The paper discusses some changes in Bolzano's definition of mathematics attested in several quotations from the Beyträge, Wissenschaftslehre and Grössenlehre: is mathematics a theory of forms or a theory of quantities? Several issues that are maintained throughout Bolzano's works are distinguished from others that were accepted in the Beyträge and abandoned in the Grössenlehre. Changes are interpreted as a consequence of the new logical theory of truth introduced in the Wissenschaftslehre, but also as a consequence of the overcome of Kant's (...) terminology, and of the radicalization of Bolzano's anti‐Kantianism. Bolzano's evolution is understood as a coherent move, once the criticism expressed in the Beyträge on the notion of quantity is compared with a different and larger notion of quantity that Bolzano developed already in 1816. This discussion is enriched by the discovery that two unknown texts mentioned by Bolzano in the Beyträge can be identified with works by von Spaun and Vieth respectively. Bolzano's evolution is interpreted as a radicalization of the criticism of the Kantian definition of mathematics and as an effect of Bolzano's unaltered interest in the Leibnizian notion of mathesis universalis. As a conclusion, the author claims that Bolzano never abandoned his original idea of considering mathematics as a scientia universalis, i.e. as the science of quantities in general, and suggests that the question of ideal elements in mathematics, apart from being a main reason for the development of a new logical theory, can also be considered as a main reason for developing a different definition of quantity. (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)

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)

This article proposes an analysis of the sources used in Albert the Great’s De vegetabilibus to explain the compilative strategy with which the Dominican master tries to reconstruct the botanical science. Following a classification of the genres, the main encyclopaedic, lexicographical, medical and pharmacological sources that influenced the work of the doctor universalis are listed. It is also aimed at analysing the interest that motivates Albert’s content choices. In this way, the work is put in relation to the main (...) texts that constituted the botanical panorama, outlining the innovation of its contribution. (shrink)

본고는 라이프니츠의 기호 언어와 그의 세 가지 철학적 구상들의 연관성을 연구한 논문 이다. 라이프니츠는 논리적인 기호 언어가 우리가 진리를 파악할 수 있게 해주는 이성적인 도구이기 때문에 그것을 사용한다면 우리의 학문들이 안전하게 발전할 것이라 믿었다. 이 렇게 볼 때 그의 기호 언어는 학문 전체에 무모순적 진리들을 공급하도록 고안된 보편학의 도구라 할 수 있다. 그리고 진리의 내용은 그것을 표시한 기호들의 의미들에만 의존하기 때문에, 기호들의 일의성을 기반으로 한 그의 기호 언어는 기호들의 조합을 통해 새로운 진리를 발견할 수 있는 새로운 발견법의 토대라 할 수 (...) 있다. 또한 일상 언어의 애매성에서 벗어난 그의 기호 언어는 개념들 간의 내포적 포함 관계와 동일성의 원리에 따라 인간의 추리를 기호 조작을 통해 형식적으로 연산하는 새로운 증명법의 토대이다. (shrink)

본고는 라이프니츠의 논리학과 직접적으로 연관되어 있는 증명을 위한 사고의 조합에 관해 연구한 논문이다. 그의 논리학 개념에는 다양한 측면들이 있으며, 그러한 측면들은 그의 여러 가 지 논리학적 구상들과 기획들에 반영되어 있다. 이 논문은 이러한 구상들과 기획들이 중요한 철 학적 함의들을 갖고 있음을, 그리고 그것들이 보편학으로서의 그의 논리학을 구성함을, 그리고 그의 논리학이 기호들의 조작과 사고의 조합에 근거한 증명법임을 보이고자 한다. 라이프니츠는 우선적으로 자신의 논리 체계를 정립하기 위해 다양한 수학적인 방법들을 도입했다. 그리고 그 는 인간 사고의 알파벳과 새로운 백과사전과 같은 예비적인 논리적 기획들을 (...) 감행했다. 여기서 전자는 복합 개념들을 분석하여 도달한 단순 개념들의 기호들의 목록이며, 후자는 모든 학문들 에 포함된 단순 개념들과 정의들을 산출한다. 이러한 기획들을 토대로, 그는 그의 논리 체계를 건설하고자 하였다. 이러한 논리 체계는 모든 학문적인 명제들을 증명하며 보편 기호법과 추리 계산법의 사용을 가능하게 한다. 비록 라이프니츠가 명확한 논리 체계를 건설하는 데 실패했다 할지라도, 그는 최초로 논리적 기호법을 고안하고자 노력했으며, 그의 논리학 개념은 부울과 프 레게에게 현대 논리학을 형성하도록 영감을 주었다. (shrink)

