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Leibniz’s Science of Forms as a Structural Science and Similarity as its Central Concept1. Oscar M. Esquisabel UNQ-CONICET-UNLP
Introduction
This paper is concerned with an aspect of combinatory science or art2 that, in our opinion, has so far been relatively neglected in the attempts to clarify the nature and goals of this science. This aspect is the presentation of combinatory science as the science of the similar and dissimilar, which nevertheless plays a central role in the determination of its function and scope, particularly in its connection with the other sciences and with the leibnizian program of a new kind of ars inveniendi. Usually, combinatory science is associated with two main ideas: on the one hand, wiht the analysis and recombination of concepts, and, on the other, with the assignations of characters or signs to those concepts, in order to obtain their possible combinations through operations with signs. Moreover, combinatory mathematics helps us to carry out in a systematic way the calculation of combinations. Therefore, we have an orderly method of finding new truths and of judging or evaluating the truths that we possess beforehand. The fact that combinatory science operates by means of characters gives to it a “symbolic” dimension. In this way, it becomes for Leibniz not only a science that manipulates signs, but also a science that has complete symbolic systems as objects. Hence, combinatory science is promoted to the status of a characteristic combinatory science or characteristic combinatorics simpliciter. Its scope is so general that all the other sciences are subordinated to it. In words of Leibniz:
“There are other completely different kinds of analysis or combinations, about the order of things, about qualities, actions and an infinity of other things, which have to be expressed in completely different ways (but also through letters or other simple signs) other than the quantities [...] But the universal analysis depends on a universal character [...] Hence, the great utility of analytical or combinatory art will be apparent, once it is carried out, when the formulae and equations that now represent only numbers, lines and other sterile and dry things will show the spaces and movements, the times, the forces and effects [...]”3
1 This work has been founded by the research projects PIP 5615 (CONICET), PICT Redes 2006 nº 2007 (ANPCyT) and 11/H 378 (UNLP). 2 We leave open the important question about the distinction between these two characterizations for the combinatory science. Nevertheless, we should differentiate what is the combinatory (either science or art) from the science of combinations, which pertains to the domain of mathematics. 3 Analysis ad Alias Res quam Quantitates Applicata, A VI 3 413-414. For the works of Leibniz we use the following abbreviations: A: Leibniz, G.W., Sämtliche Schriften und Briefe, ed. by the Preussische Akademie der Wissenschaften (after 1945 the Deutsche Akademie der Wissenschaften and in the last times the Berliner Akademie der Wissenschaften), Darmstadt, Leipzig and Berlin 1923-1999. Cited by series, volume and page.GM: Leibnizens mathematische Schriften, ed. by C.I. Gerhardt, Berlin, A. Asher, and Halle, H.W. Schmidt, 1849-1863. Cited by volume and page (Reprint by Georg Olms, 1971). 2
Indeed, we have presented a rough description of what combinatory or combinatorics intends to be. In addition, we must confess that this characterization raises a lot of questions with which we can not deal here. What’s more, there is a presentation of the combinatory science that was passed over in this description. In fact, we refer to the combinatory science as the science of form or, what seems to be the same for Leibniz, the science of the similar and dissimilar. In a letter to Tschirnhaus from May 1678, Leibniz makes a difference between combinatory mathematics and combinatory art. About the latter, he says:
“But as for me, combinatory art is a very different thing, namely the science of forms, that is, of the similar and dissimilar, in the same way as algebra is the science of magnitude or of the equal and unequal. Indeed, combinatory art seems to differ scarcely from the general characteristic science, with whose help are or can be devised characters that are appropriate for algebra, for music and even for logics”.4
The recurrent characterization of combinatory art as the science of form or of the similar and dissimilar5 allows us to think that the concepts of form and similarity are at the core of the design of this science. Notwithstanding, although much attention has been paid to the combinatory and symbolic aspects of combinatory science, not much has been said in connection with this rather enigmatic association with similarity6. In spite of this, we postulate that there is a strong link between the idea of similarity and form on the one hand, and the symbolic nature of combinatory science. Moreover, the generality that Leibniz claims for combinatory science arises precisely from this link. Thus, if our interpretation is right, the representation of combinatory science perhaps must be changed: it has to be taken as a formal science rather than as a material science. In this sense, it should be a science that compares structural properties of different objective domains and abstracts what they have in common, in order to exhibit those results through general formulae. Similarity constitutes the filum Ariadnae to discover structural analogies that not always are before the eyes. Hence, a way to clarify the structural nature of the science of forms can be found in
