1“For even if they are not extended, monads have a certain type of situation (situs) in extension, that is, they have a certain ordered relation of coexistence to other things, namely, through the machine in which they are present.” [1] In this famous letter of June 20, 1703 to de Volder, Leibniz recalls that substances can never be conceived without an organic body. This body is, at once and the same time, that which makes possible the expression of the world that goes by the name of perception, and that which explains the partial character of that expression, its relative obscurity.

2How does perception allow us to understand the relation between the singularity of a substance, grasped as a positioning in the world, and divine perfection?

3My hypothesis is that perception is the means to grasp the relation between the infinite and singularity, on condition that we understand perception on the basis of the notion of action and, more specifically, the conceptualisation of action in the Dynamics.

4I would therefore like to start with a motif that is at the heart of this conceptualization of action, that of the measurement of the real that allows one to distinguish degrees of reality in things, and thereby to differentiate things in relation to each other. This motif constitutes a principle of differentiation of the real that sets the stage for the complex relation that links perception to perfection. It enables singularity to be conceived as the assignment or identification of a degree of perfection in each reality, through which is expressed the order of the world in the infinity of its relations.

5I would therefore like to retrace the path that leads us to identify, at the heart of the ambivalence [2] of action, at once dynamic and metaphysical, the possibility of “mathematicizing ontological concepts” [3] so as to take up once more a celebrated formula—that is to say, to measure and to differentiate “the degree of reality in things,” as Leibniz writes to one of his correspondents, Denis Papin, in a 1699 letter. [4]

I. Measuring the Real in the Middle Ages

6We hypothesize that the understanding of action as ambivalent is a way to grasp the expressive variations at work in substance.

7The correspondence that unfolds between Leibniz and the Dutch physicist Burcher de Volder is the theatre for the explication of formulas that we see reiterated many times from Leibniz’s pen, in texts published in the learned journals of the epoch. The idea, in a word, that dynamics allows us to understand and to refound substance.

8And this indeed is the case in 1694’s Reform of Metaphysics, in 1695’s New System of the Nature and Communication of Substances, and again in 1698’s De Ipsa Natura. But, as Michel Fichant has emphasized, he never explains in these published (and public) texts exactly how dynamics is to allow this refoundation. This is precisely, however, what he does in the correspondence with de Volder. It is, in this regard, interesting to note that the correspondence functions (as is so often is the case with Leibniz) as a means not only for the explication, but often even for the justification of publicly formulated theses.

9In this article, we should like to understand the ambivalence at work in Leibniz’s thinking of action; or, in other words, the functional reason for the use of the same word, for its acceptation at once as dynamical object (since it is the object of a new principle of conservation) and as metaphysical object (since action is the essence of substance).

10The major characteristic of the new principle of conservation of motive action is its proposing a twofold resolution of action according to intension and extension. [5] This twofold resolution corresponds explicitly to the reprise of a motif from medieval physics: that of the latitude of forms, or the quantification of qualities. In brief, this question, formulated by the philosophers of Merton College at the beginning of the fourteenth century, concerns an analogical transposition that allows the domain of validity of quantification to be extended to quality—from piety (in the theological domain) to movement (in the physical). Peter Lombard, in the first book of his Sentences (distinction XVII), asked whether or not the charity or grace brought about by the Holy Spirit can grow in man, since it was customary at the time to consider that charity was a constant spiritual entity possessed more or less by each individual according to how much or how little he participated in it (its variation therefore depending, for example in Thomas Aquinas, on the degree of participation of each individual). However, with Duns Scotus, the idea first appeared that the quality itself, and not the degree of participation, could be considered as a variable. In this regard, it thus seemed, at one time at least, that the augmentation of a quality came from the addition of new parts to an already-existent form or quality; and inversely that a quality could diminish with the removal of distinct parts of it. Thus, quality, considered as susceptible to augmentation and diminution, was compared to a magnitude of variable intensity. As J.-P. Solère notes, “The object of the problem is to know whether a form is necessarily fixed in nature […] or whether, on the contrary, it allows of degrees in its essence.” And he formulates the problem in the following terms: “How can the same property change, becoming greater or lesser, when a form is supposed to be an invariant structure that imposes a precise determination?” [6]

11The question of the latitude of forms has been the object of important commentaries, whether we think of those, already of quite a vintage, of Pierre Duhem in his Studies on Leonardo da Vinci, or those, elaborated from quite a different perspective, of Annaliese Maier in An der Grenze von Scholastik und Naturphilosophie. [7] We draw on these interpretations to try and understand the introduction and the conceptualization of the question of degrees of perfection in the domain of fourteenth-century physics, and consequently, the significance of the metaphysical stakes of this thematic that were imported, along with the vocabulary of the latitude of forms, into the heart of the natural philosophy that produced Leibniz’s Dynamics.