El punto de partida del calculus universalis leibniziano es la teoría aristotélica del silogismo, pero Leibniz se independiza de las ideas de Aristóteles para desarrollar su propio sistema lógico, mucho más general, aplicando el instrumento combinatorio a la silogística. Lo que propone es una importante modificación del modelo demostrativo axiomático, mediante la creación de cálculos lógico-simbólicos que no se limitan a los ámbitos tradicionales de la deducción, sino que admiten procedimientos discursivos más complejos que los de la lógica clásica, (...) ampliando con ello la noción de racionalidad. (shrink)

In this paper I argue that differences between the ‘new moral science’ of the seventeenth century and scholastic natural law theory originated primarily from the skeptical challenge the former had to face. Pufendorf’s project of a scientia practica universalis is the paramount expression of an anti-skeptical moral science, a ‘science’ that is both explanatory and normative, but also anti-dogmatic insofar as it tries to base its laws on those basic phenomena of human life which, supposedly, are immune to (...) skeptical doubt. The main scholastic legacy to the new moral science is the dichotomy between an ‘intellectualist’ and a ‘voluntarist’ view of natural law (or between lex immanens and lex imposita). Voluntarism lies at the basis of both theological views, such as Calvinism, and political views, such as those of Hobbes and Locke. The need to counterbalance the undesirable implications of extreme voluntarism may account for much of the developments in ethics and politics during the seventeenth and eighteenth centuries.Scottish natural jurisprudence, which tried to find a middle way between skepticism and extreme voluntarism, is less secular and more empirical than received wisdom admits. There emerged, as one of its ‘accidental’ outcomes, a systematic, self-contained and empirical economic theory from the search for an empirically based normative theory of social life. The basic assumption of such a theory, namely, the notion of societal laws as embedded in trans-individual mechanisms, derives from the voluntarist view of natural law as ‘imposed’ law.Later discussions of social issues in terms of ‘economic’ and ‘ethical’ reasons originated partly from a misreading ofthe Scottish natural jurisprudential framework of economic theory. Starting with this reconstruction, I try to shed some light on recent discussions about the role of ethics in economics. (shrink)

I argue that the difference between the 17th century new moral science and Scholastic Natural Law Theory derived primarily from the skeptical challenge the former had to face. Pufendorf's project of a 'scientia practica universalis' was the paramount expression of an anti-skeptical moral science, a «science» both explanatory and normative, but also anti-dogmatic in so far as it tried to base its laws on those basic phenomena of human life that supposedly were outside the scope of skeptical doubt. (...) Of the Scholastic legacy to the new moral science, a dichotomy between an «intellettualistic» and a «voluntaristic» view of natural law (or between lex immanens and lex imposita). Voluntarism lays at the root both of theological views such as those of Calvinism and of political views such as those of Hobbes and Locke. A need to counterbalance undesirable implications of estreme voluntarism may account for much of 17th and 18th centuries developments in ethics and politics. Scottish natural jurisprudence, an expression of such quest for a third way between scepticism and extreme voluntarism, is less secular and more empirical than received wisdom admits of. One of its side-effects, namely a systematic, self-contained, and empirical economic theory, results from the search for a normative theory of social life on an empirical basis. The main tool for building such a theory, namely a view of societal laws as embedded in trans-individual mechanisms, derives from the voluntarist view of natural law as «imposed» law. Subsequent discussions of social issues based on the opposition of economic and ethical reasons originated partly from gross misreading of the Scottish natural jurisprudential framework for economic theory. (shrink)