4 Leibniz to Tschirnhaus, 1678, A II 1 412. 5 A VI 4 346, A VI 4 545, GM VII 159, A VI 4 723, VE 6 1335-1336. 6 Herbert Breger analyzes the concept of similarity in “Der Ähnlichkeitsbegriff bei Leibniz”, in: Mathesis rationis. Festschrift für Heinrich Schepers, Münster, 1990, 223-232, but not in connection with the science of the forms. Martin Schneider deals with similarity as a main concept of the general mathematics and of the combinatory science that –in opinion of the author- is subordinated to the general mathematics. We are strong indebted to the analysis of Prof. Schneider, although we maintain a different point of view about the relations between general mathematics and the science of the forms. See Martin Schneider, “Funktion und Grundlegung der Mathesis Universalis im Leibnizschen Wissenschaftsystem”, in: Albert Heinekamp (ed.), Leibniz: questions de logique. Symposion organisé par la Gottfried-Wilhelm-Leibniz-Gesellschaft e.V. Hannover, Studia Leibnitiana, Sonderheft 15, 1988, 162-182. Sybille Krämer refers also to the concept of similarity in relation with combinatory science, but her interpretation restricts the application of the similarity to the manners in which signs are combined. See Sybille Krämer, Berechenbare Vernunft. Kalkül und Rationalismus im 17. Jahrhudert, Berlin, 1991, 276-278. Nevertheless, in other places, she moves closer to our interpretation. See op. cit., 346-347. 3 the principle that allows structural comparisons, and that is the concept of similarity. We follow then its path, in order to establish some basic concepts.
1. Similarity and geometry. The “procedural” definition.
Before we examine the concept of similarity, we have to differentiate between two concepts of it that can be found in the Leibniz’s treatment of these questions. Similarity can be understood as empirical or sensuous similarity, the mere qualitative resemblance or likeness that a thing maintains with another (in colour, for example). On the other hand, similarity can be considered as the relation of structural analogy that holds between two things that do not resemble (in the first sense) each other. In the following pages, we restrict our analysis to the second sense of similarity. Moreover, it is even possible that Leibniz considered this sense as the fundamental one7. Leibniz perceived at an early stage the importance of the concept of similarity for the foundation of the Ars inveniendi. Thus, he says in De Arte Combinatoria, in speaking of Bisterfeld’s work Phosphorus Catholicus, that this autor had already seen the scope of similarity and dissimilarity, whose principles are found in relationships, for the art of meditation8. But the full realization of the concept came to Leibniz when he acquired a deepest mathematical knowledge during his stay in Paris. It was then that he understood that the utilization of similarity makes it possible to determine formal general laws that at the same time will give a methodology for invention. For this reason, he considered it necessary to formulate a general concept of similarity. His efforts in this direction were stimulated by his desire to improve geometrical demonstrations, so his point of departure was a criticism to the usual notion of similarity as used in geometry, whose definition lacked, according to Leibniz, the required generality. In this way, Leibniz prepared himself to make the step from a concept of similarity based on geometrical intuition to a more formal and abstract concept of it, by virtue of which similarity finds its general application. A first formulation of the concept of similarity can be found in a letter (not sent) to Gallois from 1677, where he defines it as discernibility only by co-presence or co- perception:
“After having searched carefully, I have found that two things are perfectly similar, when they could be discriminated by no other means than by per compraesentiam, for example, two unequal circles of the same material could only be discriminated by seeing them conjointly, since in this way it can be seen that one is bigger than the other” 9
From this definition there follow, in Leibniz’s words, consequences of large scope for metaphysics as well as for mathematics. Leibniz emphasizes particularly this
7 We owe this distinction to Herbert Breger, cfr. Breger, Herbert, “Der Ähnlichkeitsbegriff bei Leibniz”, in Mathesis rationis. Festschrift für Heinrich Schepers, Nodus Publikationen, Münster, 1990, 224. See too Poser, Hans, “Vom Denken in Analogien” Berichte zur Wissenschaftsgeschichte, 12, 1989, 147-148. 8 A VI 1 199, cfr. Massimo Mugnai, “Der Begriff der Harmonie als metaphysische Grundlage der Logik und Kombinatorik bei Johann Bisterfeld und Leibniz”, SL V 1 43, 1973. 9 Leibniz to Gallois, 1677, GM I 180. 4 proposition: “if two things are similar according to an operation or consideration, they are also similar according to all the others”10. Indeed, it is hard to understand the connection between this concept of similarity and the concept required by the combinatorial science, since at least in this formulation there appears to be a notion that applies only to geometrical structures. At first sight, this definition can be interpreted as an empirical and psychogical characterization of similarity, since the definition makes the similarity too dependent on the factual conditions of perception, imagination and memory. Anyway, since this concept is based on operations of comparing, let’s call it “procedural concept” of similarity.
