Kant's 1768 attack on Leibniz' conception of space

DOI 10.1515/kant-2013-0011 Kant’s 1768 attack on Leibniz’ KANT-STUDIEN conception2013; of space 104(2): 145–166 145 Abhandlungen Stefan Storrie Kant’s 1768 attack on Leibniz’ conception of space Abstract: This paper examines two features of Kant’s 1768 critique of Leibniz’ conception of space. Firstly, Leibniz’ proposed geometrical calculus called ‘analysis situs’; secondly, Leibniz’ relational conception of space. The main thesis of the paper is that Kant’s arguments are more powerful than generally recognized. With regard to the analysis situs, I will show that Kant was quite well informed about this proposed science and that his arguments severely undermine Leibniz’ claims to what it could perform. With regard to the relational theory of space, Kant’s argument would require Leibniz to present a complex story about the relation between God’s act of creation and our spatial experience to defend his relational view, rather than using the simple principle of sufficient reason. Keywords: Leibniz, Geometry, Space Stefan Storrie: Dublin; [email protected] Introduction The last of Kant’s properly so called pre-Critical works appeared as a series of three instalments in Königsberger Frag- und Anzeigungsnachrichten in 1768 and is known as Von dem ersten Grunde des Unterschiedes der Gegenden im Raume [Directions]. In recent years this work has become recognised as pivotal to Kant’s Critical doctrine of the nature of space and therefore to the Critical philosophy in general.1 The issues Kant raises in this work and in particular his ‘discovery’ of in- 1 “The importance of Directions in Space for an understanding of the development of Kant’s views on space and time, and therefore for an understanding of the emergence of the critical philosophy itself, can scarcely be exaggerated.” Theoretical Philosophy, 1755–1770. Ed. by David Walford and Ralf Meerbote. Cambridge 1992, lxx. In a recent paper Matthew Rukgaber has argued that the findings in the Directions “are at the very foundation of the ideality of space and time and Brought to you by | Trinity College Dublin Library Authenticated | 10.248.254.158 Download Date | 9/1/14 10:26 AM 146 Stefan Storrie congruent counterparts (objects that have the same internal relations but cannot occupy the same place, such as our right and left hand) has for a long time been of interest not only to Kant scholars, but generally for philosophers concerned with theories of space and time. In the early 1990s scholarly work on the Directions was set on a new course after Walford and Meerbote published a new translation of that work in 1992. They explained that the central concept ‘Gegend’ had up until then, with very few and rather insignificant exceptions, been deeply misunderstood. The general understanding of the word both by German native speakers and in English translations had up until then been ‘region’, whereas it should be taken to mean ‘direction’.2 With such a change in meaning of the central concept of the paper the work had to be reinterpreted. This later led Walford to state in his seminal paper on the Directions that, Although some two-and-a quarter centuries have elapsed since the original publication of the 1768 Essay, and although the last forty years have seen an extraordinary eruption of philosophical interest in the incongruent counterparts argument, the exact nature of Kant’s treatment of the theme has either remained largely unrecognised or been radically misunderstood.3 One thing that is clear is that in Kant’s 1768 paper Euler is the hero and Leibniz is the villain. Kant commends both Euler’s “so to speak a posteriori” (Directions, AA 02: 378) method and his doctrine, the reality of absolute space.4 Kant criticises two crucial features of Leibniz’ view on space by means of the manifestation of directionality exhibited by incongruent counterparts. Firstly, Leibniz’ project of a genuinely geometrical calculus, the ‘analysis situs’; secondly, Leibniz’ cosmological relationalism with regard to space. so of Kant’s transcendental philosophy.” “‘The Key to Transcendental Philosophy’: Space, Time and the Body in Kant”. In: Kant-Studien 100, 2009, 166–186, 172. All English translations of Kant’s works are from The Cambridge Edition of the Works of Immanuel Kant. 2 For more discussion of this translation see Paul Rusnock and Rolf George: “Snails Coiled Up Contrary to All Sense”. In: Philosophy and Phenomenological Research 54, 1994, 459–466 and David Walford: “Review of Van Cleve & Frederick (1991)”. In: Kant-Studien 85, 1994, 100–104. 3 David Walford: “Towards an interpretation of Kant’s 1768 Gegend im Raume Essay”. In: KantStudien 92, 2001, 408. 4 The method gets this name because it is neither a straightforward application of metaphysical principles nor an attempt to establish a metaphysical proof by appeal to experience. Rather, it uses a concrete example from experience and then indirectly, through several steps that are increasingly more abstract, approaches the metaphysical issue at hand. More specifically, Kant begins with the phenomena of incongruent counterparts and then seeks the basis for their difference in increasingly more abstract conceptions of directionality that ends with the notion of absolute space. Brought to you by | Trinity College Dublin Library Authenticated | 10.248.254.158 Download Date | 9/1/14 10:26 AM Kant’s 1768 attack on Leibniz’ conception of space 147 Kant’s appropriation of Euler’s method and doctrine has been treated in detail elsewhere.5 In this paper I will consider Kant’s confrontation with Leibniz in the Directions. Kant’s attack consists of two stages. In the first stage, he shows that there is a serious problem in Leibniz’ basic assumption about certain fundamental geometrical concepts. This has the consequence that Leibniz cannot account for the direction and therefore the complete determination of position. The second stage of the attack is to argue that this problem is not only an issue in Leibniz’ view of certain geometrical concepts but that it is ‘built into’ Leibniz’ relational conception of space, which therefore is impoverished in an important way. In § 1–§ 3 I will consider the first stage and show how Kant’s carefully chosen examples of incongruent objects in nature strikes at the heart of Leibniz’ explicit hopes and aspirations for the analysis situs. In § 4 I will turn to the second stage. Here my main claim is again that Kant’s argument is more powerful than previously held. 1 Leibniz’ analysis situs In the introductory paragraph Kant announces that the essay has two aims. He first proposes to investigate the foundations of the basic concepts of Leibniz’ new geometrical discipline, which is called the ‘analysis situs’. [T]o judge by the meaning of the term [situs], what I am seeking to determine philosophically here is the ultimate ground of the possibility of that of which Leibniz was intending to determine the magnitudes mathematically.6 I will give a brief account of the main idea of Leibniz’ new form of geometry in § 1, explain Kant’s argument against it in § 2 and establish to what extent Kant was familiar with the new science in § 3. The thought of the analysis situs occupied Leibniz in one form or another over four decades (1660s to 1710s). However, like so many of Leibniz’ ideas it was never fully worked out. I will here focus on the most well known and perhaps clearest statements that Loemker collectively named “Studies in a geometry of 5 David Walford “The Aims and Method of Kant’s 1768 Gegenden im Raume Essay in the Light of Euler’s 1748 Réflexions sur L’Espace”. In: British Journal for the History of Philosophy 7, 1999, 305–332. 