STUDIES IN LOGIC, GRAMMAR AND RHETORIC 1 (14)

Adam Drozdek

LEIBNIZ: STRUGGLES WITH INFINITY

In his insightful paper on Leibniz, Witold Marciszewski raises an interesting problem of reconciling finitism with infinity of nature^{1}. If nature is assumed to be infinite, then how our finite mind can explain anything, how the mind’s finite reasoning faculties can match the unboundedness of universe? It is a problem which Leibniz wrestled with all of his life and solved it by assuming infinity as the foundation of both scientific and philosophico-theological considerations.

1.

The concept of infinity is already used prominently in Leibniz’ juvenile Ars combinatoria (1666). Definition 1 says that “God is incorporeal substance of infinite power” and axiom four states that “every body whatsoever has an infinite number of parts” (Ars, L 73-74). Hence, the concept of infinity underlies the assumptions of his system and is used in all subsequent proofs. Importantly, the two senses of infinity are used: infinitely large and infinitely small. This understanding of infinity was far from new – both senses can be found at least since Anaxagoras.

Infinity was a concept assumed to be understood in Ars combinatoria and as such it was used in proving, among other things, God’s existence. Also, if only in passing, Leibniz used the concept of decreasing infinity when pronouncing in Aristotelian spirit that a continuum is infinitely divisible (L 74). As rightly observed by Kabitz, the statement on infinite divisibility of a continuum should be understood literally, otherwise the following proof concerning God would be unintelligible^{2}. But what are components of the

1 Witold Marciszewski, Why Leibniz should not have believed in ‘filum cogitationis’, “Studies in Logic, Grammar and Rhetoric” 12//13 (1993//94), 5-16.

2 Willy Kabitz, Die Philosophie des jungen Leibniz, Heidelberg: Carl Winter 1909 [reprint Hildesheim: Georg Olms 1974], p. 54.

Adam Drozdek

continuum? Are they themselves divisible? If yes, would this contradict Leibniz’ early atomism? Are these components dimensionless? These types of questions made Leibniz soon realize that a continuum is truly a labyrinth.

Infinity was also a link between the natural and supernatural. In The confession of nature against atheists, he admits that explanations in science should not constantly resort to supernatural causes; however, this should not mean that a reference to such causes is avoidable, and Leibniz shows that a natural body “is not self-sufficient and cannot subsist without incorporeal principle” (L 110). In his proof, which is “without obscurity and detours,” he refers to the chain of causes of motion, and the full reason of motion cannot be given if one body is considered a cause of motion of another body, since such a reply “will be followed by a question through all infinity” (L 111). Hence, an infinite regress is a mark of impossibility in giving a full, natural explanation. Also, the cohesion of the body can be explained, in the spirit of Democritus, by saying that atoms composing this body have hooks which hold the body together, but the hooks must be tenacious enough to enable this. “Whence this tenacity? Must we assume hooks on hooks to infinity?” (L 112). Hence, infinity is used here to demolish naturalist explanations. This infinity is assumed as obvious. However, infinity is worth studying in its own right, and natural sciences are not a proper tool for this study. Mathematics, on the other hand, is.

Leibniz: struggles with infinity

But even before his Paris period, Leibniz tried to come to grip with infinity, and probably the most serious attempt before 1672 was undertaken in Theoria motus abstracti (1671). The problem was that, as we observe in nature, each body and motion have a beginning and end in time and space. Also, as expressed in the first two principles, continuum has actual parts which are actually infinite (L 139). But if each interval can be infinitely divisible, then a beginning or an end of motion or a body would be impossible. Leibniz uses here, very unconvincingly, continued bisection of an interval as a proof, since such a bisection is supposed to leave us with nothing. Therefore, as expressed in the fourth principle: “there exist indivisibles or unextended entities.” Hobbes solved this problem by having the smallest magnitudes of time, space, and motion: conatus (tendency) was a motion taking place in the smallest imaginable space (punctum) and smallest time (instans). Leibniz refers here to Cavalieri, who said that there is a spirit more powerful than ours that can number elements of a continuum and thus isolate its constituent parts. We may suppose that this spirit would reach the level of indivisible atoms. However, these parts are indivisibles and yet they are not minimal, since these indivisibles have parts – in direct opposition to the notion of points in Euclidean geometry. So Leibniz also says in the third principle, “there is no minimum either in space or in body.” The troubling point that Leibniz addressed here was that if there were minimal parts (of magnitude zero), then “there are as many minima in the whole as in the part, which implies a contradiction” – a contradiction with the unspoken assumption that a part is always smaller than the whole^{3}.

In his discussion of indivisibles, Leibniz relied on their use by Cavalieri, or rather on the aura of reliability of mathematics in which Cavalieri applied this concept. Cavalieri, however, gives no explanation of the indivisible in his Geometria indivisibilibus (1635), and stating that certain non-Euclidian facts about the point are “obviously demonstrated” by Cavalieri (principle 5) only betrays that Leibniz knew Cavalieri’s method second hand^{4}.

