Mathesis Universalis: the Search for a Universal Science

Index of the Section: "Pages on the History of Philosophy"

Lives and Opinions of Eminent Philosophers by Diogenes Laërtius. A bibliography
History of the Ontological Argument for the Existence of God
Entia Rationis. History of the Theories on Non-existent Objects
Mathesis universalis: the Search for a Universal Science

INTRODUCTION

"Preliminary remark: It is never quite clear what (the modern concept of mathesis universalis as such exactly signifies, let alone how it may be defined. The expression itself (1) is a composite of the Greek μάθεσις latinized by transcription to mathesis, and the Latin universalis. The latinized mathesis, generally meaning, according to the dictionaries, learning / knowledge / science (= disciplina or scientia), (2) more specifically designates mathematic (= scientia mathematica), though it can even mean astrology. Hence the first and general sense of mathesis universalis signifies no more than universal science (disciplina universalis or scientia universalis). However – and this will be very important – since this "science" has a rather mathematical ring to it, we should on second thoughts take it to be an equivalent of scientia mathematica universalis (3) (or generalis or communis: due to the underlying Greek terminology, there is no difference between universal, general, or common in antiquity – things will have changed by Leibniz' time at the latest, of course).

This more specific meaning, i.e., universal (or general or common) mathematical science or universal mathematic, is essential, and more or less the bottom line for most occurrences of the expression, though it still remains very vague. However, the emphasis of this paper lies, with regard to the concept of mathesis universalis not so much on the historical details as on the more general systematical outlines. Therefore it should suffice to begin our work with an understanding of mathesis universalis that implies not much more than universal (or general or common) mathematical science, which of course still allows for a range of diverse meanings. What matters is to remain true to the sense of mathesis universalis while not confusing the two very different notions somehow inherent in the Latin, i.e., that of universal mathematic on the one hand and that of universal science on the other. A clear line should be drawn between these two concepts, of which the former is mathematical (even though sometimes in a wider sense), the latter not. I trust that it will become clear in this paper that both for historical and systematical reasons it is not only justified, but even necessary, to draw this general distinction between universal mathematic and universal science in this way." pp. 129-130

(1) The major work of reference with regard to the Renaissance and Early Modern history of mathesis universalis (mainly in the context of Paduans, Jesuits, and Humanists/Ramists) remains G. Crapulli, Mathesis universalis. Genesi di un'idea nel XVI secolo, Roma 1969 (as his focus is on the sixteenth century, Crapulli treats neither Descartes nor Leibniz, but only their predecessors). For the history of the term as such cf. R. Kauppi, "Mathesis universalis", in: J. Ritter/K. Gründer (ed.), Historisches Worterbuch der Philosophie, vol. 5: L-Mn, Basel/Stuttgart 1980, col. 937-938 and also J. Mittelstrass, "Die Idee einer Mathesis universalis bei Descartes", Perspektiven der Philosophie: Neues Jahrbuch 4 (1978), 177-178.

(2) Descartes himself was clear about the fact that not much can be gained from the word itself: hic enim vocis originem spectare non sufficit; nam cum Matheseos nomen idem tantum sonet pod disciplina (Regula IV, Oeuvres X, 377,16-18).

(3) D. Rabouin, "La 'mathematique universelle' entre mathematique et philosophie, dAristote a Proclus", Archives de Philosophie 68 (2005), 249-268, discusses the concept's ambiguous character between philosophy and mathematics. Cf. also his paper "Les interpretations renaissantes de la 'mathematique generale' de Proclus", to be published in the proceedings (ed. A. Lernould and B. Vitrac) of the International Conference on Le Commentaire de Proclus au premier livre des Elements d'Euclide, forthcoming from Septentrion (Presses Unversitaires de Lille).

From: Gerald Bechtle - How to apply the modern concepts of Mathesis Universalis and Scientia Universalis to ancient philosophy, Aristotle, Platonisms, Gilbert of Poitiers, and Descartes. In Platonisms: ancient, modern, and postmodern. Edited by Corrigan Kevin and Turner John D. Leiden: Brill 2007. pp. 129-154

"The design of mathesis universalis, for short MU, was stated in the 17th century as part of the rationalistic philosophy of this time including a program of mathematization of sciences (see Weingartner, 1983). However, the significance of MU is not restricted to that period. It belongs to main ideas of Western civilization, its beginnings can be traced to Pythagoreans and Plato.

Immediate sources of the 17th century MU are found in the 15th century revival of Platonism whose leading figure was Marsilio Ficino (1433-1499), the author of "Theologia Platonica". He was accompanied by Nicholas of Cusa (1401-1464), Leonardo da Vinci (1452-1519), also by Nicolaus Copernicus (1473-1543). All of them may have taken as their motto the biblic verse, willingly quoted by St. Augustine, Omnia in numero et pondere et mensura disposuisti, Sap. 11, 21. The core of their doctrine was expressed in Ficino's statement that the perfect divine order of the universe gets mirrored in human mind due to mind's mathematical insights; thus mathematics proves capable of the role of an universal key to the knowledge; hence the denomination mathesis universalis.

