quite independent of any extensive viz. quantitative treatment’ (Ranea 1989: 57). These two different senses of velocity, Ranea argues, are present in Leibniz’s a priori argument: while, in the first premise of his syllogism, velocity should be understood as a quotient of space and time, in the second premise it assumes this metaphysical or quasi-physical sense (Ranea 1989: 57).

The perfection of reality to which Leibniz refers would, on such a reading, amount to the intensity of motion as an intrinsic, non-extensive accident imbued with a dynamical sense. Leibniz’s metaphysical mathematics could, accordingly, be taken as a heuristic tool affording us indirect access to the ‘principles of nature’ or ‘the causes of things’, that is to say, to the forces or substantial forms, by measuring degree or perfection of motion as a quasi-physical index of their activity. Leibniz’s mobilization of the Calculatores’ vocabulary would therefore serve a strategic function in bolstering his ontological position, according to which the physical features of corporeal substances are irreducible to the modes of extension.

Couching Leibniz’s argument, as Ranea does, within the context of the medieval problem of intensio et remissio formarum, seems justified, especially given Leibniz’s professed admiration for the Calculatores’ tradition. We know, indeed, that Leibniz held the Calculators, and especially Swineshead (whom he calls Johannes Suisset⁴²), in high esteem, before he even had the opportunity to read Swineshead’s famous Liber calculationum (or simply Calculationes). For Leibniz, in fact, Swineshead’s name embodied the Calculatores’ tradition, even if Leibniz seems to have operated with his own ‘personal myth’ of Swineshead, based on the account of others, without being familiar with the precise details of the Calculator’s work (see Lourié 2012: 59).⁴³ Leibniz regarded Swineshead as the leader, or the first, (princeps) of the Calculatores, tributing him, rather inaccurately, with having introduced mathematics into scholastic philosophy (A II, 2, 555; A, VI, 4, 965) and with pioneering the application of mathematics to physics (A II, 1, 65).⁴⁴ So strong was Leibniz’s admiration for what he perceived to be Swineshead’s work, that it influenced his thinking in areas lying outside the prerogative of physics, such as logic⁴⁵ and, perhaps, the mathematics of infinitesimals.⁴⁶ When Leibniz finally had the chance to look at Swineshead’s Liber Calculationum in 1689 at the monastery of San Marco in Florence, his admiration for the Calculator was only strengthened.⁴⁷

The fact that he saw a copy of Swineshead’s Calculationes during his sojourn in Florence, shortly before setting out to work on the manuscript of Dynamica, is probably of considerable significance, given that Swineshead was working on problems similar to the ones occupying Leibniz’s attention at the time (the estimation of intensity of motion in relation to force).⁴⁸ It is more than likely that Swineshead’s masterpiece had some influence on the development of the conceptual apparatus Leibniz deployed in his analysis of actio. Fichant has even claimed that it is possible to hold, as ‘more than a hypothesis, that Leibniz’s direct acquaintance with the calculationes intensionum in Florence, […] in the process of developing the Dynamica, must have had a direct effect on the formation of the conceptual system of the new science’ (Fichant 1998: 229–30). Moreover, Swineshead’s influence, Fichant adds, must have had important consequences for Leibniz’s metaphysical project. Leibniz’s rehabilitation of the scholastic forms, already advanced before his mature dynamics, would now be complemented with ‘a resumption of the scholastic mathematization of the intensive variation of forms’ (Fichant 1998: 230). By this complementary resumption, Leibniz, Fichant claims, ‘radicalized his critique of the Cartesian reduction of physical reality to extensio’ (ibid). We should only add that Leibniz had already, prior to reading Swineshead, employed the distinction between intensio and extensio to analyze different elements of the ‘form’ of motion in a short manuscript titled ‘Specimen de motus causa et de corporum qualitatibus’, written sometime between 1678 and 1681.⁴⁹ Leibniz’s reading of Swineshead’s Calculationes in Florence, then, might have prompted him to redeploy this distinction at the time he was seeking an adequate vocabulary for a formal analysis of action.