In any event, Leibniz is convinced that by using this approach he is able to escape the labyrinth of continuum composition (e.g., to de Carcavy, June 22, 1671, AA ii i 126; to van Velthuysen, early May 1671, AA ii i 97), and his solutions have bearing not only on explaining natural phenomena, as exemplified in Theoria motus concreti, but also, and foremost, on psychological and theological issues. For example, conatus can last only for a moment – except in the mind, otherwise memory would be impossible. Using this concept, he defines body as a momentaneous mind (mens momentanea) (principle 17, also to Oldenburg March 11, 1971, AA ii i 90), which quite elegantly goes beyond Cartesian mind-body dualism^{5}. The nature of conatus, or, more generally, the nature of the indivisible, is also to be used as a launching pad for proving the immortality of the soul, the existence of God, and the defense of such mysteries of faith as the Eucharist (to Oldenburg, 1670, GM i 46, to Arnauld, GP i 71, cf. iv 225). For example, it can be said that “the mind (gemüth) exists in one point, so that it is indivisible and indestructible” (to Johann Friedrich, May 21,

3 Thus the reference in Ars combinatoria to Cardan’s statement that “one infinite is greater than another” can be considered spurious, all the more because this reference is removed from the revised version of this dissertation.

4 Probably from Hobbes’ De corpore 1.15.2; and De principiis 1 or from “Angelus’ preface to Digby’s Demonstration immortalitatis animae” (1664), cf. Joseph E. Hofmann, Leibniz in Paris 1672-1676. His growth to mathematical maturity, Cambridge University Press 1974, p. 8; Kurt Ufermann, Untersuchungen über das Gesetz der Kontinuität bei Leibniz, Berlin: Funk 1927, p. 27.

5 As pointed out by Rudolf Hahn, this definition of the body “seems to indicate that conatus should be understood as some spiritual force,” whereby a way for Leibniz’ later spiritualism is opened”, Rudolf Hahn, Die Entwicklung der Leibnizischen Metaphysik und der Einfluss der Mathematik auf dieselbe, bis zum Jahre 1686, Halle 1899, pp. 16, 20.

1671, AA ii i 108).

Philosophical analyses alone did not seem to resolve the problem of infinity. It was possible to escape the labyrinth of continuity because some assumptions were made about infinitely small quantities, hence the problem of infinity was solved only because infinity was accepted at the beginning. Even if it was not an actual infinity, at least an assumption was made that, actually or potentially, infinitely small quantities can be obtained, and, therefore, infinitistic thinking precedes the solution of the problem of infinity. Is there a better way out of the labyrinth? It is true, “the whole labyrinth about the composition of the continuum has to be unraveled as soon as possible... We must see whether it can be demonstrated that there is something infinitely small yet not indivisible,” jotted Leibniz down in his Paris notes, “from the existence of such a being there follow wonderful things about infinity” (Feb. 11, 1676, L 159). Interestingly, this note was made after some results of differential calculus had already been obtained. However, it is mathematics which should shed some light on this problem. After all, as Leibniz wrote, “it is not possible to get a thread through the labyrinth concerning the composition of the continuum or concerning the greatest (maximum) and the least (minimum) and the unnameable and the infinite unless geometry gets it; in fact, no one arrives at a sound metaphysics except the man who comes over to it by that way” (GM vii 326). But did mathematical analysis solve the problem of infinity?

In Nova methodus pro maximis et minimis (1684), his first publish account of differential calculus, Leibniz introduces basic rules of differentiation based on the definition of a differential – dy is a differential which is to some arbitrary dx, as ordinate y is to subtangent x; however, this definition relies on the definition of a tangent which is “a line that connects two points of the curve at an infinitely small distance or the continued side of a polygon with an infinite number of angles”, the polygon taking the place of the curve. “This infinitely small distance can always be expressed by a known differential like dy” (Struik 272, 276; GM v 220, 223). This is clearly a circulus vitiosus in definition which will plague Leibniz’ attempts to build a solid foundation for calculus – and the attempts of his successors until Cauchy. Also, it is interesting to observe that, at least in Nova methodus, differentials are finite intervals, and yet Leibniz refers in the definition of tangent to an infinite polygon and to points placed infinitely close to each other, which in turn is to substantiate differentials. Infinity thus precedes the finite; infinity is an assumption on which to build the foundation of the calculus. Although mathematically pregnant, Leibniz’ mathematical analyses did not solve the problem of infinity; at best they showed its depth and richness, at worst they indicated that without assuming it even mathematical problems concerning finite elements are unmanageable.