This line of thought was continued in the 16th century by Galileo Galilei (1564-1642) and Johannes Kepler (1571-1630); it penetrated not only mechanics and astronomy but also medical sciences as represented by Teophrastus Paracelsus of Salzburg (1493-1541). No wonder that in the 17th century the community of scholars was ready to treat the idea of MU as something obvious, fairly a commonplace, before Descartes made use of this term in his "Regulae ad directionem ingenii". "Regulae" did not appear in print until 1701, hence the term itself could not have been taken from this source. In fact, it was used earlier by Erhard Weigel, a professor of mathematics in Jena (Leibniz's teacher) who wrote a series of books developing the program of universal mathematics: "Analysis Aristotelis ex Euclide restituta", 1658 (an interpretation of Aristotle's methodological theory in the light of Euclid's practice); "Idea Matheseos Universae", 1669; "Philosophia Mathematica: universae artis inveniendi prima stamina complectens", 1693 (see Arndt - Einführung des Herausgegebers - in: Christian Wolff - Vernünftige Gedanken - Halle (1713) edited by H. W. Arndt, Hildesheim, Georg Olms, 1965).

The last of the listed titles involves one of the key concepts of the MU program: ars inveniendi, i.e. the art of discovering truths in a mathematical way. There were two approaches to this art, differing from each other by opposite evaluations of formal logic. According to Descartes, formal logic of Aristotle and schoolmen was useless for the discovery of truth; according to Leibniz, ars inveniendi was to possess the essential feature both of formal logic and of mathematical calculus, viz. the finding of truths in formae (in virtue of form)."

From: Witold Marciszewski - "The principle of comprehension as a present-day contribution to mathesis universalis," Philosophia Naturalis 21: 523-537 (1984). pp. 525-526.

MATHESIS UNIVERSALIS IN HUSSERL

"Apophantics as a doctrine of sense and a logic of truth. From the above said it emerges that formal logic, as classically conceived, reflects the attitude of that person who performs the critique but whose judging is not a direct one but a judgement about judgements. Formal logic is constituted like an apophantic logic, whose object is the predicative judgement. This should not constitute a limitation for logic - as in fact has been the case so far - says Husserl - for apophansis contains all the categorical intentional entities. In other words, classical formal logic kept on the apophantic level, abandoning the very aim of knowledge comprised in the "intentionality" of the judgement. However, says Husserl, judgements conceived of as "intentional entities" pertain to the region of sense. The phenomenological analysis of the sense-directed attitude leads Husserl to the following conclusions: there is a region of sense wherein a judgement is meaningful irrespective of whether or not it is exact. This shows that sense transcends the act of referring to the given subjects, sense is "transcendental" and senses are ideal poles of unity (Formale und Transzendentale Logik, p. 119). Hence it follows that pure logic has the following divisions: the doctrine of sense and the doctrine of truth, for we have seen that the sense of a judgement and its truth are two different things.

Having thus examined the whole content of classical analytics, a content which though implied is yet not explicit, in his opinion, Husserl concludes that analytics, thus conceived, represents that Mathesis Universalis i.e. that universal science dreamt of by Leibniz which has four levels:

(a) as Mathesis Universalis, the systematic form of theories; (b) as pure Mathesis, of non-contradiction; (c) as Mathesis o f the possible truth; (d) as Mathesis of pure senses."

From: Anton Dumitriu - History of logic - Volume III - Tubridge Wells, Abacus Press - 1977, p. 367.