12Duhem dates the introduction of the idea of degrees of perfection to a “disputation” between Thomas Aquinas and Giles of Rome. For Saint Thomas, in his Commentary on the Sentences, [8] intensity is defined as a greater or lesser proximity to perfection, so that a quality (whether charity, whiteness, or heat) grows or diminishes in intensity not through an addition or subtraction of intensity, but by “perfecting its proper essence.” In this framework, it is in the very essence of the thing that its latitude resides—that is to say, the reason for its augmentation. This means that in its essence a form is “capable of many degrees,” and “each inferior degree is then a potentiality of a higher degree,” so that it is possible to make each degree correspond with what he designates as a quantity of form. Against this conception of intensity that resides in the very essence of form as the mark of its degree of perfection, Giles of Rome opposes a conception according to which “there is but one sole degree, which is more or less completely realized in the subject wherein it resides.” In any given subject, the possibility of variation in a quality is not possible. In this conception, the form is determined and thus cannot be the object of variations; each form possesses an invariable degree. In a second stage, the work of Richard of Middleton will introduce a comparison between the measurement of quantity and that of quality. In 1281, in his own Commentary on the Sentences of Peter Lombard, he explicitly proposes an analogy between the intensity of a quality and the magnitude of a quantity, arguing that a qualitative form can grow through the addition of parts. He argues that one must distinguish extended quantity as quantity of mass (quantitas molis) from another quantity, quantity of force (quantitas virtutis), which takes account of the intensity of the quality. For the first time, the intensity of a quality is thought in terms of quantity. Now, in order to measure this quantity of force that takes account of the intensity of a quality, Richard of Middleton proposes an extensive measurement and an intensive measurement; and it is in terms of the second that quality effectively augments. [9] This conception is taken up again by Jean de Bassols, who makes explicit the distinction between quantity of mass and quantity of force, by comparing quantity of mass to a relation of extension, and quantity of force to “a quantity of perfection in essence” or “the quantity of force in action” [10] (that is to say, intensity). He thus takes up the idea that intensity can be quantified, but specifies the nature of this quantification: it allows the evaluation of a quantity of perfection in the essence of a thing.

13The analogy thus established between the augmentation of a quantity and the tension of a qualitative form leads to the suggestion that the intensity of a quality is henceforth susceptible to measurement. Now, this measuring of intensity immediately finds a privileged site of application in the notion of speed and in the examination of local movement.

14At the beginning of the fourteenth century, several important figures of Merton College—William Heytesbury, John Dumbleton and Richard Swineshead—were to extend the domain of the question of the latitude of forms to the definition of movement and, in doing so, would seek to approach the notion of speed. As a result, the notion of speed [11] comes to be grasped as a “permanent, but variable quality.” [12] Oresme, for example, considers that to produce a measure or a total quantity in a movement, one must take into account both quantity and quality. As for speed, the uniform change brought about by a constant speed is distinguished from a change in speed as a function of time, so that one can speak of uniform quality and uniformly difform quality. Thus in his Tractatus de Difformitate Qualitatum, in chapter 2, entitled “De figuratione et potentiarum successivarum uniformitate et difformitate,” Oresme uses these now equally measurable categories of intension and extension to apprehend the phenomenon of speed. He distinguishes between two forms of extension in movement: one involves “the distribution of speed to different points of the subject, that is to say the moving thing” and the other concerns change in speed over time. As to the intensity of degree of speed, he characterizes it in terms of the distance travelled by a moving thing over the same period of time. [13]

15Without giving all the details of the path by which this motif of the quantification of quality runs through the Galilean corpus, to be found once again in the Dynamica de Potentia, [14] we might retain three fruitful proximities. Firstly, the idea that intensity measures degrees of perfection. Secondly, the idea that there is a correlation between the intensity of a quality and the extension of a quantity, so that, between them, they produce a total estimation. Thirdly, the idea that this intensity finds its primary physical expression in the definition of the notion of speed.