Leibniz did not build a solid mathematical theory of the infinite. He explored the foundations of mathematics as a philosopher and theologian and laid foundations for calculus which were developed in a mathematically much sounder fashion by Bernoulli brothers and de l’Hospital. But although Leibniz did not solve the problem of infinity, since he treated infinity as a datum, and did not find an escape from the labyrinth of continuum, mathematics, and especially his own contributions, made him keenly aware of the place of continuum in the whole of philosophical system.

2.

Leibniz never had any doubt about an orderliness of the universe and about God being its source. He says, for example, that “a beautiful order” arises in nature “because it is the timepiece of God” (to Thomasius, April 20 1669, L 101). Incidentally, he mentions some traditional expressions of the order in nature, one being that “nature strives for continuity.” He rejects them, since they smack of deism or pantheism, by attributing wisdom to nature, not God. In 1699, he passes lightly over this principle concerning continuity, but he returns to it later, when he does not ascribe it to nature but sees in it a foundation of the order of nature. This principle is the principle of continuity.

In earliest form it appears as an unnamed principle in a 1679 letter in the context of critiquing Descartes’ dynamics: “when causes differ from one another as little as you would wish, and so approach each other that one stops at another, so their effects approach each other also indefinitely so that the difference would become smaller than any assignable value, and one stops at another” (to Craanen, AA ii i 470). However, this principle was given for the first time in print in 1687 under the name of a principle of general order: “when the differences between two instances in a given series of that which is presupposed can be diminished until it becomes smaller than any given quantity whatever, the corresponding difference in what is sought or in their results must of necessity also be diminished or become less than any given quantity whatever” (GP iii 51; L 351)^{6}. Hence, the effects

6 If this wording is expressed in a formal fashion, as in: if for any x and y, |x−y|→0 then |f(x)−f(y)|→0, then we see that the principle of continuity resembles very closely the definition of uniform continuity of function f.

are ordered since the causes are ordered. Predictable behavior of causes also makes effects predictable. Orderliness is both in causes and effects, whereby chaos is excluded and any semblance of chaos in nature and society is only what it is – a semblance, an appearance, a result of insufficient knowledge, of missing data, or of imperfect perspicuity of the observer.

Continuity is built into the world as a guiding principle which enables the world to develop, to progress. “The present is full of the past and is pregnant with the future” (to Arnauld, July 1686), and in Alexander the Great we can find “marks of all that had happened to him and evidences of all that would happen to him” (Discourse §8). The future is fully predictable, although only God can do this. The history of the world in its entirety and of each substance individually hides no surprises for God, for the infinite God, for the God of order, who – because of his perfection – could not create the world other than perfect, that is, orderly, and the primary means of ensuring this orderliness is the infinity of the world and the principle of continuity that rules in it. God would not have been almighty if the world he created had not been orderly; it would not have been orderly if the principle of continuity had not been the principle of order whose manifestation is a part of each substance. The principle of continuity could not have worked if the world had not been infinite; therefore, the world has to be infinite. Infinity is more perfect than the finite; true finitude is a distortion of the infinite. The infinite is and has to be primary in metaphysical and epistemological order. The finite is a mark of imperfection and everything is as imperfect as it is finite.

The validity of the principle of continuity is also acknowledged in biology. Since no jumps should occur in nature, one can expect transitions between species to be blurred to the extent that the dividing line between plants and animals does not exists (to Bourguet, Aug. 5, 1715, L 664; NE 4.16.12). Also, there is no dividing line between life and death. Death is infinitely small life^{7} so as rest is infinitely small movement or equality is infinitely small inequality. “Generation and corruption are nothing but transformations from small to great or the reverse,” and because “all matter must be filled with ... living substances,” yet “there is no particle of matter which does not contain a world of innumerable creatures”; a ram burned for offering is transformed into another form, like a caterpillar into a butterfly, and not annihilated (to Arnauld, October 9, 1687, L 345-7; also, L 455, 557). In Pacidius Philalethi, Leibniz uses another analogy: nature is like

7 This view is by means new; a comentator says that according to Psalm 143:3 “death is thought of as a gradual ebbing of life which continues even in the grave,” J. H. Eaton, Psalms: Introduction and commentary, Bloomsbury: SCM Press 1967, p. 307.

a tunic or a shell with an infinite number of folds, and these folds are also folds. Folds never become flat (AA vi iii 535). This principle can be found today in the theory of fractals. Fractals repeat themselves indefinitely and any level of magnification reveals the same pattern as the one the process of generating the fractals has begun.