I contend, however, that a reading of Leibniz’s a priori argument, pursued by Ranea and Fichant, against the backdrop of the Calculatores’ mathematization of intensities, while justified, reveals only half of the story, and only one aspect of the scholastic legacy in Leibniz’s thought. His correspondence with De Volder, as we shall see in the next section, allows us to discern a more abstract, more fundamental metaphysical concept of intensio. Intensity, in this thicker sense, should, I argue, be understood in the way it was employed by Scotus in his theological metaphysics—as a trans-categorial or transcendental term which no longer concerns the variation of qualities but denotes an ontological difference of a particular being. Placing the notion of intensio, understood in this way, at the heart of Leibniz’s analysis of actio would make sense in light of our conclusions in the previous section. Whereas, in its application to physics, the notion of intensity concerns alteration of qualities, in its thicker metaphysical sense it denotes, on the contrary, an invariable degree of metaphysical perfection. Accordingly, intensity, when applied to a substantial form, should be understood not in terms of its potentially variable latitude (which holds only in the case of accidental forms), but in terms of its rank in the order of beings (in ordine entium). Insofar as Leibniz complements his rehabilitation of substantial forms with a resumption of the medieval paradigm of mathematization of intensities, this mathematization should not, contrary to Fichant’s insight, be understood as quantification of the variation of qualitative forms, but as a measure of the latitude of being. Given the indivisibility and persistence of substantial forms (a view held by the schoolmen and shared by Leibniz), intensity as a degree of their metaphysical perfection takes us away from a physical problem of qualitative alteration to a more abstract, Augustinian problem of the ontological structure of the universe.⁵⁰ In Leibniz’s system, the notion of intensity, therefore, is not limited to a purely physical sense but, when understood in a more primitive, trans-categorial sense, concerns the intrinsic, unremitted activity of primitive force.

6. Intensity as a Degree of Ontological Perfection

In his correspondence with Papin, as we have seen, Leibniz employs the distinction between extensio and intensio to describe, respectively, the formal effect, i.e., the distance traversed, and the velocity with which it is completed. Leibniz, as Ranea correctly notes, aims to shift Papin’s focus from the notion of velocity, present in the first premise, as a kinematic factor (s=vt), to the factor of velocity, in the second premise, as an intrinsic component of action. In this way, Leibniz aims to bring the axiom (majus sit efficere idem minore tempore), in which the minor premise of the a priori argument is grounded, into sharper relief. However, in his correspondence with De Volder, Leibniz, in addition to the difference between intensio as velocity and extensio as distance, distinguishes between two different ways in which the notions of intensio and extensio themselves ought to be understood. Extension, Leibniz tells De Volder, can be considered either as space or as time. If extension is considered as space, then intensity is velocity, but if extension is considered as time, then intensity is the square of velocity, i.e., force (A II, 3, 622/DV 149). Here, Leibniz introduces this additional distinction because De Volder’s objections to his argument require a somewhat different approach than Papin’s critique did. De Volder, unlike Papin, eventually accepts Leibniz’s axiom. Yet De Volder still refuses to accept Leibniz’s measure. In an exchange of letters leading up to Leibniz’s introduction of the distinction between the different senses of intensity and extension, he writes to De Volder:

[Y]ou understand well enough that there is some cause or perfection in the agent [causam vel perfectionem in agentem], which makes it so that a free action is brought about [praestantur] more promptly, and that this perfection is measured by time, as if a posteriori, since there is no other way of grasping it. (A II, 3, 593/DV 119)

De Volder responds that he has ‘never doubted’ that there is a cause or perfection in the agent which makes it so that a free action is brought about more promptly, and he concedes that this perfection ought to be measured by time (A II, 3, 608/DV 136). However, the following train of thought prevents him from accepting Leibniz’s measure. A free action, taken in itself, and not with respect to a determinate time, ought to be considered, De Volder holds, according to promptitude, which he calls praestantia (efficacy or productivity) of action, and which he identifies with force. So, even though he grants that the perfection of action must be measured by the time in which it completes a certain distance, he holds, however, that this perfection does not depend on time, given that it is the same at each moment of a uniform motion. Two actions, completed in a certain interval of time, should therefore first be compared according to their respective praestantiae, and then according to time. But in that case, an action traversing a certain distance in one hour will be equal to the action traversing the same distance in two hours, for even though the first action is twice as prompt or perfect as the second, the second action is twice the first action insofar as time is considered. The actions themselves, which are calculated as the products of force and time, will thus be equal, contrary to Leibniz’s calculation (A II, 3, 609/DV 136–37).

