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What Does God Know but can’t Say? Leibniz on Infinity, Fictitious Infinitesimals and a Possible Solution of the Labyrinth of Freedom

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Abstract

Despite his commitment to freedom, Leibniz’ philosophy is also founded on pre-established harmony. Understanding the life of the individual as a spiritual automaton led Leibniz to refer to the puzzle of the way out of determinism as the Labyrinth of Freedom. Leibniz claimed that infinite complexity is the reason why it is impossible to prove a contingent truth (in a finite number of steps). But by means of Leibniz’ calculus, it actually can be shown in a finite number of steps how to calculate a summation of infinite parts. It appears that the analogy Leibniz drew between the mathematics of infinite series and the logic of contingent truths did more harm than good. A solution consistent with Leibniz’ perception of infinity is proposed. Alongside the existence of the aforementioned analogy, it is based on a disanalogy between the mathematics of infinite series and the logic of infinitely complex truths. This is a disanalogy that Leibniz had already used to solve the Labyrinth of Continuum which he declared more than once could, like the Labyrinth of Freedom, be solved by means of the nature of infinity. The solution of both labyrinths is based on the fictitiousness of the infinitesimal.

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Notes

  1. 1689?, On Freedom, FC 179, AG 95.

  2. 1686, Discourse on Metaphysics §13; 1714, The Monadology §46, §§54–55; 12 Feb. 1716, Leibniz to Clark, 5th letter §9, G VII 390, L 697). Indeed, Leibniz’ basic principle that “nothing is without a reason” (1686?, Primary Truths, C 519, AG 31) is related to the analytics of truth – and as such it is relevant to both the necessary and contingent truth (Couturat 1994 (1902), 3; Parkinson 1995, 207–208). But it appears that without the contingent domain, it is doubtful whether Leibniz would have bothered mentioning the principle of sufficient reason in relation to necessary truth.

  3. Hobbes, De Corpore (1966, vol. 1), Book I, ch. 3.

  4. 1686, Discourse on Metaphysics §8, G IV 433, AG 41; August 1677, Dialogue, G VII 193, L 185.

  5. “[A]lways, in every true affirmative proposition, necessary or contingent, universal or particular, the concept of the predicate is in a sense included in that of the subject [praedicatum insest subjecto]; or else I do not know what truth is. Now, I do not ask for more of a connection here than that which exists objectively [a parte rei] between the terms of a true proposition […] since there must always be some basis for the connection between the terms of a proposition, and it is to be found in their concepts.” (4/14 July 1686, Leibniz to Arnauld, letter X, G II 56, LA 63-64). As a result, Bertrand Russell and Louis Couturat claimed early in the twentieth century that Leibniz’ overall philosophy should be viewed on the basis of his logic, i.e., the logical nature of a true proposition (Couturat 1994 (1902), 2–8; Russell 1900, 40–53). Many have criticized Russell and Couturat’s sweeping claim, but no one would disagree that Leibniz never changed his mind about the the crucial importance of his predicate-in-notion principle.

  6. August 1677, Dialogue, G VII 192-193, L 184-185. The relation that Leibniz is referring to in understanding truth is not the extensional relation with which one can sort concepts into groups and types but rather an intensional relation by means of which one can assemble concepts. According to Leibniz, he preferred the intensional approach since he sought to deal with concepts in isolation from the existence of things (April 1679, Elements of a Calculus, C 53, PLP 20). The intensional approach, which deals with concepts without regard to the number of their practical examples, is consistent with Leibniz’ demand that the proof of the possibility of a concept will serve as a necessary (though not sufficient) condition for its actual realization, i.e. its existence. From a more general perspective, one can see the reason for preferring the intensional viewpoint of Leibniz as related to his logical foundation of the metaphysical harmony between infinitely many things in the world and his perception of rationality as completely analytical (Yakira 1988, 31).

  7. 4/14 July 1686, Leibniz to Arnauld, letter X, G II 55, LA 62; 1714, Monadology, §43.

  8. Nachtomy 2007, 25.

  9. 1679–1686, An Introduction to a Secret Encyclopaedia, C 513-514, trans. by Parkinson 1966, xxvii-xxviii, MP 7-8; cf. 1682–1684, On the Elements of Natural Science, LH XXXVII, vi, 3, L 283; Nov. 1684, Meditations on Knowledge, Truth, and Ideas, G IV 425, AG 26.

  10. 1679–1686, An Introduction to a Secret Encyclopedia, C 518, trans. by Parkinson 1966, xxix.

  11. 1686, General Inquiries about the Analysis of Concepts and Truths, C 358, PLP 49.

  12. 1689? On Freedom, FC 179, AG 95.

  13. Leibniz identified infinite analysis as an expression of non-provability of contingent truth. However, Blumenfeld argues that the inability to complete the proof (due to its infinite length) is not a reasonable solution to the contingency problem. Assuming that the analytical cannot necessarily be demonstrated, it is possible to come to one of the following conclusions: One can conclude—as Leibniz did—that there is contingent analytics (which cannot be demonstrated), but it is absolutely possible to simply think that there is a necessary truth that cannot be demonstrated. Leibniz did not justify why he began from the assumption that the necessary can be demonstrated with certainty and therefore the analytical is the source for the contingent, instead of presenting a much more reasonable argument: that not all necessary truths can be demonstrated (Blumenfeld 1985, 499–500). To this I would respond that Blumenfeld’s proposition sounds reasonable only after the famous proof of Gödel that necessary truths in mathematics are not provable (or that an infinite number of steps is required to complete such a proof). It seems to me that there is no point in expecting that Leibniz would hold such a position when at the beginning of the twentieth century mathematicians such as Hilbert still believed that one could prove any true argument in mathematics with a finite number of steps. Even if the alternative chosen by Leibniz is equivalent to that which he rejected, we must refrain from anachronistically viewing the dilemma which he faced (see Lison 2006–7).