16But the whole question, then, is that of whether the lexical and thematic proximity is strong enough to attest to a convergence. A. Maier’s work [15] undertakes precisely to place this motif within the complex intellectual context of the fourteenth century. The motif outlines an attempt at the mathematicization of physics on the basis of the question of the variation of degrees of speed. Maier renders more precise the framework within which this enterprise is undertaken: on one hand, the idea of an exact measurement is considered a priori as impossible but, on the other, everything must be the object of a measure and, more precisely, even every intensive magnitude must be expressed quantitatively. These two positions taken together explain the success of the Calculationes. It is a matter of providing an a priori way to account for the real (but in part inaccessible) possibility of quantifying qualities. It is in this context that we must understand the Leibnizian fortunes of the thematic of the latitude of forms.

17It seems to me that this brief recollection of the medieval origin of the twofold resolution of action according to intension and extension allows us to understand, more closely, in what way this measuring of the real involved in the principle of the conservation of motive action mobilizes a reflection on degrees of perfection. For to choose the conceptual field of evaluation in the medieval period, is to install at the heart of the conceptualization of action the possibility of measuring degrees of reality. One action is distinct from another according to its degree of reality.

II. The Meaning of the Use of the Motif of Latitude of Forms in the Dynamics of Action

18We should like to try and make sense here of the reprise of this medieval motif of the latitude of forms at the heart of the conceptualization of the Dynamics of action. What is at stake here is to do justice to what Leibniz designates, for de Volder, in a letter dated September 6, 1700, as a “mathesis metaphysica.” [16]

19The question of the transmission of the motif of the latitude of forms is a relatively well-known one. As Michel Fichant has shown, Leibniz, in any event, knew of the motif at the time of writing his Dynamica de Potentia. For during his voyage to Italy, at the moment when he drafted the Dynamica de Potentia, he had direct access to the text of Suisset (Swineshead) Calculationes de Motu et Intensionibus et Remissionibus Formarum seu Qualitatum, as is attested by a letter to Alberti in January 1690. [17] In this work consulted by Leibniz, Suisset does indeed distinguish quantity of force or power on one hand, from intensity, on the other. But in distinguishing them, he indicates at the same time their close correlation, so that at a given level of intensity, a mass possesses more force than at a lesser level. Part VIII of Suisset’s Calculationes, “De potential rei,” brings out both the distinction and the intrication between intensity and extension, or, otherwise formulated, between quality and quantity. [18]

20What is at stake in the elucidation of the enigmatic Leibnizian formula of mathesis metaphysica is to determine in what sense that which Leibniz understands by mathesis is involved in the treatment of certain metaphysical objects. In the Elementa Rationis, [19] Leibniz indicates that the exigency proper to mathesis, which has until now been applied to things that are visible or fall under the imagination, must now be applied to the task of grasping the reality present beneath “the aggregate of images,” on the basis of abstract notions, for they constitute the root of knowledge. This new usage of mathesis would permit one to grasp the principles of human knowledge. We must therefore try to understand what this usage of mathesis, in which Dynamics is so invested, might actually be. For, in his text Mathesis Universalis, [20] Leibniz excludes the possibility that Dynamics, in so far as it applies to the causes, forces and actions of substances, could be a matter for Mathesis understood as a logic of the imagination. It is indeed to this other sense of mathesis universalis, the sense of a true estimation, that Leibniz leads us in this text. Now, as David Rabouin recalls in his dissertation, what is singular about mathesis universalis in its Leibnizian acceptation is that it brings together a reflection on quantity and quality in so far as it is considered as a logic of the imagination, and that it thereby opens up mathesis to a formal dimension. Meaning that it allows the apprehension of quantity as measurement and quality as form, making sense of a recurrent expression in Leibniz’s work: that of quantity of reality—which is a customary and proven definition of perfection. [21] Now, if we follow what Leibniz says about quantity of reality in the Twenty-Four Metaphysical Theses, we can remark that if perfection is identified with a quantity of reality, this is the case insofar as it is located in form, for the precise reason that form is susceptible to variation. For Leibniz writes in theses thirteen and fourteen [22] of the Twenty-Four Metaphysical Theses that the variety of phenomena expressed in the notion of form can be perceived distinctly according to a relation of ordered prevalence. What produces the variation in the degrees of reality or perfection that we measure, is the more or less distinct variation that we have of the order of the world—that is to say, of the relations that exist within that order. Now, in Leibniz’s thought, it seems to us that the notion of action, understood as actio in se ipsum, expresses precisely this more or less distinct perception of relations. In the texts relating to dynamics, action is called formal action, the measure or estimation of which, precisely, allows one to produce and to evaluate degrees of reality. [23] We thus understand well that here it is not a question of making of this new mathesis a “universal calculus” capable of resolving metaphysical questions on the model of the resolution of problems in mathematics, since precisely what is at stake in this mathesis is the estimation of the degree of reality in perceiving substances, in order to be able to detect in any appearance the reality that underlies it. Thus we should not understand mathesis in the sense of mathematical quantity, but in the sense of a gradation of values, or a qualitative hierarchy; it is in this sense, no doubt, that it is possible to place confidence in the elements of order and measurement, which are proper to mathesis, but only in conferring upon them a non-extensive status. It is in this interpretative perspective that we should now like to try and make sense of the notion of “metaphysical mathesis” present in the correspondence with de Volder. This notion is introduced within the perspective of characterizing a “metaphysical reason of estimation.” How are we to understand this expression?