6 GUGR, AA 02: 377: “[A]llein nach der Wortbedeutung zu urtheilen, suche ich hier philosophisch den ersten Grund der Möglichkeit desjenigen, wovon er [Leibniz] die Größen mathematisch zu bestimmen vorhabens war.” Brought to you by | Trinity College Dublin Library Authenticated | 10.248.254.158 Download Date | 9/1/14 10:26 AM 148 Stefan Storrie situation with a letter to Christian Huygens” (Studies from here on).7 The Studies are comprised of a letter and accompanying supplement that Leibniz sent to Huygens on the 8th of September 1679 and the paper On Analysis Situs from 1693.8 In part three of the Studies Leibniz explains that he wants to found a new form of geometrical analysis because of the shortcomings that he perceived in mathematical analysis (that is, arithmetic and algebra), which concerns itself exclusively with magnitudes or quantities. He thinks that the latter form of analysis is not sufficiently fundamental because it presupposes certain theorems from geometry.9 These theorems are in turn not derived exclusively from a consideration of magnitudes but also involve figures. Unlike the mathematical forms of analysis a consideration of figure involves not only quantity but also quality (Studies, 254). It follows that the concept of figure cannot be fully determined by mathematical analysis. Leibniz therefore suggests that if quality can be as precisely determined as quantity has been in mathematical analysis we could come into possession of a more powerful and presupposition-less form of calculation. The new form of analysis requires a new kind of ‘fundamental operation’. For equality, the fundamental operation of arithmetic and algebra, only express a relationship of magnitude and therefore of quantity. Leibniz proposes to work out an analysis of figure that has a fundamental operation that will express ‘similarity’ as “sameness of form” (Studies, 254). The question is then, according to Leibniz, how to define ‘similarity’. His answer is “the matter reduces to this: things are similar which cannot be distinguished when observed in isolation from each other.” (Studies, 255) For example, all equiangular triangles are similar (Studies, 256). Since the only difference that can exist between different equiangular triangles is the length of its sides we can only tell the difference between two such triangles by either comparing them to each other or by comparing them each to a common instrument of measurement, for example a ruler. Internally, similar objects have the same relations between its parts; it is only by comparing the object to some other thing external to them that the difference between them can be discerned. 7 Leroy Loemker: Gottfried Wilhelm Leibniz: Philosophical Papers and Letters. Second edition. Dordrecht 1969, 248–258. As I will show in § 3 the letter to Huygens was probably the main source of information about the analysis situs in Kant’s time. However, the On Analysis Situs is a clearer and more detailed account of Leibniz’ project and therefor I will mainly refer to that paper in the exposition of the theory. 8 For a chronological chart of all of Leibniz writings on the analysis situs see Vincenzo De Risi: Geometry and Monadology. Basel 2007, 122–126. 9 See Studies, 258n6 and Letter to Walter von Tschirnhaus, May 1678, in: Loemker: op. cit., 192–195 for two examples. Brought to you by | Trinity College Dublin Library Authenticated | 10.248.254.158 Download Date | 9/1/14 10:26 AM Kant’s 1768 attack on Leibniz’ conception of space 149 To illustrate his point Leibniz invites the reader to consider an example. Imagine that two temples are constructed. They are built in the same material and the parts are arranged in the same way so that the walls and columns and all the rest are in the same place down to the smallest angle. If we were led into the temple blindfolded we would not be able to tell which one we were in when the blind was taken off. They could only be distinguished if we were at a place where both could be perceived at the same time. Now let us imagine that the temples are of different magnitude. In that case we could distinguish them by comparing their respective size with a measuring instrument, such as a ruler or our body. However, if there is no third thing like our body or a ruler, if we consider the spectator as a mere seeing intellect “concentrated at a point, as it were” (Studies, 255), then the difference of magnitude will not be discerned by considering one of the temples in isolation.10 The two temples are therefore similar or have the same form. Leibniz then concludes by defining congruence, [A] true geometric analysis ought not only to consider equalities and proportions which are truly reducible to equalities but also similarities and, arising from the combination of equality and similarity, congruence. (Studies, 255) The analysis situs can therefore be defined as the analysis that combines both equality and similarity into the fundamental operation of congruence.11 Two simi- 10 It is a consequence of Kant’s argument in the Directions that we could not form a concept of an object’s direction and therefore of situation in general without reference to our own body. “It is, therefore, not surprising that the ultimate ground, on the basis of which we form our concept of directions in space, derives from the relation of these intersection planes [the three dimensions of physical space] to our bodies. [… so ist kein Wunder, daß wir von dem Verhältniß dieser Durchschnittsflächen zu unserem Körper den ersten Grund hernehmen, den Begriff der Gegenden im Raume zu erzeugen.]” (GUGR, AA 02: 378–379) Therefore, contrary to what Leibniz holds, we would not be in a position to say whether the two temples are similar or not without our body. It therefore seems that Kant has the resources to reject Leibniz’ phenomenological criteria for distinguishing between situation and magnitude. The issue of embodiment will however not be discussed here as Kant does not explicitly link his criticism of Leibniz in the Directions to the issue of embodiment. 11 There is a difficulty in the interpretation of Leibniz analysis situs that concerns the question of whether the analysis is supposed to deal exclusively with quality (similarity) or with both quality and quantity (magnitude). For example, Walford (2001, 416) writes “Leibniz had conceived the analysis situs as a form of geometrical analysis from which considerations of magnitude were to be excluded […] Unfortunately, Leibniz defined the concept of congruence in terms both of similarity and equality.” (See also Walford, 1999, 328n47) This in fact points to a genuine difficulty for Leibniz’ project of the analysis situs. The problem is that to the extent that equality is defined at all, it is natural to define the equality between figures in terms of congruence. As Leibniz defines congruence in terms of equality this definition appears circular. Since the concept of equality Brought to you by | Trinity College Dublin Library Authenticated | 10.248.254.158 Download Date | 9/1/14 10:26 AM 150 Stefan Storrie lar figures (for example, two equiangular triangles) which also have the same magnitude (whose sides are of the same length) are congruent. Leibniz compliments this functional definition with a kinaesthetic definition by stating that two congruent figures can be made to occupy the same place (at different times of course) without changing the internal constitution of the triangles (Studies, 251). Leibniz expected that the invention of the new analysis would instigate a great leap in the progression of human knowledge. He therefore finds occasion to end the On Analysis Situs on a characteristically positive note. The analysis situs, he tells us, will explain the situation of things immediately and in more precise ways than the mathematical analysis. The former needs to concern itself exclusively with the symbols that are related by congruence. The latter must take recourse in imprecise representations, such as actual figures that are perceived and understood by our imperfect “empirical imagination” (Studies, 257). Leibniz writes, Therefore this calculus of situation which I propose will contain a supplement to sensory imagination and perfect it, as it were. It will have applications hitherto unknown not only in geometry but also in the invention of machines and in the description of the mechanisms of nature. (Studies, 257) It seems that Leibniz believed that ultimately all extended objects, as such, could be given a complete account based on the concept of congruence. In the supplement to the letter to Huygens Leibniz expresses similar hopes and explains the practical application of the analysis situs in more detail than anywhere else. He again uses the example of machines and living things in nature. Not only will it be applied to the invention and construction of machines, but also generally to the understanding of how an already existing machine works. In Leibniz’ words the new analysis “could provide the mind with a method of knowing the machine and all its parts, their motion and use.” (Studies, 250) Leibniz continues, “I