LEIBNIZ, GOTTFRIED WILHELM. - Born in Leipzig on 21 June 1646, died in Hanover on 14 Nov. 1716: philosopher, mathematician, scholar; one of the greatest minds in the history of humanity.
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 showed a very precocious inclination for study. Still a child, exploiting his father's rich library, he taught himself Latin and Greek and rapidly extended his interest, through letters, to philosophy and theology. Of philosophy he knew first of all the scholastic tradition, on sources of nominalistic inspiration. Soon afterwards he came into contact with modern scientific culture, and at the age of 15 he found himself facing his first crisis. He himself recounts that at that age he posed the problem whether he ought to preserve substantial forms, or reject them in favor of the mechanistic conception, which was inspired by the new science. At that time he decided in favor of the latter sense: and with that decision he marked the starting point of his intellectual career. The point of arrival, on the other hand, would be constituted by a sort of reconciliation, of a constructive and ingenious character, between the new point of view and the traditional one. That he was, and had to be, in this sense, and in even broader senses, a reconciler, he would become ever more clearly aware in the subsequent maturing of his thought; just as he would be fully conscious of the originality of his synthesis.
He was driven along this path by the extraordinary variety of his interests and competences, which he gradually acquired with an impressive alacrity. At twenty he took his degree in law. His interest in juridical disciplines suggested to him various writings relating to them, up to the Nova methodus discendae docendaeque iurisprudentiae (1667), in which he intended to confer upon them a new systematization, inspired in turn by his studies of logic. He aimed, essentially, at a logical organization of law, constituted by fundamental definitions and deductive concatenation, and to be pursued, in substance, by means of the analysis of juridical language (it is no wonder that L., who could immediately be introduced into the highest circles of German culture, received in these same years from a prince the task of coordinating Roman law with the law in force). Law, however, interested L. more than anything else as a field of application of the "new logic" that he was already planning, and to which he devoted even then an essay De arte combinatoria (1666). Logic would remain permanently one of his fundamental preoccupations, the object of a great many writings. When, after a few years, and precisely during his stay in Paris (1672-76), he devoted himself to mathematics - and did so with such commitment as to arrive in a few years at the invention of infinitesimal calculus! - he himself would declare that he had done so only to perfect his own art of thinking. From this came a mathematical logic - L. can be considered the progenitor of this discipline - in which much account is taken of Aristotelian logic (its themes will be taken up and developed extensively) but at the same time an integration of it is aimed at in a sort of universal grammar (L. also dreamed of a universal language, logically constructed), an inventory of all elementary ideas, whose combinations would permit the solution of every problem, thought being reduced to a sort of calculation. The analytic character of Leibnizian logic deserves to be emphasized, that is, its intent to resolve all notions into their simple components, by means of the use of definition. Connected to this conception is the famous principle of the identity of subject and predicate (praedicatum inest subiecto). Many interpreters of Leibnizian thought have wished to discern in logic, and in particular in the aforementioned principle, the basis of the metaphysical conception itself (which will be expressed, finally, in the doctrine of monads). Since other interpreters intend to identify the structure of Leibnizian thought differently, centering it on other motifs, it must be noted that, in the current state of studies, while, on the one hand, the problem appears ever more difficult of solution (and this especially in relation to the fragmentary and reticent character of Leibniz's own writings), on the other hand there is gaining credit - at least methodically - the tendency to recognize in this philosopher's system a convergence of multiple motifs constituting an original intuition of reality. With regard to logic it must be observed that its very strictly analytic character, if on the one hand it may have inspired the conception of the monad (as subject or substance containing in itself the infinite series of its predicates), on the other hand forbade any claim to a constructive deductivity. It is interesting to consider instead how this logical analyticism marked out for L. the paths of the aforementioned convergence.