3. The “substantive concept” of similarity.
But if we examine closer the assumptions of the definition, we find that it is based on a set of concepts through which Leibniz tries to overcome the restriction to merely psychological and empirical conditions. In this sense, he construes an argument in order to emphasize that undiscernibility by imagination and memory are grounded on the very nature of the geometrical object. The outline of the argument supposes a complete reduction of the world dimensions, including the observer as such, so that it is impossible for him to discern the difference between the state prior to the reduction and the state after it. We can find this argument in an essay on geometrical characteristics from 1679 (there is an outline too in the cited letter to Gallois):
“But if we imagined that God has reduced everything that is presented to us and around us in a room, maintaing the same proportions, everything would appear to us in the same way and we could not discern the former state from the latter state, unless we left the field of the proportionaly reduced things, that is, our room, since then the difference would be apparent, by doing the co-perception with the things that were not reduced”.11
The impossibility to discern arises from the fact that, since the internal relations of the world remain the same, it is not possible to establish differences of magnitude between the former and the latter states. This was only possible through conjoint perception of both states or through intermediation of a common measure that would maintain its size in spite of the reduction. To summarize, since the undiscernibility is connected with the structural identity of the geometrical object, similarity is based not on the way we exercise our cognitive faculties, but on the structure of things. Discernibility by co-perception becomes a requirement for similiarity as far as magnitude is an external feature which is known by perception:
“[Magnitude]...is what can be distinguished in things only through co- perception, that is, by the application either immediate (through an actual
10 Leibniz to Gallois, ibidem. 11 Characteristica Geometrica, 1679, GM V 154. 5
congruence or through coincidence) or mediate, namely, by intervention of a measure that is applied first to one and then to the other”12.
Notwithstanding, the concept of similarity understood as discernibility by co- perception possesses still a procedural nature and, as such, it is not primary, but it depends on more elaborate concepts. As in some way it was anticipated by the leibnizian argument of the total reduction of the world, there is a substantive notion of similarity that departs from the idea of the formal identity of the things that are similar. To differentiate this concept from the notion of similarity based on co-perception, that has a procedural nature, let’s call it “substantive concept” of similarity. This “substantive concept” of similarity which allows us to make the transition to the general notion of similarity expresses the fact that similar things have the same intrinsic properties:
“Similar are these things that could not be discerned, if each one of them is considered separately and by itself, such as two equilateral triangles: in fact, we can not find any attribute, any property in one of them that could not be found also in the other [...] But if they were perceived conjointly, then there would appear the distinction that one is bigger than the other [...] Hence, I usually say that similar things are only discerned by co-perception [...]”13
Therefore, similarity as undiscernibility by co-perception, an procedural concept, must be retraced to more basic conditions that are linked to the intrinsic properties of the similar things. From this point of view, it can be said that similarity entails identity of form. Hence, an elucidation of the concept of similarity compels us to clarifying the notion of form. To this conclusion arrives Leibniz in another essay on geometrical characteristic, De Analysi Situs:
“[...] Consequently, it is not sufficient to say that similar are the things whose form is the same, unless we have the general notion of form. It is for me an assured issue that, once the elucidation of quality or form is established, the question is finally reduced to this: similar are those things that can not be discerned if they are contemplated separately [...]”14
In this way, the substantive concept of similarity depends on concepts such as form, intrinsic property, consideration by itself and identity. In fact, similarity as discernibility by co-perception appears as the result or consequence of the impossibility to discern two geometrical forms, when it is found that their intrinsic properties are the same when considering each of them separately by itself. In this way, similar things are those that possess the same form. In short, the explication of the substantive concept of similiarity confronts us with the elucidation of the notions that are included in its formulation.