  14. 1686?, Primary truths, C 518-519, AG 31.

  15. 1689?, On Freedom, FC 179, AG 95. Starting from the second half of the 1680s, and until the end of his career, Leibniz was consistent in the use of infinity to define contingent truths. See, for example, 5 August 1715, Leibniz to Bourguet, G III 581, L 664.

  16. 1679–1686?, A Specimen of the Universal Calculus, A 6.4 280-289, G VII 200, W 98-99.

  17. 1686, Necessary and Contingent Truths, C 17-18, MP 97.

  18. “It is perfectly correct to say that there is an infinity of things, i.e. that there are always more of them than one can specify. But it is easy to demonstrate that there is no infinite number, nor any infinite line or other infinite quantity, if these are taken to be genuine wholes. The Scholastics were taking that view, or should have been doing so, when they allowed a ‘syncategorematic’ infinite, as they called it, but not a ‘categorematic’ one. The true infinite, strictly speaking, is only in the absolute, which precedes all composition and is not formed by the addition of parts.” (1709, New Essays, book II, ch. xxvii [of infinity], NE 157. Also: 2 Feb. 1702, Leibniz to Varignon, GM IV 93, L 543; 1 Sep. 1706, Leibniz to Des Bosses, G II 314, LR 53. For a logical formulation of the argument, see: Arthur 2001b, 107; Levey 2008, 109; Arthur 2015, 145–146).

  19. However, there is no perfect symmetry between large infinity and small infinity. Only large infinity leads to a problem of the whole being equal to its parts (the infinite number is the number of all numbers and therefore constitutes a part of itself since it is included, along with all the other numbers, within the number of all numbers, which is identical to it). Such a contradiction does not exist in the definition of an infinitesimal number and this is perhaps the reason why Leibniz does not prove that the infinitesimal is not possible. As shown by Bassler, Leibniz indeed provided a proof that the infinitesimal is not possible but never came back to it. This proof is based on the fact that an infinite size cannot be limited (Bassler 2008, 146–148). Therefore, it does not relate directly to the infinitesimal as an impossible size. Levey claims that Leibniz never proved that the infinitesimal is not possible even though he declared the existence of such a proof in his correspondence with Bernoulli (“As concerns infinitesimal terms, it seems to me not only that we cannot penetrate to them but that there are none in nature, that is, that they are not possible. Otherwise, as I have already said, I admit that if I could concede their possibility, I should concede their being.” 18 Nov. 1698, Leibniz to Bernoulli, GM III 551, L 511). Although Leibniz already proved at the end of 1672 that a minimal size is not possible (by proving that a line cannot be composed of undivided points), in that same correspondence with Bernoulli he also emphasized that ruling out a minimum is not the same as ruling out an infinitesimal: “I admit that the impossibility of our infinitesimals does not follow directly from this [...] since a minimum is not the same thing as an infinitesimal“(1699, Leibniz to Bernoulli, GM III 535-536, Levey 1998, 56 n7).

  20. “Although it is not at all rigorously true that rest is a kind of motion or that equality is a kind of inequality, any more than it is true that a circle is a kind of a regular polygon, it can be said, nevertheless, that rest, equality, and the circle terminate the motions, the inequalities, and the regular polygons which arrive at them by a continuous change and vanish in them. And although these terminations are excluded, that is, are not included in any rigorous sense in the variables which they limit, they nevertheless have the same proportions as if they were included in the series, in accordance with the language of the infinities and infinitesimals, which takes the circle, for example, as a regular polygon with an infinite number of sides. Otherwise, the law of continuity would be violated, namely, that since we can move polygons to a circle by a continuous change and without making a leap, it is also necessary not to make a leap in passing from the properties of polygons to those of a circle.” (Jan. 1701, Justification of the Infinitesimal Calculus by that of Ordinary Algebra, GM IV 106, L 546 (my bold); cf. 1701?, Cum Prodiisset, Child 148-149). See Ishiguro 1990, 79–100.

  21. “The differential calculus could be employed with diagrams in an even more wonderfully simpler manner than it was with numbers, because with diagrams the differences were not comparable with the things which differed; and as often as they were connected together by addition or subtraction, being incomparable with one another, the less vanished in comparison with the greater; and thus irrationals could be differentiate no less easily than surds, and also, by the aid of logarithms, so could exponents. Moreover, the infinitely small lines occurring in the diagrams were nothing else but the momentous differences of the variable lines.” (1714, Historia et Origo Calculi Differentialis, Child 53-54).