believe that by this method one could treat mechanics almost like geometry, and one could even test the qualities of materials, because this ordinarily depends on certain figures in their sensible parts.” (Studies, 250) Similarly, he believes that the analysis will be able to give exact descriptions of the structure of plants and animals (Studies, 250). depends on the concept of quantity we could expect Leibniz to present a definition of quantity and develop his account of equality out of that. However, Leibniz could never produce a clear definition of quality and therefore problems with the notion of equality remained (see De Risi: Op. cit., 143ff, for a discussion of this issue). Brought to you by | Trinity College Dublin Library Authenticated | 10.248.254.158 Download Date | 9/1/14 10:26 AM Kant’s 1768 attack on Leibniz’ conception of space 151 2 Kant’s criticism of the analysis situs In the Directions Kant considers a certain “well-known characteristic” that a wide variety of objects have.12 As examples of the kinds of objects he is referring to he mentions machines (or parts thereof, nuts and screws, GUGR, AA 02: 381), plants (hops and beans, GUGR, AA 02: 380) and animals (different kinds of snails, GUGR, AA 02: 380). Kant observes that the threads of a screw and the threads of a nut run from either left to right or from right to left. Even if the screw and the nut have the size required for a fit, whether they will actually fit also depends on the direction of the threads. The species of plants and animal that he mentions are classified, sometimes in part, sometimes exclusively, by the direction that the parts are turned, so that, in virtue of this distinctive characteristic, two creatures may be distinguished from each other, even though they may be exactly the same in respect to size, proportion and even the relative position of their parts.13 Hops wind around their poles from left to right, whereas beans wind the opposite way. Some species of snails have shells that curve in one direction while others curve in the opposite direction. The certain “distinctive characteristic” is the manner in which the objects are organised in particular directions. Kant believes that his foregoing observations of the importance of directionality has established that the direction of the different parts of an object, whether from left to right, up to down, in front to behind, and all their opposites, are essential to determine something as basic as whether a screw will fit a nut and what kind of species a particular plant or animal belong to. If we recall Leibniz’ hopes for his analysis situs, where he states that it will give “exact descriptions” of plants and animals and enable us to “carry out the description of a machine, no matter how complicated” (Studies, 250), we can conclude that the analysis situs must be able to account for the direction of the parts of an object in order to be successful in these respects. However, Kant argues that similarity and equality, in the case of three-dimensional objects, is not sufficient for congruence; and by implication that Leibniz’ analysis situs cannot account for the phenomena of direction. Similarity consists of the identity of the internal relations between the parts of an object. Equality is 12 GUGR, AA 02: 380: “namhaftes Kennzeichen”. 13 GUGR, AA 02: 380: “[…] wornach die Ordnung ihrer Theile gekehrt ist und wodurch zwei Geschöpfe können unterschieden werden, obgleich sie sowohl in Ansehung der Größe als auch der Proportion und selbst der Lage der Theile unter einander völlig überein kommen möchten.” Brought to you by | Trinity College Dublin Library Authenticated | 10.248.254.158 Download Date | 9/1/14 10:26 AM 152 Stefan Storrie defined as sameness of magnitude. If we consider a right winding and a left winding screw or our right and our left hand, it is clear that they can both be similar and equal and yet the difference between the pairs do not vanish. In the case of human hands, one example of the internal relations is the length of one finger in relation to another finger. Another example is the angle between the lines drawn from the tip of one finger to some point on the palm of the hand and the tip of any other finger to the same point in the palm compared to any other angle that is generated by drawing similar lines on other fingers on the same hand. Our left and our right hand are almost similar, and if they were exactly similar there is no reason to believe that the difference in direction would vanish. Further, our right and our left hand have almost the same magnitude, that is, the corresponding fingers and parts of the palm are of the same length, thickness and broadness. Again, if they were of exactly the same magnitude there is no reason to think that the difference in direction would disappear. There is therefore a difference between the two hands or between a right and a left winding screw that is not accountable by the minimal differences in shape or size. This is the very ‘left-ness’ and ‘right-ness’, the direction in which the object is placed in space. This difference is confirmed by simple experiments, such as trying to fit a right and a left winding screw on the same nut or trying to fit a left and a right hand in the same glove. These objects, which can be similar and equal, cannot be made to coincide, i.e. take up exactly the same place. Kant calls “a body that are exactly equal and similar to another, but which cannot be enclosed by the same limits as that other, its incongruent counterpart.”14 Since Leibniz defined congruence as similarity and sameness of magnitude and yet Kant has discovered a great number of objects that fulfil this criterion but are incongruent, Leibniz’ definition of congruence is insufficient. Congruence also requires sameness of direction. Contrary to what Leibniz claimed the analysis situs can describe neither the direction in which the parts of an object are organised, nor can it describe the movements of the parts of, for example, a machine. 14 GUGR, AA 02: 382: “Ich nenne einen Körper, der einem andern völlig gleich und ähnlich ist, ob er gleich nicht in eben denselben Grenzen kann beschlossen werden, sein incongruentes Gegenstück.” Brought to you by | Trinity College Dublin Library Authenticated | 10.248.254.158 Download Date | 9/1/14 10:26 AM Kant’s 1768 attack on Leibniz’ conception of space 153 3 Kant’s knowledge of the analysis situs How much Kant knew about the analysis situs it is a matter of some dispute. Some commentators claim that Kant had no knowledge of Leibniz’ theory but only knew of the name ‘analysis situs’.15 Walford claims that Kant might have read Leibniz’ letter to Huygens but provides no positive reason for his claim beside the suggestion that Kant was interested in such matters and therefore ought to have read it.16 I will go further in this direction and see Leibniz as the direct target of Kant’s argument. I will argue that not only is it plausible to assume that Kant would have read Leibniz’ letter on these grounds but that Kant’s actual argument in the first two thirds of the Directions appears to be directed specifically at Leibniz’ claims in his letter to Huygens. There are three main sources of information about the analysis situs that would have been available to Kant by 1768: (1) Leibniz’ letter to Huygens from 1679, (2) brief and vague references to the analysis situs by prominent mathematicians, logicians and scientists such as Euler, Lambert and Buffon (to this category I also include the rumours about the analysis situs that were to a large part spread by Leibniz himself in private correspondences in the 1690s), and (3) Wolff’s many discussions of the concept of similarity. In this section I will show that it is probable that Kant availed himself of all these sources and therefore had a far wider knowledge of the analysis situs than is generally recognized. (1) The only published account where Leibniz goes into detail about his project of the analysis situs in Kant’s time was Leibniz’ letter to Huygens dated 8th of September 1679, including the supplement. Though a private letter, it was 15 Daniel Sutherland: “Kant on Fundamental Geometrical Relations”, in: Archiv für Geschichte der Philosophie 87, 2005, 117–158, 136, simply states that “Leibniz’ analysis situs was not available to Kant”, the same view appears to be maintained in Daniel Sutherland: “Philosophy, Geometry, and Logic in Leibniz, Wolff, and the Early Kant”, in: Discourse on a New Method, ed. by Mary Domski and Michael Dickson, Chicago 2010, 155–192, 157. Unfortunately Sutherland does not present any argument in favour of this claim. A similar view is held by Rusnock and George, who state that a “number of people, including Kant (GUGR, AA 02: 377) had heard the name of Leibniz’s proposed science, but had seen nothing of it.” Paul Rusnock and Ralf George: “A Last Shot at Kant and Incongruent Counterparts”. In: Kant-Studien 86, 1995, 257–277, 262n. Their reference to the first paragraph of the Directions suggests that they take Kant’s introductory remarks to indicate that Kant had no knowledge of the analysis situs beside the name. I will argue for a different reading of the relevant passage in the section. 