Theory and History of Ontology

by Raul Corazzon | e-mail: rc@ontology.co

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Selected Bibliography on the History of Mathesis Universalis

Studies in English

GENERAL STUDIES

  1. Buzon, Frédéric de. 1998. "Mathesis Universalis." In La Science Classique: Xvie-Xviiie Siècle. Dictionnaire Critique, edited by Blay, Michel and Halleux, Robert, 610-621. Paris: Flammarion.

  2. Charrak, André. 2009. Empirisme Et Théorie De La Connaissance. Réflexion Et Fondement Des Sciences Au Xviiie Siècle. Paris: Vrin.

  3. Cohen, Jonathan. 1954. "On the Project of a Universal Character." Mind no. 63:49-63.

    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.

  4. Desanti, Jean-Toussaint. 1972. "Réflexions Sur Le Concept De "Mathesis"." Bulletin de la Société Française de Philosophie no. 77:1-22.

    Reprinted in: J. T. Desanti - La philosophie silencieuse ou Critique des philosophies de la science - Paris. Le Seuil, 1975, pp. 196-219.

  5. Dumoncel, Jean-Claude. 2002. La Tradition De La Mathesis Universalis. Platon, Leibniz, Russell. Paris: Unebévue.

  6. Knobloch, Eberhard. 2004. "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, 77-90. Stuttgart: Franz Steiner Verlag.

  7. Marciszewski, Witold. 1984. "The Principle of Comprehension as a Present-Day Contribution to Mathesis Universalis." Philosophia Naturalis no. 21:523-537.

  8. Martineau, Emmanuel. 1976. "L'ontologie De L'ordre." Études Philosophiques:475-494.

  9. Mittelstrass, Jürgen. 1979. "The Philosopher's Conception of "Mathesis Universalis" from Descartes to Leibniz." Annals of Science no. 36:593-610.

    "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."

  10. Poser, Hans. 1998. "Mathesis Universalis and Scientia Singularis. Connections and Disconnections between Scientific Disciplines." Philosophia Naturalis no. 35:3-21.

  11. Rabouin, David. 2009. Mathesis Universalis. L'idée De "Mathématique Universelle" D'Aristote À Descartes. Paris: Presses Universitaires de France.

    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.

THE GREEK ORIGINS

  1. Bechtle, Gerald. 2007. "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., 129-154. Leiden: Brill.

  2. Klein, Jacob. 1968. Greek Mathematical Thought and the Origin of Algebra. Cambridge: MIT Press.

    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: Chapter 11.3 "The reinterpretation of the katholou pragmateia as Mathesis Universalis in the sense of a rs analytice" pp. 178-185, And Chapter 12 "The cconcept of number: A. In Stevin (186), B. In Descartes (197), C. In Wallis (211-224)".

  3. Napolitano Valditara, Linda M. 1988. Le Idee, I Numeri, L'ordine. La Dottrina Della Mathesis Universalis Dall'accademia Antica Al Neoplatonismo. Napoli: Bibliopolis.

  4. Ortiz de Landázuri, Carlos. 2000. "Mathesis Universalis En Proclo De Las Aporias Cosmologicas Al Universo Euclideo." Anuario Filosofico no. 33:229-257.

    "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."

  5. Rabouin, David. 2005. "La 'Mathématique Universelle' Entre Mathématique Et Philosophie, D'Aristote À Proclus." Archives de Philosophie no. 68:249-268.

    "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."

  6. Rabouin, David. 2010. "Le Rôle De Proclus Dans Les Débats Sur La ‘Mathématique Universelle’ À La Renaissance." In Études Sur Le Commentaire De Proclus Au Premier Livre Des Eléments D’euclide, edited by Lernould, Alain, 217-234. Villeneuve d'Ascq: Press Universitaires du Septentrion.

THE RENAISSANCE PERIOD

  1. Bockstaele, Paul. 2009. "Between Viète and Descartes: Adriaan Van Roomen and the Mathesis Universalis." Archiv for History of Exact Sciences no. 63:433-470.

    "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."

RENÉ DESCARTES

GOTTFRIED WILHELM LEIBNIZ

  1. Berlioz, Dominique. 1993. "Langue Adamique Et Caractéristique Universelle Chez Leibniz." In Leibniz and Adam, edited by Dascal, Marcelo and Yakira, Elhanan, 153-168. Tel Aviv: University Publishing Projects.

  2. Cardoso, Adelino. 1996. "Mathesis Leibniziana." Philosophica: Revista do Departamento de Filosofia da Faculdade de Letras de Lisboa:51-77.