Psychology is not free from the rule of the continuity principle either; in particular, our perceptions are under this rule. Each perception is a ground of the next perception; the sequence of perceptions has no gaps. Although the number of perceptions is infinite, we cannot – because we are finite beings – be conscious of all of them at the same time. But thereby they do not disappear. They still exist as “little perceptions,” as the domain of the subconscious^{8}. Conscious acts are only a proverbial tip of the iceberg, the iceberg itself being the subconscious acts. Consciousness exists thanks to subconsciousness; subconscious acts constitute a glue for conscious acts without which the latter acts would be a disordered and hence incomprehensible heap of mental events. So, the infinity of the world and the infinity of our mental structure in conjunction with the principle of continuity leads to the discovery of the subconscious, since the conscious cannot by itself bear the burden of the infinite^{9}. Had the subconscious not existed, the conscious would have collapsed. The orderliness of cognition is due mainly to the underlying current of the subconscious events. Only in God is there no need for the subconscious, but it takes an infinite supreme being for this to be possible.

The principle of continuum can be also used as a methodological guideline: If there exist two different events, then we are bound to find an event that mediates them, either synchronically (e.g., a new biological genus between two existing genera) or diachronically (e.g., finding a historical event between two other events). This is possible since, in Leibniz’ system, there are two kinds of continuum, one being an ideal continuum and one being its faint reflection in the real world^{10}. Ideal continuum has no parts, or rather its parts are indeterminate; it is infinitely divisible, but not divided. The actual continua are the aggregates of substances.

8 In the words of Ufermann, this understanding of the subconscious was a trump-card Leibniz used against Locke’s psychology, op. cit., p. 66.

9 Leibniz’ psychology “stems from the infinity and the continuum”, Ufermann, op. cit., p. 72.

10 Hermann Weyl writes about perceptual and mathematical continua (1918), Friedrich Kaulbach writes about abstract and concrete continua (Das Prinzip der Kontinuität bei Leibniz, in P. Schneider, O. Saame (eds.), Das Problem der Kontinuität, Mainz: Krach 1970, p. 12), but cf. Herbert Breger, Leibniz, Weyl und das Kontinuum, in Leibniz. Werk und Wirkung. IV Internationaler Leibniz-Kongreß. Vorträge, Hannover: Gottfried-Wilhelm-Leibniz-Geselschaft 1983, pp. 80-81.

The world’s continuum would be the same as a continuum defined in a standard set theory according to which the continuum is a dense and ordered collection of points. The ideal continuum, although much less frequently used in today’s mathematics, can still be very rigidly defined, as exemplified by Brouwer’s and Hermann Weyl’s intuitionistic mathematics or Paul Lorenzen’s constructivist approach.

The ideal continuum is a top-down entity, which encompasses an infinite number of indefinite parts. The continuum itself precedes the parts, the subcontinua. The real world continuity is bottom-up, it is being produced from an infinity of elements to result in a dense set. A similar distinction is true also about infinity. A true infinity, a perfect infinity, which can be found also in God, is characterized by its wholeness, by “being anterior to all composition,” by “not being formed by the addition of parts” (NE 2.17.1). Infinity of points or of instants, on the other hand, is an infinity of a lesser kind. It is like an infinity of numbers which is an accumulation or aggregate of numbers, “not a whole any more than the infinite number itself, whereof one cannot say whether it is even or uneven” (Theodicy §195). In this sense, an actual infinity encountered in the world is less perfect than the ideal infinity found in God and in his ideas, the infinity which contains parts only potentially.

Both real and ideal worlds are continuous, but continuity is understood differently. Therefore, the difference is made not between discrete and continuous, but between two kinds of continuity. This should be clear from the statement that “all repetition is either discrete as where parts are discriminated ... or continuous when the parts are indeterminate and can be assumed in infinite ways” (1702, GP iv 394). Hence, the ideal continuum is undiscriminating, the actual is discriminating. We can describe matter as discrete, meaning thereby not a discrete set but a set composed of parts, and each part is, to be sure, different than another part, and in this sense discrete or discriminate. So the gap between the ideal and actual worlds is not unbridgeable, since continuum can be found in both of them^{11}.

“Uniformly ordered continuity, although it is nothing but supposition and abstraction, forms the basis of the eternal truths and of the necessary sciences ... Matter appears to us [to be] a continuum, but it only appears so, just as does actual motion” (to princess Sophia, Nov. 30, 1701, GP vii 564).

11 Therefore, the is no need to limit the number of objects in the world to the countable. Even if the set of objects in the world were not dense, but discrete, it would not imply that “there are at most countable number of objects in the universe”, as claimed by Monika Osterheld-Koepke, Der Ursprung der Mathematik aus der Monadologie, Frankfurt: Haag u. Herchen 1984, p. 80, since the number of discrete objects can well surpass any cardinality (the sequence of all cardinals being a good example).

