Fondée sous les auspices du père de notre modernité philosophique Descartes, puis consolidée par des penseurs aussi importants que Leibniz, Bolzano ou Husserl, la mathesis universalis paraît représenter à elle seule l'ambitieux programme du « rationalisme classique ». Des philosophes tels que Husserl, Russell, Heidegger ou Cassirer ont pu s'accorder en ce point. Le développement de la « science moderne » aurait porté ce grand « rêve dogmatique » pour mener vers son terme le destin de la métaphysique (...) occidentale. Pourtant les recherches historiques récentes ont montré que l'idée de « mathématique universelle » existait bien avant Descartes, que ce dernier ne revendiquait d'ailleurs aucune rupture sur ce point et que sa réflexion se situait même assez clairement dans l'héritage des Anciens. Comment dès lors justifier que les Anciens, avec lesquels le programme des Classiques était censé rompre, aient pu déjà se préoccuper de « mathématique universelle »? Plus simplement encore, de quoi se préoccupaient donc ces philosophes sous ce concept? Le regain d'intérêt pour la mathesis universalis à la fin du XIXe siècle n'avait-il pas conduit paradoxalement à la perte de son sens comme problème? Cette étude a pour but de suivre ces questions jusqu'à leur origine et de montrer leur importance dans le dialogue entre mathématique et philosophie. Pages de début Remerciements Introduction La constitution de la « mathématique universelle » comme problème philosophique I. Aristote II. « Mathématique universelle » et théories mathématiques : Aristote, Euclide, Epinomis III. Le moment néo-platonicien Vers la science de l'ordre et de la mesure IV. La renaissance de la mathématique universelle V. La mathesis universalis cartésienne Conclusion Annexe I. La quaestio de scientia mathematica communi Annexe II. Essai bibliographique sur la mathesis universalis chez Descartes etLeibniz Bibliographie Index nominum Pages de fin. (shrink)

The paper’s central theme is the link between phenomenology and the notion of the mathesis universalis, a link articulated by Husserl in the third volume of the Ideas: “My way to phenomenology was essentially determined by the mathesis universalis.” The paper suggests three interpretations of the phenomenology—mathesis universalis nexus: the first is related to the development of Husserl’s conception of the foundations of arithmetic; the second is based on the role of the theory of (...) manifolds in Husserl’s Logical Investigations; and the third reflects the importance of the distinction between “generalization” and “formalization” for phenomenology. After examining these interpretations, the paper explores which is most helpful for understanding why Husserl distanced himself from Bolzano, arguing that the third interpretation provides the most edifying answer. (shrink)

Zwrot \"Mathesis Universalis\" (MU) oznacza projekt unifikacji całości wiedzy za pomocą metody matematycznej. Powstał on pod tą nazwą u progu nowożytności, a częściowo miał antycypacje w starożytności i średniowieczu. Głównymi jego rzecznikami byli Kartezjusz i Leibniz. Podejście Leibniza jest radykalnie formalistyczne, a przez to nadające sie do realizacji maszynowej, podczas gdy Kartezjusza - zdecydowanie antyformalistyczne. Artykuł koncentruje się na projekcie Leibniza jako tym, który ma kontynuację w nauce współczesnej. Zamysł, żeby narzędziem realizacji projektu była uniwersalna symbolika i rachunek (...) logiczny jest obecnie realizowany w odniesieniu do całości matematyki za pomocą środków, które jako pierwszy stworzył Gottlob Frege (1879). Istotny wkład ma Georg Cantor jako autor teorii mocy zbiorów, która wprowadza odróżnienie zbioru nieskończonego przeliczalnie od zbioru o mocy kontinuum. Umożliwia to dowód fundamentalnego dla współczesnej MU twierdzenia o istnieniu problemów nierozstrzygalnych w arytmetyce, co jest dziełem Kurta Gödla, oraz w logice. To drugie wykazał Alan Turing, dowodząc, że zbiór realizowalnych przez maszynę programów do dokonywania obliczeń, czyli do rozstrzygania problemu, jaka jest wartość określonej funkcji, jest tylko przeliczalny, podczas gdy zbiór problemów do rozstrzygnięcia ma moc kontinuum; stąd wynika, że pewne problemy są nierozstrzygalne w sposób formalny czyli maszynowy, co z kolei implikuje nierozstzygalność logiki. Analogiczny wynik Gödla ma ważne dopełnienie w jego twierdzeniu, że ograniczenia rozstrzygalności są tylko względne (w stosunku do określonego systemu formalnego) i mogą być przezwyciężane przez konstrukcję nowych pojęć pierwotnych, aksjomatów i metod postępowania badawczego. Rysuje się więc przed nauką możliwość nieustannego postępu. Takie dynamiczne ujęcie stanowi rys istotnie różniący współczesną postać MU od klasycznej, którą cechowała statyczność. (shrink)

