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The Concept of Similarity in “Der Ähnlichkeitsbegriff Bei Leibniz”, In: Mathesis Rationis

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The Concept of Similarity in “Der Ähnlichkeitsbegriff Bei Leibniz”, In: Mathesis Rationis 1 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.
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