SELECTED BIBLIOGRAPHY

Arenas Luis, "Matematicas, metodo y "mathesis universalis" en las "Regulae" de Descartes," Revista de Filosofia (Spain) 8: 37-61 (1996).
Arndt Hans Werner, "Die Semiotik Christian Wolffs als Propädeutik der ars characteristica combinatoria und der ars inveniendi," Zeitschrift für Semiotik 1 (1979).
"The central thesis in Wolff's approach towards semiotics is that a semiotically classified representation of philosophical sciences is a prerequisite to the development of an ars inveniendi. Assuming that an isomorphic relationship between concepts, signs, and things as well as between their differences and relations exists, Wolff develops a system of concepts resulting in a real Organon for philosophy. Wolff's method follows the ideal of explicating concepts originating in ordinary language, which, because of this origin, become lexicographically applicable, even independently of the theoretical context. While here (and this is true to Daries) all content of consciousness is assumed to be accessible to an analysis notionum and to be solely conveyed by signs, later on, language and signs are regarded as media capable of evoking their own effects."
Bechtle Gerhard. How to apply the modern concepts of Mathesis Universalis and Scientia Universalis to ancient philosophy, Aristotle, Platonisms, Gilbert of Poitiers, and Descartes. In Platonisms: ancient, modern, and postmodern. Edited by Corrigan Kevin and Turner John D. Leiden: Brill 2007. pp. 129-154
Beck Leslie John. The method of Descartes. A study of the Regulae. Oxford: Clarendon Press 1952.
See in particular Chapter XII. The science of order pp. 190-202.
Berlioz Dominique. Langue adamique et caractéristique universelle chez Leibniz. In Leibniz and Adam. Edited by Dascal Marcelo and Yakira Elhanan. Tel Aviv: University Publishing Projects 1993. pp. 153-168
Bockstaele Paul, "Between Viète and Descartes: Adriaan van Roomen and the Mathesis Universalis," Archiv for History of Exact Sciences 63: 433-470 (2009).
"Adriaan van Roomen published an outline of what he called a Mathesis Universalis in 1597. This earned him a well-deserved place in the history of early modern ideas about a universal mathematics which was intended to encompass both geometry and arithmetic and to provide general rules valid for operations involving numbers, geometrical magnitudes, and all other quantities amenable to measurement and calculation. 'Mathesis Universalis' (MU) became the most common (though not the only) term for mathematical theories developed with that aim. At some time around 1600 van Roomen composed a new version of his MU, considerably different from the earlier one. This second version was never effectively published and it has not been discussed in detail in the secondary literature before. The text has, however, survived and the two versions are presented and compared in the present article. Sections 1-6 are about the first version of van Roomen's MU the occasion of its publication (a controversy about Archimedes' treatise on the circle, Sect. 2), its conceptual context (Sect. 3), its structure (with an overview of its definitions, axioms, and theorems) and its dependence on Clavius' use of numbers in dealing with both rational and irrational ratios (Sect. 4), the geometrical interpretation of arithmetical operations multiplication and division (Sect. 5), and an analysis of its content in modern terms. In his second version of a MU van Roomen took algebra into account, inspired by Viète's early treatises; he planned to publish it as part of a new edition of Al-Khwarizmi's treatise on algebra (Sect. 7). Section 8 describes the conceptual background and the difficulties involved in the merging of algebra and geometry; Sect. 9 summarizes and analyzes the definitions, axioms and theorems of the second version, noting the differences with the first version and tracing the influence of Viète. Section 10 deals with the influence of van Roomen on later discussions of MU, and briefly sketches Descartes' ideas about MU as expressed in the latter's Regulae."
Burkhardt Hans. The Leibnizian Characteristica Universalis as link between grammar and logic. In Speculative grammar, universal grammar, and philosophical analysis of language. Edited by Buzzetti Dino and Ferriani Maurizio. Amsterdam: Benjamins 1987. pp. 43-63
Buzon Frédéric de. Mathesis universalis. In La science classique: XVIe-XVIIIe siècle. Dictionnaire critique. Edited by Blay Michel and Halleux Robert. Paris: Flammarion 1998. pp. 610-621
Cajori Florian. A history of mathematical notations. Chicago: The Open Court Publishing Company 1928.
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Cardoso Adelino, "Mathesis Leibniziana," Philosophica: Revista do Departamento de Filosofia da Faculdade de Letras de Lisboa: 51-77 (1996).