16 Walford claims that “It is improbable that Kant, with his strong interest in such matters, would not have read the book [Vollständige Sammlung aller Streitschriften]. The selections from Leibniz’ letters to Huygens on the analysis situs are described by Köning (Part III p. 88, Note A) “as more important than anything else in the book.” (2001: 416n15) Brought to you by | Trinity College Dublin Library Authenticated | 10.248.254.158 Download Date | 9/1/14 10:26 AM 154 Stefan Storrie published as a consequence of the Köning-Maupertuis dispute in S. Köning’s Appel au Public (Leyden: 1752) and the Defense (Leyden: 1753), which was translated into German as Vollständige Sammlung aller Streitschriften (1753).17 In the letter to Huygens Leibniz specifically emphasizes the wide range of practical applications that he envisaged for the analysis situs. This is natural given that it is an early paper on the new calculus. Leibniz had not developed the theoretical aspects of it in great detail yet and does not have any impressive mathematical results to give Huygens at that point. Instead, Leibniz’ aim is to convince Huygens that the analysis situs is an idea worth pursuing and to this end he emphasizes the tremendous practical use he envisages for it. As shown in § 2, in the Directions Kant drew our attention to exactly the same classes of beings (machines, plants and animals) as Leibniz had singled out in his letter to Huygens and he argues specifically for these things that we can have no practically useful knowledge of them with the Leibnizian conception of congruence. This means that the Directions strikes at the core of what Leibniz actually said in that letter. Given that (a) the letter was published as part of a dispute that was widely followed in intellectual circles in the 1750s and (b) was available both in French and in a German translation, and finally and most decisively (c) because Kant’s argument targets the very things Leibniz mentions in the letter, I conclude that it is highly probable that Kant was thoroughly acquainted with the letter to Huygens. (2) Huygens was one of the leading mathematicians of the time and he completely dismissed Leibniz’ effort.18 This blunt rebuke dissuaded Leibniz from making any further attempts to present his ideas to the public. Despite Leibniz decision not to go public, the proposed science was to become a rather well known secret by the middle of the 1700s. In the early 1690s Leibniz informed a number of influential thinkers about his analysis situs, including Arnauld, Mencke and the Marquis de l’Hospital.19 He also hints at this project in letters to Bernoulli, whose correspondence with Leibniz was published in 1745.20 In the mid-1700s great scientists (such as Lambert, Buffon, Euler) rou- 17 De Risi offers a succinct account of the Köning affaire (op. cit., 104–105). 18 “I will tell you candidly, in my opinion all these are but beautiful dreams, and I would need more proofs to believe that there is something real in what you suggest.” GM, II, 28, transl. in De Risi (op. cit., 71). 19 By the end of the 1690s the rumour of the analysis situs had astonishingly spread to China. “A zealous missionary father in China happened to be asked to illustrate the geometria situs to the Son of Heaven – or at least to some of his inquisitive eggheads.” De Risi (op. cit., 83). See Leibniz korrespondiert mit China. Ed. by Rita Windmaier. Frankfurt 1990, 87–89. 20 See De Risi (op. cit., 83). Brought to you by | Trinity College Dublin Library Authenticated | 10.248.254.158 Download Date | 9/1/14 10:26 AM Kant’s 1768 attack on Leibniz’ conception of space 155 tinely refer to the analysis situs as an idea of Leibniz’ that had not yet been realized as a science.21 In the Directions Kant explicitly mentions Buffon’s reference to the analysis situs and refers to unspecified others who also “lamented” [bedauert] Leibniz’ failure to realise it.22 It is reasonable to assume that Kant has at least Euler and Lambert in mind. It is hard to think that Kant would not be familiar with Euler’s solution to the ‘Seven bridges of Königsberg’ problem and Kant had been in correspondence with Lambert between 1765–66, just at the time when the latter had published his reference to Leibniz’ analysis situs. In the Directions Kant’s own assessment echoes the sentiment of the times, “it looks as if this discipline was never more than a thought [Gedankending] in Leibniz’s mind”.23 We might think that this casual dismissal of the analysis situs indicates that neither Kant nor his great contemporaries were familiar with the letter to Huygens. But here it must be remembered that Huygens himself, after having “attentively examined” what Leibniz had proposed to him, stated that it was only a “beautiful dream”. Kant’s comment is therefore consistent with him having read the Huygens letter and is similar to Huygens’ own reaction to the analysis situs. (3) Finally, the greatest exponent of the as yet non-existent science was Wolff. In his Elementa Matheseos universae (1713) he tells us that Leibniz had conceived of a new analysis of situation, but Wolff himself does not elaborate on the theory.24 However, in his 1715 paper Meditatio de similitudine Figurarum he presented a general definition of one of the central concepts of the analysis situs, that of ‘similarity’.25 Here Wolff explains that he had learned of this definition from Leibniz (though the analysis situs is not mentioned) “about four years ago”.26 In 21 Leonhard Euler: Solutio problematic ad geometriam situs pertinentis, 1735, 1; Johann Lambert: “De universaliori Calculi Idea”. In: Nova Acta Eruditorum, 1765, 441–473; Georges-Louis Leclerc, Comte de Buffon Histoire Naturelle générale et particulière. Paris 1740–88, IV, ix, 73. 22 GUGR AA 02: 377. 23 Ibid. 24 “In effect, what we have said about the elements of Geometry does not entirely originate from modern analysis, primarily if it depends on the situation of lines and surfaces. This is why Leibniz […] excogitated a certain new analysis of situation, constructed upon a peculiar kind of calculus (which he called calculus of situation), completely different from the calculus of magnitude that we make use of in our own analysis.” Wolff: Elementa Matheseos universae, 1713–41, vol. 1, 296 translated in De Risi (op. cit., 101–102). De Risi states that Wolff said nothing about the analysis situs beyond this quote because “Wolff himself knew nothing more.” De Risi (op. cit., 102). 25 Christian Wolff: “Meditatio de similitudine Figurarum”. In: Acta Eruditorum, May issue, 1715, 213–218. 