    "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."

  3. Couturat, Louis. 1901. La Logique De Leibniz: D'aprés Des Documents Inédits. Paris: Felix Alcan.

    Reprinted: Hildesheim, Olms, 1961 e 1985.

  4. Danek, Jaromit, and Möckel, Christian. 2000. "Idee Der Mathesis Universalis - Die Logische Vernunft: Leibniz Und Husserl." In Phänomenologie Und Leibniz, edited by Cristin, Renato and Kiyoshi, Sakei, 88-121. Freiburg: Alber.

  5. Dascal, Marcelo. 1987. Leibniz. Language, Signs and Thought. Amsterdam: Benjamins Publishers.

  6. Gérard, Vincent. 2006. "Leibniz Et La Mathématique Formelle." Philosophie no. 92:29-55.

  7. Hayashi, Tomohiro. 2002. "Leibniz's Construction of Mathesis Universalis: A Consideration of the Relationship between the Plan and His Mathematical Contributions." Historia Scientiarum:121-141.

  8. Heinekamp, Albert. 1975. "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, 257-286. Wiesbaden: F. Steiner.

  9. Knecht, Herbert. 1981. La Logique Chez Leibniz. Essai Sur Le Rationalisme Baroque. Lausanne: L'Age d'Homme.

    See Chapter III: La mathématique universelle pp. 91-123.

  10. Mittelstrass, Jürgen. 1985. "Leibniz and Kant on Mathematical and Philosophical Knowledge." In The Natural Philosophy of Leibniz, edited by Okruhlik, Kathleen and Brown, James Robert, 227-261. Dordrecht: Reidel.

    See in particular § 2 Mathesis universalis pp. 232-239.

  11. Pombo, Olga. 1987. Leibniz and the Problem of a Universal Language. Münster: Nodus Publikationen.

  12. ———. 2002. "Leibniz and the Encyclopaedic Project." In Ciência, Tecnologia Y Bien Comun: La Actualidad De Leibniz, 267-278. Valencia: Editorial de la Universidas Politecnica de Valencia.

    "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."

  13. Poser, Hans. 1979. "Signum, Notio Und Idea. Elemente Der Leibnizschen Zeichentheorie." Zeitschrift für Semiotik no. 1:309-324.

    "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."

  14. Rabouin, David. 2012. "Interpretations of Leibniz’s Mathesis Universalis at the Beginning of the Xxth Century." In New Essays on Leibniz Reception. In Science and Philosophy of Science 1800-2000, edited by Krömer, Ralf; Chin-Drian, Yannick, 187-201. Dordrecht: Springer.

    "In his doctoral dissertation, completed in 1922 under the direction of Edmund Husserl and published in 1925 in the Jahrbuch für Philosophie und Phänomenologische Forschungen, Dietrich Mahnke proposed a very valuable overview of the so-called “Leibniz Renaissance”. As indicated by the choice of his title: Leibnizens Synthese von Universalmathematik und Individualmetaphysik, this renaissance was seen by Mahnke as marked by a tension between two Leibnizian programs: that of a “universal mathematics” and that of a “metaphysics of individuation”. His agenda was to propose a way of reconciling these two programs through a point of view inspired by the development of Husserlian phenomenology. In this paper, I will concentrate on the first program, “universal mathematics” or mathesis universalis, and see how the interpretation of this Leibnizian theme was indeed a key point in the demarcation between different ways of articulating logic, mathematics and philosophy at the beginning of the XXth century. "

  15. Rauzy, Jean-Baptiste. 1995. "Quid Sit Natura Prius? La Conception Leibnizienne De L'ordre." Revue de Métaphysique et de Morale no. 98:31-48.

    " 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."

  16. Risse, Wilhelm. 1969. "Die Characteristica Universalis Bei Leibniz." Studi Internazionali di Filosofia no. 1:107-116.

  17. Schmitz, François. 2000. "La Pyramide De Leibniz. Note Sur Le Logiquement Possible Et La Logique Modale." Cahiers de Philosophie du Langage no. 4:63-99.

  18. Schneider, Martin. 1988. "Funktion Und Grundlegung Der Mathesis Universalis Im Leibnizschen Wissenschaftsystem." Studia Leibnitiana.Sonderheft no. 15:162-182.

  19. Weingartner, Paul. 1983. "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., 151-195. Dordrecht: Reidel.

    Papers derived from the Third International Conference on the History and Philosophy of Science, Montreal, Canada, 1980, Vol. 1.