Matter has an appearance of ideal continuity since it is already actually continuous. And so does motion. It is like watching a movie that invokes in us an illusion of continuous motion although the movie is just a sequence of discrete frames shown at the rate of 24 frames per second. However, the actual motion does not have 24 frames per second but a dense set of frames. Motion so understood arouses in us an impression of continuity. But this is only continuity of a lesser kind. Our reality is perceived as truly continuous since continuity is within us, in our minds, as an abstraction, as an idea. “Space, just like time, is a certain order ... which embraces not only actuals, but possibles also” (to des Bosses 1709, GP ii 379, also 336). “Space is something continuous but ideal” (ib.), or rather it is continuous because it is ideal. What is actual, can only be, so to speak, a breakable continuity, since true continuity would defy the reality of the phenomenal world. “In the real world, if matter were not divided up, there would be no distinct things” and this actual division and discrimination presupposes simple substances (to de Volder, GP ii 276). The order in the real world can be preserved if the mind sees the world through the mirror of continuity, if the law of continuity is assumed to be effective. The mind requires perfect continuity, and yet the nature of the world refuses it. In this, atomism was never truly abandoned by Leibniz, since to create real unities, a “real and animated point, or an atom of substance” is needed (Système nouveau (1695), L 460 note 3).

These atoms of substance are the only true realities since even matter has a borrowed reality. Matter is a phenomenon and is something between the mental things and the real things. There is an infinite number of these things, and hence, matter can be infinitely divisible. At the bottom of this division, which we never reach, there are real things. In the order of nature, these things constitute matter; these atoms of substance are primary, and only thanks to them matter emerges as a phenomenon. In the realm of mental things, continuum, a whole, is primary, and parts are secondary. These parts are in potentia; they are indefinite.

These atoms of substance also indicate that the two kinds of continuum are not separate entities, real continuum being an approximation of the ideal continuum. We should remember that the monad is without windows, and as such its knowledge comes from within, it is inscribed, if only in the form of little perceptions, in the monad itself. Leibniz ascribes actual infinity even to the smallest elements as mirrors of the universe. “The present is full of the past and is pregnant with the future” also in the form of the knowledge the monad possesses, so that the knowledge of the present is full of the past knowledge and is pregnant with the future knowledge^{12}. That is, ideal continuum is within us; it is inscribed in us. The atoms of substance which constitute a continuum of substances are based on the ideal continuum; the ideal continuum is within them. So a real continuum is an infinite and dense aggregate of entities, each one of them including ideal continuum. Continuum within continuum, infinity within infinity. It is somewhat analogical to an infinite set of real numbers from the interval (0,1), each number defined as an infinite sequence of digits following zero and the decimal point. Or, in Brouwer’s mathematics, it is similar to an infinite set of points, each point being defined as an infinite and converging sequence of rectangles. In fact, a similar approach is also found in Leibniz, for whom the irrational numbers are an infinite series of rational numbers (Nova algebrae promotio, GM vii 156, v 308), whereas for Descartes they are segments of a line (hypotenuses).

It can be claimed now that such understanding of irrationals is a predecessor of the monad concept. The concept of the irrational incorporates the concept of infinity, and so does the concept of a windowless monad which possesses the concept of infinity and continuity. This influence of mathematics on philosophical concepts is magnified by an impact of the infinitesimal concept on that of the monad. An infinitesimal also included an infinity, as exemplified already in Theoria motus abstracti. Infinitesimal can be smaller than any number, hence smaller than an infinity of numbers. Consequently, although implicitly, infinity is included not only in the name of infinitesimal but also in its concept. In this sense we may agree that infinitesimal “became a foundation of the world in the concept of monad”^{13}.

Although the real world is not truly continuous, it is nevertheless infinite, and so is the ideal continuum. The infinity constitutes the link between the real and ideal worlds, and it allows us to apply the continuity principle in our world as well. Infinity can be ordered or disordered, it can be continuous or discrete; therefore, infinity by itself is an insufficient key to the universe. An order has to be added to it, and the order is supplied by the continuity principle, by the principle of general order. This order, the continuous order, on the other hand, would be impossible if it were not for an underlying infinity since finite universe cannot be ordered in the sense required by this principle. After all, the principle of continuity “has

12 Cf. Kaulbach, op. cit., p. 14: The fact that each being “reflects in itself the totality of the world with all its content” was to be an exit from the labyrinth of continuum.

13 Miodrag Cekić, Infinitezimalni račun i monadologia, Niš 1980, p. 49; cf. review by Radmila Šajković in “Studia Leibnitiana” 13 (1981), pp. 151-154.

its origin in the infinite” (1687, L 351), and consequently in God, because only he is truly infinite. God’s infinity is, therefore, poured into the world’s infinity. Our human task is exploration of the world. This exploration is the means of glorifying God through recognizing his greatness in the greatness of the universe. This exploration requires sharp tools, and God’s means of creations indicate what tools they should be. One tool necessary for exploring the world is infinity; thus, the better we know the latter, the fuller our knowledge can be. Therefore, an exploration of infinity in its own right has its merit. Furthermore, since God “acts as a perfect geometrician” (L 351), since it is true that “God uses geometry and that mathematics makes up a part of the intelligible world,” mathematics “is therefore more fit to be an entrance into it” (1702, L 585) and this knowledge of the world, both real and ideal, can be acquired best through geometry, or, more generally, mathematics.