26 C. Wolff: “Meditatio de similitudine Figurarum”, 214. Brought to you by | Trinity College Dublin Library Authenticated | 10.248.254.158 Download Date | 9/1/14 10:26 AM 156 Stefan Storrie his Vernüfftige Gedancken von Gott, der Welt und der Seele des Menschen, auch allen Dingen überhaupt [Rational Thoughts] first published in 1720, Wolff proceeds to incorporate the definition of similarity into his general ontology.27 In § 18 of Rational Thoughts he presents his definition of similarity and in § 20 he explains how similar things are distinguished. This definition and accompanying explanation is then used in his highly generalized metaphysical explanation of how there can be different kinds of things. As Wolff explains, “the similarity between essences is the ground of the kinds of things.” (§ 177) In this way the concept of similarity is a central building block in Wolff’s philosophy and this takes the use of the concept far beyond what Leibniz envisaged.28 It is clear that Kant was aware of Wolff’s use of the concept of similarity. Already in the 1764 work Untersuchung über die Deutlichkeit der Grundsätze der natürlichen Theologie und der Moral [Inquiry] Kant uses Wolff’s definition of similarity as an example of a methodological mistake in metaphysics.29 Rusnock and George (1995) have, primarily on the basis of Kant’s remarks on Wolff’s notion of similarity in the Inquiry, concluded that Kant’s argument in the Directions is aimed at Wolff’s conception of similarity and not at the distinctly Leibnizian analysis situs. On their view Kant’s admonishment of Wolff is Kant’s first attempt to explain how incongruent pairs of things can be both similar and equal (264). As they explain, Kant’s first reference to a possible problem with the LeibnizianWolffian concept of congruence is found in the Metaphysik Herder (V-Met/Herder AA 28: 15), lecture notes from Kant’s metaphysics lectures taken down by Herder between 1762–64.30 Kant’s ‘second’ attempt, that is, the Directions paper should therefore be seen as a continuation of Kant’s dispute with the Wolffian conception of similarity. This view is at odds with my conclusion in (1) and I therefore want to consider their arguments for this position in some detail. 27 Christian Wolff: Vernünfftige Gedancken von Gott, der Welt und der Seele des Menschen, auch allen Dingen überhaupt (Halle 1751). In: Kant’s Critique of Pure Reason – Background Source Materials. Transl. Eric Watkins. Cambridge 2009. 28 Leibniz notices Wolff’s wide application of his definition already in a letter dated the 2d of April 1715. Referring to Wolff’s 1715 article in Acta Eruditorum Leibniz remarks, “I have noticed from the Acts that, having accepted certain things from me, specially things concerning the definitions of similitude and of use, and things concerning the analysis of axioms into identical propositions and definitions, you have used them in your own distinguished work, which pleases me so much that I offer you my thanks on that account.” Roger Ariew and Daniel Garber: G.W. Leibniz: Philosophical Essays. Indiana 1989, 230. 29 Though the Inquiry was published in 1764 it was written and submitted for the prize-essay competition organised by the Prussian Royal Academy at the latest on the 31st of December 1762. 30 “[Objects] equal and similar are not congruent unless they lie in a plane.” Translated in Rusnock and George (1995, 263). Brought to you by | Trinity College Dublin Library Authenticated | 10.248.254.158 Download Date | 9/1/14 10:26 AM Kant’s 1768 attack on Leibniz’ conception of space 157 They present an auxiliary, historical argument and a main, textual argument for their view. The auxiliary argument is supposed to bolster both the claim that Kant’s target in the discussion of congruence in the Directions is Wolff and the claim that the relevant discussion of Wolff’s theory of similarity in the Inquiry concerns the issue of incongruence. They suggest that Kant discovered the existence of incongruent counterparts at the time when he was writing the Inquiry. With this discovery fresh in mind it would then be natural for Kant to attack the notion of congruence in terms of similarity and equality in the Inquiry.31 If Kant is attempting to use his discovery of incongruence to attack Wolff’s concept of similarity in the Inquiry, then the relevant Metaphysik Herder note must have been written some time prior to 31 December 1762, which is when Kant submitted the Inquiry for the prize competition. But that is far from certain. As Herder attended Kant’s 1762 summer semester class towards the end of the course and the discovery concerning incongruence was made in connection to an entry in the beginning of Baumgarten’s book it is implausible that Herder heard of Kant’s reservation about congruence that term. Herder could have heard of this early in the winter semester 1762/1763 and so shortly before the Inquiry was submitted. However, Kant could also have ‘noticed the problem’ about incongruence in any of the subsequent lecture courses on metaphysics that Herder attended. That is, Kant could have made this discovery in the 1763/1764 winter semester, 1764 summer semester or 1764/1765 winter semester.32 Accordingly, more work on the dating of particular sections of the Metaphysik Herder must be done before we can state with any certainty whether Kant made his discovery of incongruent counterparts before or after submitting the Inquiry. Rusnock’s and George’s main argument turns on their interpretation of the argument against Wolff’s definition of similarity that Kant presents in the Inquiry. On their view the argument is directed against the view that congruence is exhaustively defined in terms of equality and similarity. However, this requires a 31 It is not entirely clear if Rusnock and George want to claim that Kant definitely discovered the problem of incongruence while writing the Inquiry or whether they regard this as a mere possibility. A claim in the main body of their text suggest the former, “Kant noticed the problem that had escaped Euclid, Leibniz, and Wolff around 1762/3.” (1995, 263) In a footnote their use of the word “may” suggest the latter, “Herder attended his first lecture on 21 August 1762, when Kant was finishing his metaphysics course with reflections on witches, kobolds, and poltergeists (§ 796 of Baumgarten’s Metaphysica, AA 28: 148). Incongruence is treated in § 70 of Baumgarten, and may have come up later in 1762, in the next cycle of lectures. At that time Kant was writing the Prize Essay.” (1995, 236n30) 32 See Steve Naragon: “The Metaphysics Lectures in the Academy Edition of Kant’s gesammelte Schriften”. In: Kant-Studien 91, 2000 (Sonderheft), 189–215, for details of Herder’s attendance at Kant’s lectures on metaphysics. Brought to you by | Trinity College Dublin Library Authenticated | 10.248.254.158 Download Date | 9/1/14 10:26 AM 158 Stefan Storrie reading of Kant’s pre-critical conception of the method of ‘synthesis’ that goes well beyond Kant’s explicit argument against Wolff in the Inquiry. Kant’s 1763 paper is an essay on philosophical method. The main contrast in this work is between the methodology proper to philosophy, which Kant calls ‘analysis’, and the method proper to mathematics, which he calls ‘synthesis’. Analysis and synthesis are for Kant two different ways that one can arrive at knowledge of a general concept. The characteristic mark of the analytic method is that “the concept of the thing is always given”.33 The task for the philosopher is then to separate out the essential constituents of the concept so that we can arrive at an adequate definition of the things (UD, AA 02: 276–277). The synthetic method, on the contrary, is that “the concept which I am defining is not given prior to the definition itself; on the contrary, it only comes into existence as a result of that definition.”34 The concepts that come about by synthesis are said to be the product of an “arbitrary combination” [willkürliche Verbindung].35 As an example of such a concept construction Kant says, “think arbitrarily of four straight lines bounding a plane surface so that the opposite sides are not parallel to each other. Let this figure be called a trapezium.”36 Here the concept of ‘trapezium’ is first generated through the synthesis of thinking arbitrarily of four lines in accordance with a rule. In this way the mathematician creates the concept through a construction of the object. Kant’s argument against Wolff in this context is that he attempted to give an analytic definition of the concept of similarity in geometry (UD, AA 02: 277). This, on Kant’s view, is a methodological confusion. Instead, Kant argues that a synthetic definition is sufficient for all purposes. In the case of similarity the synthetic method for reaching a definition of the concept would be to, [T]hink of figures, in which the angles enclosed by the lines of the perimeter are equal to each other, and in which the sides enclosing those angles stand in identical relations to each other – such a figure could always be regarded as the definition of similarity between figures, and likewise with the other similarities between spaces.37 33 UD, AA 02: 276: “Es ist hier der Begriff von einem Dinge schon gegeben”. 34 UD, AA 02: 276: “Der Begriff, den ich erkläre, ist nicht vor der Definition gegeben, sondern er entspringt allererst durch dieselbe.” 