  20. Westerhoff, Jan C. 1999. "'Poeta Calculans': Harsdorffer, Leibniz, and the "Mathesis Universalis"." Journal of the History of Ideas no. 60:449-467.

    "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."

Mathesis Universalis AFTER LEIBNIZ

  1. Arndt, Hans Werner. 1971. Methodo Scientifica Pertractatum. Mos Geometricus Und Kalkülbegriff in Der Philosophischen Theorienbildung Des 17. Und 18. Jahrhunderts. Berlin: Walter de Gruyter.

    Inhaltsverzeichnis: Vorwort: V; Einleitung 1; I. Zur Herausbildung der "Matematischen Methode" im Zusammenhang der Entwicklung des Begriff der Methode 15; II. Descartes' Begriff der Methode im Verhältnis zu seiner Konzeption einer "Mathesis Universalis" 29; III. Zur Auffassung der "Mos Geometricus" un der "Mathesis Universalis" in der zweiten Hälfte des 17. Jahrunderts 69; IV. Das Verhältnis von matematischer Methode und "Mathesis Universalis" in der Philosophie von Leibniz 99; V. "Methodo Scientifica" und "Mathesis Universalis" in der Metodenlehre Christian Wolffs 125; VI. Johann Heinrich Lamberts Konzeption einer Wissenschaflichen Grundlehre 149; Literaturverzeichnis 161; Sachregister 166; Namenregister 169-170.

  2. ———. 1979. "Die Semiotik Christian Wolffs Als Propädeutik Der Ars Characteristica Combinatoria Und Der Ars Inveniendi." Zeitschrift für Semiotik no. 1.

    "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."

  3. Ciafardone, Raffaele. 1971. "Il Problema Della "Mathesis Universalis" in Lambert." Il Pensiero no. 16:171-208.

  4. Knowlson, James. 1975. Universal Language Schemes in England and France 1600-1800. Toronto: University of Toronto Press.

  5. Maat, Jaap. 2004. Philosophical Languages in the Seventeenth Century: Dalgarno, Wilkins, Leibniz. Dordrecht: Kluwer.

  6. Peckhaus, Volker. 1997. Logik, Mathesis Universalis Und Allgemeine Wissenschaft. Leibniz Und Die Wiederentdeckung Der Formalen Logik Im 19. Jahrundert. Berlin: Akademie Verlag.

    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.

  7. Thiel, Christian. 1982. "From Leibniz to Frege: Mathematical Logic between 1679 and 1879." In Logic, Methodology and Philosophy of Science, Vi, edited by Cohen, Jonathan, 755-770. Amsterdam: North-Holland.

    Proceedings of the Sixth International Congress of logic. methodology and philosophy of science, Hannover 1979.

EDMUND HUSSERL

  1. Gané, Gilles. 1971. L'idée De La "Mathesis Universalis". Essai Sur La Doctrine De La Science D'edmund Husserl.

    Thèse inédite presentée à l'Université d'Ottawa.

  2. Gérard, Vincent. 2001. Mathématique Universelle Et Métaphysique De L’individuation. L’élaboration De L’idée De Mathesis Universalis Dans La Phénoménologie De Husserl.

    Thèse inédite.

    Résumé: "La mathesis universalis est-elle l'ontologie formelle? Telle est la question à laquelle nous nous proposons de répondre dans ce travail. Dans la première partie, on trouve la genèse de l’idée de mathesis universalis comme ontologie formelle. Dans la deuxième, les délimitations ontologiques de la mathesis universalis par rapport a la géométrie et l'axiologie formelle. dans la troisième, l'élucidation phénoménologique de la mathesis universalis comme théorie des sens apophantiques purs. Dans la quatrième, son articulation sur une métaphysique formelle ou théorie de l'individuation: la mathesis universalis est alors réarticulée sur l'ontologie formelle, mais en un autre sens de l'ontologie formelle. Les résultats auxquels nous sommes parvenu sont les suivants : 1) Husserl emprunte son concept de mathesis universalis, non pas à la Règle IV-b de Descartes, soit pour en accomplir le sens, soit pour la détourner de son sens, mais a la tradition arithmétisante de Van Schooten, Wallis, Newton et du Leibniz de 1695; 2) l'élaboration husserlienne de l'idée de mathesis universalis est une tentative pour identifier un ensemble de noyaux régulateurs (principe de permanence de Hankel, etc.) qui norment les possibilité d'admission d'objets dans le champ analytique formel; 3) la géométrie comme science de l'espace est exclue de ce champ; 4) il existe en revanche une analogie radicale entre l'axiologie formelle et la mathesis universalis; 5) Husserl n'est pas seulement redevable a Leibniz de l'idée de mathesis universalis, mais également de sa conversion philosophique; 6) la mathesis philosophique pensée a la lumière de la théorie de la connaissance telle qu'elle est élaborée par Leibniz vers 1684 n'est, ni ne veut être, une théorie de l' être, mais une théorie pure de la signification; 7) cette théorie de la signification s'articule sur une métaphysique formelle dont Husserl emprunte le concept a Lotze. Elle a pour tâche de décrire les formes idéales auxquelles doivent correspondre les relations entre les éléments d'un monde, quel qu'il soit."

