## Leibniz: Physics, Logic, Metaphysics

# Leibniz: Physics, Logic, Metaphysics

The special problems for any comprehensive treatment of the scientific investigations of Leibniz arise, on the one hand, from the fact that essential parts of his work have not been edited and, on the other hand, from the universality of his scientific interests. In view of this diversity of interests and the fragmentary, or rather encyclopedic, character of his achieving, at least in part, what Leibniz himself, following architectonic principles (within the framework of a *scientia generalis*), was unable to accomplish.

Leibniz is a striking example of a man whose universal interests (in his case ranging from physics through theory of law, linguistic philosophy, and historiography to particular questions of dogmatic theology) hindered rather than aided specialized scientific work. On the other hand, this broad interest, insofar as it remained oriented in architectonic Principles, led to a concentration on methodological questions. In relation to the structure of a science, these are more important than concrete results. The position of Leibniz at the beginning of modern science is analogous to that of Aristotle at the beginning of ancient science. Leibniz’ universality is comparable with that of Aristotle, different only in that it did not, as Aristotle’s remain grounded in essentially unchanged metaphysical distinctions but evolved by degrees from an encyclopedic multiplicity of interests. Consequently, in the strict sense in which there is an Aristotelian system, there is no Leibnizian system but rather a marked metaphysical and methodological concern that systematically expresses variations on the same theme in the various special fields (such as physics and logic) and underlies Leibniz’ quest to establish a unified system of knowledge.

Leibniz’ autobiographical statement in a letter to Rémond de Montmort in 1714 explains how, at the age of fifteen, though accepting the mechanical philosophy, his search for the ultimate grounds of mechanism led him to metaphysics and the doctrine of entelechies. This indicates the early orientation of Leibniz’ thought towards the ideas of the *Monadologie*. Instead of setting out his philosophy systematically in *a magnum opus*, Leibniz presented piecemeal clarifications of his views in works that, in various ways, were inspired by the publications of others. After reading the papers of Huygens and Wren on collision and the *Elementorum philosophiae* of Hobbes, Leibniz composed his *Hypothesis physica nova*, consisting of two parts, *Theoria motus abstracti* and *Theoria motus concreti*, which in 1671 he presented respectively to the Paris Academy of Sciences and the London Royal Society. At this time, Leibniz owed more to Descartes, whose work he knew only at second hand, than he was later willing to admit. Closer study of the philosophy of Descartes led Leibniz to a more decisive rejection.

In 1686 he published in the *Acta eruditorum* a criticism of Descartes’s measure of force, *Brevis demonstratio erroris memorabilis Cartesii et aliorum circa legem naturae*, which started a controversy with Catalan, Malebranche, and Papin lasting until 1691. Also in 1686 Leibniz sent to Arnauld, for his comments, an essay entitled *Discours de métaphysique*, in which he developed the ideas of the later *Théodicée*. The tracts entitled *De lineis opticis, Schediasma de resistentia medii*, and *Tentamen de motuum coelestium causis*, published in the *Acta eruditorum* in 1689, were hurriedly composed by Leibniz after he had read the review of Newton’s *Principia* in the same journal, in an attempt to obtain some credit for results which he had derived independently of Newton. In 1692, at the instigation of Pelisson, Leibniz wrote an *Essay de dynamique*, which was read to the Paris Academy by Philippe de la Hire, and in 1695 there appeared in the *Acta eruditorum* an article entitled *Specimen dynamicum*, which contained the clearest exposition of Leibniz’ dynamics.

Leibniz’ *Nouveaux essais sur l’entendement humain*, completed in 1705 but not published during his lifetime, presented a detailed criticism of Locke’s position. By adding *nisi ipse intellectus* to the famous maxim, *Nihil est in intellectu quod non prius fuerit in sensu* (wrongly attributed to Aristotle by Duns Secotus^{1}), Leibniz neatly reversed the application of the principle by Locke. According to Leibniz, the mind originally contains the principles of the various ideas which the senses on occasion call forth.

In 1710 Leibniz published his *Essais de Théodicée sur la bonté de Dieu, la libereté de l’homme et l’origine du mal*, a work composed at the instigation of Sophia Charlotte, with whom Leibniz had conversed concerning the views of Bayle. In response to a request from Prince Eugene for an abstract of the *Théodicée*, Leibniz in 1714 wrote the *Principes de la nature et de la grâce fondées en raison and the Monadologie*. When in 1715 Leibniz wrote to Princess Caroline of Wales, criticizing the philosophical and theological implications of the work of Newton, she commissioned Samuel Clarke to reply. The ensuing correspondence, containing Leibniz’ most penetrating criticism of Newtonian philosophy, was published in 1717.

**Rational Physics (Protophysics)** . In his efforts to clarify fundamental physical principles, Leibniz followed a plan which he called a transition from geometry to physics through a “science of motion that unites matter with forms and theory with practice.” ^{2} He sought the metaphysical foundations of mechanics in an axiomatic structure. The *Theoria motus abstracti*^{3} offers a rational theory of motion whose axiomatic foundation (*fundamenta praedemonstrabilia*) was inspired by the indivisibles of Cavalieri and the notion of *conatus* proposed by Hobbes. Both the word *conatus* and the mechanical idea were taken from hobbes,^{4} while the mathematical reasoning was derived from Cavalieri. After his invention of the calculus, Leibniz was able to replace Cavalieri’s indivisibles by differentials and this enabled him to apply his theory of *conatus* to the solution of dynamical problems.