"Dans cet article, l'auteur essaie de montrer qu'on trouve chez Leibniz une "mathesis", c'est-a-dire une conception du savoir et de l'organisation des savoirs, originale, laquelle est entierèment discernable d'autres "mathesis" qui ont été proposées par ses contemporains du XVIIe siecle. Du point de vue thématique, l'auteur croit que cette " mathesis" reçoit son intelligibilité de la relation que Leibniz établit entre la métaphysique et les mathématiques. Sous ce rapport, on constate des vraies transformations dans la pensée de Leibniz, dès le moment où il fait son adhesion au mécanisme (1668) jusqu'à la formulation de sa dernière pensée. Dans cette évolution, la correspondance avec de Volder joue un role décisif."
Cobb-Stevens Richard. La géométrie des Regulae: mathesis et ontologie. In Lire Descartes aujourdh'hui. Edited by Depré Olivier and Lories Danielle. Louvain-Paris: Éditions Peters 1997. pp. 85-107
Cohen Laurence Jonathan, "On the project of a Universal Character," Mind 63: 49-63 (1954).
Reprinted in: Knowledge and language. Selected essays of L. Jonathan Cohen - edited and with an introduction by James Logue - Dordrecht; Kluwer, 2002 pp. 1-14.
Couturat Louis. La logique de Leibniz: d'aprés des documents inédits. Paris: Felix Alcan 1901.
Reprinted: Hildesheim, Olms, 1961 e 1985.
Couturat Louis and Leau Leopold. Histoire de la langue universelle. Paris: Hachette 1903.
Crapulli Giovanni. Mathesis universalis. Genesi di un'idea nel XVI secolo. Roma: Edizioni dell'Ateneo 1969.
Danek Jaromir and Möckel Christian. Idee der Mathesis universalis - die logische Vernunft: Leibniz und Husserl. In Phänomenologie und Leibniz. Edited by Cristin Renato and Kiyoshi. Freiburg: Alber 2000. pp. 88-121
Dascal Marcelo. Leibniz. Language, signs and thought. Amsterdam: Benjamins Publishers 1987.
Desanti Jean-Toussaint, "Réflexions sur le concept de "mathesis"," Bulletin de la Société Française de Philosophie 77: 1-22 (1972).
Reprinted in: J. T. Desanti - La philoosophie silencieuse ou Critique des philosophies de la science - Paris. Le Seuil, 1975, pp. 196-219.
Dumitrescu Marius, "Le défi cartésien par l'idée de mathesis universalis," Revue Roumaine de Philosophie 42-43: 25-32 (1999).
Dumoncel Jean-Claude. Le jeu de Wittgenstein. Éssai sur la mathesis universalis. Paris: Presses Universitaires de France 1991.
Dumoncel Jean-Claude. La tradition de la mathesis universalis. Platon, Leibniz, Russell. Paris: Unebévue 2002.
Eco Umberto. The search for the perfect language. Oxford: Blackwell 1995.
Translated by James Fentress from the Italian: La ricerca della lingua perfetta nella cultura europea - Bari, Laterza, 1993
Gérard Vincent, "La mathesis universalis est-elle l'ontologie formelle?," Annales de Phénomenologie 1: 61-98 (2002).
Gérard Vincent, "Leibniz et la mathématique formelle," Philosophie 92: 29-55 (2006).
Ha Byung-Hak. Das Verhältnis der Mathesis universalis zur Logik als Wissenschaftstheorie bei E. Husserl. Bern: Peter Lang 1997.
Hayashi Tomohiro, "Leibniz's construction of Mathesis universalis: a consideration of the relationship between the plan and his mathematical contributions," Historia Scientiarum: 121-141 (2002).
Heinekamp Albert. Natürliche Sprache und Allgemeine Charakteristik bei Leibniz. In Akten des II Internationalen Leibniz-Kongresses. Hannover, 17-22. Juli 1972. Vol IV: Erkenntnislehre, Logik und Sprachphilosophie. Wiesbaden: F. Steiner 1975. pp. 257-286
Jamart Géraldine, "Logique, mathématique et ontologie: La Ramée, précurseur de Descartes," Études Philosophiques: 17-28 (1996).
Klein Jacob. Greek mathematical thought and the origin of algebra. Cambridge: MIT Press 1968.
Translated from the original German: Die griechische Logistik und die Entstehung der Algebra, published in Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik, Abt. B, Studien, v. 3, 1934.
Reprinted: New York, Dover Publications, 1992.
See in particular: "The reinterpretation of the katholou pragmateia as Mathesis Universalis in the sense of ars analytice" pp. 178-185.
Knecht Herbert. La logique chez Leibniz. Essai sur le rationalisme baroque. Lausanne: L'Age d'Homme 1981.
See Chapter III: La mathématique universelle pp. 91-123.
Knoboloch Eberhard. Mathesis - The idea of a universal science. In Form, Zahl, Ordnung. Studien zur Wissenschafts- und Technikgeschichte. Festschrift für Ivo Schneider zum 65. Geburtstag. Edited by Seising Rudolf, Folkerts Menso, and Hashagen Ulf. Stuttgart: Franz Steiner Verlag 2004. pp. 77-90
Knowlson James. Universal Language Schemes in England and France 1600-1800. Toronto: University of Toronto Press 1975.
Kuster Friederike. Wege der Verantwortung. Husserls Phänomenologie als Gang durch die Faktizität. Dodrecht: Kluwer 2008.
See Chapter 1: Eine mathesis der Geist und de Humanität.
Loi Maurice. La mathesis universalis aujourd'hui. In L'esprit cartésien. Quatrième centenaire de la naissance de Descartes. Edited by Bourgeois Bernard and Havet Jacques. Paris: Vrin 2000. pp. 231-233
Maat Jaap. Philosophical languages in the Seventeenth century: Dalgarno, Wilkins, Leibniz. Dordrecht: Kluwer 2004.