12 Characteristica Geometrica, ibidem. It is pertinent to point out that there is a remarkable difference between form and magnitude. Magnitude pertains to perception, so that it can be established and discriminated only be perceptive comparison. On the contrary, form can be known by itself, without comparison. See Definitiones: Ens, Possibile, Existens, A VI 4 868. 13 Characteristica Geometrica, GM V 153. 14 De Analysi Situs, GM V 180. 6
Nevertheless, when we try to analize these notions, we realize that the attempt to separate the procedural notion from the substantive notion seems at first sight doomed to failure. In fact, Leibniz follows two argumentative strategies to explain these conceptual components. According to one of them, the determination of concepts such as form, quality and qualitative identity depends on notions that have epistemic nature. The second kind of analysis leads us to the formulation of the same concepts in terms that could be called “logical”. In our view, this second type tries to overcome the limitations of the former. From the epistemic point of view, the explication of the notion of form is undertaken starting from its qualitative nature. Because of this, quality, that becomes the central concept, is defined as a predicate that is distinguishable by the faculties of intellection and memory, when the thing is considered by itself:
“Namely, quality is in general sense, every predicate that can be conceived about something that can be considered by itself. On the other hand, quantitiy is what is perceived in a co-perception with something else”.15
“[...] quantity is only recognized by co-perception. On the contrary, quality is recognized by memory and intelligibility [...] Quality is a distintion that happens in thinking from the thing [...] Quantity is a distintion that happens in perceiving from the thing [...]”16
On the other hand, in order to clarify how the epistemic concept of quality operates, through which the formal identity that underlies to similarity relationship is determined, the notion of identity understood as substituibility salva qualitate is introduced:
“Similar are those things that can be substituted reciprocally, saving their quality; that is, so that they can not be discerned, unless they were considered conjointly”.17
Starting from this condition it can be justified that similar things are those that can be discerned only by co-perception or co-presence. In fact, two things are undiscernible from the point of view of quality or form, which is object of intellection and memory, if they are interchangeable salva qualitate. In this case, they will be distinguishable only by magnitude and position, whose recognition is only possible by perception. So if similarity is defined as a relation of formal identity, then there follows the notion of similarity as discernibility by co-perception. But if the “substantive concept” of similarity is elucidated in this way, there arises the problem that by clarifying the quality concept in epistemic terms we run two alternative risks: either we relapse in the psychologism from which we have tried to escape or we commit a circulus in definiendo, when we make the attempt to fix the notion of similarity whithout appealing to epistemic concepts. These problems become evident when we examine closer the nature of differential properties and the requirement that these properties must be discovered by the separate
15Divisio Terminorum ac Enumeratio Attributorum, A VI 4 564-565. 16 Definitiones: Ens, Possibile, Existens, A VI 4 868-870. 17 Definitiones, A VI 4 406. 7 consideration of each similar thing by itself. Formal identity of similar things is grounded on the fact that their qualitative properties are identical; that is, undiscernible. Now, the way in which is established the undiscernibility of the corresponding qualitative properties appeals to recognition and remembering acts. So the formulation of the definition must resort necessarily to epistemic notions: intellection, recollection and imagination. In this way, the menace of psychologism appears again. The same happens with the requisite that each thing must be considered separately and by itself. The final outcome of the analysis is the conclusion that the difficulties of the attempt to define the “substantive concept” of similiarity arise from the utilization of epistemic and procedural notions in order to establish the undiscernibility of the differential properties. To avoid the circulus in definiendo or the psychologization of the similarity concept, it is required that the fundamental concepts such as ´qualitative undiscernibility´ and the ´separate consideration by itself´ will be redefined so that they do not entail epistemic considerations. To achieve this goal it is necessary to formulate the characterization of intrinsic qualitative properties of the thing in such a way that such formulation will be connected with the very nature of the thing and what can be deduced from it. So Leibniz is compelled to elaborate a concept of form and of formal identity in a purely ´objective´ way.