The concept of *conatus* provided for Leibniz a path of escape from the paradox of Zeno. Motion is continuous and therefore infinitely divisible, but if motion is real, its beginnning cannot be a mere nothing.^{5}*Conatus* is a tendency to motion, a mind-like quality
having the same relation to motion as a point to space (in Cavalieri’s terms) or a differential to a finite quantity (in terms of the infinitesimal calculus). *Conatus* represents virtual motion; it is an intensive quality that can be measured by the distance traversed in an infinitesimal element of time. A body can possess several *conatuses* simultaneously and these can be combined into a single *conatus* if they are compatible. In the absence of motion, *conatus* lasts only an instant,^{6} but however weak, its effect is transmitted to infinity in a plenum. Leibniz’ doctrine of *conatus*, in which a body is conceived as a momentary mind, that is, a mind without memory, may be regarded as a first sketch of the philosophy of monads.

Mathematically, *conatus* represents for Leibniz accelerative force in the Newtonian sense, so that, by summing an infinity of *conatuses* (that is, by integration), the effect of a continuous force can be measured. Examples of *conatus* given by Leibniz are centrifugal force and what he called the solicitation of gravity. Further clarifications of the concept of *conatus* are given in the *Essay de dynamique* and *Specimen dynamicum*, where *conatus* is compared with static force or *vis mortua* in contrast to *vis viva*, which is produced by an infinity of impressions of *vis mortua*.

**Physics (Mechanical Hypothesis)** . Leibniz soon recognized that the idea of *conatus* could not by itself explain the results of the experiments of Huygens and Wren on collision. Since in the absence of motion *conatus* lasts only an instant, a body once brought to rest in a collision, Leibniz explains, could not then rebound.^{7} A new property of matter was needed and this was provided for Leibniz by the action of an ether. As conceived by Leibniz in his *Theoria motus concreti*,^{8} the ether was a universal agent of motion, explaining mechanically all the phenomena of the visible world. This essentially Cartesian notion was adopted by Leibniz following a brief attachment to the doctrine of physical atomism defended in the works of Bacon and Gassendi. Leibniz did not, however, become a Cartesian, nor did he aim to construct an entirely new hypothesis but rather to improve and reconcile those of others^{9}.

A good example of the way in which Leibniz pursued this goal is provided by the planetary theory expounded in his *Tentamen de motuum coelestium causis*. In this work, Leibniz combined the mechanics of *conatus* and inertial motion with the concept of a fluid vortex to give a physical explanation of planetary motion on the basis of Kepler’s analysis of the elliptical orbit into a circulation and a radial motion. Leibniz’ harmonic vortex accounted for the circulation while the variation in distance was explained by the combined action of the centrifugal force arising from the circulation and the solicitation of gravity. This solicitation he held to be the effect of a second independent vortex of the kind imagined by Huygens, to whom he described Newton’s attraction as “an immaterial and inexplicable virtue,” a criticism he made public in the *Theéodicée* and repeated in the correspondence with Clarke.

Although Leibniz’ planetary theory could be described as a modification of that of Descartes, he did not acknowledge any inspiration from this source. Attributing the idea of a fluid vortex to Kepler and also, but incorrectly, the idea of centrifugal force, Leibniz claimed that Descartes had made ample use of these ideas without acknowledgement. Leibniz had already, in a latter to Arnauld,^{10} rejected the Cartesian doctrine that the essence of corporeal substance is extension. One reason that led him to this rejection was the theological problem of transubstantiation which he studied at the instigation of Baron Boyneburg,^{11} but the most important dynamical reason was connected with the relativity of motion. As explained by Leibniz in the *Discours de métaphysique,*^{12} since motion is relative, the real difference between a moving body and a body at rest cannot consist of change of position. Consequently, as the principle of inertial motion precludes an external impulse for a body moving with constant speed, the cause of motion must be an inherent force. Another argument against the Cartesian position involves the principles of the identity of indiscernibles. From this principle it follows that, besides extension, which is a property carried by a body from place to place, the body must have some intrinsic property which distinguishes it from others. According to Leibniz, bodies possess three properties which cannot be derived from extension: namely, impenetrability, inertia, and activity. Impenetrability and inertia are associated with *materia prima*, which is an abstraction, while *materia secunda* (the matter of dynamics) is matter endowed with force.

Since, for Leibniz, force alone confers reality to motion, the correct measure of force becomes the central problem of dynamics. Now Descartes had measured what Leibniz regarded as the active force of bodies, that is, the cause of their activity, by their quantity of motion. But, as Leibniz remarked in a letter of 1680,^{13} Descartes’s erroneous laws of collision implied that his basic principle of the conservation of motion was false. In 1686 Leibniz published his criticism of the Cartesian principle of the conservation of motion in his *Brevis demonstratio erroris mirabilis Cartesii*, thereby precipitating the *vis viva* controversy. According to Leibniz, Descartes had incorrectly generalized from statics to dynamics. In statics or
virtual motion, Leibniz explains, the force is as the velocity but when the body has acquired a finite velocity and the force has become live, it is as the square of the velocity. As Leibniz remarked on several occasions, there is always a perfect equation between cause and effect, so that the force of a body in motion is measured by the product of the mass and the height to which the body could rise (the effect of the force). Using the laws of Galileo, this height was shown by Leibniz to be proportional to the square of the velocity, so that the force (*vis viva*) could be expressed as *mv ^{2}*.

Since *vis viva* was regarded by Leibniz as the ultimate physical reality,^{14} it had to be conserved throughout all transformations. Huygens had shown that, in elastic collision, *vis viva* is not diminished. The *vis viva* apparently lost in inelastic collision Leibniz held to be in fact simply distributed among the small parts of the bodies.^{15}

Leibniz discovered the principle of the conservation of momentum, which he described as the “quantité d’avancement.” ^{16} Had Descartes known that the quantity of motion is preserved in every direction (so that motion is completely determined, leaving no opportunity for the directing influence of mind), Leibniz remarks, he would probably have discovered the preestablished harmony. But in Leibniz’ view, the principle of the conservation of momentum did not correspond to something absolute, since two bodies moving together with equal quantities of motion would have no total momentum. Leibniz’ discovery of yet another absolute quantity in the concept of *action* enabled him to answer the Cartesian criticism that he had failed to take time into account in his consideration of *vis viva*. In his *Dynamica de potentia et legibus naturae corporeae*, Leibniz made an attempt to fit this new concept into his axiomatic scheme.