  22. According to Leibniz, “The doctrine of irrational numbers, like what is contained in book X of the Elements [of Euclid],” is compared to “knowledge by intuition”; whereas “common arithmetic” is equalized to “knowledge of simple understanding”. Thus “it is impossible to give [a] demonstration of contingent truths” because “irrational proportions [cannot] be understood arithmetically, that is, they cannot be explained through the repetition of a measure.” (1685–1689?, The Source of Contingent Truths, C 1-3, AG 98-100)

  23. This can be illustrated by means of the following table:

     

    Commensurable quantities

    Incommensurable quantities

    Necessary truth

    Contingent truth

    Analytical a priori

    ✓

    ✓

    ✓

    ✓

    Provable

    ✓

    ✘

    ✓

    ✘

  24. “…every series, complex and irregular as it may be, and even including miracles, can be reduced to a unitary rule. To claim this, Leibniz plausibly had in mind his mathematical experience in the study of functions. In his texts on infinite analysis, however, the analogy in not usually drawn from some new discovery in infinitesimal calculus, but from the polarity between rational and irrational numbers, already known from antiquity.” (Di Bella 2005, 356). Leibniz himself relates explicitly to Euclid’s Elements: the shift of the Greeks from arithmetic to geometry allowed them to consider irrational sizes such as π, without relating to the fact that they are incommensurable quantities.

  25. “Leibniz thinks for example that we must understand clearly that we can rigorously prove that 1/2 + 1/4+ 1/8 + … = 1, although the number of terms of the series is not finite. But to give a proof of a truth involving infinitely many terms is surely not to give a proof which itself has ‘infinitely’ many steps.” (Ishiguro 1990, 194)

  26. The problem can be demonstrated by means of the table presented above:

     

    Commensurable quantities

    Incommensurable quantities

    Necessary truth

    Contingent truth

    Analytical a priori

    ✓

    ✓

    ✓

    ✓

    Provable

    ✓

    ✓

    ✓

    ✓

  27. “Terminating [Leibniz’] series after some very large finite number of terms gives us a good value for π/4, but even then there will be ‘a new remainder that results in a new quotient’, which itself is explicable as a series of infinitely many further quotients. This, Leibniz held, is analogous to a contingent truth: a contingent truth ‘involves infinitely many reasons, but in such a way that there is always something that remains for which we must again give some reason’ (On the Origin of Contingent Truths; A 6.4 1662, AG 99). But given Leibniz’s doctrine explain above that all demonstrations must involve only a finite number of steps, this means it is impossible for there to be a demonstration of a contingent truth, since such a demonstration would require an infinite analysis.” (Arthur 2014, 95–96)

  28. Russell 1972 (1903), 378 n. 8; Lovejoy 1936, 174–175; Curley 1972, 80.

  29. More on the infinite complexity and the problems in identifying infinite complexity as the source of the solution, in section 6 below.

  30. Rescher 1967, 46; Adams 1982, 258. According to Sleigh, there is a difference between an infinite analysis and an analysis that has no end (Sleigh 1982, 227–236). This difference in his opinion can be used to explain how God knows contingent truth: God can complete an infinite analysis. The problem is that this interpretation is not consistent with a syncategorematic view of infinity which Leibniz held.

  31. “But in contingent truths [...] the resolution proceeds to infinity, God alone seeing, not the end of the resolution, of course, which does not exist, but the connection of the terms or the containment of the predicate in the subject, since he sees whatever is in the series.” (1689?, On Freedom, FC 181-182, AG 96)

  32. The solution can be illustrated using the above table:

     

    Commensurable quantities

    Incommensurable quantities

    Necessary truth

    Contingent truth

    Analytical a priori

    ✓

    ✓

    ✓

    ✓

    Provable

    ✓

    ✓

    ✓

    ✘

  33. “For my part I confess that there is no way that I know up till now by which even a single quadrature can be perfectly demonstrated without an inference ad absurdum. Indeed, I have reasons for doubting that this would be possible through natural means without assuming fictitious quantities, namely infinite and infinitely small ones.” (Fall 1675-Summer 1676, De Quadratura Arithmetica, 35, trans. by Arthur 2008, 25 n16).

  34. “Just as I have denied the reality of a ratio, one of whose terms is less than zero, I equally deny that there is properly speaking an infinite number, or an infinitely small number, or any infinite line, or a line infinitely small [...]. The infinite, whether continuous or discrete, is not properly a unity, nor a whole, nor a quantity, and when by analogy we use it in this sense it is a certain facon de parler; I should say that when a multiplicity of objects exceeds any number, we nevertheless attribute to them by analogy a number, and we call it infinite. And thus I once established that when we call an error infinitely small, we wish only to say an error less than any given, and thus nothing in reality. and when we compare an ordinary term, an infinite term, and one infinitely infinite, it is exactly as if we to compare, in increasing order, the diameter of a grain of dust, the diameter of the earth, and that of the sphere of the fixed stars.” (1712, Acta Eruditorum; GM V 389, trans. by Jesseph 2008, 231 n30)

  35. “It would suffice here to explain the infinite through the incomparable, that is, to think of quantities incomparably greater or smaller than ours [...] But at the same time we must not consider that these incomparable magnitudes themselves are not at all fixed or determined but can be taken to be as small as we wish” (2 Feb. 1702, Leibniz to Varignon, GM IV 91, L 543). As mentioned, it appears that Leibniz already in 1676 arrived at the conclusion that the infinitesimal can be viewed spatially rather than quantitatively and continued to hold this view from that point onward. (11 Feb. 1676, On the Secrets of the Sublime, A 6.3 475, LLC 49; cf. Feb. 1689, Tentamen De Motuum Coelestium Causis, GM VI 150-151)