In the first part of this paper a short review of the recently published 4th volume of Series 6 of the Akademie-Ausgabe of Leibniz’s Sämtliche Schriften und Briefe is given. This 3,000-page volume was edited by the Leibniz-Forschungsstelle in Münster, Germany. It contains unsurpassable, text-critical versions of more than 500 pieces which Leibniz composed between 1677 and 1690. One major topic dealt with in these essays is "Scientia Generalis, Characteristica, Calculus universalis". Here we find in particular various fragments (...) of a logical calculus that Leibniz developed around 1687. The main part of this paper presents a detailed reconstruction of this so-called "Plus-Minus-Calculus" which, by way of its somewhat unorthodox operators of "addition" and "subtraction", inclusion and identity, "communication", "commune" and "Nothing", provides an interesting alternative for the Boolean algebra of sets. (shrink)

Leibniz hat eine große Anzahl von Fragmenten zur Logik verfaßt. Der umfangreichste dieser Texte sind die Generales inquisitiones (1982 als Band 338 der PhB erschienen). Leibniz hat jedoch verschiedene Systeme der Logik entworfen, von denen das in den Generales inquisitiones dargestellte nur eines ist, an dem er zudem später noch weitergearbeitet hat. Es lag daher nahe, weitere Texte zur Logik von Leibniz in einer ähnlichen Ausgabe zu bearbeiten, in denen einerseits diese Weiterentwicklungen exemplifiziert und andererseits weitere Systementwürfe vorgestellt werden. Diese (...) neue Ausgabe enthält zehn Texte aus den Jahren 1683-1690 zur Allgemeinen Wissenschaft (Scientia generalis), zur Allgemeinen Wissenschaftssprache (Characteristica universalis) und zum Logischen Kalkül (Calculus logicus). Im Bereich der Logikkalküle werden Texte zum sogenannten Identitätskalkül vorgestellt, die eine Weiterentwicklung der Generales inquisitiones darstellen, sowie Texte zum sogenannten Plus-Minus-Kalkül. Innerhalb beider Gruppen ist jeweils ein Text enthalten, der eher als Vorstufe angesehen werden kann, der also Leibniz "bei der Arbeit" zeigt. (shrink)

The disciplinary differentiation of sciences attracted Leibniz’s attention for a long period of time. From nowadays prospects it looks very well grounded as soon as in Leibniz’s manuscripts a modern scholar finds clue ideas of any research field which would tempt him to consider Leibniz as one of the founders of this particular discipline. We argue that this is possible only in retrospection and would significantly distort the essence of Leibniz’s epistemology. Our approach implies, in contrary, the investigation of the (...) Leibniz doctrine of signs on the background of the related philosophical problem, that of expression. The choice of semiotics is justified by the fact that it took a central place in his theoretical constructions, both those of natural sciences and of philosophy. In Leibniz system of knowledge the concept of notes (notae) and sings (signa) served a theoretical foundation of his most important and long-life aspiration to build up practical science of universal characteristics (characteristica universalis). In his eyes this practical science was the science of sciences (Scientia scientiarum), and we can consider it as the matrix for all possible scientific knowledge. (shrink)

On trouvera ici l’édition critique de la conclusion générale, restée inédite, des deux tomes de La Philosophie de l’algèbre. Vuillemin y décrit d’abord les deux révolutions successivement survenues en algèbre : le passage, décrit dans le tome I, de l’Algèbre cartésienne des équations à l’Algèbre des structures exemplifiée dans la théorie de Galois, puis le passage de celle-ci à l’Algèbre de l’algèbre chez Dedekind et Birkhoff. Un renouvellement parallèle peut donc être attendu dans la philosophie théorique : après la mathesis (...) universalis de l’âge classique, qui transposait les méthodes des mathématiques classiques, puis celle de la phénoménologie, qui importait en philosophie l’idée de structure, il reste à mener en philosophie une nouvelle révolution, qui fonde l’analyse des structures dans une critique préalable de la raison. Ce texte est suivi d’une notice de Gudrun Vuillemin-Diem décrivant les manuscrits de la deuxième partie de La Philosophie de l’algèbre. (shrink)

Descartes's conception of matter changed the account of physical nature in terms of extension and related quantitative terms. Plants and animals were turned into species of machines, whose natural functions can be explained mechanistically. This article reflects on the consequences of this transformation for the psychology of human soul. In so far the soul is rational it lacks extension, yet it is also united with the body and affected by it, and so it is able to act on extended matter. (...) The article examines Descartes's concept of scientia and his different uses of nature, and argues that there is much more continuity between Aristotelian and Cartesian psychology than is usually recognized when it comes to an explanation of the functions of the embodied human soul. If this makes psychology unfit for inclusion in the new science of nature, its object is still a natural phenomenon and has an important place within scientia as Descartes conceived of it. (shrink)