Meanwhile, already in the academic dissertation (1663) Disputatio metaphysica de principio individui, this theme characteristic of his highest speculation is kindled in L. Immediately afterwards he is grappling with theological problems, on the one hand (Confessio naturae contra atheistas; Defensio Trinitatis) and with problems of physics on the other (Hypothesis physica nova [1671], comprising a Theoria motus concreti and a Theoria motus abstracti). This last essay is particularly important for recognizing the evolution of L.'s thought, and in it one can discern the delineation of the point of view from which he will subsequently develop his critique of current mechanism. It is from this critique that dynamism will first emerge and, as a subsequent deepening, monadologism. If among the many lines of development of Leibnizian thought one can be chosen as principal - even if not such as to resolve into itself or generate from itself the others - it is this one of dynamics. "The notion of force, to whose explication I destined a particular science, Dynamics, brings much light to the understanding of the true notion of substance" (De primae philosophiae emendatione et de notione substantiae, ed. Gerhardt, IV, p. 469). L. always takes care to distinguish his concept of force from the scholastic one 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 needs an excitation from outside and, as it were, a stimulus, in order to pass into act. While active force contains in a certain way the act or entelechy and is something intermediate between the faculty of acting and the action itself, and involves the conatus (inter facultatem agendi actionemque ipsum media est, et conatum involvit) and thus passes of itself to operation, nor does it need aids, but only the removal of impediments" (ibid.). The opposition to scholasticism, here as in other places, therefore has neither historical nor critical value (L., moreover, is among the great moderns one of the most favorable to medieval philosophy, as well as one of the best informed in the matter), but serves only to illustrate his own point of view.
The dominant mechanism had received different forms or expressions, but all agreed in assigning matter and movement as the explanatory principles of physical phenomena. L. would attack mechanism above all in its Cartesian formulation: intending to demonstrate, also by mathematical considerations, that movement presupposes force, and that extension cannot be conceived as substance. Extension, in fact, presupposes the reality that extends itself: it, in itself, is constituted by parts external to one another, without end (one will never succeed in reaching the limit of divisibility of matter insofar as it is extended). If one says that extension is formed of extended parts, there still remains to be explained analytically the constitution of such parts; if, on the other hand, of unextended parts, then one does not see how from a sum of unextended things an extension can result homogeneously. From this L. infers, pressing on with many other considerations, that extension is not substance. It must be noted here that, also according to L., the reality that underlies the extended, and which is force as the spontaneous principle of action, and, in the final analysis, the monad, is unextended: how is it, then, that against it the objection that from the unextended the extended cannot be born will not hold?
Here there intervenes, in its speculative function, the concept of phenomenon, as opposed to substance or "thing in itself." This concept was already furnished to L. by the epistemological current, which, for example, recognized the phenomenality of the so-called secondary or sensible qualities, such as color, taste, etc. The novelty of L. consists precisely in degrading extension too to phenomenon, analytically elevating to the rank of thing in itself another determination. Now between the thing in itself and the phenomenon, although there runs a relation of dependence and, in the final analysis, of explication, and although the phenomenon expresses the substance, there is nevertheless no homogeneity: that is, the force of L., or the monad as metaphysical point, is not, with respect to the extended, in the same position in which the physical parts are, whether they be extended or unextended; force or the monad are evidently not parts of extension. How extension results from them, L. does not feel obliged to explain determinately, insofar as he can be content with the quia of the phenomenality of extension itself. In this way the distinction of phenomenon and substance comes to assume the importance of a limit distinction, of a fundamental framework of Leibnizian theoretic thought. In the philosopher's writings such a distinction is not extensively thematized; but its preponderant functionality emerges from an elemental structural analysis of the doctrine. When Kant entered into his explicit and vast thematization of it, he had by no means broken the continuity of this philosophical tradition.
L.'s commitment, on the other hand, would be accentuated in the determination of substance and of the system of substances. The results of his systematic elaboration, which can be considered established in their fundamental lines around 1685, are set forth in numerous writings, which run down to the year of his death. Some variation, some deepening is also found in this period of maturity and old age, but it is not very prominent.