  36. “The consideration of differences and sums in number sequences has given me my first insight, when I realized that differences correspond to tangents and sums to quadratures.” (28 May 1697, Leibniz to Wallis; GM IV 25, trans. by Bos 1974, 13)

  37. The infinite division of the differential makes possible various level of differentials, which can be ordered by means of a fixed ratio. This fixed ratio of differentials of different orders is important to Leibniz to oppose Cavalieri, Wallis and other mathematicians at that time that believe the differential should be considered as indivisible and the difference between infinite series and its sum should be considered as null. (“For my quantity a is the same as your dx, except that my a is nothing and your dx infinitely small. Then when those things are neglected which I hold should be neglected in order to abbreviate the calculation, that which remains is your minute triangle, which according to you is infinitely small, but according to me is nothing or evanescent”, 30 July 1697, Wallis to Leibniz, GM IV 37, trans. by Jesseph 1998, 24). However, Leibniz argued, if the differential is divisible, then there is no necessary reason why the differential of the differential will not be divisible either. In Leibniz’ view, Cavalieri’s method of indivisibles relates to a special situation in which dx = 1 and ddx = 0; But this limitation necessarily prevents recognizing the existence of differentials of higher orders and limits the generality of calculus (Late 1675 – summer 1676, De Quadratura Arithmetica 24, trans. by Arthur 2008, 25). For this reason, Leibniz claimed in an official publication ten years later that the omission of differentials from the equation (or their identification as zero) are appropriate only in the special case in which the rate of the variable’s progress is fixed (Acta Eruditorum, June 1686, De Geometria Recondita et Analysi, GM VI 233, trans. by Bos 1974, 79).

  38. Already at the beginning of 1676, Leibniz raised for the first time the possibility that the Cartesian identity between space and matter is mistaken because space is a whole continuum while matter is discrete and aggregative (11 Feb. 1676, On the Secrets of the Sublime; A 6.3 473-474, LLC 47). Leibniz wondered about this since he found it difficult to explain the continuity of matter. Mathematically, the spatial definition of the infinitesimal by means of an “incomparable” ratio points to the fact that this is an undifferentiated quantity which is defined only as a derivative of the whole that preceded it. The idea that the ideal whole is precedent to its parts (in contrast to the priority of the parts to the whole on the real level), leads to the precedence of the summation over the infinite series and makes it possible to ignore the difference between them. However, in this manner, the way out of the Labyrinth of Continuum is founded on an unbridgeable gap between reality and mathematics and infinity is perceived as only potential. Indeed, later in his career, Leibniz claimed that the solution of the Labyrinth of Continuum lies in the gap between mathematics which is based on the whole being precedent to the parts and physics which is based on the simple preceding the complex (“For space is something continuous but ideal, whereas mass is discrete, indeed an actual multitude, or a being by aggregation, but one of infinite unities. In actual things, simples are prior to aggregates; in ideal things, the whole is prior to the part. Neglect of this consideration has produced the labyrinth of the continuum.” 6 Sep. 1709, Leibniz to Des Bosses, G II 379, LR 245; cf. 19 Jan. 1706, Leibniz to De Volder, G II 282, L 539, AG 185). At first glance, this view of infinity as potential is not consistent with Leibniz’ view of infinity as syncategorematic and actual. This is because potential division does not have to take into account that a whole can be divided into half or into thirds but never into halves and also into thirds. Potential parts can be the result of all the possible divisions since they are undifferentiated. In contrast, actual parts must be differentiated since they together constitute the actual whole. The conclusion that we cannot show how the actual continuum is constituted and that the infinitesimal parts are fictitious spills over to the view of infinity as only potential and is no different from that of Aristotle. But on second glance and despite the separation between ideal mathematics and actual existence that enables us to describe the infinitesimal as fictitious, Leibniz was careful to emphasize that actual reality nonetheless operates according to the eternal rules of mathematics (1702, Leibniz’ Reply to Bayle’s article ‘Rorarius’, G IV 569, L 583). Therefore, Leibniz’ position is more complex than simply restricting the infinitesimal to the status of an undifferentiated fiction, and this is manifested in the metaphysical meaning of his Principle of the Continuum (for more on that, see Levey 2008, 128–132).

  39. Levey criticized Leibniz’ “model of folds” as incoherent (Levey 1999, 148), whereas Arthur (2001a, lxxvi; Arthur 2014, 84–85) saw it, alone with the important role of force, as successful solution to the Labyrinth of Continuum.

  40. “It is just as if we suppose a tunic to be scored with folds multiplied to infinity in such a way that there is no fold so small that it is not subdivided by a new fold […] And the tunic cannot be said to be resolved all the way down into points; instead, although some folds are smaller than others to infinity, bodies are always extended and points never become parts, but always remain mere extrema.” (Pacidius; A 6.3 554-555, LLC 185-187) Leibniz remained faithful to his model of folds throughout his life and infinite flexibility indeed lies at the foundation of his later dynamics (1695, Specimen of Dynamics, GM VI 234-254, AG 117-138)