In a fragment entitled Elementa Nova Matheseos Universalis Leibniz writes “the mathesis [...] shall deliver the method through which things that are conceivable can be exactly determined”; in another fragment he takes the mathesis to be “the science of all things that are conceivable.” Leibniz considers all mathematical disciplines as branches of the mathesis and conceives the mathesis as a general science of forms applicable not only to magnitudes but to every object that exists in (...) our imagination, i.e. that is possible at least in principle. As a general science of forms the mathesis investigates possible relations between “arbitrary objects”. It is an abstract theory of combinations and relations among objects whatsoever. In 1810 the mathematician and philosopher Bernard Bolzano published a booklet entitled Contributions to a Better-Grounded Presentation of Mathematics. There is, according to him, a certain objective connection among the truths that are germane to a certain homogeneous field of objects: some truths are the “reasons” of others, and the latter are “consequences” of the former. The reason-consequence relation seems to be the counterpart of causality at the level of a relation between true propositions. Arigorous proof is characterized in this context as a proof that shows the reason of the proposition that is to be proven. Requirements imposed on rigorous proofs seem to anticipate normalization results in current proof theory. The contributors of Mathesis Universalis, Computability and Proof, leading experts in the fields of computer science, mathematics, logic and philosophy, show the evolution of these and related ideas exploring topics in proof theory, computability theory, intuitionistic logic, constructivism and reverse mathematics, delving deeply into a contextual examination of the relationship between mathematical rigor and demands for simplification. (shrink)

이 글은 데카르트의 정신지도를 위한 규칙에서 보편 수리학의 관점에서 비례의 개념을 해명하는 것을 목표로 한다. 보편 수리학이란 데카르트 자신의 설명에 따르면, “순서와 척도에 관해 연구될 수 있는 것을 모두 설명하는 어떤 일반적인 학문”이다. 구체적으로 말하자면 측량할 수 있는 모든 크기를 그 차원이나 존재 방식과 무관하게 어떤 보편적 인식 원리에 따라 규정하고 인식하는 것이 보편적 수리학의 과제인 것이다. 그런데 이런 보편 학문의 이념을 실현하는 것을 불가능하게 만들었던 장벽이 있었는데, 그것이 수와 도형의 통약 불가능성 그리고 같은 도형에서도 차원이 다른 도형들 사이의 연산 (...) 불가능성이다. 그런데 데카르트는 『정신지도를 위한 규칙』에서 수와 도형 사이의 장벽과 도형의 상이한 차원에 따른 연산 불가능성을 비례의 개념을 통하여 극복한다. 그리고 이를 통해 대수와 기하의 장벽을 허물고 보편적 수리과학의 기초를 마련했다. 이 글은 『정신지도를 위한 규칙』에서 비례의 개념이 어떻게 보편적 수리과학의 기초가 되는지를 주요 구절들을 통해 간단 명료하게 보이려는 시도이다. (shrink)

The author shows how Proclo is a precursor of 'Mathesis universalis' concept, without admiting 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.

데카르트는 학문들의 통일된 체계를 보편적 방법을 통해 확립하고자 하였다. 이러한 그의 기획은 그의 초기 미완의 저술 『정신지도규칙』에 이미 주제로 나타난다. 본 논문은 「규칙4-B」의 보편수리학과 「규칙4-A」의 보편 방법사이의 관계를 살펴봄으로써, 보편 방법의 일차적인 관심이 존재론적인 것이라기보다 논리학적인 것임을 보일 것이다. 보편 방법은 학문적 증명들을 순수 지성의 근원적 자율성 위에 정초함으로써 학문들의 통일된 체계를 확립하고자 하는 기획이다. 데카르트는 이러한 기획을 실현하기 위해 두 가지 중요한 방법을 사용한다. 첫째, 「규칙6」에서 그는 관계들을 정신활동들로 이해한다는 전제 하에서, 학문적 물음들을 관계들을 통해 정식화할 것을 권한다. 둘째, (...) 「규칙7」에서 그는 연속적이고 부단한 사유의 반복을 통해서, 학문적 증명이 정신의 자율성 위에 정초되는 방식을 동시적으로 직관하라고 권한다. 그렇지만 『규칙』에서의 자율성은 데카르트가 『성찰』에서 주장하게 될 “자유의지”와 구별되어야 한다. 왜냐하면 『규칙』의 자율성은 규칙을 생산하는 근원적 능력을 의미하지 아직 절대적 선택의 능력을 의미하지는 않기 때문이다. (shrink)

This paper seeks to indicate some connections between a major philosophi- cal 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.