At this point, Leibniz realizes where the root of De Volder’s intransigence lies. The reason why De Volder fails to accept his measure of action is the fact that he has conflated velocity and force. What De Volder calls praestantia of an action of a body completing a certain distance in one hour is indeed twice as perfect as that of the praestantia an action of a body completing the same distance in two hours, Leibniz agrees, if by praestantia we understand intensity of motion (velocity). However, when understood as intensity of force, the praestantia of the first action is four times that of the second, because force, according to Leibniz, is calculated as the product of time and velocity squared (A II, 3, 622/DV 149). It is to make De Volder grasp the difference between the two analyses of actio, and between two different senses of praestantia which De Volder failed to distinguish, that Leibniz presents him with the two corresponding analyses of intensio and extensio. The neglect of the two analyses, Leibniz claims, ‘has brought a great deal of confusion with respect to the doctrine [of actio]. For it is quite clear that intensity taken in one sense, [as velocity], must be different from intensity taken in the other sense, [as force]’ (A II, 3, 622/DV 149). Taken in the latter sense, intensity refers not to perfectio motus, but to perfectio agentis which makes it so that a free action is brought about (praestantur) more promptly.

Leibniz’s distinction between the two senses of intensity, corresponding to two different senses of praestantia or perfectio, should allow us to gain a clearer understanding of Leibniz’s claim that ‘the principles of nature’ or ‘the causes of things are hidden in a certain metaphysical mathematics which measures perfections or degrees of reality’ (A II, 3, 656/DV 181). By the principles of nature or the causes of things, Leibniz, as we pointed out, has in mind the metaphysical constituents of corporeal substance—primitive forces. When Leibniz refers to perfections or degrees of reality which are measured by metaphysical mathematics, he should, then, be understood as referring not only to perfection of motion, or perfection of physical activity, but to perfection of primitive activity which is the source of the violent action of derivative force. His metaphysical mathematics, accordingly, is meant to furnish a quantitative language for these ontologically primitive differences in intensity, or degrees, of perfection.

Ranea, to be sure, is correct to insist on the two senses of velocity, a kinematic and a metaphysical one, present in Leibniz’s a priori argument. As Leibniz writes to De Volder, of the three ‘truths’: 1) s = vt; 2) a = vs; 3) a = vvt, the first one is geometrical, while the other two are metaphysical (A II, 3, 669/DV 197). This supports Ranea’s distinction between velocity as a quotient of space and time (hence the ‘geometrical’ truth of the first premise), and velocity as a metaphysical sign of perfection of motion understood as accidens intrinsecus. Yet the second of the two ‘metaphysical truths’ (a = vvt), present in the conclusion, is marked by a different sense from that of the metaphysical notion of velocity. It indicates the inner perfection of primitive activity presupposed by motion. It should, therefore, be taken in a more abstract metaphysical sense of gradus perfectionis which we find in Leibniz’s other texts (most notably, De rerum originatione radicali), as a quantity of essence or reality with which every possible tends towards existence and strives to persevere in it (G VII 303/AG 150).⁵¹ Leibniz, in fact, explicitly associates, in a letter to Johann Christian Schulenburg, the ‘degree of a created perfection’ with ‘the force of acting’ which ‘constitutes a substantial nature’ (A II, 3, 427).