  41. As Arthur pointed out, when Leibniz discovered the Principle of the Conservation of Force in 1678, he now possessed a metaphysical principle that could be used to distinguish between different bodies and protect the consistent existence of the body over time. Leibniz could finally exploit the potential of his model of folds and relate to motion as something imaginary, without thereby denying the existence of matter. In fact, the discovery of force created an inversion in Leibniz’ theory of physics, although the solution of the Labyrinth of the Continuum using the model of folds remained intact from November 1676 onward (Arthur 2001a, lxxxv-lxxxvii; Arthur 2014, 88–89). In March 1678, Leibniz solved the paradox of the heap that had bothered him in his Pacidius Dialog eighteen months earlier. The paradox makes it difficult to distinguish between poverty and wealth (or between single grains and a heap), since it assumes that the difference is based on a certain point, starting from which poor becomes rich and grains become a heap, or else change does not take place in the world at all. In November 1676, Leibniz arrived at the conclusion that change must be a connection of the last point of the previous situation to the first point of the subsequent situation, namely by means of a leap (29 Oct. – 10 Nov. 1676, Pacidius to Philalethes, A 539-540, LLC 155-157). In the absence of a metaphysical principle that preserves the stability of matter, there is no other way to explain change except by means of a leap from one extreme point to another. Leibniz’ conclusion in Pacidius is that there is no actual continuum and that the material world is always in a chaotic situation of unceasing leaps. However, in March 1678, one month after the discovery of the Law of Conservation of Force, Leibniz claimed that there is a totally different solution to the paradox: a change in quantities or actual magnitudes cannot occur at a particular point in a process since quantities or magnitudes are imaginary and are not in any way differentiated. As a result, the actual continuum appears to be possible and the material world is always only in a situation of continuous changes without an end point (March 1678, Chrysippus’s heap, A 6.4 69-70, LLC 229-231).

  42. 30 April 1687, Leibniz to Arnauld, letter XX, G II 98, AG 86, LA 122. Also: “There is no determinate shape in actual things, for none can be appropriate for an infinite number of impressions. And so neither a circle, nor an ellipse, nor any other line we can define exists except in the intellect.” (1689, Primary Truths, A 6.4 1648, AG 34; 1686, There is no perfect shape in bodies, A 6.4. 1613, LLC 297; 1702, Letter to Queen Sophia Charlotte of Prussia, On What is Independent of Sense and Matter, G VI 500, AG 187-188)

  43. Arthur 2001a, lxxxv-lxxxviii; Levey 2005, 75–76. Following his criticism of Leibniz’ model of folds, Levey believes that the model shows a picture of extreme discontinuity (for details, see Levey’s interpretation of Leibniz’ model of folds in his 2003 article). But I think, like Arthur’s position, that this assessment falls in interpreting Leibniz’ mathematics from Cantorian point of view. This is because the infinite division of each piece of matter should not yield an infinite number of singular points. The idea of the folds was translated by Levey into fractals, assuming a Non-divisible singular point to be skipped by an inevitable leap. Leibniz did mention such a leap, but it was only before he identified the force as the critical factor that enables the stability and continuity of matter over time. The infinite flexibility of matter, as can be seen from Leibniz’ later physics, does not converge into singular points and cannot be represented by a fractal structure.

  44. 1689?, On Freedom, FC 179-180, AG 95; 1704, Theodicy (preface), G VI 29, H 53.

  45. “[H]aving contended himself with saying that matter is actually divided into parts smaller than all those we can possible conceive, [Descartes] warns that the things he thinks he has demonstrated ought not to be denied to exist, even if our finite mind cannot grasp how they occur. But it is one thing to explain how something occurs, and another to satisfy the objection and avoid absurdity.” (29 Oct. – 10 Nov. 1676, Pacidius to Philalethes, A 553-554, LLC 183-185)

  46. “That same distinguished philosopher I cited a short while ago [Descartes] preferred to slash through both of these knots with a sword since he either could not solve the problems, or did not want to reveal his view. For in his Principles of Philosophy I, art. 40–41, he says that he can easily become entangled in enormous difficulties if we try to reconcile God’s preordination with freedom of the will; but, he says, we must refrain from discussing these matters, since we cannot comprehend God’s nature. And also, in Principles of Philosophy II, art. 35, he says that we should not doubt the infinite divisibility of matter even if we cannot grasp it. But this is not satisfactory, for it is one thing for us not to comprehend something, and quite something else for us to comprehend that it is contradictory.” (1689?, On Freedom, FC 180-181, AG 95)

  47. “For everything is ordered in things once and for all, with as much order and agreement as possible, since supreme wisdom and goodness can only act with perfect harmony: the present is pregnant with the future; the future can be read in the past; the distant is expressed in the proximate. One could know the beauty of the universe in each soul, if one could unfold all its folds, which only open perceptibly with time. But since each distinct perception of the soul includes an infinity of confused conceptions which embrace the whole universe, the soul itself knows the things it perceives only so far as it has distinct and heightened perceptions; and it has perfection to the extent that it has distinct perceptions. Each soul knows the infinite – knows all – but confusedly.” (1714, Principles of Nature and Grace Based on Reason, §13, G VI 603, AG 211; 1714, The Monadology, §60)

  48. It is important to mention that the relations that exist between all of the individual concepts that make up the concept of the best of all possible worlds (i.e. the concept of the actual world) exist whether or not the individual concepts are complete. These relations also exist between incomplete concepts but in that case, we are prevented from knowing this from the analysis of only a single individual concept. This is because when we speak of the analysis of a contingent truth, it is our intention to reduce it to an identity argument and therefore we are in fact talking about a complete concept “which is so distinct that all its components are distinct” (1709, New Essays, book II, ch. xxxi [of complete and incomplete ideas], NE 266; November 1684, Mediation on Knowledge, Truth and Ideas, G IV 423, L 292). In contrast, an incomplete concept can be “distinct, and does contain the definition or criteria of the object”; however, “these criteria or components are not all distinctly known as well.” (ibid., NE 267). We will discuss the distinct knowledge of the concept in detail in the next section.