  3. ———. 2002. "La Mathesis Universalis Est-Elle L'ontologie Formelle?" Annales de Phénomenologie no. 1:61-98.

  4. ———. 2012. " Mathesis Universalis Et Géométrie: Husserl Et Grassmann." In Philosophy, Phenomenology, Sciences. Essays in Commemoration of Edmund Husserl, edited by Ierna, Carlo, Jacobs, Hanne and Mattens, Filip, 255-300. New york: Springer.

  5. Ha, Byung-Hak. 1997. Das Verhältnis Der Mathesis Universalis Zur Logik Als Wissenschaftstheorie Bei E. Husserl. Bern: Peter Lang.

  6. Hopkins, Burt C. 2011. The Origin of the Logic of Symbolic Mathematics: Edmund Husserl and Jacob Klein. Bloomington: Indiana University Press.

  7. Kuster, Frederike. 2008. Wege Der Verantwortung. Husserls Phänomenologie Als Gang Durch Die Faktizität. Dordrecht: Kluwer.

    See Chapter 1: Eine mathesis der Geist und de Humanität.

  8. Milkov, Nikolay. 2006. "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, 119-135. Dordrecht: Springer.

  9. Olvera Mijares, Raul. 1994. "Some Historical Remarks on Husserl's Theory of Multiplicity." Axiomathes no. 5:385-394.

  10. Rabouin, David. 2006. "Husserl Et Le Projet Leibnizien D'une Mathesis Universalis." Philosophie no. 92:13-28.

    "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."

  11. Safou, Jean-Bernard. 2002. Husserl Et La Métaphysique De Descartes: Essai D'une Interprétation Phénoménologique Du Projet De La Mathesis Universalis. Thèse inédite available au Atelier National de reproduction des Thèses - ANRT (Ref. 31760).

  12. Tito, Johannes Maria. 1990. Logic in the Husserlian Context. Evanston: Northwestern University Press.

  13. Wiegand, Olav K. 1998. Interpretationen Der Modallogik. Ein Beitrag Zur Phaenomenologischen Wissenschaftstheorie. Dordrecht: Kluwer.

    "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)".

  14. ———. 2000. "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., Dostal, Robert J., Embree, Lester, Kockelmans, Joseph and Mohanty, Jitendra Nath, 101-132. Dordrecht: Kluwer.

    See in particular § 2. Husserl's phenomenological analysis of the mathesis universalis pp. 105-111.

    "Husserl's analyses of the mathesis universalis, in keeping with their detailed presentation in FTL [Formal and Transcendental Logic 1929], continue to offer a durable foundation for more extensive phenomenological investigations of the formal sciences. Here Husserl makes a particularly important distinction, one of exemplary significance for the whole of phenomenological description. In the first place, the mathesis universalis understood as objectively existing science -- in Husserl's terminology as objective logic (26) -- is to be phenomenologically-descriptively analyzed. In the second place, these investigations directed toward objective logic are to be supplemented through a subjective logic i. e., (27) through analyses of the cognitive structures of mathematical or logical knowing. The problems Husserl takes on in FTL according to these terms are particularly, (i) the relation between formal logic and mathematics (their co-extension and distinguishability), and (ii) the inner structure of the mathesis universalis. Both problems will be briefly addressed in what follows." p. 105

  15. Winance, Eleuthère. 1966. "Logique, Mathématique Et Ontologie Comme ' Mathesis Universalis' Chez Edmund Husserl." Revue Thomiste no. 66:410-434.

LUDWIG WITTGENSTEIN

  1. Dumoncel, Jean-Claude. 1991. Le Jeu De Wittgenstein. Éssai Sur La Mathesis Universalis. Paris: Presses Universitaires de France.

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