Marciszewski Witold, "The principle of comprehension as a present-day contribution to mathesis universalis," Philosophia Naturalis 21: 523-537 (1984).
Marciszewski Witold, "The Principle of Comprehension," Philosophia Naturalis 24: 523-537 (1984).
Marion Jean-Luc. On Descartes' metaphysical prism: the constitution and the limits of onto-theo-logy in Cartesian thought. Chicago: University of Chicago Press 1999.
Translated by Jeffrey L. Kosky.
Original edition: Sur le prisme métaphysique de Descartes: constitution et limites de l'onto-théo-logie dans la pensée cartésienne - Paris, Presses Universitaires de France, 1986.
Marion Jean-Luc. Descartes's grey ontology. South Bend: St. Augustine's Press 2005.
Original edition: Sur l'ontologie grise de Descartes. Science catrésienne et savoir aristotelicien dans les Regulae - Paris, Vrin, 1975.
Martineau Emmanuel, "L'ontologie de l'ordre," Études Philosophiques: 475-494 (1976).
Milkov Nikolay. The formal theory of everything: exploration of Husserl's theory of manifolds. In Logos of phenomenology and phenomenology of the Logos. Book One. Edited by Tymieniecka Anna-Teresa. Dordrecht: Springer 2006. pp. 119-135
Mittelstrass Jürgen, "Die Idee der Mathesis universalis bei Descartes," Perspektiven der Philosophie 4: 177-192 (1978).
Mittelstrass Jürgen, "The philosopher's conception of "Mathesis Universalis" from Descartes to Leibniz," Annals of Science 36: 593-610 (1979).
"In Descartes, the concept of a 'universal science' differs from that of a 'mathesis universalis', in that the latter is simply a general theory of quantities and proportions. Mathesis universalis is closely linked with mathematical analysis; the theorem to be proved is taken as given, and the analyst seeks to discover that from which the theorem follows. Though the analytic method is followed in the Meditations, Descartes is not concerned with a mathematisation of method; mathematics merely provides him with examples. Leibniz, on the other hand, stressed the importance of a calculus as a way of representing and adding to what is known, and tried to construct a 'universal calculus' as part of his proposed universal symbolism, his 'characteristica universalis'. The characteristica universalis was never completed-it proved impossible, for example, to list its basic terms, the 'alphabet of human thoughts'-but parts of it did come to fruition, in the shape of Leibniz's infinitesimal calculus and his various logical calculi. By his construction of these calculi, Leibniz proved that it is possible to operate with concepts in a purely formal way."
Mittelstrass Jürgen. Leibniz and Kant on mathematical and philosophical knowledge. In The natural philosophy of Leibniz. Edited by Okruhlik Kathleen and Brown James Robert. Dordrecht: Reidel 1985. pp. 227-261
Se in particular § 2 Mathesis universalis pp. 232-239.
Muñoz-Alonso Lopez Gemma. El legado de Descartes. Methodo y "mathesis universalis". Edited by Grupodis. Madrid: 1985.
Napolitano Valditara Linda M. Le idee, i numeri, l'ordine. La dottrina della mathesis universalis dall'Accademia antica al neoplatonismo. Napoli : Bibliopolis 1988.
Olivo Gilles, "L'évidence en règle: Descartes, Husserl et la question de la mathesis universalis," Études Philosophiques: 189-221 (1996).
Olivo Gilles. La sagesse des principes: la mathesis universalis dans les Principiae philosophiae de Descartes. In Lire Descartes aujourd'hui. Edited by Depré Olivier and Lories Danielle. Louvain-Paris: Éditions Peters 1997. pp. 69-84
Olivo Gilles. Descartes et l'essence de la vérité. Paris: Presses Universitaires de France 2005.
Tanle des matières: Introduction: La clarté transcendantale de la vérité - Vérité de la méthode ou vérité de la philosophie première? - La question du fondement;
Section I -- LA CERTITUDE DE LA SAGESSE UNIVERSELLE
Chapitre 1: La sagesse universelle des Regulae ou la primauté de l'entendement; Chapitre 2: Les Regulae, La recherche de la vérité et la mathesis universalis; Chapitre 3: Les Regulae et La recherche de la vérité présupposent-elles une philosophie première?
Section II -- LA CERTITUDE EN VERITE
Chapitre 4: La règle générale et l'hypothèse du Dieu trompeur; Chapitre 5: La règle générale entre évidence et certitude; Chapitre 6: Des natures simples à l'idée vraie; Chapitre 7:- La IVe Méditation et la création des vérités éternelles
Conclusion; Index
Olvera Mijares Raul, "Some historical remarks on Husserl's theory of multiplicity," Axiomathes 5: 385-394 (1994).
Ortiz de Landázuri Carlos, "Mathesis universalis en Proclo de las aporias cosmologicas al universo euclideo," Anuario Filosofico 33: 229-257 (2000).
"The author shows how Proclo is a precursor of 'Mathesis universalis' concept, without admitting the aporetic method of mathematics which is in Plato, Aristotle and Euclides thought. Today, his paradigm is rejected but it is a decisive factor to understand the sources of Western thought. This study deals with the works of Brisson, Cleary, Trudeau, Beierwaltes and Schmitz."