Such a stance is, by the way, a break with the tradition of Cartesianism. For Descartes, infinity was a sacred property ascribable only to God. Mathematics was separate from theology; it investigated its own world of numbers and figures; therefore, infinity, as a sacred property, did not belong to mathematics. Thus, because Descartes did not consider infinity to have a legitimate place in mathematics, he did his best to avoid using it in his proofs. Occasionally, Descartes showed that he can be quite proficient with the use of infinity in proofs (cf. his proofs of de Beaune problem), but such proofs were to him inadmissible. For Leibniz, mathematics was an extension of theology, a field allowing for deeper investigation and gaining better understanding of concepts that theology also may ponder upon, in particular, the concept of infinity.

However, Leibniz agrees with Descartes in the priority of the idea of infinity in epistemological order. According to Descartes, because “there is more reality in an infinite substance than in finite substance,” “there is in me somehow in the first place understanding of the infinite before the finite, i.e., [understanding] of God before myself” (third Meditation, AT vii 45). This principle was not expressed more explicitly by anybody before Descartes; consequently, it can be called Descartes’ principle^{14}.

A similar thought is also found in Leibniz: Infinity can be understood as an infinity that unfolds itself, as in numerical series. But it can also be understood as a complete whole, a positive infinity. “The positive infinity is nothing but the absolute,” hence we have an idea of positive infinity, and

14 Adam Drozdek, Descartes: Mathematics and sacredness of infinity, “Laval théologique et philosophique” 52 (1966), 167-178.

“this [idea] precedes the [idea] of the finite” (Sur l’Essay de l’entendement de Monsieur Locke, GP v 17). “The idea of infinity does not come from stretching the finite ideas” (NE ii 23). We have an idea of this infinity, since we know about the absolute, because we simply participate in it, and thereby we possess some measure of fullness (Entretien de Philarete et d’Ariste, GP vi 592).

In the light of Descartes’ principle and of the infinity built into each monad, we have to disagree with the statement that according to Leibniz, “our finite mind can take it [infinity] only as a sign abbreviating some operation which is impossible for us” to conduct^{15}. If infinity played only this symbolic role in the human mind, then such an infinity would not be infinite any longer, all the more that an operation for which it stands is undoable. If it is reduced to a mere sign, then no cognition is possible since finite can be understood only in the light of the infinite. However, because the “intelligible world” of ideas “is in God and in some way also in us” (L 585)^{16}, there is much more to the concept of infinity than a symbolic side. Infinity is in us not only as an idea, but also in the form of sequence of perceptions. Furthermore, if we refer to the latter, then this gives us a meaning in which infinity can be understood as a sign, namely as a blind or symbolic intuition (e.g., Meditationes de cogitatione, veritate et ideis 1684, L 291-292). We are not aware of everything at the same time, but this does not preclude the subconscious area of thinking from existence. Infinity is very real in us although, because the mind is finite, it cannot make the full use of infinity. But it is indispensable, otherwise, by Descartes’ principle, no cognition would be possible. We can do a great deal with it, and calculus is but one proof of that.

Descartes’ principle also has a methodological offshoot in Leibniz. To him, an entity can be perfectly distinguished from another entity if it is completely described, but such a description is nothing short of an infinite list of characteristics. “Individuality involves the infinite, and only he who understands the latter can have first-hand knowledge of the principle of individuation of this or that thing; this arises from the influence of all things in the universe on one another” (NE 3.3.6). Full description of an entity requires giving all relationships of this entity with everything else in the

15 Gilles-Gaston Granger, Philosophie et mathématique leibnizienne, “Revue métaphysique et morale” 86 (1981), p. 29.

16 Cf. also “insofar as our intellect is a reflection of his, we may say that God has an intellect similar to ours and that God understands things as we do; but there is this difference, namely, that he understands them simultaneously in an infinite number of ways, but we only in one” (1676, AA vi iii 400, cf. p. 523).

world, this world which is infinite. In this way, infinity again has to precede the finite.

3.

The real world is but one realized possibility; the real is a manifestation of the possible. The possible precedes the real, and in this sense the possible can be considered more real than the real itself. All possibilities are in God’s mind, and although they become instantiated to be real, real to us, they are real as ideas in God’s mind, they are not illusions or figments of an imagination. The law of continuity can be fully applied to the ideal, and although it is not fully applicable to the actual, it is not thereby suspended; only its manifestation is limited, and in our eyes it may appear to have limited validity. This law is as valid in the ideal as in the real, but because the real is just a carved-out portion of the immensity of the possible, manifestation of this law is not full; it is also carved out^{17}. After all, it is “quite true ... that the existence of intelligible things ... is incomparably more certain than the existence of sensible things and it would thus not be impossible ... that there should exist at bottom only intelligible substances, of which sensible things would be only the appearances. Instead, our lack of attention causes us to take sensible things for the only true ones” (1702, L 549).