35 Ibid. 36 UD, AA 02: 276: “Man gedenkt sich z. E. willkürlich vier gerade Linien, die eine Ebene einschließen, so daß die entgegenstehende Seiten nicht parallel sind, und nennt diese Figur ein Trapezium.” 37 UD, AA 02: 277: “[…] wenn ich mir Figuren denke, in welchen die Winkel, die die Linien des Umkreises einschließen, gegenseitig gleich sind, und die Seiten, die sie einschließen, einerlei Verhältniß haben, so kann dieses allemal als die Definition der Ähnlichkeit der Figuren angesehen werden, und so mit den übrigen Ähnlichkeiten der Räume.” Brought to you by | Trinity College Dublin Library Authenticated | 10.248.254.158 Download Date | 9/1/14 10:26 AM Kant’s 1768 attack on Leibniz’ conception of space 159 Rusnock and George take Kant’s point here to be the following. First they state that according to Kant there is “a general concept or family of concepts of similarity in geometry” (263). They continue by stating that the synthetic or “cumulative” (263) procedure for reaching this general concept or family of concepts is by giving accounts of similarity of different kinds of figures. Kant can then tackle the issue of similar and equal figures not being congruent. The ‘solution’ is to state that there is no general concept of similarity; therefore there cannot be a theorem that similar and equal things are congruent. This theorem, as Kant states in the relevant passage from the Metaphysik Herder is true in some cases (that is, in plane geometry) but not in others (geometry of solids). This rather complex solution reads in more than Kant actually states in his argument. The idea of generating general concepts or family resemblances between concepts by a cluster of definitions is entirely lacking from Kant’s general account of the synthetic method and from the particular examples of the method in use. The synthetic method is not cumulative in this sense at all. What are generated by the synthetic method in geometry are not groups of definitions but diagrammatical representations of geometrical objects that are adequate to the definition. For example, in the case of similarity Kant explains that the definition should be a figure displaying two geometrical objects that are identical in certain ways. For Kant this one figure (which could be constructed in any number of ways and that is therefore ‘arbitrary’) “could always be regarded as the definition of similarity between figures”. Kant does not say anything about the value of having a large number of definitions for similarity, or to have different definitions for similarity in plane and solid geometry, respectively. Instead, Wolff’s mistake is to take a definition of similarity in general and apply it to geometry. In Kant’s words, “The general definition of similarity is of no concern whatever to the geometer.”38 In other words, the generality of the concept is such that it is impossible to generate the concept through construction and therefore it has no role to play in any geometrical procedure. Accordingly, Kant’s synthetic definition of similarity does not address the issue of incongruent counterparts. Therefore there is no reason to believe that his dissatisfaction with Wolff’s definition of similarity is connected with Kant’s discovery of incongruent counterparts (even though he might have made the discovery around this time) or with his argument against the analysis situs. 38 UD, AA 02: 277: “Dem Geometra ist an der allgemeinen Definition der Ähnlichkeit überhaupt gar nichts gelegen.” Brought to you by | Trinity College Dublin Library Authenticated | 10.248.254.158 Download Date | 9/1/14 10:26 AM 160 Stefan Storrie 4 Inner ground and creation in Kant’s ‘lone hand’ argument Kant’s second stage in his criticism of Leibniz’ conception of space is directly concerned with the nature of space itself. Kant presents a thought experiment (that I will call his ‘lone hand’ argument) that aims to show an absurdity in Leibniz’ relational theory of space. Kant presents the thought experiment with the lone hand in the first paragraph on page (UGUR, AA 02: 383). However, in the last paragraph on the previous page Kant has indicated that he is done with his preliminary part of the essay, where it is shown that incongruent counterparts are possible and that their difference lies in the primitive spatial property of direction. In this paragraph he will begin with the “philosophical application of these concepts [similarity, equality and congruence]”.39 At this point Kant introduces the new idea of an “inner ground” [inneren Grunde] and this is crucial for the argument in the next paragraph as it supplies the key insight for the most forceful interpretation of the ‘lone hand’ argument.40 Kant explains that the difference between two incongruent counterparts exists independently of these objects’ relation to other objects. Instead, as the difference depend on how they are originally placed (through a process of generation, creation or manufacture), the difference between pairs of incongruent counterparts can be characterised as an “inner difference” [innerer Unterschied].41 In the earlier part of the Directions Kant noted that “the cause of the curvature [from east to west or west to east] of the natural phenomena just mentioned [hair on the crown of a human head, hops, beans and snail shells] is to be found in the seeds themselves.”42 In modern terms this just means that the direction of these things depends on biological generative processes. For example, the cells of the organism are built up according to certain instructions in the organism’s DNA. If the DNA contained instructions that were different in specific ways or if it was interpreted in a different way, then the relevant part the organism would curve in the opposite direction. Similarly, the manufacture of gloves and screws depends on how the various instruments are applied to the material out of which the object is made. If the instruments were applied in a quite different way, the result would be a glove or screw that would be directed in the opposite direc- 39 GUGR, AA 02: 382: “philosophische[.] Anwendung dieser Begriffe”. 40 Ibid. 41 Ibid. 42 GUGR, AA 02: 380: “[…] bei den angeführten Naturproducten die Ursache der Windung in den Samen selbst liegt.” Brought to you by | Trinity College Dublin Library Authenticated | 10.248.254.158 Download Date | 9/1/14 10:26 AM Kant’s 1768 attack on Leibniz’ conception of space 161 tion. In all cases, the object’s direction depends on which one of two distinctly different methods of its production is used. Kant generalises this claim about the connection between inner grounds for spatial differences in objects and differences in creative causes. In Kant’s words, “The action of the creative cause in producing the one would have of necessity to be different from the action of the creative cause producing its counterpart.”43 Kant new point here is that not only are incongruent counterparts different in ways that can be empirically tested; they are also necessarily created in different ways. Therefore, if there is an inner ground for the difference between objects then the act of making them must also necessarily be different. Kant then applies this finding to the case of an act of divine creation. If the first thing created was a human hand, it would be either a right or a left directed hand. As the difference between pairs of incongruent counterparts is based on a difference in the cause of their creation, the act of creating one hand is a different act from that of creating the other. Next, Kant puts forward his much discussed argument against the relational concept of space “entertained by many modern philosophers, especially German philosophers”.44 Kant characterises the relational theory as that which takes the nature of space to consist of “the external relation of the parts of matter which exist alongside each other”.45 If this theory is correct then the fact of incongruent counterparts must be explained by an appeal to the external relation between different objects or parts thereof. As these relations are identical for right and left hands, the hand created must on the relational view be indeterminate with regards to its direction. Kant concludes the argument by stating that “[i]n other words, the hand would fit equally well on either side of the human body; but that is impossible.”46 As a first step for assessing this argument I will note that there are philosophical, textual and historical reasons to see it as a response to a particular argument for the relational theory of space that Leibniz presents in the Controversy between Leibniz and Clarke (Controversy).47 Kant’s thought experiment describes a situ- 43 GUGR, AA 02: 382–383: “[…] ist es nothwendig entweder eine Rechte oder eine Linke, und um die eine hervorzubringen, war eine andere Handlung der schaffenden Ursache nöthig, als die, wodurch ihr Gegenstück gemacht werden konnte.” 