LEIBNIZ, GOTTFRIED WILHELM. — Born in Leipzig on 21 June 1646, died in Hanover on 14 November 1716: philosopher, mathematician, and scholar; one of the greatest minds in the history of humanity.
SUMMARY:
I. The formation of Leibnizian thought
II. The monadological system
III. The historical limitation of Leibnizianism
IV. L. and mathematics.I. THE FORMATION OF LEIBNIZIAN THOUGHT
The son of a professor of moral philosophy at the University of Leipzig and of the daughter of a professor of law, he displayed an exceptionally early inclination toward study. While still a child, making use of his father’s rich library, he taught himself Latin and Greek and rapidly directed his interest, through literature, toward philosophy and theology. In philosophy he first became acquainted with the scholastic tradition, drawing on sources of nominalist inspiration. Soon afterward he came into contact with modern scientific culture, and at the age of fifteen faced his first crisis. He himself relates that at that age he posed the question whether he should retain substantial forms or reject them in favor of the mechanistic conception inspired by the new science. At that time he decided in the latter sense, and with this decision marked the starting point of his intellectual career. Its point of arrival, on the other hand, would consist in a kind of constructive and ingenious reconciliation between the new point of view and the traditional one. He would become increasingly aware that he was, and ought to be, in this sense—and in still broader senses—a conciliator, just as he would be fully conscious of the originality of his synthesis.What drove him along this path was the extraordinary variety of his interests and competencies, which he gradually acquired with impressive diligence. At the age of twenty he earned a law degree. His interest in juridical disciplines led him to write various works relating to them, culminating in the Nova methodus discendae docendaeque iurisprudentiae (1667), in which he sought to give them a new organization, itself inspired by his studies in logic. Essentially, he aimed at a logical organization of law, constituted by fundamental definitions and deductive concatenation, and to be pursued, in substance, through an analysis of juridical language (it is not surprising that L., who was soon able to gain entry into the highest circles of German culture, received from a prince during these same years the commission of coordinating Roman law with the law then in force). Law interested L., however, above all as a field of application for the «new logic» that he was devising, and to which he had already devoted an essay, De arte combinatoria (1666). Logic would remain permanently one of his fundamental concerns and the subject of a great many writings. When, a few years later, and specifically during his stay in Paris (1672–76), he devoted himself to mathematics—and did so with such commitment that within a few years he arrived at the invention of infinitesimal calculus!—he himself declared that he had done so solely to perfect his art of thinking. The result was a mathematical logic—L. may be considered the progenitor of this discipline—in which Aristotelian logic is given considerable weight (its themes would be extensively taken up and developed), but which at the same time sought its integration into a kind of universal grammar (L. also dreamed of a universal, logically constructed language), an inventory of all elementary ideas whose combinations would permit the solution of every problem, reducing thought to a kind of calculation. The analytical character of Leibnizian logic deserves emphasis, that is, its intention of resolving all notions into their simple components through the use of definition. Connected with this conception is the famous principle of the identity of subject and predicate (praedicatum inest subiecto). Many interpreters of Leibnizian thought have sought to discern in logic, and particularly in the principle just mentioned, the basis of the metaphysical conception itself (which would ultimately be expressed in the doctrine of monads). Since other interpreters seek to identify the structure of Leibnizian thought differently, centering it on other motifs, it must be pointed out that, in the present state of scholarship, while on the one hand the problem appears increasingly difficult to resolve (especially in view of the fragmentary and reticent character of Leibniz’s own writings), on the other hand the tendency is gaining ground—at least methodologically—to view this philosopher’s system as a convergence of multiple motifs forming an original intuition of reality. With regard to logic, it should be observed that its strictly analytical character, while on the one hand it may have inspired the conception of the monad (as a subject or substance containing within itself the infinite series of its predicates), on the other hand precluded any claim to constructive deductivity. It is interesting, rather, to consider how this logical analytic character marked out for L. the paths of the aforementioned convergence.