  49. 1714, The Monadology §57, G VI 617, AG 220.

  50. “And since everything is connected because the plenitude of the world, and since each body acts on every other body, more or less, in proportion to its distance, and is itself affected by the other through reaction, it follows that each monad is a living mirror or a mirror endowed with internal action, which represents the universe from its own point of view and is as ordered as the universe itself. […] Thus there is perfect harmony between the perception of the monad and the motions of bodies, pre-established from the first between the system of efficient causes and that of final causes. And in this consists the agreement and the physical union of soul and body, without the one being able to change the laws of the other.” (1714, Principles of Nature and Grace Based on Reason, §3, G VI 600, AG 207-208)

  51. In view of this, it would have been logical for Leibniz to attribute the infinite force that implements the spatial plan of the world to the world itself, namely to God who operates as the soul of the world. However, Leibniz refrained from doing so. The question of why is the subject of a debate in The Leibniz Review between Laurence Carlin (1997), Gregory Brown (1998, 2000), and Richard Arthur (1999, 2001b). I personally believe that the identification of God as the soul of the world is in fact totally consistent with Leibniz’ philosophy, though Leibniz refrained from stating it out of a desire to distance himself from Spinozian pantheism. Ambivalence on this issue indeed characterizes the metaphysics of Leibniz from February 1676 until the discovery of the metaphysical force in material bodies in February 1678.

  52. 1686, Necessary and Contingent Truths, C 18-19, MP 98; also 1689?, On Freedom, FC 181, AG 95.

  53. In his article, Carriero shows that it is not possible to create an algorithm for infinite complexity (Carriero 1993, 24–25). However, I am not certain that such an argument is consistent with Leibniz’ philosophy. I think that Leibniz in fact believed that at the foundation of every infinitely complex individual there is an algorithm that organizes the entirety of the attributes and relations of that individual. Since there is compatibility between the definition of the individual and the conceptual space of the world, it is reasonable to assume that the spatial plan of the world can also be formulated using an algorithm: “[E]verything is in conformity with respect to the universal order. This is true to such an extent that not only does nothing completely irregular occurs in the world, but we would not even be able to imagine such a thing. Thus, let us assume, for example, that someone jots down a number of points at random on a piece of paper […]. I maintain that it is possible to find a geometric line whose notion is consistent and uniform, following a certain rule, such that this line passes through all the points in the same order in which the hand jotted them down.” (1686, Discourse on Metaphysics §6, G IV 432, AG 39; 1686?, A Specimen of Discoveries, A 6.4 1619, LLC 310-311) A different question, which is related to the nature of demonstration in Leibniz’ philosophy, is whether such an algorithm can be demonstrated and in fact it cannot be.

  54. “It should be no cause for astonishment that I endeavour to elucidate these things by comparisons taken from pure mathematics, where everything proceeds in order, and where it is possible to fathom them by a close contemplation which grants us an enjoyment, so to speak, of the vision of the ideas of God. One may propose a succession or series of numbers perfectly irregular to all appearance, where the numbers increase and diminish variably without the emergence of any order; and yet he who knows the key to the formula, and who understands the origin and the structure of this succession of numbers, will be able to give a rule which, being properly understood, will show that the series is perfectly regular, and that it even has excellent properties. One may make this still more evident in lines. A line may have twists and turns, ups and downs, points of reflexion and points of inflexion, interruptions and other variations, so that one sees neither rhyme nor reason therein, especially when taking into account only a portion of the line; and yet it may be that one can give its equation and construction, wherein a geometrician would find the reason and the fittingness of all these so-called irregularities.” (1704, Theodicy §242, H 277)

  55. “An analysis of concepts by which we are enabled to arrive at primitive concepts, i.e. at those which are conceived through themselves, does not seem to be in the power of man. But the analysis of truths is more in human power, for we can demonstrate many truths absolutely and reduce them to primitive indemonstrable truths.” (1679–1686, An Introduction to a Secret Encyclopaedia, C 513-514, MP 8). Note that Leibniz stated this after completing the development of his Infinitesimal calculus.

  56. 1686, General Inquiries about the Analysis of Concepts and Truths §§135–136; C 388, PLP 77-78.

  57. “[R]est. can be considered as an infinitely small velocity or as an infinite slowness. Therefore whatever is true of velocity or slowness is in general should be verifiable also of rest taken in this sense, so that the rule for resting bodies must be considered as a special case of the rule for motion [...]. Likewise equality can be considered as an infinitely small inequality, and inequality can be made to approach quality as closely as we wish.” (July 1687, Letter of Mr. Leibniz on a General Principle Useful in Explaining the Laws of Nature through a Consideration of Divine Wisdom – Reply to Father Malebranche, G III 51, L 351)

  58. “Since the Cartesians don’t understand the use of elastic force in the collision of bodies, they also err in thinking that changes happen through leaps, as if, for example, a body at rest could, in a moment, pass into a state of determinate motion, or as if a body placed in motion could suddenly be reduced to rest, without passing through intermediate degrees of velocity. If elastic force were lacking, then, I confess, what I call the law of continuity, through which leaps are avoided, would not be observed, nor would there be place for other excellent contrivances of the Architect of Nature, contrivances by which the necessity of matter and the beauty of form are united. Moreover, this very elastic force, inherent in every body, shows that there is internal motion in every body as well as a primitive and (so to speak) infinite force, although in collision itself it is limited by derivative force as circumstances demand.” (May 1702, On Body and Force, Against the Cartesians, G IV 399, AG 255)

  59. 1709, New Essays, book II, ch. xxix [of clear and obscure, distinct and confused ideas], NE 255.