“There is an intelligible world in the divine mind,” “the region of ideas.” Minds “are produced as images of divinity. The mathematical sciences, which deal with eternal truths rooted in the divine mind, prepare us for knowledge of substances” (1707, L 592). Thus, although it is true that “spatial concepts have become prior in knowledge to one’s concept of bodily extension,” this, in Leibniz’ view, cannot be reconciled with the view that “bodily extension is metaphysically more fundamental than space”^{18}. The order of the mental is prior to that of the material; consequently, the concept of space and time, i.e., of continuum, precedes the concept of extension and duration. “[T]he seeds of the things we learn are within us – the ideas and the eternal truths which arise from them.” The innate ideas are much to be preferred over the concept of tabula rasa (1707, L593), and Plato’s reminiscence thesis is “a well-founded doctrine” (Discourse 26). Abstraction

17 “The ideal is inherent both to the possible and the actual – so far as the latter can be considered the possible”, Alexandru Giuculescu, Der Begriff des Unendlichen bei Leibniz und Cantor, in Leibniz 1983, p. 880.

18 Glenn A. Hartz and J. A. Cover, Space and time in the Leibnizian metaphysics, “Nous” 22 (1988), p. 511.

of space and time, therefore, does not consist of creating these concepts from observations of extension and duration. The latter may sharpen the former, may make us realize their existence, but not create them. They may accompany them since “there is never an abstract thought which is not accompanied by some images or material traces” (1702, L 556; also 551), but in no wise are they “metaphysically more fundamental.”

Although “continuity is something ideal ... the real never ceases to be governed perfectly by the ideal and the abstract. ... This is because everything is governed by reason” (1702, L 544). The abstract determines the structure of the universe, of the possible, and of the real. The essence of being can be found in the abstract, hence the abstract is the basis of all its possible instantiations, including the real world^{19}. Without reason the world would not exist; the abstract precedes its manifestations in chronological and metaphysical order, and therefore it can be considered more real than reality itself. The abstract encompasses infinity and is embodied in an infinite number of possibilities.

Leibniz is careful in distinguishing these two different modes of reality. God’s understanding, the source of essences, contains ideas of possibilities, whereas his will, which chooses the best possible world, is the domain of existences (Theodicy §7). The modes of existence of possibilities and their actualizations are different, but both possibilities and existences are nevertheless real.

What is thus the difference between these two realities? “[T]here can be nothing real in nature except simple substances and the aggregates resulting from them” (1706, L 539). Having discriminate parts is the distinctive feature of this world of ours. On the other hand, “continuity is something ideal”; there is nothing perfectly uniform in nature (1702, L 544). Our reality is the realm of the discrete, the continuous is something ideal and is related to the possible, since it is indefinite and indeterminate, whereas there is nothing indefinite in actual things (to de Volder, Jan. 19, 1706, L 539). But although true continuity cannot be found in this world, the principle of continuity is applicable in it, since infinity permeates both the world of ideal and the world of existent and makes both these worlds real. So, for example, aggregates can have any number of objects, hence the number of objects in the world is infinite, and there is an infinity of planets like ours in the universe (Theodicy 2.19). Infinity is a platform on which these two

19 Cf. Osterheld-Koepke, op. cit., 77: “The abstract being is the structure of being, the structure of both possible and real being; it includes all possibilities of thought. So it is not independent, but it is an essence of being. The abstract being can recur in infinitely many appearances.”

worlds meet; it enables possibilities to be realizable as existences, it enables the existences to be considered as emerging from possibilities. Therefore, “the knowledge of the continuous (scientia continuorum) ... contains eternal truths which are never violated by actual phenomena, since the difference is always less than any given assignable amount” (1706, L 539); this situation would have been impossible had the world been finite.

Infinity is also a platform enabling an order in these two worlds. First, the realms of the possible and the actual are ordered through continuum; and the principle of continuity expressly states the nature of this orderliness of continuum. A harmony is only possible through continuum, and hence, through infinity. Continuum is order; it is harmony^{20}. So, for instance, “space and time taken together constitute the order of possibilities of the one entire universe.” Because the actual is secondary to the possible, it cannot break out of the laws to be found in the realm of the possible, and “the actual phenomena of nature are arranged, and must be, in such a way that nothing ever happens which violates the law of continuity” (L 583). The world could not exist if it were not ordered, i.e., if it were not under the rule of the law of continuity. However, the world can be under the rule of this law only because the actual world is infinite, and so is the world of the possible. Infinity is an underlying assumption of this orderliness: there can be an unordered infinity, but not a non-infinite continuum. Infinity is, so to speak, a foundation upon which the order worthy of the infinite author can be built. The world exists because it comes from an author of order; it exists because it is ordered, and because it is infinite. Infinity of continuum enables this order, and since the actual world is not continuous, the order is ascertained through its infinity. An infinite author, God, is its source, and its infinity is the surest link between the world and God. Actual infinity affects nature everywhere “to better mark the perfection of its author. I also believe that there is no part of matter which would not be, I don’t say divisible, but actually divided, and consequently the smallest particle should be considered as a world full of an infinity of diverse creatures” (to Fouchet, 1693, GP i 416).