Among the aforementioned systematic writings may be mentioned: 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). Let there be added the two works of broader scope, 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), where the polemic is against Bayle, and one will have a reduced picture of L.'s philosophical production. The immense epistolary — correspondence with almost all the learned men of his time — is for the most part philosophical.
L. carried on his enormous scientific and literary activity in the midst of a heap of political-diplomatic occupations. He went to Paris for reasons of this type, with the commission from a German prince to persuade Louis XIV to undertake an expedition to Egypt, by which he would have been diverted from attacking Holland. Having returned to Germany, L. found a post as librarian and historiographer of the dukes of Hanover, and in that office he remained all his life. It was not a sinecure for him; for he had to lose much of his time to the reconstruction of the history of the house of his lords. Naturally, even in this mercenary activity L. brought his personal stamp, and, contemporaneously with Muratori, with whom he was also in correspondence, he set historical studies on the basis of the valorization of documents. To complete the picture of this phenomenal industriousness, it must be recalled that L. occupied himself a great deal with religious problems, and that, inspired by his harmonious universalism, he worked for the reunion of all the Christian churches (correspondence with Bossuet) and, once the possibility of agreement with the Catholic Church had failed, he set his hopes on the union of all the Protestant sects. These too were threads that he pulled toward the unity of his system. Against the Cartesians who, in the face of interest in physico-mathematical science, despised all the rest of knowledge as negligible erudition, L. used to preach that everywhere there is something to be valued. The structure of his philosophical system was particularly suited to absorb this cultural multiplicity.
II. THE MONADOLOGICAL SYSTEM
While the concept of force was gaining credit — as has been seen — in opposition to simple mechanism, on account of its theoretical functionality (that is, insofar as it was translated into equations, which could give a broader and more coherent explanation of phenomena); such functionality amounting, in the end, to a network of relations, did not by itself legitimize equating force with the thing in itself. The scientific criticism of experience, therefore, while it came to demolish the Cartesian identification of extension with substance, was not then capable of replacing that identification with another, that is, it was not capable of touching, by its own means alone, the depths of being. To meet this requirement, a consideration of another order enters into the Leibnizian system: namely, the elevation of the other type of Cartesian substance — the res cogitans — to the status of paradigm of all being. Undoubtedly force belongs to the thing in itself, but it cannot adequately define its essence. Undoubtedly « the primitive active force, which Aristotle calls 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; which 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 attain substance, and such experience offers us the only model by which it is given to us to understand the in-itself of beings other than ourselves (C'est partout comme ici). Once materialism is dissolved, one cannot even imagine a substance that is something other than a center of representation and appetition. These — representation and appetition — are the two essential characteristics of the monad.
The monad is simple substance, that is, without parts. The world is, in itself, an infinity of monads. The infinite is one of the concepts that most attracted L.'s meditation, in mathematics and in metaphysics; both under the species of the infinitely great and under that of the infinitely small. He was in favor of the actual infinite. Every machine constructed by human industry, he said, is composed of parts ordered to an end: so too is it in the entities of nature; except that while in artificial machines the parts are finite in number, in nature every part consists of other infinite parts, each of which is in turn actually constituted by other infinite parts, and so on, without limit. Now this conception is supported, on the one hand, by the principle of the best (v. below); but on the other it finds its incentive in the concept of the infinite divisibility of matter, insofar as the order of monads (the metaphysical points) must also correspond («express», says L.) in some way to that infinity, since it is through it that one arrives at the monad. L. cannot be satisfied with a simple infinite in potency (extension can be divided beyond every limit), because he intends to speak of substance and of the thing in itself as it is in act, whereas the infinite in potency (or indefinite, that is, greater than every assigned finite) concerns not the thing in itself but our way of considering it. The in-itself is either infinite or finite, with no middle term (it is from this way of considering things that one will be able to understand Kant's antinomies relative to this concept of the infinite). The monads, although conceived and defined in analogy to the conscious I, and therefore all being of spiritual and psychic essence (panpsychism), differ in perfection; and thereby 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 passage to the absolute monad or God, but 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 comes immediately after the first principles. These are two: the principle of contradiction and that of the best or of sufficient reason.