  60. 1686, Discourse on Metaphysics §24, AG 56.

  61. Ibid., AG 57.

  62. Ibid., AG 56.

  63. Nov. 1684, Meditation on Knowledge, Truth and Ideas, G IV 24, AG 25.

  64. Interestingly, Leibniz defined a concept as imaginary when it is possible but does not belong to the conceptual space of the actual world: “[P]ossible ideas become merely chimerical when the idea of actual existence is groundlessly attached to them – as is done by those who think they can find the Philosopher’s Stone, and would be done by anyone who thought that there was once a race of centaurs.” (1709, New Essays, book II, ch. xxx [of real and chimerical ideas], NE 265-266) However, such a situation cannot be part of a process of infinite analysis of contingent truth. The analysis of concepts related to one possible world cannot involve concepts related to a different possible world. Therefore, the analysis of contingent truths about the world cannot include an ‘infinitesimal-fictitious’ stage that is, by its definition as imaginary, not included in these truths.

  65. 4/14 July 1686, Leibniz to Arnauld, letter X, G II 53, LA 59.

  66. In a paper from the early 1680s, Leibniz allegedly considered fictitiousness as a basis for the solution of the problem of contingency and divine freedom, though not in the form we discussed above (1680–1682?, On Freedom and Possibility, Grua 291, AG 22). In that paper, he stated that contingent truths cannot be demonstrated, even though there is no mention of a solution to the Labyrinth of Freedom by means of infinity (the solution took shape for Leibniz during 1686 or somewhat prior to that). To illustrate that contingent truths cannot be demonstrated, Leibniz used the imaginary solution of a quadratic equation. Leibniz pointed to the difference between an impossible solution which must necessarily be ruled out and an imaginary solution which can possibly, and therefore freely, be ruled out. As a result, contingent truths may be unprovable without becoming impossible.

  67. Arthur refrained from arguing this claim and emphasizes that there is a complete analogy between infinite analysis of contingent truth and the mathematics of infinite series (Arthur 2014, 96). For this reason, his explanation does not answer the fact that Leibniz’ calculus actually allowed him to prove, in a finite number of steps, that there is a sum of infinite series and a way to calculate the size of an incommensurable figure.

  68. “But in contingent propositions one continues the analysis to infinity through reasons for reasons, so that one never has a complete demonstration, though there is always, underneath, a reason for the truth, but the reason is understood completely only by God, who alone traverses the infinite series in one stroke of mind.” (1686?, On Contingency, Grua 303, AG 28).

  69. 1686, Necessary and Contingent Truths, C 17, MP 96.

  70. Ishiguro 1990, 196.

  71. 1714, Principles of Nature and Grace Based on Reason, §13, G VI 603, AG 211.

  72. August 1677, Dialogue, G VII 193, L 185.

  73. As argued by Nachtomy, the complete concept of a contingent subject, which is composed of an infinite set of its predicates, is defined by a production rule or an algorithm. This allows Leibniz to avoid the paradox related to an infinite number. Due to his intensional perspective on his concept-containment principle, Leibniz can claim that the complete concept of a contingent subject does not include an infinite number of characteristics but rather expresses the regularity or order that generates a syncategorematic infinity of characteristics and ratios (Nachtomy 2007, 63–67).

  74. 1685–1689?, The Source of Contingent Truths, C 1-3, AG 98-100.

  75. “For necessary truths can be resolved into such as are identical, as commensurable quantities can be brought to a common measure; but in contingent truth, as in surd numbers, the resolution proceed to infinity without ever terminating. And so the certainty of and perfect reason for contingent truths is known only to God, who grasps the infinite in one intuition.” (Circa 1686–1688, A Specimen of Discoveries, A 6.4 1615, G VII 309, trans. by Russell 1900, 221–222, LLC 305).

  76. 1709, New Essays, book IV, ch. ii [The degrees of our knowledge], NE 367.

  77. It seems to me that only God can know a priori that a contingent truth is possible, i.e. that it does not include internal contradiction, since only he sees it as a whole, while no one, including God, can analyze all its components one after another. Nevertheless, inspired by an overly strong analogy between mathematics and logic, Arthur declares that Leibniz has proved that contingent truth is possible: “by defining a contingent truth as one in which, although the predicate is in the concept of the subject, it is impossible for this to be demonstrated, Leibniz has proved its possibility, a possibility revealed to him by the analogy with incommensurables.” (Arthur 2014, 96) In fact, precisely because of this analogy, Leibniz concluded that the mathematical demonstration of the calculation of incommensurable sizes must be different from the demonstration of contingent truths. Otherwise contingency will collapse into mathematical necessity.