Meanwhile, already in the academic dissertation (1663) Disputatio metaphysica de principio individui, this characteristic theme of his highest speculation is found fully active in L. Soon afterward he was grappling, on the one hand, with theological problems (Confessio naturae contra atheistas; Defensio Trinitatis) and, on the other, with problems of physics (Hypothesis physica nova [1671], comprising a Theoria motus concreti and a Theoria motus abstracti). This latter essay is particularly important for recognizing the development of L.’s thought, and one may discern in it the outlining of the standpoint from which he would subsequently undertake his critique of prevailing mechanism. From this critique would emerge dynamism first and, as a subsequent deepening of it, monadology. If, among the many lines of development in Leibnizian thought, one can be chosen as principal—even though it is not such as to resolve the others within itself or generate them from itself—it is the line of dynamics. «The notion of force, to whose elucidation I devoted a particular science, Dynamics, sheds much light on the understanding of the true notion of substance» (De primae philosophiae emendatione et de notione substantiae, ed. Gerhardt, IV, p. 469). L. always took care to distinguish his concept of force from the scholastic concept of potency: (differt vis activa a potentia unda vulgo scholis cognita) since «the active potency of the Scholastics, or faculty, is nothing other than a proximate possibility of acting, which nevertheless requires an excitation from outside and, as it were, a stimulus in order to pass into act. Whereas active force in a certain way contains act or entelechy and is something intermediate between the faculty of acting and action itself; it involves conatus (inter facultatem agendi actionemque ipsum media est, et conatum involvit) and thus passes by itself into operation, requiring no aids, but only the removal of impediments» (ibid.). The opposition to scholasticism, here as elsewhere, therefore has neither historical nor critical value (L., moreover, is among the great modern thinkers one of the most favorable to medieval philosophy, as well as one of the best informed about it), but serves solely to elucidate his own point of view.
The dominant mechanism had received various forms or expressions, but all agreed in assigning matter and motion as the explanatory principles of physical phenomena. L. would attack mechanism above all in its Cartesian formulation, intending to demonstrate, including by mathematical considerations, that motion presupposes force, and that extension cannot be conceived as substance. Extension, in fact, presupposes the reality that is extended: in itself, it consists of parts external to one another, without end (the limit of the divisibility of matter insofar as it is extended will never be reached). If one says that extension is formed of extended parts, it still remains to explain analytically the constitution of those parts; if, on the other hand, they are unextended parts, then it is not clear how an extension could result homogeneously from a sum of unextended elements. From this L. infers, pressing the point with many other considerations, that extension is not substance. It must be noted here that, also according to L., the reality underlying the extended, and which is force as the spontaneous principle of action, and ultimately the monad, is unextended: how, then, will the objection that the extended cannot arise from the unextended not apply to it?
Here the concept of phenomenon intervenes, in its speculative function, as opposed to substance or the “thing in itself.” This concept had already been supplied to L. by the epistemological current, which, for example, recognized the phenomenal character of the so-called secondary or sensible qualities, such as color, taste, and so on. L.’s novelty consists precisely in reducing extension also to a phenomenon, while analytically elevating another determination to the rank of the thing in itself. Now, between the thing in itself and the phenomenon, although there is a relation of dependence and, ultimately, of explication, and although the phenomenon expresses the substance, there is nevertheless no homogeneity: that is, L.’s force, or the monad as a metaphysical point, does not stand in relation to the extended in the same position as do physical parts, whether extended or unextended; force or the monad are evidently not parts of extension. L. does not feel obliged to explain determinately how extension results from them, insofar as he can remain content with the quia of the phenomenal character of extension itself. In this way, the distinction between phenomenon and substance comes to assume the importance of a limiting distinction, of a fundamental framework of Leibnizian theoretical thought. In the philosopher’s writings this distinction is not extensively thematized; but its preponderant function becomes clear upon a more careful structural analysis of the doctrine. When Kant undertakes its explicit and extensive thematization, he will in no way have broken the continuity of this philosophical tradition.
L.’s effort will instead become more focused on determining substance and the system of substances. The results of his systematic elaboration, which may be considered established in their fundamental outlines around 1685, are set forth in numerous writings extending
to the year of his death. Some variations and further developments are also found in this period of maturity and old age, but they are not particularly significant.
Among the aforementioned systematic writings may be recalled: De primae philosophiae emendatione et de notione substantiae (1684); Discours de métaphysique (1686); Système nouveau de la nature et de la communication des substances (1695); Principes de la nature et de la Grâce (1714); Monadologia (1714). One should add the two more substantial works, the Nouveaux essais sur l'entendement humain (1704, but published posthumously in 1765), which are a critical examination of Locke’s essay, and the Essai de théodicée sur la bonté de Dieu, la liberté de l'homme et l'origine du mal (1710), in which the polemic is directed against Bayle; together they provide a condensed picture of L.’s philosophical production. His immense correspondence—correspondence with almost all the learned men of his time—is largely philosophical.