The principle of contradiction is, for L., the principle of truths of reason, that is, of those truths whose negation implies precisely contradiction. They are established a priori, insofar as they are identical or tautological (A is A), or insofar as they are reduced to identical ones by means of the definition of terms (e.g., the proposition «the whole is greater than the part» is reduced to a tautology, if in place of the terms whole, part and greater their definitions are put). The principle of reason is instead the principle of truths of fact or contingent truths, which are known by experience: as, e.g., that Caesar crossed the Rubicon. The negation of such truths does not imply contradiction; on the other hand they too affirm an identity (because the very form of truth in general consists, according to L., in the identity of subject and predicate, as has already been mentioned). L. explains that, if one possessed the complete notion of Caesar, one would see in it contained the note of the crossing of the Rubicon: our limited knowledge, on the other hand, allows us only to apprehend these implications a posteriori. It must be said, then, that the distinction of the two greatest principles subsists only from the point of view of the finite intellect? This does not seem to be L.'s thought, 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 subsists, so to speak, from the divine point of view itself: since while, e.g., God cannot not be what he is, he could instead, de potentia absoluta, not create the world: he creates it, and creates it as it is, only because this is better. A structural observation deserves to be made here, which allows one to penetrate critically into the organism of Leibnizian thought: the principle of contradiction is at one and the same time principium cognoscendi and *principium eses-sendi** of truths of reason; on the contrary, 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 knowing is constructed only by means of the principle of contradiction and experience, both because some scientific truths, such as, e.g., Archimedes' laws on the equilibrium of bodies, can be founded, that is, known, with the principle of sufficient reason; and because there are speculative principles, such as, for example, the aforementioned principle of continuity, which likewise can be founded only on the principle of reason. Which, therefore, returns to concur 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, under the aspect of representative power, have only perception, and monads which also have apperception. Perception is an unconscious representation; apperception has consciousness in addition: the apperceiving being not only has representations, but is conscious of having them. The perceiving monad is properly called soul. In it perception is accompanied by memory. The grounds on which L. bases his thesis of unconscious representations (which he attributes also to man) are in part psychological (e.g., one remembers having had a sensation to which, at the moment, one had paid no attention) but above all metaphysical (since the monad is essentially perceiving, if it were without perception it would be as if annihilated; therefore the soul always thinks, even in sleep). The principle of continuity also supports this conception; it "makes it possible to establish that even evident perceptions derive by degrees from those that are too small to be observed. To judge otherwise means not to know sufficiently the immense subtlety of things, which always and everywhere implies an actual infinite" (Now. Essais, preface). To speak in any case of unconscious perceptions, with regard to the lower monads, suggests that the philosopher arrived at their existence only by the negative reason that extension cannot be conceived as substance. Yet this critique of experience is converted in L. into a positive doctrine, which seems to have much that is arbitrary or dogmatic, and above all theoretically ambiguous. L. in fact admits that the monad, a simple substance, comes to constitute composite substances. Such are animals. Every animal has a soul, which is the central monad, and a body, which is constituted by infinite aggregated and subordinate monads (what L. calls second matter, while first matter is the ontological limit inherent in every single monad, insofar as it is finite). How that subordination of body to soul is realized L. sought to explain with the not easy theory of the vinculum substantiale. It suffices to mention that with this theory L. believed he had absorbed the corresponding Aristotelian conception of matter and form (v. ILENORPIEMO) with the sole difference that the Aristotelians do not recognize the monad (which here exercises the function of form). This doctrine on monads — which L. then develops in many particulars — presents, as was said, a certain theoretical ambiguity: L. in fact, after having raised himself onto a metaphenomenal plane, returns, as is clear, to the phenomenon, draws from it, and discourses