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
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"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
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both geometry and arithmetic and to provide general rules valid for operations
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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
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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
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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
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de sa dernière pensée. Dans cette évolution, la correspondance avec de Volder
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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
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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
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multiplicity," Axiomathes 5: 385-394 (1994).
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(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."
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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.
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"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
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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."
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disconnections between scientific disciplines," Philosophia Naturalis 35:
3-21 (1998).
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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).
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