L. carried out his enormous scientific and literary activity amid a multitude of political and diplomatic occupations. He went to Paris for reasons of this kind, charged by a German prince with persuading Louis XIV to undertake an expedition to Egypt, thereby diverting him from attacking Holland. On returning to Germany, L. found a position as librarian and historiographer to the dukes of Hannover, and remained in that office throughout his life. It was not a sinecure for him, since he had to devote much of his time to reconstructing the history of the family of his lords. Naturally, even in this remunerated activity L. brought his personal imprint; and, simultaneously with Muratori, with whom he was also in correspondence, he established historical studies upon the use of documents. To complete the picture of this extraordinary industry, it must be recalled that L. devoted considerable attention to religious problems and that, inspired by his harmonious universalism, he worked for the reunification of all the Christian churches (correspondence with Bossuet) and, when the possibility of agreement with the Catholic Church disappeared, turned his efforts toward the union of all the Protestant sects. These too were threads that he drew toward the unity of his system. Against the Cartesians who, faced with their interest in physico-mathematical science, despised all the rest of knowledge as negligible erudition, L. was accustomed to preach that everywhere there is something worthy of being valued. The structure of his philosophical system was particularly well suited to absorb this cultural multiplicity.
II. THE MONADOLOGICAL SYSTEM
While the concept of force was gaining recognition — as we have seen — over simple mechanism, because of its theoretical functionality (that is, insofar as it was translated into equations capable of providing a broader and more coherent explanation of phenomena), such functionality, ultimately amounting to a network of relations, did not in itself justify equating force with the thing in itself. The scientific critique of experience, therefore, while demolishing the Cartesian identification of extension with substance, was not then able to replace that identification with another; that is, it was unable, by its own means alone, to reach the foundation of being. To meet this requirement, a consideration of another order enters the Leibnizian system: namely, the elevation of the other type of Cartesian substance — the res cogitans — to the paradigm of all being. Undoubtedly force belongs to the thing in itself, but it cannot adequately define its essence. Undoubtedly “primitive active force, which is called by Aristotle the first entelechy (ἐντελέχια ἡ πρώτη) and in common language the form of substance, is the natural principle which, together with matter or passive force, constitutes corporeal substance; and this is in itself a unity, that is, not a mere aggregate of several substances.” “This entelechy — L. adds — is either a soul or something analogous to a soul” (fragment of 1702). The determination of force thus passes into that of the soul or monad.In inner experience, that is, in the consciousness of our personality, we directly apprehend substance, and this experience offers us the only model by which we are able to understand the in-itself of entities other than ourselves (C'est partout comme ici). Once materialism has been dissolved, it is likewise impossible to imagine a substance that is something other than a centre of representation and appetition. Representation and appetition are the two essential characteristics of the monad.
The monad is simple substance, that is, substance without parts. The world is, in itself, an infinity of monads. Infinity is one of the concepts that most attracted L.’s meditation, in mathematics and metaphysics alike, both in the form of the infinitely great and in that of the infinitely small. He upheld actual infinity. Every machine constructed by human industry, he said, is composed of parts ordered to an end: the same is true of natural entities; except that, whereas in artificial machines the parts are finite in number, in nature every part consists of infinitely many other parts, each of which in turn is actually constituted by infinitely many other parts, and so on, without limit. Now this conception is supported, on the one hand, by the principle of the best (v. appresso); but on the other it finds its incentive in the concept of the infinite divisibility of matter, insofar as the order of the monads (the metaphysical points) must also correspond (“express,” L. says), in some way, to this infinity, since it is through it that one arrives at the monad. L. cannot be content with a mere potential infinity (extension can be divided beyond every limit), because he intends to speak of substance and of the thing in itself as it actually is, whereas potential infinity (or the indefinite, that is, greater than any given finite quantity) concerns not the thing in itself but our manner of considering II. The in-itself is either infinite or finite, with no intermediate possibility (it is from this manner of considering things that Kant’s antinomies concerning this concept of infinity can be understood). Although the monads are conceived and defined in analogy with the conscious self, and therefore are all spiritual and psychic in essence (panpsychism), they differ in perfection; and thereby they constitute a hierarchy. From the less perfect monads one passes to the more perfect through a gradation of infinitesimals: the hierarchy, that is, knows no discontinuity (except in the transition to the absolute monad or God, though this discontinuity was not felt by L.). The principle of continuity (natura non facit saltus; non datur hiatus; non datur vacuum formarum) is one of those that L. regarded with particular satisfaction as his own: it is not a first principle, but follows immediately after the first principles. These are two: the principle of contradiction and that of the best, or of sufficient reason.