Although succeeding generations described the *vis viva* controversy as a battle of words, there can be no doubt that Leibniz himself saw it as a debate about the nature of reality. Referring to his search for a true dynamics, Leibniz remarked in 1689 that, to escape from the labyrinth, he could find “no other thread of Ariadne than the evaluation of forces, under the supposition of the metaphysical principle that the total effect is always equal to the complete cause.” ^{17}

**Scientia Generalis** . According to the usual distinctions, the position that Leibniz took in physics, as well as in other fundamental questions, was rationalistic, and to that extent, despite all differences in detail, was related to Descartes’s position. Evident confirmation of this may be seen in Leibniz’ controversy with Locke; although he does not explicitly defended the Cartesian view, he uses arguments compatible with this position. It is often overlooked, however, that Leibniz was always concerned to discuss epistemological issues as questions of theoretical science. For example, while Locke speaks of the origins of knowledge, Leibniz speaks of the structure of a science which encompasses that particular field. Thus Leibniz sees the distinction between necessary and contingent truths, so important in the debate with Locke, as a problem of theoretical science which transcends any consideration of the historical alternatives, rationalism and empiricism. Neither an empiricist in the sense of Locke nor yet a rationalist in the sense of Descartes, Leibniz saw the refutation of the empiricist’s thesis (experience as the nonconceptual basis of knowledge) not as the problem of a rational psychology as it was then understood (in Cartesian idiom, the assumption of inborn truths and ideas) but as a problem that can be resolved only within the framework of a general logic of scientific research. Nevertheless, he shares with Descartes one fundamental rationalistic idea, namely the notion (which may be discerned in the Cartesian *mathesis universalis*) of a *scientia generalis*. In connection with his theoretical linguistic efforts towards a *characteristica universalis*, this thought appears in Leibniz as a plan for a *mathématique universelle*.^{18}.

Inspired by the ideas of Lull, Kircher, Descartes, Hobbes, Wilkins, and Dalgarno, Leibniz pursued the invention of an alphabet of thought (*alphabetum cogitationum humanarum*)^{19} that would not only be a form of shorthand but a formalism for the creation of knowledge itself. He sought a method that would permit “truths of reason in any field whatever to be attained, to some degree at least, through a calculus, as in arithmetic or algebra.” ^{20} The program of such a *lingua philosophica* or *characteristica universalis* was to proceed through lists of definitions to an elementary terminology encompassing a complete encyclopedia of all that was known. Leibniz connected this plan with others that he had, such as the construction of a general language for intellectual discourse and a rational grammar, conceived as a continuation of the older*grammatica speculativa*.

While Leibniz’ programmatic statements leave open the question of just how the basic language he was searching for and the encyclopedia were to be connected (the *characteristica universalis*, according to other explanations, was itself to facilitate a compendium of knowledge), a certain *ars combinatoria*, conceived as part of an *ars inveniendi*, was to serve in the creation of the lists of definitions. As early as 1666, in Leibniz’ *Dissertatio de arte combinatoria*, the *ars inveniendi* was sketched out under the name *logica inventiva* (or *logica inventionis*) as a calculus of concepts in which, in marked contrast to the traditional
theories of concepts and judgments, he discusses the possibility of transforming rules of inference into schematic deductive rules. Within this framework there is also a complementary *ars iudicandi,* a mechanical procedure for decision making. However, the thought of gaining scientific propositions by means of a calculus of concepts derived from the *ars combinatoria* and a mechanical procedure for decision making remained lodged in a few attempts at the formation of the “alphabet.” Leibniz was unable to complete the most important task for this project, namely, the proof of its completeness and irreducibility, nor did he consider this problem in his plans for the *scientia generalis,* a basic part of which was the “alphabet,” the *characteristica universalis* in the form of a *mathématique universelle.*

The *scientia generalis* exists essentially only in the “tables of contents,” which are not internally consistent terminologically and thus admit of additions at will. Nevertheless, it is clear that Leibniz was thinking here of a structure for a general methodology, consisting, on the one hand, of partial methodologies concerning special special sciences such as mathematics, and on the other hand, of procedures for the *ars inveniendi,* such as the *characteristica universalis;* taken together, these were probably intended to replace traditional epistemology as a unified conceptual armory. This was by no means impracticable, at least in part. For example, the analytical procedures in which arithmetical transformations occur independently of the processes to which they refer, employed by Leibniz in physics, may be construed as a partial realization of the concept of a *characteristica universalis.*

**Formal Logic.** Leibniz produced yet another proof of the feasibility of his plan for schematic operations with concepts. Besides the infinitesimal calculus, he created a logical calculus *(calculus ratiocinator, universalis, logicus, or rationalis)* that was to lend the same certainty to deductions concerning concepts as that possessed by algebraic deduction. Leibniz stands here at the very beginning of formal logic in the modern sense, especially in relation to the older syllogistics, which he succeeded in casting into the form of a calculus. A number of different steps may be distinguished in his program for a logical calculus. In 1679 various versions of an arithmetical calculus appeared that permitted a representation of a conjunction of predicates by the product of prime numbers assigned to the individual predicates. In order to solve the problem of negation—needed in the syllogistic modes—nagative numbers were introduced for the nonpredicates of a concept. Every concept was assigned a pair of numbers having no common factor, in which the factors of the first represented the predicates and the factors of the second represented the nonpredicates of the concept. Because this arithmetical calculus became too complex, Leibniz replaced it in about 1686 by plans for an algebraic calculus treating the identity of concepts and the inclusion of one concept in another. The components of this calculus were the symbols for predicates, a, b, c, … *(termini)*, an operational sign—*(non)*, four relational signs ⊂, ⊄, =, ≠ (represented in language by *est non est, sunt idem* or *eadem sunt, diversa sunt)* and the logical particles in vernacular form. To the rules of the calculus *(principia calculi)*—as distinct from the axioms *(propositions per se verae )* and hypotheses *(propositions positae)* which constitute its foundation—belong the principles of implication and logical equivalence and also a substitution formula. Among the these *(propositions verae)* that can be proved with the aid of the axioms and hypotheses, such as *a* ⊂ *a* (reflexivity of the relation ⊂), and *a* ⊂ *b* et *b* ⊂ *c* implies *a* ⊂ *c* (transitivity of ⊂), is the proposition *a* ⊂ *b* et *d* ⊂ *c* implies *ad* ⊂ *bc*. This was called by Leibniz the “admirable theorem” *(praeclarum theorema)* and appears again, much later, with Russell and Whitehead.^{21}