Este artigo problematiza a concepção de mathesis universalis na filosofia de Edmund Husserl a partir de uma discussão sobre a noção de Mannigfaltigkeitslehre. Inicialmente, retoma-se a questão dos números “imaginários” e depois é repassado o contexto matemático que influenciou o desenvolvimento filosófico husserliano. Finalmente, a concepção de mathesis universalis é enquadrada nas referências históricas de Descartes e Leibniz para, finalmente, mostrar as implicações fenomenológicas próprias do pensamento de Husserl.

Dans La Philosophie de l’algèbre (1962), Jules Vuillemin présente sa démarche comme une manière d’instruire « le problème, si important et si négligé aujourd’hui, de la mathesis universalis dans ses rapports à la philosophie ». Il intitule d’ailleurs la seconde partie du traité « mathématique universelle », titre qu’il reprend pour la conclusion. Présentant le projet du second tome, il avance que cette étude devait le conduire « aux questions concrètes de la mathématique universelle ». Pourtant, à aucun (...) moment, on ne voit Vuillemin expliciter la nature de ce problème. Le but de cet article sera de produire une telle explicitation en s’appuyant, en particulier, sur certains éléments avancés dans le manuscrit préparatoire du second tome. (shrink)

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. (shrink)

The notion of mathesis universalis appears in many of Edmund Husserl’s works, where it corresponds essentially to “a universal a priori ontology”. This paper has two purposes; one, largely exegetical, of clarifying how Husserl elaborates on Leibniz’ concept of mathesis universalis and associated notions like symbolic thinking and symbolic knowledge filtering them through the lesson of the so called “bohemian Leibniz”, Bernard Bolzano; another, more properly philosophical, of examining the role that the universal mathesis is (...) allowed to play, and the space it occupies in Husserl’s intuition-based epistemology. (shrink)

Tomar a Regra IV como a junção ? sem o aval de Descartes ? de dois textos distintos passou a ser um lugar comum na literatura sobre aquele texto. Seria necessário explicar a distinção entre os textos recorrendo à outra importante distinção: entre mathesis universalis e método. Com efeito, nosso artigo tenta pôr em questão essa interpretação clássica na medida em que pretende mostrar que a revisão que se opera no seio da matemática torna-a menos uma ciência das (...) grandezas abstratas que um método que constrange as ciências a instituir a ordem e a medida em seus estudos. Em outras palavras, pode-se pensar que a discussão sobre a matemática em que se ocupa a Regra IV tem como objetivo uma universalização das operações de ordem e medida, em detrimento de uma compreensão da matemática como ciência das grandezas abstratas. Assim, defenderemos que a Regra IV ocupa-se de um único problema: a instituição de um método universal. Devemos considerá-la então como um único texto. (shrink)

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. (shrink)

O objetivo deste artigo consiste em aprofundar a intuição foucaultiana no livro As palavras e as coisas a respeito da máthêsis universalis como ciência geral da ordem e da medida, para tratar da Idade Clássica. Essa caracterização de Michel Foucault lhe permite afastar-se da percepção desta época pela lente do racionalismo, mecanicismo e matematização da natureza. Ao refletir sobre a ordem empírica dos seres, palavras e coisas pela relação entre máthêsis, taxinomia e teoria do signo, Foucault relativiza também certa (...) percepção da fenomenologia (husserliana) segundo a qual o racionalismo seria o solo fundante da Modernidade. Palavras-chave: Máthêsis. Taxinomia. Arqueologia. Fenomenologia. Michel Foucault. Descartes. (shrink)

ABSTRACT : On many accounts, Rule IV appears to be composed of two distinct texts, and this should be justified by the difference between the mathesis universalis and the Cartesian method. This article runs counter to the usual interpretation by showing that the discussion on mathematics in Rule IV has enabled the introduction of method constraining scientific research to operate on grounds of order and measurement. The mathesis universalis is not so much a science of higher (...) mathematics as a universal method by the means of which one can find truth. (shrink)

Book review of John Schuster: Descartes-agonistes: Physico-mathematics, method and corpuscular-mechanism 1618-33. (Studies in History and Philosophy of Science, Volume 27.) Dordrecht: Springer, 2013, xix + 631pp. Descartes-Agonistes is the magnum opus of John Schuster, formerly of the University of New South Wales, honorary fellow at the University of Sydney. Its roots go back to the dissertation he wrote 35 years ago under Thomas Kuhn at Princeton University. As Schuster correctly remarks, some regard his dissertation as an underground classic. I count (...) myself among them: Schuster’s work has been directional in my work on the history of early modern science. Schuster himself prefers to regard his dissertation as ‘a vestige’ of a history of science of the 1970s that he has since gone beyond. In the 1980s, he elaborated and published on some central insights in a series of seminal articles, but his work on Descartes really got into its stride again around the turn of the millennium (Schuster 1980, 1990; Gaukroger et al. 2000). This book is the acme of this work, and grand it is: clocking 600 pages (excluding appendices) it extends, deepens and reinforces his analysis of Descartes into a dense and .. (shrink)