For L., the principle of contradiction is the principle of truths of reason, that is, of those truths whose negation precisely implies a contradiction. They are established a priori, insofar as they are identical or tautological (A is A), or insofar as they are reduced to identical propositions through the definition of the terms (for example, the proposition “the whole is greater than the part” is reduced to a tautology if, in place of the terms whole, part, and greater, one substitutes their definitions). The principle of reason, on the other hand, is the principle of truths of fact, or contingent truths, which are known through experience: for example, that Caesar crossed the Rubicon. The negation of such truths does not imply a contradiction; on the other hand, they too affirm an identity (because, according to L., as has already been mentioned, the identity of subject and predicate constitutes the very form of truth in general). L. explains that, if someone possessed the complete notion of Caesar, he would see in it the note of his crossing the Rubicon; our limited knowledge, on the other hand, permits us only to apprehend a posteriori these implications. Is it then to be said that the distinction between the two supreme principles exists only from the standpoint of the finite intellect? This does not appear to be L.’s view, for whom the difference between necessary and contingent truths is objective. The distinction between the order of non-contradiction and the order of the best exists, so to speak, from the divine standpoint itself: for whereas, for example, God cannot fail to be what he is, he could instead, de potentia absoluta, refrain from creating the world; he creates it, and creates it as it is, only because this is better. A structural observation should be made here, one that allows us to penetrate critically into the organism of Leibnizian thought: the principle of contradiction is at once principium cognoscendi and principium es-
es- sendi of truths of reason; by contrast, truths of fact find their principium cognoscendi in experience, and therefore the principle of reason is, with regard to them, only the ratio propter quid. It does not follow that our knowledge is constructed only through the principle of contradiction and experience, both because certain scientific truths, such as Archimedes’ laws concerning the equilibrium of bodies, can be founded, that is, known, through the principle of sufficient reason, and because there are speculative principles, such as the aforementioned principle of continuity, which likewise can be founded only on the principle of reason. The latter, therefore, comes once again to cooperate with the principle of non-contradiction in the role of foundation of knowledge.
It is, as has been said, on the basis of the principle of continuity that L. establishes the hierarchy of monads. There are monads which, from the standpoint of representational power, possess only perception, and monads which also possess apperception. Perception is an unconscious representation; apperception adds consciousness: the apperceiving monad not only has representations, but is conscious of having them. The perceiving monad is properly called a soul. In it, perception is accompanied by memory. The reasons on which L. bases his thesis of unconscious representations—which he also attributes to man—are partly psychological (e.g., one remembers having had a sensation to which, at the time, one had paid no attention), but above all metaphysical (since the monad is essentially perceiving, if it were without perception it would be as though annihilated; hence the soul is always thinking, even in sleep). The principle of continuity also supports this conception; it “makes it possible to establish that even distinct perceptions arise gradually from those that are too small to be observed. To judge otherwise is to lack sufficient knowledge of the immense subtlety of things, which always and everywhere implies an actual infinite” (New Essays, Preface). In any case, speaking of unconscious perceptions with regard to the lower monads makes it clear that the philosopher arrived at their existence only by the negative reasoning that extension cannot be conceived as substance. Yet this critique of experience is transformed in L. into a positive doctrine, which seems to have much that is arbitrary or dogmatic about it, and above all theoretically ambiguous. L. in fact admits that the monad, a simple substance, enters into the constitution of composite substances. Such are animals. Every animal has a soul, which is the central monad, and a body, which is constituted by infinitely many aggregated and subordinate monads (what L. calls second matter, whereas first matter is the ontological limit inherent in every individual monad insofar as it is finite). L. sought to explain how this subordination of the body to the soul is realized through the difficult theory of the vinculum substantiale. It is enough to mention that, by means of this theory, L. believed he had assimilated the corresponding Aristotelian conception of matter and form (v. ILENORPIEMO), with the sole difference that Aristotelians do not recognize the monad (which here performs the function of form). This doctrine of monads—which L. subsequently develops in many particulars—presents, as was said, a certain theoretical ambiguity: L., having moved onto a metaphenomenal plane, clearly returns to the phenomenon, draws upon it, and discusses it extensively, taking his inspiration from its suggestions. This is a doctrine of substance and of the in-itself, in L., which seems to seek to make its way through the thickness of the phenomenon, considered as bene fundatum.