21In the letter addressed to de Volder on 6 September, 1700, Leibniz describes it in the following way:

22

[…] the principles of nature are no less metaphysical than mathematical, or rather the causes of things are hidden in a certain metaphysical mathesis, which estimates the perfections or degrees of reality. [24]

23First of all, if Leibniz evokes this metaphysical reason of estimation, it is in order that one consider that “the principles of nature are no less metaphysical than mathematical.” He thus closely associates in his Dynamics—since Dynamics is the science that must rationally account for the principles of nature—the two dimensions of mathematics and metaphysics. But what form does this association take? It does not seem to us, as is proposed by André Robinet in his work Architectonique disjonctive…, that the homogeneity of “true metaphysics” in the field of universal mathematics leads us to recuse or surpass a metaphysics of reality. [25] On the contrary, indeed, we hypothesize that, if mathesis metaphysica allows us to estimate perfections or degrees of reality, it is because, in doing so, we accede to a higher level of reality, thanks to the estimation of action, and because this level is higher than others since it allows us to consider the “causes of things.” It is in this sense that the comprehension of the complexity of the notion of action (originally action in se ipsum, transposed into Dynamics to become a formal action that integrates time into its estimation since, with this integration of time, once more transposed into the properly metaphysical field of the intelligibility of the concept of substance) allows us to accede to the ultimate level of legitimate reality: [26] that of the causes of things which are enveloped in a certain metaphysical mathesis. In this case, it will be a case of an action, identified this time with perception.

24If mathesis is metaphysical here, it is insofar as it allows the estimation of the perfection of a reality, but also insofar as it contributes to identifying its cause.

25To understand that the conceptualization of substance in the 1690s was fostered by the dynamics of action leads us to grasp the ontological stakes of action: namely, the measurement of degrees of reality. This measurement can be read as a principle of differentiation that articulates singularity and perfection by distribution expressive degrees.

26It seems to me that on the basis of this understanding it is possible to grasp the meaning of perceptual variations.

III. Perceptual Variations

27It is, we believe, on the basis of the explicit reprising of the medieval motif of the quantification of qualities that it is possible to understand how dynamics is helpful in understanding the new definition of substance: thanks to dynamics, substance comes to be understood on the basis of its action over time. But in what way is perception an action?

28

I recognize that monads are active in themselves, but in them all that can be understood is the perception that envelops action. [27]