44 GUGR, AA 02: 383: “[…] den Begriff vieler neueren Philosophen, vornehmlich der Deutschen”. 45 GUGR, AA 02: 383: “[…] der Raum nur in dem äußeren Verhältnisse der neben einander befindlichen Theile der Materie bestehe […].” 46 GUGR, AA 02: 383: “[…] d.i. sie würde auf jede Seite des menschlichen Körpers passen, welches unmöglich ist.” 47 Loemker: Op. cit., 675–721. Brought to you by | Trinity College Dublin Library Authenticated | 10.248.254.158 Download Date | 9/1/14 10:26 AM 162 Stefan Storrie ation where the universe has a particular shape as opposed to its incongruent counterpart. In the Controversy Leibniz’ and Clarke’s shared assumption of the irrelevance of the directionality of the shape of the universe is the occasion for one of Leibniz’ arguments (which I will call the ‘mirror world’ argument) against the absolute view. Kant, like virtually all philosophers and scientists at the time was well acquainted with the issues raised in the Controversy, particularly the debate about the nature of space and time.48 Kant is most probably referring to the Controversy at the beginning of the Directions when he motivates his choice of method for proving the existence of absolute space, the “so to speak a posterior method”, on the ground of the failure of attempts to settle the question with metaphysical arguments. “Everybody knows how unsuccessful the philosophers have been in their efforts to place this point [the existence of absolute space] once and for all beyond dispute, by employing the most abstract judgments of metaphysics.”49 The only reasonable referent to a dispute about which “Everybody knows” in this context is the Controversy. In his third letter of the Controversy Leibniz presents his mirror world argument against “those who take space to be a substance or at least an absolute being.” (Controversy, 682) The argument employs the principle of sufficient reason and claims to derive an absurdity out of the absolute conception of space. If space was an absolute being, there would be something happen for which it would be impossible there should be a sufficient reason. Which is against my axiom. […] Space is something absolutely uniform, and, without the things placed in it, one point of space does not absolutely differ in any respect whatsoever from another point of space. Now from hence it follows (supposing space to be something in itself, beside the order of bodies among themselves) that ‘tis impossible there should be a reason why God, preserving the same situations of bodies among themselves, should have placed them in space after one certain particular manner and not otherwise; why everything was not placed the quite contrary 48 Kant referred to the Controversy already in his 1755 work Principiorum primorum cognitionis metaphysicae nova dilucidatio [New Elucidation]. In his discussion of Leibniz’ principle of the identity of indiscernibles he mentions the fact that no two leaves from the same tree are completely alike. This is most certainly a reference to Leibniz’ delightful anecdote in the Controversy about an “ingenious gentleman” of his acquaintance who tried but failed to disprove Leibniz’ principle by finding “two leaves perfectly alike” (Controversy, 687). In his private notes Kant later, sometime in 1775–1777, referred specifically to Leibniz’ and Clarke’s dispute about the nature of space and time (Refl, AA 17: 699). As an indication of the general familiarity with the Controversy in Kant’s time, see Lambert’s casual reference to it in his letter to Kant, dated Oct. 13, 1770 (Br, AA 10: 108). 49 GUGR, AA 02: 378: “Jedermann weiß, wie vergeblich die Bemühungen der Philosophen gewesen sind, diesen Punkt vermittelst der abgezogensten Urtheile der Metaphysik einmal außer allen Streit zu setzen […].” Brought to you by | Trinity College Dublin Library Authenticated | 10.248.254.158 Download Date | 9/1/14 10:26 AM Kant’s 1768 attack on Leibniz’ conception of space 163 way, for instance, by changing east into west.50 […] Their [the two systems of similarly placed particles of matter] difference therefore is only to be found in our chimerical supposition of the reality of space in itself. But in truth the one would be exactly the same things as the other, they being absolutely indiscernible […]. (Controversy, 682) The argument could be represented in the following way. L1. Nothing happens without a sufficient reason. L2. There is no difference between this world and a reflection of this world where east has become west and vice versa. L3. If absolute space exists, then God can create either this world or its reflection. L4. When God creates this world something would happen without a sufficient reason. This argument aims to show that the existence of absolute space entails that there is an absurdity in the very creation of the world. God has created this world rather than its mirror reflection for no sufficient reason. In one way the discussion on the nature of space between Leibniz and Clarke is disappointing. The reason for this is that Clarke’s arguments against the relational view simply presuppose the absolute view. The clearest case is Clarke’s third reply, where he attempts to reply specifically to Leibniz’ mirror world argument. Clarke claims that the relational view entails an absurdity. If the totality of objects that make up the world were created some distance from where it is now, but maintaining the same relations between each other, then the relational view must hold that they would be created in the same place as they are now. This is absurd because then the relationalist must say both that the alternative world would be created in a different place (as stipulated by Clarke’s argument) and that it was created in the same place at the actual world (as the relational view entails) (Controversy, 685). The problem with this argument, as Leibniz points out, is that the relationalist position does not accept what Clarke stipulates. For the relationalist the two alternative worlds are the same world, so one cannot say that 50 As Jeremy Byrd: “A Remark on Kant’s Argument From Incongruent Counterparts” in British Journal for the History of Philosophy 16, 2008, 789–800, 792f. notes, to change “east into west” could be interpreted in two ways. Either it could refer to a 180 degree rotation of the universe along one axis of the three dimensions of space. Or it could mean the recreation of the universe into a mirror image of itself. Leibniz’ formulation and use of the argument allows for either interpretation. However, if the universe has the shape of a hand, the former would merely represent the movement of a hand whereas the latter would be changing the ‘handedness’ of the hand which is the basis of that argument. I will therefore only consider the latter conception of changing “east to west”. Brought to you by | Trinity College Dublin Library Authenticated | 10.248.254.158 Download Date | 9/1/14 10:26 AM 164 Stefan Storrie one would be created at a distance from the other (Controversy, 687). Therefore, Clarke’s argument presupposes an absolute space in which it is possible for either world to be placed, in order to arrive at the alleged absurdity. Instead, the more fruitful part of Leibniz’ and Clarke’s discussion about this argument centres on the status of the principle of sufficient reason. This principle states “that nothing happens without a reason why it should be so rather than otherwise” and it is a metaphysical principle that is necessary for “natural philosophy” (Controversy, 677). Clarke repeatedly states (Controversy, 680, 684) that he agrees with Leibniz that nothing happens without a sufficient reason but qualifies this by stating that the will of God by itself can be the sufficient reason. Clarke argues that if God could not determine a course of action by his mere will alone, then God would to this extent be restrained and dependent on outside influences. God would be as determined in his choices as the movement of a