Leibniz extended this algebraic calculus in various ways, first with a predicate-constant *ens* (or *res)*, which may be understood as a precursor of the existential quantifier, and secondly with the interpretation of the predicates as propositions instead of concepts. Inclusion between concepts becomes implication between propositions and the new predicate-constant *ens* appears as the truth value (*verum*), intensionally designated as *possibile*. These discourses were concluded in about 1690 with two calculi^{22} in which a transition is made from an (intensional) logic of concepts to a logic of classes. The first, originally entitled *Non inelegans specimen demonstrandi in abstractis* (a “plus-minus calculus” ), is a pure calculus of classes (a dualization of the thesis of the original algebraic calculus) in which a new predicate-constant *nihil* (for *non-ens*) is introduced. The second calculus (a “plus calculus” ) is an abstract calculus for which an extensional as well as an intensional interpretation is expressly given. Logical addition in the “plus-minus calculus” is symbolized by +. In the “Plus calculus,” logical addition, as well as logical multiplication in the intensional sense, is symbolized by ⊕, while the relational sign = (*sunt idem* or *eadem sunt*) is replaced by ∞ and the sign ≠ by *non A ∞ B.* Furthermore, subtraction appears in the “plus-minus calculus” symbolized by—or ⊖, and also the relation of incompatibility (*incomemunicantia sunt*) together with its negation (*communicantia**sunt or compatibilia sunt)*. For example, one of the propositions of the calculus states: *A* - *B* = *C* holds exactly, if and only if *A* = *B* + *C*, and *B* and *C* are incompatible; in modern notation

(The condition *b* ⊂ *a* is implicit in Leibniz’ use of the symbol—).

If we disregard a few syntactical details and observe that Leibniz’ work gives an approximation to a complete interpretation of the elements of a logical calculus (including the rules for transformation), we see here for the first time a formal language and thus an actual successful example of a *characteristica universalis*. It is true that Leibniz did not make sufficient distinction between the formal structure of the calculus and the interpretations of its content; for example, the beginnings of the calculus are immediately considered as axiomatic and the rules of transformation are viewed as principles of education. Yet it is decisive for Leibniz’ program and our appreciation of it that he succeeded at all, in his logical calculus, in the formal reconstruction of principles of education concerning concepts.

**General Logic** . Leibniz’ general logical investigations play just as important a role in his systematic philosophy as does his logical calculus. Most important are his analytical theory of judgement, the theory of complete concepts on which this is based, and the distinction between necessary and contingent propositions. According to Leibniz’ analytical theory of judgement, in every true proposition of the subject-predicate form, the concept of the predicate is contained originally in the concept of the subject (*praedicatum inest subiecto*). The *inesse* relation between subject and predicate is indeed the converse of the universal-affirmative relation between concepts, long known in traditional syllogisms (*B inest omni A* is the converse of *omne A est B*).^{23} Although it is thus taken for granted that subject-concepts are completely analyzable, it suffices, in a particular case, that a certain predicate-concept can be considered as contained in a certain subject-concept. Fundamentally, there is a theory of concepts according to which concepts are usually defined as combinations of partials, so that analysis of these composite concepts *(notiones compositae)* should lead to simple concepts (*notiones primitivae or irresolubiles*). With these, the *characteristica universalis* could then begin again. When the predicate-concept simply repeats the subject-concept, Leibniz speaks of an identical proposition; when this is not the case, but analysis shows the predicate-concept to be contained implicitly in the subject-concept, he speaks of a virtually identical proposition.

The distinction between necessary propositions or truths of reason (*vérités de raisonnement*) and contingent propositions or truths of fact (*vérités de fait*) is central to Leibniz’ theory of science. As contingent truths, the laws of nature are discoverable by observation and induction but they have their rational foundation, whose investigation constitutes for Leibniz the essential element in science, in principles of substitution of equivalents to reduce the classical syllogism, as a principle of deduction, by the principle of substitution of equivalents to reduce composite propositions to identical propositions. Contingent propositions are defined as those that are neither identical nor reducible through a finite number of substitutions to identical propositions.

All contingent propositions are held by Leibniz to be reducible to identical propositions through an infinite number of steps. Only God can perform these steps, but even for God, such propositions are not necessary (in the sense of being demanded by the principle of contradiction). Nevertheless, contingent propositions, in Leibniz’ view, can be known *a priori* by God and, in principle, also by man. For Leibniz, the terms *a priori and necessary* are evidently not synonymous. It is the principle of sufficient reason that enables us (at least in principle) to know contingent truths *a priori*. Consequently, the deduction of such truths involves an appeal to final causes. On the physical plane, every event must have its cause in an anterior event. Thus we have a series of contingent events for which the reason must be sought in a necessary Being outside the series of contingents. The choice between the possibles does not depend on God’s understanding, that is to say, on the necessity of the truths of mathematics and logic, but on his volition. God can create any possible world, but being God, he wills the best of all possible worlds. Thus the contingent truths, including the laws of nature, do not proceed from logical necessity but from a moral necessity.