29Thus, in the monad, all action supposes perception, and perception appears as the monadic form of action. If the notion of expression presupposes in its definition the existence of action, and is thus a higher form of it, it is correlatively the sole level of intelligibility to which we can accede when we try to grasp the monad. Now, the function of perception as sole access to the intelligibility of the monad seems to exceed the meaning of the monad alone: for it is also, in a certain sense, all of the relations that monads entertain with each other and with the phenomena that are conjointly involved. What constitutes substance is the fact that there is one and the same law that persists and that contains in it all future states. Definitively, what Leibniz seeks to establish is simply, on one hand, that one can perceive infinite things in which “there is a determinate law of the progress of phenomena and the reason for the conspiracy between the phenomenon of diverse things” and, on the other, that the reason of their existence and their conspiracy is common to them all: God. But “to seek why there is perception or appetition in simple substances, is to seek something ultra-worldly; it is to convoke God to explain why he wished there to be something at the origin of our conceptions.” [28] The fact is that a law of order is accessible to us, because this law of order relates to the intelligibility of the rationality at work in the world. However, to understand why and how this particular law of order has been chosen relates to the choice of the best possible world, which it is impossible to account for rationally any further. But if the active principle is, through its action, the expression of the law of order, then what, ultimately, is its function in the definition of substance?

30It is taken as read that the new intelligibility of substance rests on the understanding of dynamical action. Moreover, when we speak of an aggregate of substances, we speak of real phenomena. On this basis, it seems to us that the whole intelligibility of the hierarchy of degrees present in the accidental unity that is the aggregate, can only be understood by way of that which founds the hierarchy of substances, that is, by taking into account a quantity of perfection that defines the essence of the thing on the basis of its capacity to express relations. Here we hypothesize that it is only by taking account of the important dimension of the invention of Dynamics and of the conceptualization of motive action that one can understand the notion of aggregate. Paul Lodge, in his article “Leibniz’s Notion of an Aggregate,” writes: “Leibniz’s explanation of the notion of an aggregate depends essentially on the activity of perceivers.” [29] We agree with Lodge when he writes that the meaning and the ontological status of the aggregate depend on the activity of perceivers, on condition that we understand whence comes this perceptive activity—namely, from a capacity to express relations which is proportionate to the quantity of perfection or of reality present in the essence of a being. Now, this quantity can only be thought as measurable on condition that we understand the procedure of the mathematicization of ontological concepts that draws the measurement of this perfection from the estimate of the motive action of dynamics. Without this, we cannot understand what subtends the hierarchy of substances, and correlatively, that of aggregates. Leibniz, indeed, indicates in his letter to de Volder of June 30, 1704, [30] how action is that which produces change on itself, by itself. Such is the proper meaning that he attributes to formal action: this capacity to produce, from itself, an action upon itself, that overcomes the natural resistance it might possess. Moreover, he specifies the nature of this internal principle of action: not a reason of being [raison d’être], but a reason of order [raison d’ordre], we might say—that is, a capacity to explicate, on the basis of action itself, how it progresses from one action to another, from one perception to another. This reason of progression can be explained firstly by reinstituting the framework in which perception takes place: it is a temporal framework, in which the passage from one perception to another is the passage from a present state to a future state, effectuated according to a law of development which is at once internal to substance, but which simultaneously realizes, through this very development, and to its own extent, the harmony of the world. This is what Leibniz indicates when he writes that the reason at work in the internal principle “consists in the progress of perceptions of each monad.” Perception is indeed the capacity of the monad to express the universe according to its own point of view. It is more or less distinct according to whether it is endowed with a greater or lesser quantity of perfection or of reality—that is to say, also, a greater or lesser capacity to express distinctly relations with other substances, and with the harmony of the world. It is in this sense that we can understand the formula in the last letter addressed to de Volder in 1706. If the only existence that can be proved is that “of perceiving things and perceived things,” which have in common the perception of the progress of perceptions and the perception of the reason of this progress,

31

[w]e do not have, nor ought we hope for, any other mark of reality in phenomena than the fact that they correspond with each other and with eternal truths as well. [31]

32In other words, the mark of reality in things lies in their capacity to express the harmony of the world—to express, to infinity, the relations between substances. Now, this expression is the development of the law of series, which contains in it the series of expressive relations. Each expression then appears as the singular mark of a degree of perfection through which the perfection of God and the world manifest themselves.

33* * *

34The analysis of the medieval origin of the question of the latitude of forms, and with it, that of the estimation of degrees of reality, has allowed us to bring to light the importance of the intensive aspect of reality. To our mind, it is this that allows us to understand the full conceptual import of action, and hence the singularity of perception. Thus, perception, on condition that it is understood, by way of the conceptualization that makes the dynamics of action possible, as the expression of a level of reality, allows us to think in one and the same place—the same situs—singularity and the expression of perfection.