The scholastic concept of intensio which Leibniz redeploys to denote the perfectio agentis is no longer, on my reading, restricted to the problem of quantification of motion. Rather, Leibniz here resumes the Scotistic notion of intensio qua gradus perfectionis conceived in terms of an intrinsic degree of a being’s perfection. The intensive difference of being, on Scotus’s account, is not a formally distinct feature determining a being from without but is an intrinsic mode (modus intrinsecus) of being. God, according to Scotus, enjoys the highest, infinite degree of perfection, while finite beings are limited in their own respective degrees of perfection. In a famous passage, he explains this idea by invoking the intensive differences of white. ‘Infinite being’, Scotus tells us, is not a concept composed of a subject and an attribute, like ‘good being’ or ‘true being’, because ‘infinity’ is not an attribute of being but rather its intrinsic mode (Ord. I, d. 3, p. 1, q. 2, n. 58 [Scotus 1950–: 48)]). Similarly, ‘intense white’ is not an accidental concept like ‘visible white’ but designates (dicit) an intrinsic degree (gradum intrinsecum) of white (Scotus 1950–: 48). Although Scotus, in articulating the metaphysical notion of an intrinsic mode of being, draws a parallel with a degree of a physical quality, color, the notion of intensive magnitude in its metaphysical sense is no longer taken in a categorial, but in a trans-categorial or a transcendental sense, that is, as transcending qualities and quantities, as well as other categories, and thus as attributable to being in general. Although ‘more and less’, Scotus argues, are ‘proper attributes [passiones propriae] of quantity, nonetheless, taken in a transferred sense [translative], they are transcendentals [transcendentia] and the attributes of the entirety of being [passiones totius entis]’ (Quod., q. 6 [Scotus 1950–: 144]). This distinction between a categorial and a transcendental sense of ‘great’, according to Scotus, corresponds to the distinction, invoked by Augustine, between the dimensive sense of ‘great’ (‘magnum’ mole) and its virtual sense, or a sense denoting perfection (‘magnum’ virtute vel perfectione [Scotus 1950–: 144]).⁵² When Leibniz invokes the notion of intensity to denote perfectio agentis (i.e., primitive force) rather than perfectio motus (velocity), we should take it in such a transcendental, or virtual, sense of the term. That Leibniz was familiar with Scotus’s modal distinction is indeed apparent from passages in which he mobilizes the distinction between infinite and finite as degrees of perfection of being when describing God as a ‘being of the greatest intensity of perfection’ (ens maximum intensione perfectionis, G IV 511/AG 162).⁵³ Under such a transcendental import, Leibniz’s notion of intensity qua perfectio agentis denotes the degree of primitive force.⁵⁴ Since, in the a priori argument, the domain of sensible matter, in which the flux of derivative forces—their intension and remission—plays out, and which is susceptible to a categorial, qualitative and quantitative analysis, is bracketed, what else can the perfection of being denote, other than the intensity of entelechy with which a substance acts sine intermissione?

Conclusion

In this paper, I have argued that Leibniz’s a priori demonstration of conservation of action can be understood as both advancing his metaphysics of substance and grounding the principle of conservation of living force in a more abstract, primitive element of substantial activity. Leibniz brackets the contingent aspects of the world and, by means of a combination of abstract phoronomic terms, brings into relief the activity of corporeal substances, reduced to the factor of actio, which, however, transcends geometric intuition and is only accessible to reason. Leibniz thematizes the concept of actio as an immanent activity—actio in se ipsum—an activity of primitive force or entelechy on the complementary metaphysical element of primary matter or primitive passive force. Key to Leibniz’s strategy, as I was trying to show, is his redeployment of the medieval notion of intensio as a degree of ontological perfection, a notion that can be traced to Scotus’s theological metaphysics. Understood in such a metaphysically inflected sense, intensio no longer denotes a degree, or latitude, of a physical quality, as it did for the Calculatores, but a degree of perfection within the latitude of being. In his metaphysics, Leibniz uses this concept to denote the quantity of essence or reality with which every possible tends towards existence and strives to persevere in it. It is, I have argued, precisely in this sense that Leibniz invokes the notion of intensio as degree of perfection in his analysis of actio.

Acknowledgements

I would like to thank the anonymous reviewers for their thoughtful and instructive feedback, which greatly helped me improve this paper.

Competing Interests

The author has no competing interests to declare.