abundantly, inspired by its suggestions. It is a doctrine of substance and of the in-itself, this of L., which seems to want to open 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 draws its representations from itself, from its own depth; it means that knowledge is not determined by an action (influens physician) of the object. Between such an action and knowledge there is in fact no analogy. The said physical action belongs, like extension, to the phenomenal order. If therefore the monad represents to itself, more or less perfectly, reality — which is constituted by monads — it means that there exists among the monads a pre-established harmony. "Being therefore compelled to admit the impossibility that the soul or any other real substance can receive some impulse from outside, ... I was led insensibly to an idea which ... presents very great advantages ... One must say, that is, that God has from the beginning created the soul, ... in such a way that everything in it springs from its own depth, by a perfect spontaneity with respect to itself, and yet with a perfect conformity with external things. (Système nouveau de la nature et de la communication des substances, § 14). This harmony supposes a harmonizer, who is God, the Monad of Monads. L. also demonstrates God by other ways: he admits the Cartesian proofs — as well as that "a contingentia mundi" — but with characteristic perfections. One can, he says, deduce the existence of God from his idea, provided that it is previously demonstrated that this is possible, that is, does not imply contradiction. Which is what he undertakes to do. The spirit of this proof is given by the principle that the possible tends of itself to be real, and is real without qualification, if anything (that is, a contrary possible) does not hinder it. In God such an obstacle is not conceivable; therefore God is real. And, by the principle of the best, God will create the whole sum of possible being, that is, the best among all abstractly possible worlds (Leibnizian optimism). ... That cause which brings it about that something exists, that is, that possibility demands existence, also brings it about that every possible has a tendency to existence; since in general no reason of restriction to the existence of possibles can be found. Thus one can say that every possible is a beginning of existence, insofar as it is founded on an actually existing necessary being, without which there would be no way by which it could come and be actualized. But from this it does not follow that all possibles exist: this would happen if all possibles were compossible. There exists therefore the maximum perfection; and it consists in nothing but 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 runs counter to the critique of experience. It presents an affinity with the principle of classical metaphysics that deserves to be fathomed.
III. THE HISTORICAL LIMIT OF LEIBNIZIANISM
That of L. is, above all, an analyticism, an analytics of knowledge that resolves every datum into its principles. In this resolution, however, one does not arrive at absolutely grounded principles, but rather stops at a sort of postulates. The demand for grounding would shortly thereafter be taken up again by Wolff; but, rather than finding itself satisfied, it would make felt the fundamental limit of this direction of thought, namely that simple analysis serves indeed to illustrate logical structures, but in the end presupposes them. With this, the very concept of grounding 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 it. Nevertheless, in Kantianism itself there are many and fundamental Leibnizian suggestions. Nor would L. cease to exert an influence even further afield, not so much by virtue of his system as by virtue of certain aspects of it, of certain of his ingenious intuitions. And this is particularly notable in French philosophy in the 19th century.
Bibliography19 works
IV. LEIBNIZ AND MATHEMATICS
L.'s activity has been much discussed, as it seemed that he attributed to himself results found by others. To him is owed the art of combinatorics, which studies the various ways of grouping objects and offers results that still have wide application in mathematics; to him is owed the fairly convenient series expansion of frac{π}{4} = 1 - frac{1}{3} + frac{1}{5} - frac{1}{7} + dots
He opened the way to geometric calculus, developed about two centuries later by Hamilton and then by Grassmann; he set out the elements of infinitesimal calculus, availing himself of the ideas of Mengoli and Newton; he gave the formula that goes under his name and that provides the derivative of a product of two functions, and he studied the problem of the brachistochrone (the line of descent of a body from one position to another not situated on the same vertical, and traversed in the minimum time). Having recognized the importance of base 2 in numeration, he also devised a calculating machine that, however, does not seem to have had any application.