**Methodological Principles** . Logical calculi and the notions mentioned under “General Logic” belong to a general theory of foundations that also encompasses certain important Leibnizian methodological principles. The principle of sufficient reason (*principium rationis sufficientis, also designated as principium nobilissimum*) plays a special role. In its simplest form it is phrased “nothing is without a reason” (*nihil est sine ratione*), which includes not only the concept of physical causality (*nihil fit sine causa*) but also in general the concept of a logical antecedent-consequent relationship. According to Leibniz, “a large part of metaphysics [by which he means rational theology], physics,and ethics” may be constructed on this
proposition.^{24} Viewed methodologically, this means that, in the principle of sufficient reason, there is a teleological as well as a causal principle; the particular import of the proposition is that both principles may be used in the same way for physical processes and human actions.

Defending the utility of final causes in physics (in opposition to the view of Descartes), Leibniz explained that these often provided an easier path than the more direct method of mechanical explanation in terms of efficient causes.^{25} Leibniz had himself in 1682 used a variation of Fermat’s principle in an application of his method of maxima and minima to the derivation of the law of refraction. Closely associated with the principle of sufficient reason is the principle of perfection *(principium perfectionis or melioris).* In physics, this principle determines the actual motion from among the possible motions, and in metaphysics leads Leibniz to the idea of “the best of all possible worlds.” The clearest expression of Leibniz’ view is to be found in his *Tentamen anagoticum,*^{26} written in about 1694, where he remarks that the least parts of the universe are ruled by the most perfect order. In this context, the idea of perfection consists in a maximum or minimum quantity, the choice between the two being determined by another architectonic principle, such as the principle of simplicity. Since the laws of nature themselves are held by Leibniz to depend on these principles, he supposed the existence of a perfect correlation between physical explanations in terms of final and efficient causes.

In relation to Leibniz’ analytical theory of judgment and his distinction between necessary and contingent propositions, the principle of sufficient reason entails that, in the case of a well-founded connection between, for example, physical cause and physical effect, the proposition that formulates the effect may be described as a logical implication of the proposition that formulates the cause. Generalized in the sense of the analytical theory of truth and falsehood that Leibniz upholds, this means: “nothing is without a reason; that is, there is no proposition in which there is not some connection between the concept of the predicate and concept of the subject, or which cannot be proved *a priori.*” ^{27} This logical sense of the principle of sufficient reason contains also (in its formulation as a *principium reddendae rationis)* a methodological postulate; propositions are not only capable of being grounded in reasons (in the given analytical manner) but they must be so grounded (insofar as they are formulated with scientific intent).^{28}

In addition to the principle of sufficient reason, the principle of contradiction (*principium contradictionis*) and the principle of the identity of indiscernible *(principium identitatis indiscernibilium)* are especially in evidence in Leibniz’ logic. In its Leibnizian formulation, the principle of contradiction, ⌉(*A* ⋀ ⌉*A*), includes the principle of the excluded middle, *A* ⋁ ⌉*A* (*terbium non datur*): “nothing can be and not be at the same time; everything is or is not.” ^{29} Since Leibniz’ formulation rests on a theory according to which predicates, in principle, can be traced back to identical propositions, he also classes the principle of contradiction as a principle of identity. The principle of the identity of indiscernible again defines the identity of two subjects, whether concrete or abstract, in terms of the property that the mutual replacement of their complete concepts in any arbitrary statement does not in the least change the truth value of that statement (*salva veritate*). Two subjects *s*_{1} and *s*_{2} are different when there is predicate *P* that is included in the complete complete concept *S*_{1} of *s*_{1} but not in the complete concept *S*_{2} of *s*_{2,} or vice versa. If there is no such predicate, then because of the mutual replaceability of both complete concepts *S*_{1} and *S*_{2,} there is no sence in talking of different subjects. This means, however, that the principle of the identity of indiscernible, together with its traditional meaning ( “there are no two indistinguishable subjects” ^{30}), is synonymous with the definition of logical equality ( “whatever can be put in place of anything else, salva veritate, is identical to it” ^{31}).

**Metaphysics (Logical Atomism)** . Since the investigations of Russell and Couturat, it has become clear that Leibniz’ theory of monads is characterized by an attempt to discuss metaphysical questions within a framework of logical distinctions. On several occasions, however, Leizbniz himself remarks that dynamics was to a great extent the foundation of his system. For example, in his *De primae philosophiae emendatione et de notione substantiae,* Leibniz comments that the notion of force, for the exposition of which he had designed a special science of dynamics, added much to the clear understanding of the concept of substance.^{32} This suggests that it was the notion of mechanical energy that led to the concept of substance as activity. Again, it is in dynamics, Leibniz remarks, that we learn the difference between necessary truths and those which have their source in final causes, that is to say, contingent truths,^{33} while optical theory, in the form of Fermat’s principle, pointed to the location of the final causes in the principle of perfection.^{34} Even the subject-predicate logic itself, which forms the rational foundation of Leibniz’ metaphysics, seems to take on a biological image, such as the growth of a plant from a seed, when Leibniz writes to De Volder that the present state of a substance must
involve its future states and vice versa. It thus appears that physical analogies very probably provided the initial inspiration for the formation of Leibniz’ metaphysical concepts.