Paty Michel. Mathesis universalis e inteligibilidad en Descartes. In Memorias del Seminario en Conmemoración de los 400 Anos del Nacimiento de René Descartes. Edited by Albis Victor. Santafé de Bogotà: 1997. pp. 135-170
Peckhaus Volker. Logik, Mathesis universalis und allgemeine Wissenschaft. Leibniz und die Wiederentdeckung der formalen Logik im 19. Jahrundert. Berlin: Akademie Verlag 1997.
Contents: Vorwort VII-VIII; 1. Einleitung 1; 2. Die Idee der mathesis universalis bei Leibniz 25; 3. Die frühe Rezeption Leibnizscher mathesis universalis und Logik 64; 4. Die "logische Frage" und die Entdeckung der Leibnizschen Logik 130; 5. Leibniz und die englische Algebra der Logik 185; 6. Ernst Schröder: "Absolute Algebra" und Leibnizprogramm 233; 7. Schluss 297; Verzeichnisse 309-412.
Perini Roberto, ""Mathesis universalis" e metafisica nel metodo cartesiano," Giornale di Metafisica 28: 159-207 (1973).
Pombo Olga. Leibniz and the problem of a universal language. Münster: Nodus Publikationen 1987.
Pombo Olga. Leibniz and the encyclopaedic project. In Ciência, tecnologia y bien comun: la actualidad de Leibniz. Valencia: Editorial de la Universidas Politecnica de Valencia 2002. pp. 267-278
"My talk will have three moments. In a first moment, I will try to identify the main determinations of encyclopaedic project in its whole. Since Varro (116-24 b.c.), Rerum Divinorum et Humanorum Antiquitates, St. Isidorus (560-636) Etimologies, Alsted Encyclopaedia Omnia Scientiarum (1630), or Diderot and D'Alembert Encyclopédie ou Dictionnaire Raisonné des Sciences, des Arts et des Métiers (1751-1765), to the Internet - which constitutes (I will argue) the most recent and eloquent development of the history of encyclopaedism - the aim will be to look for what is common to all this kind of excessive works. In a second moment, I will attempt to understand how Leibniz's idea of encyclopaedia inserts itself in that project of all times, what specific place Leibniz occupies within those many attempts. In the third moment, I will try to estimate the presence of Leibniz's idea of encyclopaedia in subsequent developments of encyclopaedism, namely in the XX / XXI century. This will be my humble contribution to this Congress whose major purpose is to think out the actuality of Leibniz."
Poser Hans, "Signum, notio und idea. Elemente der Leibnizschen Zeichentheorie," Zeitschrift für Semiotik 1 (1979).
"Leibniz' approach towards a "characteristica universalis", a "universal art of signs" (Zeichenkunst), as an essential instrument of human knowledge is rooted both in the Cartesian ideal method of a universal mathesis and in the ars magna as a universal language comprising all the simple concepts and their combinations. The signum (sign vehicle) expresses a notio (concept) based on an idea fundamental to the res (object). The assumption here is that an isomorphic relationship between the logical and ontological areas is the precondition enabling denotation. However, the deficiency of human thought prevents characterization in its entirety; a multitude of sign systems - "Bereichscharakteristiken", area-specific characteristics - take the place of this ideal. Under these conditions it is also possible to transpose ordinary language into a lingua rationis. Beyond that, the importance of ordinary language consists in its correlating sign and meaning."
Poser Hans, "Mathesis universalis and scientia singularis. Connections and disconnections between scientific disciplines," Philosophia Naturalis 35: 3-21 (1998).
Rabouin David, "La 'mathématique universelle' entre mathématique et philosophie, d'Aristote à Proclus," Archives de Philosophie 68: 249-268 (2005).
"In this paper, we study the concept of 'universal mathematics' used by philosophers like Aristotle, Jamblichus and Proclus, in its relationship to mathematics. We try to show that it stands neither for a free interpretation of a mathematical datum, nor for a pure and simple reference to given mathematical theory, but is grounded on a fundamental problem which we attempt to reestablish, that of the universality of mathematics."
Rabouin David, "Husserl et le projet leibnizien d'une mathesis universalis," Philosophie 92: 13-28 (2006).
"L'auteur tente de cerner les traits d'une interprétation de la doctrine leibnizienne depuis l'idée de mathématique formelle qui se cristallise notamment chez Husserl, et ce, pour en interroger la validité et la confronter à la manière dont on peut aujourd'hui reconstituer la nature du projet leibnizien de 'mathesis universalis'. Il tâche de préciser l'écart qui sépare ces deux interprétations, ainsi que les questions philosophiques qu'il soulève."
Rabouin David. Mathesis universalis. L'idée de "mathématique universelle" d'Aristote à Descartes. Paris: Presses Universitaires de France 2009.
