Présentation, transcription et traduction de quatre manuscrits mathématiques inédits de Leibniz traitant de la loi de continuité, des logarithmes des nombres imaginaires, des axiomes du calcul des grandeurs et du calcul dyadique.

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)

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.

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)

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)

This is a study of J.G. Fichte’s philosophy of mathematics and theory of geometry. It investigates both the external, formal & internal, cognitive parallels between the axioms, intuitions and constructions of geometry and the scientific methodology of the Fichtean system of philosophy. In contrast to 'ordinary' Euclidean geometry, in his Erlanger Logik of 1805 Fichte posits a model of an 'ursprüngliche' or original geometry. – That is to say, a synthetic and constructivistic conception of geometry grounded in ideal archetypal elements (...) that are grasped through geometrical or intelligible intuition. -/- Accordingly, this study classifies Fichte’s philosophy of mathematics as a whole as a species of mathematical Platonism or neo-Platonism, and concludes that the Wissenschaftslehre itself may be read as an attempt at a new philosophical mathesis, or a “mathesis of the mind.” -/- . (shrink)

Le roman proustien est hanté par la présence du possible, qui appelle son élucidation en logique modale. Il est aussi balisé de notions mathématiques. Ces deux aspects convergent dans la psychologie amoureuse, donnant au romanesque une teneur intellectuelle qui se prolonge dans sa portée politique.

Dans le cadre du programme « Eikon : la vie des portraits grecs » financé par l’Agence nationale de la recherche, ont été poursuivis les travaux entamés à Délos en 2013 au cours de deux campagnes d’étude sur le site et au Musée. Études des supports de statues-portraits conservés sur le site (Fr. Herbin) Sur le site, l’objectif de la mission était de terminer l’inventaire des supports de statues-portraits. Seuls les blocs portant une dédicace ainsi que des semelles pour statue(s) (...) en bronze ou... (shrink)

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)

Die Forschung zu und über Leonhard Euler verweist oft darauf, dass das Entstehen und Etablieren neuer Zeichenpraktiken zum dominierenden Legitimationsmuster der übersichtlichen Mathematik in der Zeit der Aufklärung wird. Bei genauerem Hinsehen fällt jedoch auf, dass die konkreten Formen der Erscheinung und Durchsetzung dieser Praktiken mit ihren symbolischen Dimensionen über Jahrzehnte kaum ins Zentrum der Aufmerksamkeit gerückt wurden. Gerade diesen Formen der Wissensentfaltung und -vermittlung versuchen die hier versammelten Aufsätze eine zentrale Position zukommen zu lassen. Denn im Falle Eulers beschränkt (...) sich die Problematik solcher Erscheinungen nicht auf das mathematische Abstrahieren der zu lösenden Aufgaben, sind diese doch mit den graphischen Abstrahierungen vernetzt, deren Erscheinungskriterien untersucht werden. (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 paper deals with the connection of "mathesis", intuition and imagination in Descartes' "Regulae ad directionem ingenii" - young Descartes' first writing about methodology, intended to set the foundation of a "mathesis generalis" - and in some of his followers/critics, in particular Leibniz, on the background of the so-called psychology of faculties. Two are the main points at stake: the presence of mental images in thought processes, and the cognitive functions of the imagination. A thorough analysis of Descartes' (...) examples of intuitive, not-imaginative cognition shows the difficulties in his approach. Different solutions will be proposed by Descartes himself and by most cartesians, on one hand, and by Tschirnhaus on the other hand. Leibniz's peculiar solution is finally considered. (shrink)

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)

There is an infinity of figures and of movements, present and past, which enter into the efficient cause of my present writing, and in its final cause, there are an infinity of slight tendencies and dispositions of my soul, present and past.1.

Resumen: El objetivo de este artículo es mostrar la relevancia de la máthesis para la concepción de la mimesis aristotélica. A partir de las observaciones de la Poética, delimitaré las características del aprendizaje tomando como eje su objeto, modalidad y consecuencias. Para ello analizaré, en primer lugar, el objeto sobre el que recae el aprendizaje mimético, esto es, los hombres que actúan. Luego examinaré la modalidad de presentación de sus acciones para que sea posible el aprendizaje, prestando especial atención al (...) papel que desempeña el error trágico o hamartía. Finalmente, resaltaré los aspectos por los que, a mi entender, la mímesis tiene consecuencias éticas y políticas relevantes en el marco del proyecto educativo de Aristóteles, y que permiten pasar de una consideración de la mímesis en el plano individual a una consideración sobre su función en el ámbito político.: The aim of this paper is to show the relevance of mathesis for the Aristotelian concept of mimesis. From the framework of Poetics, i try to demarcate the characteristics of learning, taking into account its object, modality, and consequences. In order to do this, I will analyze, first, the object of mimetic learning, that is, men who act. Then, I will examine the modality of presentation of their actions that make learning possible, paying special attention to the role played by tragic mistake or hamartía. Finally, I will highlight the aspects on account of which, in my opinion, mimesis has relevant ethical and political consequences in the framework of Aristotle's educational project, and which allow passing from a consideration of mimesis on an individual level to a political level. (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.