In the preface to the Théodicée, Leibniz declares that there are two famous labyrinths in which our reason goes astray; the one relates to the problem of liberty (which is the principal subject of the *Théodicée*), the other to the problem of continuity and the antinomies of the infinite. To arrive at a true metaphysics, Leibniz remarks in another place, it is necessary to have passed through the labyrinth of the continuum.^{35} Extension, like other continuous quantities, is infinitely divisible, so that physical bodies, however small, have yet smaller parts. For Leibniz, there can be no real whole without real unities, that is, indivisibles,^{36} or as he expresses it (repeating a phrase used by Nicholas of Cusa^{37}), being and unity are convertible terms.^{38} Now the real unities underlying physical bodies cannot be mathematical points, for these are mere nothings. As Leibniz explains, “only metaphysical or substantial points … are expact ane real; without them there would be nothing real, since without true unities *[les véritables unités]* a composits whole would be impossible.” ^{39} Within the narrower framework of physics, such unities can be understood as the concept of mass-points, but they are meat in the broader sense of the classical concept of substance, to the consideration of which Leibniz had in 1663 devoted his first philosophical essay, *Dissertatio metaphysical de principio individui.* Leibniz’ metaphysical realities are unexpended substances or monads (a term he used from 1696), whose essence is an intensive quality of the nature of force or mind.

Leibniz consciously adheres to Aristotelian definitions, when he emphasizes that we may speak of an individual substance whenever a predicate-concept is included in a subject-concept, and this subject-concept never appears itself as a predicate-concept. A concept fulfilling this condition may thus be designated as an individual concept (*notion individuelle*) and may be construed as a complete concept, that is, as the infinite conjunction of predicates appertaining to that individual. If a complete subject-concept were itself to appear as a predicate-concept, then according to the principle of the identity of indiscernibles, the predicated individual would be identical to the designated individual. This result of Leibniz is a logical between substance and quality.

The definition of individual concepts, according to which individual substances are denoted by complete concepts, leads also to the idea, central to the theory of monads, that each monad or individual substance represents or mirrors the whole universe. What Leibniz means is that, given a particular subject, all other subjects must appear, represented by their names or designations, in at least one of the infinite conjunction of predicates constituting the complete concept of that particular subject.

Leibniz describes the inner activity which constitutes the essence of the monad as perception.^{40} This does not imply that all monads are conscious. The monad has perception in the sense that it represents the universe from its point of view, while its activity is manifested in spontaneous change from one perception to another. The attribution of perception and appetition to the monads does not mean that they can be sufficiently defined in terms of physiological and psychological processes, although Leibniz does compare them to biological organisms; he was, of course, familiar with the work of Leeuwenhoek, Swammerdam, and Malpighi on microorganisms, which seemed to confirm the theory of preformation demanded by the doctrine of the monads. Once again, the logical basis for the theory of perception is that the concept of the monad’s inner activity must occur that an individual substance encounter “is only the consequence of its notion or complete concept, since this notion already contains all predicates or events and expresses the whole universe.” ^{41}

Within the framework of this conceptual connection between a theory of perception and a theory of individual concepts, sufficient room remains for physiological and psychological discussion and here Leibniz goes far beyond the level of debate in Locke and Descarts, In particular, Leibniz distinguishes between consciousness and self-consciousness, and again, between stimuli which rise above the threshold of consciousness and those that remain below it; he even observes that the summation of sub-threshold threshold of consciousness, a clear hint of the existence of the unconscious.

Since, for Leibniz, the real unities constituting the universe are essentially perceptive, it follow that the real continuum must also be a continuum of perception. The infinite totality of monads represent or mirror the universe, of which each is part, from all possible points of view, so that the universe is at once continuous and not only infinitely divisible but actually divided into an infinity of real metaphysical atoms.^{42} As these atoms are purely intensive unities, they are mutually exclusive, so that no real interaction between them is possible. Consequently, Leibniz needed his principal of the preestablished harmony (which, he claimed, avoided the perpetual intervention
of God involved in the doctrine of occasionalism) to explain the mutual compatibility of the internal activities of the monads. Leibniz thus evaded the antinomies of the continuum by conceiving reality not as an extensive plenum of matter bound by physical relations but as an intensive plenum of force or life bound by a preestablished harmony.

The monads differ in the clarity of their perceptions, for their activity is opposed and their perceptions consequently confused to varying degrees by the *materia prima* with which every created monad is endowed. As with the *materia prima* of dynamics, that of the monads is thus associated with passivity. Only one monad, God, is free of *materia prima* and he alone perceives the world with clarity, that is to say, as it really is.

Physical bodies consist of infinite aggregates of monads. Since such aggregates form only accidental unities, they are not possessed of real magnitude. It is in Leibniz’ view, not possessed of real magnitude. It is in this sense that Leibniz denies infinite number while admitting the existence of an actual infinite. When an aggregate has one dominant monad. the Aggregates appears as the organic body of this monad. Aggregates without a dominant monad simply appear as *materia secunda*, the matter of dynamics. Bodies as such are therefore conceived by Leibniz simply as phenomena, but in contrast to dreams and similar illusions, well-founded phenomena on account of their consistency. For Leibniz then, the world of extended physical, bodies is just a world of appearance, a symbolic representation of the real world of monads.

From this doctrine it follows that the forces involved in dynamics are only accidental, or derivative, as Leibniz terms them. The real active force, or *vis primitiva*, which remains constant in each corporeal substance, corresponds to mind or substantial form. Leibniz does not, however, reintroduce substantial forms as physical causes;^{43} for in his view, physical explanations involve a *vis derivativa* or accidental force by which the *vis primitiva* or real principle of action is modified.^{44} One monad, having more *vis primitiva,* represents the universe more distinctly than another, a difference that can be expressed by the terms active and passive, thought there is, of course, no real interaction. In the world of phenomena, this relation is symbolized in the notion of physical causality. The laws of nature, including the principle of the conservation of *vis viva*, thus have relevance only at the phenomenal level, through they symbolize, on the metaphysical plane, an order manifested through the realization of predicates of individual substances in accordance with the preestablished harmony.