Table des matières: Introduction 9; La constitution de la "mathématique universelle" comme problème philosophique 33; I. Aristote 37; II. "Mathématique universelle" et théories mathématiques: Aristote, Euclide, Epinomis 85; III. Le moment néo-platonicien 129; Vers la science de l'ordre et de la misure 193; Introduction 193; IV. La renaissance de la mathématique universelle 195; V. La mathesis universalis cartésienne 251; Conclusion 347; Annexe I. La quaestio de scientia mathematica communi 363; Annexe II. Essai bibliographique sur la mathesis universalis chez Descartes et Leibniz 367; Bibliographie 375; Index nominum 397-402
Rauzy Jean-Baptiste, "Quid sit natura prius? La conception leibnizienne de l'ordre," Revue de Métaphysique et de Morale 98: 31-48 (1995).
" It is well known that Leibniz's logic is grounded in the inherence of the predicate in the subject and in the compossibility of notions. It naturally stresses, therefore, relations of equivalence, rather than of order. Nevertheless, Leibniz provided a logical analysis of order, i.e., an account of the meaning of "prior", "subsequent", "concomitant". His account comprises three points: (1) Given two beings, the one that is more simple (i.e., the one whose analysis requires less operations of the mind) is prior by nature ("natura prius"); hence, concomitant ("simul") being. (2) The degree of composition of being corresponds to its degree of perfection. Hence, prior beings being simpler, subsequent beings are more perfect. (3) Given two beings such that one is simpler and the other more perfect, they differ temporally if they also contradict each other; conversely, two compossible beings contradict each other if, and only if, they are not simultaneous (i.e., if they do not belong to the same "state of the universe"). It will be shown that this relation makes it possible to characterize the axiomatic order of incomplete notions (in the field of the "mathesis universalis"). But the attempt to explain the terms prius, posterius and simul in a metaphysical manner, i.e., by laying the stress on the order among substances, raises grave philosophical problems."
Risse Wilhelm, "Die Characteristica Universalis bei Leibniz," Studi Internazionali di Filosofia 1: 107-116 (1969).
Robinet André, "Le référent "dialectique" dans les Regulae," Études Philosophiques: 3-15 (1996).
Robinet André. Aux sources de l'esprit cartésien. L'axe La Ramée-Descartes: de la Dialectique des 1555 aux Regulae. Paris: Vrin 1996.
Robinet André, "L'axe La Ramée-Descartes: position de la "mathesis universalis"," Giornale Critico della Filosofia Italiana 17: 286-293 (1997).
Reprinted in: André Robinet - Aux sources de l'esprit cartésien: L'axe La Ramée-Descartes : de la Dialectique des 1555 aux Regulae - Paris, Vrin, 1996.
"Le quatrième des concepts de la dialectique presentée dans les "Regulae" de Descartes répond a une lourde histoire au cours de laquelle il est consideré comme étant la suite naturelle du recours à la méthode. Depuis Dasypodius jusqu'a Romanus, ces concepts, issus des travaux de La Ramée sur la dialectique et sur Euclide, ont eté longuement travaillés et discutés, notamment par les auteurs du courant ramiste. Le choix particulier fait par Descartes répond a une disposition ordinaire dans la logique du XVIeme siècle et se propage par lui a travers le XVIIeme siècle."
Rossi Paolo. Logic and the art of memory. The quest for a universal language. Chicago: University of Chicago Press 2000.
Translated from the Italian: Clavis universalis; arti mnemoniche e logica combinatoria da Lullo a Leibniz - Milano, R. Ricciardi, 1960 (second revised edition, Bologna, Il Mulino, 1983) with an introduction by Stephen Clucas.
Sagemüller Franz. La Mathesis universalis et son jeu de langage. In L'esprit cartésien. Quatrième centenaire de la naissance de Descartes. Edited by Bourgeois Bernard and Havet Jacques. Paris: Vrin 2000. pp. 264-270
Actes du 26. Congrés de l'Association des Sociétés de Philosophie de Langue Française (A.S.P.L.F.) / organisé par la Société Française de Philosophie, 30 août-3 septembre 1996.
Sasaki Chikara. Descartes as a reformer of mathematical disciplines. In Descartes et le Moyen Âge. Edited by Biard Joël and Rashed Roshdi. Paris: Vrin 1997. pp. 37-45
Sasaki Chikara. Descartes's mathematical thought. Dordrecht: Kluwer Academic Publishers 2003.
See the second Part: The concept of 'mathesis universalis' in historical perspective pp. 287-438.
Schmitz François, "La pyramide de Leibniz. Note sur le logiquement possible et la logique modale," Cahiers de Philosophie du Langage 4: 63-99 (2000).
Schneider Martin, "Funktion und Grundlegung der Mathesis Universalis im Leibnizschen Wissenschaftsystem," Studia Leibnitiana.Sonderheft 15: 162-182 (1988).
Schuster John A. Descartes' Mathesis Universalis, 1619-28. In Descartes: philosophy, mathematics and physics. Edited by Gaukroger Stephen. Sussex : Harvester Press 1980. pp. 41-96
Thiel Christian. From Leibniz to Frege: mathematical logic between 1679 and 1879. In Logic, methodology and philosophy of science, VI. Edited by Cohen Jonathan. Amsterdam: North-Holland 1982. pp. 755-770
Proceedings of the Sixth International Congress of logic. methodology and philosophy of science, Hannover 1979.