  78. A contingent truth is not necessary and therefore its negation does not imply a contradiction (1686, Discourse on Metaphysics §13, AG 45). In other words, this is a truth that can never be demonstrated to include the argument that contradicts it. But to the same extent it cannot be demonstrated that it does not include such an argument. Therefore, an analysis of contingent truth cannot guarantee that it is possible at all. Such an analysis is based on a nominal definition of the object of the investigation and we need an a posterior confirmation that a particular fact is indeed possible. Therefore, certainty with respect to the validity of a contingent truth is not due to the knowledge that this truth does not include an argument that contradicts it but rather is due to the lack of knowledge as to whether or not this truth includes an argument that contradicts it. This is because Leibniz was ready to accept that possibility is not only everything that necessarily does not imply a contradiction, but also everything that does not necessarily imply a contradiction. Such an argument also lies at the foundation of Leibniz’ ontological proof which functioned as his formal proof that the concept of God is possible (18–21 Nov.? 1676, That a Most Perfect Being Exists, A 6.3 578-579, PDSR 101-103; 1704, Theodicy §23, H 88). As we know, although Leibniz was concerned that a thought which seems possible is not really so (for example, the notion of an ‘infinite number’), he was also aware of the fact that proof of possibility was almost always limited to a nominal or “merely real and nothing more” definitions (1686, Discourse on metaphysics §24, AG 57). In light of this, it is somewhat surprising to find that certainty is related to the fact that we cannot absolutely know whether or not a concept is possible. This certainty should be referred to as an intuitive certainty that cannot be proven, since intuition perceives the entirety (the whole which has no parts) while a proof perceives the complex and consistent process of the parts. We intuitively perceive simple concepts only and we are unable to prove or justify such concepts in the same way that we do in the case of complex concepts that do not include an internal contradiction. On the other hand, God can also perceive with an intuitive glance infinitely complex concepts; in other words, he can see them as if they were simple although even he cannot prove that there is no internal contradiction in such concepts. In a similar way, I have claimed that the process of intuitive perception clearly cannot be complex but can only be simple; it therefore fails in providing a priori proof of the existence of God (Lison 2017).

Abbreviations

A:

G. W. Leibniz. Sämtliche Schriften und Briefe. Ed. by the Deutsche Akademie der Wissenschaften.Multiple vols. in 7 series. Darmstadt/Leipzig/ Berlin: Akademie Verlag, 1923-.

AG:

G. W. Leibniz. Philosophical Essays. Ed. and trans. by R. Ariew, and D. Garber. Indianapolis and Cambridge: Hackett, 1989.

C:

Opuscules et Fragments inedits de Leibniz. Ed. by L. Couturat. Paris, 1903. Reprinted Hildesheim, 1961.

Child:

The Early Mathematical Manuscripts of Leibniz. Ed. and trans. by J. M. Child. Chicago: Open Court 1920. Reprinted New York: Dover, 2005.

FC:

Nouvelles Lettres et Opuscules de Leibniz. Ed. by A. Foucher de Careil. Paris: Auguste Durand, 1857. Reprinted Hildesheim: Olms, 1962.

G:

Die Philosophischen Schriften von G. W. Leibniz. Ed. by C. I. Gerhardt. 7 vols. Hildesheim: Olms, 1875. Reprinted 1978.

GM:

G. W. Leibniz. Mathematische Schriften. Ed. by C. I. Gerhardt. 7 vols. Hildesheim: Olms, 1854. Reprinted 1975.

Grua:

G. W. Leibniz. Texts inedits. Ed. by Gaston Grua. 2 vols. Paris, 1948

H:

G. W. Leibniz. Theodicy: Essays on the Goodness of God the Freedom of Man and the Origin of Evil. Ed. by F. Austin, trans. by E. M. Huggard. New Haven, Yale University Press, 1952.

L:

G. W. Leibniz. Philosophical Papers and Letters (2nd edition). Ed. L. E. Loemker. Dordrecht: D. Reidel, 1969.

LA:

The Leibniz-Arnauld Correspondence. Ed. and trans. by H. T. Mason. Manchester: Manchester University Press, 1967.

LH:

Die Leibniz-Handschriften der koniglichen offentlichen bibliothek zu Hannover. Ed. by E. Bodemann. Hannover, 1895. Reprinted Hildesheim, 1966.

LLC:

G. W. Leibniz. The Labyrinth of the Continuum: Writings on the Continuum Problem, 1672-1686. Ed., trans. and with an introduction by R. T. W. Arthur. New Haven: Yale University Press, 2001.

LR:

The Leibniz-Des Bosses Correspondence. Ed., trans. and with an introduction by B. C. Look, and D. Rutherford. New Haven and London: Yale University Press, 2007.

MP:

G. W. Leibniz. Philosophical Writings. Ed. and trans. by M. Morris and G. H. R. Parkinson. London: Dent, 1973.

NE:

G. W. G. W. Leibniz. New Essays on Human Understanding. Ed. and trans. by P. Remnant, and J. Bennett. Cambridge: Cambridge University Press, 1981.

PDSR:

G. W. Leibniz. De Summa Rerum: Metaphysical Papers 1675-1676. Ed., trans. and with an introduction by G. H. R. Parkinson. New Haven and London: Yale University Press, 1992.

PLP:

G. W. Leibniz. Logical Papers. Ed. and trans. by G. H. R. Parkinson. Oxford: Clarendon Press, 1966.

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Lison, E. What Does God Know but can’t Say? Leibniz on Infinity, Fictitious Infinitesimals and a Possible Solution of the Labyrinth of Freedom. Philosophia 48, 261–288 (2020). https://doi.org/10.1007/s11406-019-00086-4

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