In the early stages of the formulation of his metaphysics, Leibniz located the unextended substances, that he later called monads, in points.^{45} This presupposed a real space in which the monads were embedded, a view that consideration of the nature of substance and the difficulties of the continuum caused him soon to abandon. In his *Système nouveau*,^{46} written in 1695, Leibniz described atoms of substance (that is, monads) as metaphysical points, but the mathematical points associated with the space of physics he described as the points of view of the monads. Physical space, Leibniz explains, consists of relations of order between coexistent things.^{47} This may be contrasted with the notion of an abstract space, which consists of an assemblage of possible relations between possible existents.^{48} For Leibniz makes space entirely dependent on the monads, for points are not parts of space. In the sense of distance between points, space. In the sense of distance between points, space is a mere ideal thing, the consideration of which, Leibniz remarks, is never-theless useful.^{49} Distance between points of view of the monads. Owing to the preestablished harmony, these relations are compatible, so that space is a well-founded phenomenon, an extensive representation of an intensive continuum. Similarly time, as the order of noncontemporaneous events^{50}, is also a well-founded phenomenon.

Leibniz’ objections to the kind of absolute real space and time conceived by Newton are expressed most completely in his correspondence with Clarke. Real space and time, Leibniz argues, would violate the principle of sufficient reason and the principle of the identity of indiscernible. For example, a rotation of the whole universe in an absolute space would leave the arrangement of bodies among themselves unchanged, but no sufficient reason could be found why God should have placed the whole universe in one of these positions rather that the other. Again, if time were absolution, no reason could be found why God should have created the universe at one time rather than another.^{51}

From the ideal nature of space and time, it follows that motion, in the physical sense, is also ideal and therefore relative^{52}.Against Newton,Leibniz maintainsthe relativity not only of rectilinear motion butalso of rotation.^{53}.Yet Leibniz is willing to admit that there is a difference between what he calls an absolute real motion of a body and a mere change in its position relative to other bodies. For it is the body in which the cause of motion (that is, the active force)
resides that is truly in motion.^{54} Now it is evidently the *vis primitiva* that Leibniz has in mind, since consideration of the *vis derivativa* (*conatus or vis viva*) does not serve to identify an absolute motion, so that we may interpret Leibniz as saying that true absolute motion appertains to the metaphysical plane, where it can be perceived only by God. Indeed all bodies have an absolute motion in this sense, for since all monads have activity, all aggregates have *vis viva.* The world of phenomena is therefore conceived by Leibniz as a world of bodies in absolute motion (rest being a mere abstraction) but a world in which only relative changes of position can be observed.

The theory of monads may be seen as a sustained effort to present, in “cosmological” completeness, a systematic unified structure of knowledge on the basis of a logical reconstruction of the concept of substance. Insofar as the central assertion, namely, that because there are composite “substances,” three must be simple substances, is not only a cosmological and metaphysical statement, but in addition, an assertion of the priority of synthetic over analytic procedures, his efforts retain their original methodological meaning. As revealed in his plan for a *characteristica universalis*, analysis, for Leibniz implies synthesis, but a synthesis that must begin with irreducible elements. Where there are no such elements, neither analysis nor synthesis is possible within the framework of Leibniz’ constructive methodology. This means that Leibniz, in the course of his protracted efforts to define an individual substance, moved from physical atomism to logical atomism in the modern sense, as represented by Russell. Leibniz belived that he had proved the thesis of an unambiguously defined world, in which the physico-theological theme running through the mechanistic philosophy of his age might once again be seen as a metaphysics in the classical sense.

**Influence.** The thought of Leibniz influenced the history of philosophy and science in two ways; first, through the mediocre systematization of Christian Wolff known as the “Leibnizo-Wolffian” philosophy, and secondly, through the significance of particular theories in the history of various sciences. While the tradition of the Leibnizo-Wolffian philosophy ended with Kant, the influence of particular theories of Leibniz lasted through the nineteenth and into the twentieth century.

The controversy with Newton and Clarke was not conducive to the reception of Leibniz’ work in physics and hindered the objective evaluation of important contributions such as his law of radial acceleration. The *vis viva* controversy arose as a direct result of Leibniz’ criticism of Descartes and concerned not only the measure of force but also the nature of force itself. While ’s Gravesande and d’Alembert (in his *Traiteé de mécanique*) judged the dispute to be merely a semantic argument, Kant in 1747 (*Gedanken von der wahren Schätzung der lebendigen kräfte*) made an ineffective attempt at reconciliation. Leibnizian dynamics was developed further by Bošković, who transformed the concept of dynamicforce in the direction of a concept of relational force.

Within the framework of rational physics, kant contributed some essential improvements, such as the completion of the distinction between necessary and contingent propositions by means of the concept of the synthetic *a priori* and clarification of the principle of causality. Leibniz’ protophysical plan, however, remained intact. It was continued later by Whewell, Clifford, Mach, and Dingler, to mention only a few. Insofar as the fundamental concepts of space and time were concerned, the Newtonian ideas of absolute space and time at first prevailed over the Leibnizian ideas of relational space and time. Kant also tried here to mediate between the ideas of Leibniz and Newton, but his own suggestion (space as the origin of the distinction between a nonreflexible figure and its mirror image, such as a pair of gloves) strongly resembled the Newtonian concept of absolute space. Modern relativistic physics has turned the scales in favor of the ideas of Leibniz.