scale is determined by the weight placed upon it. (Controversy, 680) According to Leibniz, Clarke’s position means that he in fact rejects the principle of sufficient reason. For if God makes a choice dependent on his will alone where there is equal good reason to do two different things then “God wills something without any sufficient reason for his will, against the axiom or the general rule of whatever happens.” (Controversy, 683) Leibniz continues by stating that to maintain that God acts according to reason is to hold that God acts in a way agreeable to his wisdom and not to introduce blind fatality or necessity devoid of all wisdom. (Controversy, 683) The dispute over this issue was then repeated in the subsequent letters. I now want to consider how the lone hand argument can work against the mirror world argument. One suggestion has recently been put forward by Byrd (2008). On his view Kant claims to have shown that “there are possible worlds which differ only in being mirror reflections of one another, such as the imagined universes occupied by a lone left or right hand.” (794). On Byrd’s view the difference between incongruent counterparts is not enough to count as a sufficient reason for God to create one world rather than its mirror image (so Byrd does not take Kant to reject L2). He further claims (795) that Kant would not opt for Clarke’s response, that God’s will is a sufficient reason for why one world is created rather than another (so Byrd does not take Kant to argue for the existence of absolute space by modifying L1). He therefore concludes (796) that Kant completely rejected the principle of sufficient reason as a substantive metaphysical principle at the time (so according to Byrd Kant rejects L1). This interpretation is problematic for two reasons. Firstly, it does not square with Kant’s argument in the first two thirds of the Directions, which aims to show that there are significant and real differences between incongruent counterparts. It would be expected from Kant’s avowal to then turn to the “philosophical appli- Brought to you by | Trinity College Dublin Library Authenticated | 10.248.254.158 Download Date | 9/1/14 10:26 AM Kant’s 1768 attack on Leibniz’ conception of space 165 cation” of incongruence that the significance of these differences be used in his argument. As Kant has shown in his argument against the analysis situs, pairs of incongruent counterparts have real and significant differences so that a universe and its incongruent counterpart would be significantly different and therefore that there would be a sufficient reason for God to create one rather than another. We should therefore expect Kant to reject the premise L2. Secondly, in Kant’s lone hand argument he claims that if the universe were comprised of a single hand it would have to be a left or a right hand. Again, it is the specific difference between the left and the right directed object that drives Kant’s argument, not a rejection of a metaphysical principle. The question is then what Kant’s best argument against L2 is and how Leibniz could respond to it. We should therefore ask what difference between a left and a right directed universe Kant takes to be important in his argument and consider how effective such argument would really be against Leibniz view. I take there to be two alternatives that can be gathered from Kant’s Directions paper. (a) There is a difference in the ‘effect’. That is to say, the created objects can be shown to spatially relate to other objects in different ways. (b) There is a difference in the ‘cause’. In other words, the creative act that brings about one object is different from the creative act that brings about the other. Considering (a) this interpretation presents itself quite naturally. Kant suggests that his lone hand argument should be interpreted in this way when he introduces an empirical test similar to fitting a glove on a hand when he claims that a direction-neutral hand would fit equally well on either side of the human body. But if this is Kant’s argument then it would hardly trouble Leibniz. Leibniz could plausibly argue for a disanalogy between left and right hands and left and right directed universes. In other words, he could accept Kant’s criticism of the analysis situs and accept that for any incongruent object in space its mere directionality makes it spatially different from its counterpart and still hold that in the case of the entirety of space things are different. Leibniz could do this because on his relational view of space there is literally no room for the kind of empirical test suggested by Kant. To project in imagination a glove that will fit a right hand shaped universe but not a left, requires that we conceive of a space outside of the totality of matter in the hand-shaped universe. Any conceivable test of the kind Kant proposes for distinguishing incongruent counterparts presupposes that we think of what is stipulated as the whole of space as a mere part of a larger space. For either Kant must hold that what we are conceiving is absolute space in which the hand shaped universe is placed in which the totality of matter is a mere part; and therefore assume the existence of absolute space in the same question begging way that Clarke did in the Correspondence. Or Kant must hold that in the thought experiment there is more relative Brought to you by | Trinity College Dublin Library Authenticated | 10.248.254.158 Download Date | 9/1/14 10:26 AM 166 Stefan Storrie space than the space occupied by the lone hand. In this case there are additional spatial relations introduced by the object by which we test the handedness of the hand, for example, a glove. But this is to directly contradict one of the premises of the lone hand argument. Instead (b) is the interpretation of Kant’s argument that gives him the most powerful reply to Leibniz’ argument. Even though the difference between a right and a left directed universe could not be shown by empirical experiments Kant could say that these universes require different acts of creation. Kant could, in accordance with his “so to speak a posteriori” method, again appeal to the many examples of incongruent counterparts that he has mentioned in the Directions and state that they are made in different ways. Since this is the case for material objects in space, why would it not be the case for the totality of matter in space? The features that made Leibniz’ rebuttal of (a) powerful are not present in this case. To speak of a difference in creative cause presupposes neither absolute space nor any spatially external object. Instead the difference appeals to the presumably unextended event of creation, which could perhaps be described as a decision, process or act, of God. Of course, a difference of kind (b) leads to the difference appealed to in (a). This two part structure of the argument accords with Kant speaking of the (b) type difference in the paragraph before the paragraph with lone hand argument, and of the (a) type difference in that paragraph. If Kant’s argument is understood in this way Leibniz would have to argue that the creative cause for producing a right and a left handed universe is the same if he is to save L2. But then he would have to explain why different creative causes are required in the cases Kant alerts us to in the first part of the Directions and not for God’s creation. Such an explanation would presumably involve the claim that spatial relations are merely ideal, man made relations; whereas God’s creations are intelligible entities that we cannot properly grasp.51 Even if such an account is coherent on its own terms, it requires Leibniz to step out of the cosy domain of ‘natural philosophy’ and the common sense sounding principle of sufficient reason that Kant and Clarke are happy to engage with and instead enter into his rather exotic metaphysics, where few are willing to follow.52 51 See Controversy, 703, 713f. for Leibniz’ claim that space is something merely ideal. 52 My work on this paper was supported by the Irish Research Council for the Humanities and Social Sciences. I would like to thank Lilian Alweiss and Manfred Baum for commenting on some ideas in this paper. I would also like to thank the two anonymous reviewers for Kant-Studien for helpful comments. Brought to you by | Trinity College Dublin Library Authenticated | 10.248.254.158 Download Date | 9/1/14 10:26 AM