4. The “substantive concept” of similarity. The logical way.
We can not affirm categorically that Leibniz was dissatisfied with the formulation of the concept of similarity as substituibility salva qualitate. Yet, we can speculate about this dissatisfaction since texts can be found where Leibniz reformulates the requisites for the definition of similarity as substituibility salva qualitate in order to transform the epistemic conditions into logical conditions. Hence, in a text that was formerly published in the Vorausedition and that is now available in the recently published fourth volume, sixth series of the Akademie Ausgabe18, Leibniz tries to formulate a notion of similarity by appealing to logical considerations about the formal structure of objects. It is worth noting that in a previous writing from ca. 1679, Leibniz defines similarity in terms of the impossibility to prove a priori the diversity of things that are considered as similar19. In other words, he applies a corollary of the principle of sufficient reason, that demands the existence of an a priori reason for the diversity between objects20, as he states it explicitly in Calculus ratiocinator. These applications of the principle of sufficient reason suggest that there must be grounds in re for discernibility between things, if they are cognitively discernible by their form. From this new point of view, what is at stake is the diversity or identity of the objects in themselves, independently of the recognition acts. Consequently, similarity is defined in terms of formal identity. Again, formal identity is now understood as the impossibility to demonstrate a priori the diversity of the objects by means of what follows from their nature.
18 Definitiones: Aliquid, Nihil, Non-ens, Ens, A VI 4 930-934. 19 Elementa ad Calculum Condendum, A VI 4 154. 20 Calculus Ratiocinator seu Artificium Facile et Infallibiliter Ratiocinandi. Res Hactenus Ignorata, A VI 4 278. 8
So it is necessary to reformulate the concepts of form and formal identity so that they will fit the new conditions that the principle of sufficient reason sets up. The change of view compels Leibniz to consider purely structural factors among which the notion of logical consequence plays a central role. Thus, if form is defined as the aggregate of the attributes of a thing, such that they are sufficient to deduce the rest of its predicates21, the following preliminary definition of similarity will result:
“Similar are those things that can not be discerned, if considered separately, through elements that are necessarily connected; that is, through demonstrable truths about those things. In other words, we can not assign to them demonstrable predicates that will be different”22
Thus, similarity can be characterized in terms of what can be deduced from the nature of the objects between which a similarity relationship can be established. In other words, similar are those things from whose corresponding natures or forms can not be deduced differential predicates. The logical definition of the substantive concept of similarity will consist in a progressive adjustment and correction of the assumptions that are entailed in this general characterization that depends on the notion of form and deducibility. To achieve this goal, Leibniz resorts to the principle of interchangeability salva veritate. Briefly, two things are similar to each other if the predicates that are deducible from each of them are mutually interchangeable salva veritate. Indeed, although it can be proved that Leibniz was not successful in his attempt to formulate a general concept of similarity by means of interchangeability salva veritate, this failure does not invalidate the idea that he tried to capture (maybe unsuccesfully) through such concept. In fact, Leibniz understood that the similarity relationship is based on shared formal properties that were interpreted by him as an identity of form or structure (perhaps a very strong condition). In his attempt to clarify conceptually the notion of structural identity, he appealed to the notion of interchangeability salva veritate, which was shaped within the frame of anlytical and predicative logics, where concepts such as the relationship between continent and content, no-relational predicates, the definition as the development of the conceptual content of the form, and the structure of the categorical proposition are determinant. Let us resume our reflections from the beginning about the science of forms. Our investigations on the concept of similarity have shown that its general formulation entails the consideration of the possibility of establishing structural identities between objects, the term ‘object’ being understood in a very general sense. If our analyses are right, the science of forms as the science of similarity has the task to search and investigate these structural identities that obtain between diverse theoretical domains and to exhibit them through the abstraction of form in the abstract sense. Since similarity is based on the identity of structure or form, it is possible, by comparison of similar objects, to abstract the form as such and to present it in its articulations. Precisely, it is the formula as a sensuous structure composed by characters that has the double function of making it possible to compare structures of objects and to exhibit their identities, once the specific contents are removed, obtaining so a structural science, the leibnizian Science of Forms or Formulae:
21 Aliquid, Nihil, Impossible, Possibile. Definitiones, A VI 4 941. 22 Definitiones: Aliquid, Nihil, Non-ens, Ens, A VI 4 931. 9
“[...] From there results a subordination, which until now has been ignored or at least neglected, of the Algebra to the Combinatory Art, that is, of figurative algebra (algebra speciosa) to the general figurative science (speciosam generalem). In other words, the science of the formulae that signify the quantity depends on the theory of formulae or expressions of order, similarity, relation etc. in general; that is, the general science of quantity depends on the general science of quality [...]” 23
23 Mathesis Universalis, GM VII 61.