**For my more recent views of the third kind of cognition, see my "Finding Oneself in God"** -/- Abstract: Cognition of the third kind, or scientia intuitiva, is supposed to secure beatitudo, or virtue itself (E5p42). But what is scientia intuitiva, and how is it different from (and superior to) reason? I suggest a new answer to this old and vexing question at the core of Spinoza’s project in the Ethics. On my view, Spinoza’s scientia intuitiva resembles (...) Descartes’s scientia more than has been appreciated. Although Spinoza’s God is not Descartes’s benevolent, transcendent God, Spinoza agrees with Descartes that the highest certainty requires that a cognizer correctly conceive of God and her causal relation to God; it is only with cognition of the third kind that a cognizer can be certain that her adequate (that is, clear and distinct) representations of extramental things agree with formally real, extramental ideata, and so are true. If this is right, a reading of Spinoza that has dominated scholarship since the Ethics’ publication is misguided: for Spinoza, it is not always the case that having (and recognizing that one has) a clear and distinct idea is sufficient for knowing that that idea is true. I end the chapter by suggesting why scientia is intuitive for Spinoza: Spinoza attempts to avoid Cartesian-Circle-style circularity by insisting that a cognizer must intuit the correct representation of God and God’s relation to things. (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)

Excerpt from Clavis Universalis Stuart, edited by Sir Wm. Hamilton, 1854, p. 349. 3 Robert Benson's Memoirs of the Life and Writings of the Rev. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection (...) in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works. (shrink)

Legal dogmatics in Continental European law (scientia iuris, Rechtswissenschaft) consists of professional legal writings whose task is to systematize and interpret valid law. Legal dogmatics pursues knowledge of the existing law, yet in many cases it leads to a change of the law. Among general theories of legal dogmatics, one may mention the theories of negligence, intent, adequate causation and ownership. The theories produce principles and they also produce defeasible rules. By means of production of general and defeasible theories, (...) legal dogmatics aims at obtaining a system of law that is both internally coherent and harmonized with its background in morality and (political) philosophy. Legal dogmatics is necessary in the context of constitutional constraints on the majority rule. Only if the courts act on the basis of Reason they can be a legitimate counterpart of the majority rule. And Reason cannot be exhausted by particular decision making. It also needs a more abstract deliberation, given by expert jurists. However, legal dogmatics has been a target of several kinds of criticism: empirical, morally-political, epistemological, logical, and ontological. The position taken in this article is to answer such criticism by mutually adjusting philosophy and the practices of the law. (shrink)

Leśniewski’s systems deviate greatly from standard logic in some basic features. The deviant aspects are rather well known, and often cited among the reasons why Leśniewski’s work enjoys little recognition. This paper is an attempt to explain why those aspects should be there at all. Leśniewski built his systems inspired by a dream close to Leibniz’s characteristica universalis: a perfect system of deductive theories encoding our knowledge of the world, based on a perfect language. My main claim is that (...) Leśniewski built his characteristica universalis following the conditions of de Jong and Betti’s Classical Model of Science (2008) to an astounding degree. While showing this I give an overview of the architecture of Leśniewski’s systems and of their fundamental characteristics. I suggest among others that the aesthetic constraints Leśniewski put on axioms and primitive terms have epistemological relevance. (shrink)