The monad has neither doors nor windows (Monadology, § 7): this means that it derives its representations from itself, from its own depths; it means that knowledge is not determined by a physical action (influens physician) of the object. Between such an action and knowledge there is in fact no analogy. That physical action belongs, like extension, to the phenomenal order. If, therefore, the monad represents reality—which is constituted by monads—more or less perfectly, this means that there exists among the monads a pre-established harmony. “Being thus compelled to admit the impossibility that the soul or any other real substance could receive any impulse from outside, ... I was imperceptibly led to an idea which ... presents very great advantages ... That is to say, one must affirm that God created the soul from the beginning, ... in such a way that everything in it springs from its own depths, with perfect spontaneity in relation to itself, and yet in perfect conformity with external things” (Système nouveau de la nature et de la communication des substances, § 14). This harmony presupposes a harmonizer, who is God, the Monad of Monads. L. also demonstrates God by other routes: he accepts the familiar Cartesian proofs—as well as that « a contingentia mundi »—but with characteristic refinements. One may, he says, deduce God’s existence from His idea, provided that one first proves that this idea is possible, that is, that it does not imply a contradiction. This is precisely what he undertakes to do. The spirit of this proof is supplied by the principle that the possible tends of itself to become real, and is real without qualification if nothing (that is, a contrary possibility) prevents II. Such an obstacle cannot be conceived in God; therefore God is real. And, by the principle of the best, God will create the entire sum of possible being, that is, the best of all the worlds that are abstractly possible (Leibnizian optimism). ... “That cause which makes something exist, that is, which makes possibility demand existence, also causes every possible thing to have a tendency toward existence; for in general no reason for restricting the existence of possibles can be found. Thus it may be said that every possible thing is a beginning of existence, insofar as it is founded upon a necessary being that actually exists, without which there would be no way by which it could come to exist and be actualized. But it does not follow from this that all possibles exist: that would occur if all possibles were compossible. There therefore exists the greatest perfection; and it consists in nothing other than the quantity of reality” (Fragment of unknown date, ed. Gerhardt, VII, pp. 289–90). This principle of “possibility-existence” has, as can be seen, a peculiar theoretical position. It governs theology and extends, through the principle of the best, into cosmology, and therefore nourishes a theory that encounters the critique of experience. It displays an affinity with the principle of classical metaphysics that deserves to be examined closely.
III. THE HISTORICAL LIMIT OF LEIBNIZIANISM
L.’s thought is above all an analyticism, an analysis of knowledge that resolves every datum into its principles. In this resolution, however, one does not arrive at absolutely founded principles, but stops at a sort of postulates. The demand for a foundation would shortly be taken up again by Wolff; but, rather than finding satisfaction, it would make apparent the fundamental limit of this direction of thought, namely, that simple analysis is indeed sufficient to elucidate logical structures, but ultimately presupposes them. With this, the very concept of foundation would be urged to move onto a higher speculative plane.Subsequent modern philosophy would clearly denounce this limit of Leibnizian philosophy and the dogmatism connected with II. Nevertheless, within Kantianism itself, Leibnizian suggestions are many and fundamental. Nor would L. cease to exert an influence even beyond this, not so much by virtue of his system as of certain aspects of it, certain of his ingenious intuitions. And this is particularly noteworthy in nineteenth-century French philosophy.
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He opened the way to geometric calculus, developed about two centuries later by Hamilton and then by Grassmann; he expounded the elements of infinitesimal calculus, making use of the ideas of Mengoli and Newton; he gave the formula that bears his name and provides the derivative of a product of two functions, and he studied the problem of the brachistochrone (the path of a falling body from one position to another not situated on the same vertical, traversed in the shortest time). Recognizing the importance of base 2 in numeration, he also devised a calculating machine, which, however, does not seem to have found any application.