Tito Johanna Maria. Logic in the Husserlian context. Evanston: Northwestern University Press 1990.
Van de Pitte Frederick, "Descartes' Mathesis Universalis," Archiv für Geschichte der Philosophie 61: 154-174 (1979).
Reprinted in: Georges J. D. Moyal (ed.) - René Descartes: Critical assessments - Vol. I - New York, Routledge, 1991, pp. 61-79.
Van de Pitte Frederick, "The dating of Rule IV-B in Descartes' Regulae ad directionem ingenii," Journal of the History of Philosophy 29: 375-395 (1991).
" A careful analysis of Rule IV requires the acceptance of a later dating for this fragment--probably as late as 1639-1640, when the Meditations were uppermost in Descartes's thought. It also permits a clarification of his terminology: Mathesis is a science of necessary relations. 'Mathesis universalis', rather than a mere extension of 'mathesis', is a distinct discipline which transforms systems of necessary relations into genuine 'scientia' by providing the underlying conditions for the very possibility of knowledge. Thus, Descartes provides not a simple mathematical method, but a very profound methodology."
Villalobos José, "Mathesis Universalis cartesiana," Cudaernos sobre Vico 5/6: 239-250 (1996).
Weber Jean-Paul, "Sur la composition de la Regula IV de Descartes," Revue Philosophique de la France et de l'Étranger: 1-29 (1964).
Weber Jean-Paul. La constitution du texte des Regulae. Paris: Société d'Édition d'Einsegnement Supérieur 1964.
Chapitre I. La Règle IV pp. 3-17.
Weingartner Paul. The ideal of the mathematization of all sciences and of "more geometrico" in Descartes and Leibniz. In Nature mathematized. Historical and philosophical case studies in classical modern natural philosophy. Edited by Shea William R. Dordrecht: Reidel 1983. pp. 151-195
Papers derived from the Third International Conference on the History and Philosophy of Science, Montreal, Canada, 1980, Vol. 1.
Westerhoff Jan C., "'Poeta Calculans': Harsdorffer, Leibniz, and the "mathesis universalis"," Journal of the History of Ideas 60: 449-467 (1999).
"This paper seeks to indicate some connections between a major philosophical project of the seventeenth century, the conception of a "mathesis universalis", and the practice of baroque poetry. I shall argue that these connections consist in a peculiar view of language and systems of notation which was particularly common in European baroque culture and which provided the necessary conceptual background for both poetry and the mathesis universalis."
Wiegand Olav K. Interpretationen der Modallogik. Ein Beitrag zur phaenomenologischen Wissenschaftstheorie . Dordrecht: Kluwer 1998.
"The author's aim is to point out interpretations of modal logic which are compatible with the phenomenological approach to mathematics. The book consists of three parts with ten chapters. In the first part (pp. 19-77) the author presents E. Husserl's conception of a ''mathesis universalis''. For Husserl, the mathesis universalis contains both, formal mathematics and formal (symbolic) logic. It has a hierarchical structure consisting of a pure logical grammar, a logic of consequences and a logic of truths. The author pays special attention to the differences between formal logic and formal mathematics which can be observed despite their extensional identity.\par In the second part (pp. 81--143) the author presents what he calls ''phenomenological semantics'', i.e. the phenomenological theory of modalization being a general analysis of intentions. The author distinguishes three levels of modalization, the level of protological passive synthesis, the level of protological active synthesis, and the level of (logical) predication.\par The third part (pp. 147--194) combines the results of the preceding parts in a phenomenological criticism of modern modal logic, especially its interpretation as possible worlds semantics. The problems of applying this semantics to natural language are seen as anchor points of phenomenological criticism. The provability interpretation of modal logic is proposed as a genetic interpretation, notwithstanding the problems which Hilbert's program and Husserl's closely related idea of definite manifolds had with Gödel's and Church's results. (Volker Peckhaus)".
Wiegand Olav K. Phenomenological-semantic investigations into incompleteness. In Phenomenology on Kant, German Idealism, hermenutics and logic. Philosophical essays in honor of Thomas M. Seebohm. Edited by Wiegand Olav K. et al. Dordrecht: Kluwe 2000. pp. 101-132
See in particular § 2. Husserl's phenomenological analysis of the mathesis universalis pp. 105-111
Winance Eleuthère, "Logique, mathématique et ontologie comme 'mathesis universalis' chez Edmund Husserl," Revue Thomiste 66: 410-434 (1966).

Language as Calculus versus language as Universal Medium

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Mathesis Universalis: the Search for a Universal Science

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