Esiste, oltre alle singole cose rosse, la proprietà universale "rosso"? O esistono invece solo le entità individuali e particolari? Come spiegare il fatto che più entità tra loro distinte possono essere tuttavia uguali per un determinato aspetto? E che cos'è che distingue due entità che condividano esattamente le stesse proprietà? Questi sono alcuni dei problemi che costituiscono la questione degli universali e dei particolari, che, a partire dai dialoghi platonici e dalla metafisica di Aristotele, ha sempre rappresentato uno dei nodi (...) centrali dell'ontologia. Di tale questione il libro ripercorre le tappe storiche principali, partendo dalla formulazione che ne fu proposta nell'antichità, esaminando gli importanti e variegati sviluppi medievali, individuandone la presenza talvolta nascosta nella filosofia moderna e ricostruendo i dibattiti della metafisica contemporanea, che ha visto una significativa rinascita di interesse per questi temi, soprattutto nell'ambito della filosofia analitica. -/- (English version) Does there exist, besides single red things, a universal red property? Or are there only individual, particular entities? How does one explain the fact that many distinct entities can be considered the same as regards a specific feature? And what is it exactly that distinguishes two things that share the exact same property? These are some of the questions that involve universals and particulars, a general issue that, ever since Plato's dialogues and Aristotle's Metaphysics, has played a crucial role in ontology. This book outlines the evolution of this issue through its main historical phases, from its initial conception in ancient times, through its medieval development, up to its sometimes hidden presence in modern philosophy. The text also addresses contemporary debates in metaphysics, in which there is a significant revival of interest for this topic, especially in analytical philosophy. (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.

R. G. Collingwood saw one of the main tasks of philosophers and of historians of human thought in uncovering what he called the ultimate presuppositions of different thinkers, of different philosophical movements and of entire eras of intellectual history. He also noted that such ultimate presuppositions usually remain tacit at first, and are discovered only by subsequent reflection. Collingwood would have been delighted by the contrast that constitutes the overall theme of the essays collected in this volume. Not only has (...) this dichotomy ofviews been one ofthe mostcrucial watersheds in the entire twentieth-century philosophical thought. Not only has it remained largely implicit in the writings of the philosophers for whom it mattered most. It is a truly Collingwoodian presupposition also in that it is not apremise assumed by different thinkers in their argumentation. It is the presupposition of a question, an assumption to the effect that a certain general question can be raised and answered. Its role is not belied by the fact that several philosophers who answered it one way or the other seem to be largely unaware that the other answer also makes sense - if it does. This Collingwoodian question can be formulated in a first rough approximation by asking whether language - our actual working language, Tarski's "colloquiallanguage" - is universal in the sense of being inescapable. This formulation needs all sorts of explanations, however. (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)

In the Fifth Meditation Descartes considers the problem of knowledge preservation : the challenge of accounting for the diachronic certainty of perfect knowledge [scientia]. There are two general solutions to PKP in the literature: the regeneration solution and the infallible memory solution. While both readings pick up on features of Descartes’ considered view, I argue that they ultimately fall short. Salvaging pieces from both readings and drawing from Descartes’ virtue theory, I argue on textual and systematic grounds for a (...) dispositionalist solution. On this view, the diachronic certainty of scientia is achieved through virtuous habits of belief. (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)

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)

Recent work in formal philosophy has concentrated over-whelmingly on the logical problems pertaining to epistemic shortfall - which is to say on the various ways in which partial and sometimes incorrect information may be stored and processed. A directly depicting language, in contrast, would reflect a condition of epistemic perfection. It would enable us to construct representations not of our knowledge but of the structures of reality itself, in much the way that chemical diagrams allow the representation (at a certain (...) level of abstractness) of the structures of molecules of different sorts. A diagram of such a language would be true if that which it sets out to depict exists in reality, i.e. if the structural relations between the names (and other bits and pieces in the diagram) map structural relations among the corresponding objects in the world. Otherwise it would be false. All of this should, of course, be perfectly familiar. (See, for example, Aristotle, Metaphysics, 1027 b 22, 1051 b 32ff.) The present paper seeks to go further than its predecessors, however, in offering a detailed account of the syntax of a working universal characteristic and of the ways in which it might be used. (shrink)

Medieval epistemology begins as ideal theory: when is one ideally situated with regard to one's grasp of the way things are? Taking as their starting point Aristotle's Posterior Analytics, scholastic authors conceive of the goal of cognitive inquiry as the achievement of scientia, a systematic body of beliefs, grasped as certain, and grounded in demonstrative reasons that show the reason why things are so. Obviously, however, there is not much we know in this way. The very strictness of this (...) ideal in fact gives rise to a body of literature on how Aristotle's framework might be relaxed in various ways, for certain specific purposes. In asking such questions, scholastic authors are in effect pursuing the project of social epistemology, by trying to adapt their ideal theory to the circumstances of everyday life. (shrink)