4.

What is the result of Leibniz’ investigations as far as the concept

20 “The activity of God’s reason established an order as continuity”, Rainer Piepmaier, Aspekte der Erinnerung bei Leibniz, in Leibniz 1983, p. 601.

of infinity is concerned? The concept led him from philosophy to mathematics and then back to philosophy. However, these philosophical and scientific peregrinations did not subdue the problem of infinity. Leibniz for a very good reason was very proud of his new calculus. This is reflected, among other things, in terminology he uses. He says namely in A new method for maxima and minima that his method “also covers transcendental curves.” As remarked by Struik, “this may be the first time that the term “transcendental” in the sense of “nonanalytical” occurs in print^{21}. Leibniz’ distinction between transcendental and algebraic curves corresponds to Descartes’ division of curves into mechanical and geometrical, but for Leibniz, unlike for Descartes, mechanical lines can be analyzed in mathematics. By naming mechanical curves transcendental, Leibniz alludes to scholastic tradition in which the concept of infinity is transcendental, since it surpassed man’s powers. From the fact that transcendental curves and functions are those for which no finite number of algebraic operations suffices, and from the fact that Leibniz’ calculus can grapple with these lines, a seemingly inescapable conclusion may be drawn, to the effect that Leibniz tamed infinity. This, however, is not the case, since the concept of infinity underlies Leibniz’ definitions as already indicated in this paper. Infinity cannot be elucidated by mathematical means unless some knowledge of the infinite is assumed. And this fact may have been one of the most important realizations to which Leibniz was led by his mathematical investigations. Infinity is unconquerable with finite means. Some understanding of infinity must precede any attempts to build an axiomatic system, and this understanding is simply a given, a part of human cognitive apparatus.

This turned out to be true in philosophy and even more so in theology. His struggles with the labyrinth of continuity could be crowned with some measure of success only if infinity underlay all his attempts. In this he is an heir of Descartes by assuming more or less explicitly that infinity is clearer than the finite, and even grappling with the finite must assume some insight of the infinite. Hence, Descartes’ principle is always used by Leibniz.

Two most difficult problems for Leibniz were the composition of a continuum and the problem of freedom. Leibniz recognized the source of this problem, which was infinity. This is an interesting tie to Kant who looked with wonderment at two things: the starry sky above him and moral law within himself. Leibniz was puzzled by the infinitely small and by

21 D. J. Struik (ed.), A source book in mathematics, 1200-1800, Cambridge: Harvard University Press 1969, p. 271, note 8. Cf. also Wilhelm Wundt, Leibniz. Zu seinem 200jährigen Todestag, Leipzig 1917, p. 30.

morality. Kant wondered about the infinitely large and also about morality. Both of them were concerned with infinity and the moral dimension of man. Philosophically, they have Pascal as an intermediary, for whom the human being, a trembling reed, is suspended between two infinities. Infinity is a part of human condition wherever human beings lay their eyes. Infinity cannot be conquered; if it can, then only partially and only because we are saturated with infinity, only because Descartes’ principle is at work, only because it is the foundation of all.

Sources

AA – Gottfried Wilhelm Leibniz Sämtliche Schriften und Briefe, herausgegeben von der Preussischen (jetz Deutschen) Akademie der Wissenschaften zu Berlin, 6 Reihen, Darmstadt 1923, Leipzig 1983, Berlin 1950 – (quoted as volumne, part, page).

AT – Oeuvres de Descartes, ed. C. Adam and P. Tannery, 12 vols. Paris: Vrin//CNRS 1964-1978, Revised edition – (quoted as volumne, page).

GM – Leibnizens Mathematische Schriften, 7 vol., ed. C. I. Gerhardt, Berlin 1875-1890, (repr. Hildesheim 1962) – (quoted as volumne, page).

GP – Die Philosophischen Schriften von G. W. Leibniz, 7 vol., ed. C. I. Gerhardt, Halle 1849-1863 (repr. Hildesheim 1965) – (quoted as volumne, page).

L – Gottfried Wilhelm Leibniz – Philosophical Papers and Letters, ed. L. E. Loemker, 2nd ed. D. Reidel Publisching Company Dordrecht 1969.

NE – G. W. Leibniz, New Essays on Human Understanding, transl. and ed. P. Remnant and J. Bennet, Cambridge University Press, Cambridge 1981 – (quoted as book, chapter and paragraph).