We develop a new method for constructing hyperimmune minimal degrees and construct an ANR degree which is a minimal cover of a minimal degree.

El concepto cartesiano de una _mathesis universalis _tiene como antecedente histórico inmediato algunas de las ideas defendidas en el contexto de las discusiones del aristotelismo renacentista acerca de la certeza matemática y el puesto de las matemáticas en la clasificación de las ciencias. Una de las figuras principales en este debate es la del jesuita valenciano Benet Perera (1535 – 1610). A través de la presentación de una serie de pasajes extraídos de los manuscritos inéditos de Perera, en este artículo (...) analizamos la defensa que este autor realiza de la existencia de una ciencia matemática común, en relación con algunas de sus ideas fundamentales, como la impugnación del carácter científico de las matemáticas, o la división de la metafísica en una filosofía primera dedicada al ente en cuanto ente y una teología natural que estudia las inteligencias y Dios. (shrink)

It may only be a handful of years before fully autonomous neurosurgical robots (ANRs) are pushed into widespread clinical adoption. Nevertheless, whether it is ethical to greenlight the development and adoption of ANRs is still up for debate. On the one hand, the widespread adoption of ANRs may lead to unprecedented therapeutic effects, increase sterility, improve pain profiles, increase precision, and reduce complications over the long term. On the other hand, ANRs may lead to human neurosurgical skill atrophy, increased legal (...) uncertainty, increased burnout rates, and may produce no significant effect on pain profiles or complication rates, all of which may put patients at novel levels of risk. At this watershed, it is critical for stakeholders to preemptively deliberate about whether they would ultimately agree to these ethical trade-offs and decide to consciously support, thus help usher in the advent of autonomous neurosurgical technology. (shrink)

The ‘mathematical imaginary’ at work in psychology is central to the contingent history of the discipline, but is also responsible for considerable confusion and ambiguity around the ontological assumptions of psychological theories and models. Rather than reject the mathematical altogether, this article argues for an alternative form of mathematical description in psychology through the use of topology. Drawing on DeLanda’s topological account of the virtual, the relationship between psychology and ontology is progressively questioned in relation to memory. Henri Bergson’s conception (...) of duration and the pure past is juxtaposed with the topological psychology of Kurt Lewin and its notion of life space as a manifold with n-dimensions. Using a diagrammatic strategy developed by Mullarkey, an actualist account of the virtual is used to hold these distinct bodies of work together. Psychological events may be given a ‘thick description’ through the identification of topological relations and invariants. The interplay between the actual entities that are afforded in experience can be seen to be virtualized as planes of irreality. A topological approach to psychology offers the opportunity of replacing the taxonomy of essences common to the discipline with a dynamic account of invariance through homeomorphism. (shrink)

In this chapter I argue that the general category of 'iatromechanics' conceals different views about how mathematics and mathematical physics should be applied in medicine, and thereby about how physiology should be quantified and mathematized. Mechanism, quantification and mathematization are indeed different, albeit interrelated, notions, which overlap without ever coming to be identical, and all of which depend upon the overall epistemological debate over the method for finding truth in science. This chapter compares the epistemological thought of the Newtonians Archibald (...) Pitcairne (1652-1713), James Keill (1673-1719) and the rationalists Yvo Gaukes (ca. 1660-1738) and Friedrich Hoffman (1660-1742). It contends that, still in the early eighteenth-century, contrasting understandings and uses of quantification in medicine related to different conceptions of the epistemological status of mathematics and its connection with experience especially. The chapter then focuses on how the experimental quantitative approach introduced in physiology by Santorio Santorio (1561-1636) is understood in these different scenarios. I argue that contrasting understandings and uses of quantification in medicine correspond not only to different conceptions about the epistemological status of mathematics, but also of its conjunction with experience. (shrink)