Among the methodological principles of Leibniz, only the principle of sufficient reason has played a prominent role in the history of philosophy. Wolff, disregarding Leibniz’ methodological intentions, tried to prove it by ontological means (*Philosophia sive ontologia*); Kant reduced it essentially to the law of causality and in 1813 Schopenhauer drew on it for the elucidation of his four conditions of verification (*Über die vierfache Wurzel des Satzes vom zureichenden Grunde*). On the other hand, the methodological project of a *characteristica universalis* together with the ensuing development of logical calculi has played a most significant role in the history of modern logic. In 1896 Frege, recalling Leibniz, described his *Begriffsschrift* of 1879 as a *lingua characterica* (not just *a calculus ratiocinator*), thus distinguishing it from the parallel efforts of Boole and Peano. De Morgan and Boole tried to carry out what Scholz has described as the “Leibniz program” of the development of a logical algebra of classes. This connection between logic and mathematics, evident also in Peirce and Schröder, was once again weakened by Frege, Peano, and Russell, whose work (especially that of Frege) nevertheless bears the inescapable influence of Leibniz; for even where the differences are greatest,
the development of modern logic can be traced back to Leibniz. In this connection, it is fortunate that (in the absence of any publications of Leibniz) there is a tradition of correspondence beginning with letters between Leibniz, Oldenburg, and Tschirnhaus. The emphasis here, as exemplified in the logic theories of Ploucquet, Lambert, and Castillon, is on the intensional interpretation of logical calculi.

While there is an affinity between the theory of monads and Russell’s logical atornism, a direct influence of the more metaphysical parts of the theory of monads on the history of scientific thought is difficult to prove. Particular results, such as the biological concept of preformation (accepted by Haller, Bonnet, and Spallanzani) or the discovery of sensory thresholds, though related to the theory of monads in a systematic way, became detached from it and followed their own lines of development. Yet the term “monad” played an important role with Wolf, Baumgarten, Crusius, and, at the beginning, with Kant (as exemplified in his *Monadologica physica* of 1756), then later with Goethe and Solger as well. Vitalism in its various forms, including the “biological romanticism” of the nineteenth century (the Schelling school), embraced in general the biological interpretation of the theory of monads, but this did not amount to a revival of the meta-physical theory. It is more likely that vitalism simply represented a reaction against mechanism, a tradition to which Leibniz also belonged.

## NOTES

1. Duns Scotus, *Questiones super universalibus Porphyrii* (Venice, 1512), Quest. 3.

2. L. Couturat, ed., *Opuscules et fragments inédits de Leibniz* (Paris, 1903; Hildesheim, 1966), p. 594.

3. G. W. Leibniz, *Sämtliche Schriften und Briefe,* VI, 2, pp. 258-276.

4. T. Hobbes, *Elementorum philosophiae* (London, 1655), sectio prima: de corpore, pars tertia, cap. 15,§2 and §3.

5.*Sämtliche Schriften und Briefe*, VI, 2, p. 264

6.*Ibid*., p. 266.

7.*Ibid*., p. 231.

8.*Ibid*., pp. 221-257.

9.*Ibid*., p. 257.

10.*Sämtliche Schriften und Briefe*, II, 1, p. 172.

11.*Ibid*., pp. 488-490

12. G. W. Leibniz, *Die philosophischen Schriften*, C. I. Gerhardt, ed., IV, p. 444; cf. p. 369.

13. Sämtliche Schriften und Briefe, II, 1, p. 508.

14. P. Costabel, *Leibniz et la dynamique* (Paris, 1960), p. 106.

15. G. W. Leibniz, *Mathematische Schriften*, C. I. Gerhardt, ed., VI, pp. 230-231. The manuscript called *Essay de dynamique* by Gerhardt is not earlier than 1698.

16. P. Costabel, *op. cit*., p. 105.

17.*Ibid*., p. 12.

18.*Sämtliche Schriften und Briefe*, VI, 6, p. 487.

19. L. Couturat, *op. cit*., p. 430.

20. Die *Philosophischen Schriften*, VII, p. 32.

21.*Principia mathematica*, *3.47.

22.*Die Philosophischen Schriften*, VII, pp. 228-247. Cf. L Couturat, *op. cit*., pp. 246-270.

23.*Sämtliche Schriften und Briefe*, VI, 1, p. 183.

24.*Die philosophischen Schriften*, VII, p. 301.

25.*Ibid*., IV, pp.447-448.

26.*Ibid*., VII, pp. 270-279.

27.*G. W. Leibniz, Textes inédits*, G. Grua, ed. (Paris, 1948), I, p. 287.

28.*Die philosophischen Schriften*, VII, p. 309. Cf. L. Couturat, *op. cit*., p. 525.

29. L. Couturat, *op. cit*., p. 515.

30.*Sämtliche Schriften und Briefe*, VI, 6, p. 230.

31.*Die philosophischen Schriften*, VII, p. 219.

32.*Ibid*., IV, p. 469.

33.*Ibid*., III, p. 645.

34.*Ibid*., IV, p. 447.

35.*Mathematische Schriften*, VII, p. 326.

36.*Die philosophischen Schriften*, II, p. 97.

37. Nicholas of Cusa, *De docta ignorantia*, Bk. II, ch. 7.

38.*Die philosophischen Schriften*, II, p. 304.

39.*Ibid*., IV, p. 483.

40.*Principes de la nature et de la grâce*, A. Robinet, ed., p. 27.

41.*Discours de métaphysique*, G. le Roy, ed., p. 50.

42.*Die philosophischen Schriften*, I, p. 416.

43.*Ibid*., II, p. 58.

44.*Mathematische Schriften*, VI, p. 236.

45.*Die philosophischen Schriften*, II, p. 372.

46.*Ibid*., IV, p. 482.

47.*Ibid*., p. 491. Cf. II, p. 450.

48.*Ibid*., p. 415.

49.*Ibid*., p. 401.

50.*Mathematische Schriften*, VII, p. 18.

51.*Die philosophischen Schriften*, VII, p. 364.

52.*Ibid*., II, p. 270. Cf. *Mathematische Schriften*, VI, p. 247.

53.*Mathematische Schriften*, II, p. 184.

54.*Die philosophischen Schriften*, VII, p. 404. Cf. IV, p. 444 and L. Couturat, *op. cit*., p. 594.

JÜrgen Mittelstrass

Eric J. Aiton

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