The paper describes the conception of logic in Polish didactics authored by the Commission of National Education (KEN), an important educational institution of the European Enlightenment. Since the documents of the Commission refer to a vision of science presented by such influential works then as the Encyclopédie ou dictionnaire universel raisonné [Great French Encyclopedia], the paper compares the requirements from the Commission’s programmer with the encyclopaedic entries that entail logical problems broadly understood. It turns out that the Commission, following the (...) Encyclopédie, not only recommended a list of textbooks of logic but also shared its eclectic vision of logic. Although it is characteristic of modernity to take a relative approach to the importance of traditional logic, transformed into science on method, or literally an outline of epistemology, understood according to É. Condillac as a specific form of metaphysics, nevertheless some elements of logic were eclectically made valid. This logic, from the times of I. Kant, has been defined as formal logic. Practical logical skills were preferred to the knowledge of logical theories. At the same time attention was paid to the meaning of natural logical skills, and drills in logical reasoningwhen studying languages and mathematics. Despite preferences for the analytical method they also noticed the importance of synthetic method. It seems also that although the documents of the Commission do not say anything about the teaching of syllogistic issues, in didactic practice inspired by the Encyclopédie in the schools controlled by the Commission, the room was made to teach these problems. Condillac’s book was preferred in the schools controlled by the Commission, nevertheless, it was not, as in the case of other textbooks, a must on the reading list, an obligatory reading matter, therefore it was not published in Poland. The conception of logic presented by the Commission as modelled on the Encyclopédie managed to avoid the one-sidedness of Condillac’s approach, the approach that in fact eliminated the teaching of logic. (shrink)

Las definiciones ocupan un lugar muy importante en la _Ética_ de Spinoza, sin embargo, su naturaleza y su función permanecen aún en gran medida bajo un manto de oscuridad. Tal es así que numerosos comentadores han abordado el tema aunque nunca se ha llegado a un acuerdo definitivo. En el presente trabajo revisitaremos esta vieja cuestión comenzando por una reconstrucción de la doctrina spinoziana de la definición expuesta en diferentes partes del _corpus_. Para ello tomaremos como punto de referencia la (...) oposición entre definición real y definición nominal tal como aparece delineada en la doctrina de la _Lógica_ de Port Royal. Luego ofreceremos un examen de la naturaleza y función de las definiciones de la _Ética_. El recorrido realizado aquí nos permitirá, por un lado, determinar cuál de las interpretaciones en juego tiene mejores fundamentos y, por el otro, arrojar luz sobre la función de las definiciones en la metodología spinoziana. (shrink)

Descartes’ Rules for the direction of the mind presents us with a theory of knowledge in which imagination, considered as an “aid” for the intellect, plays a key role. This function of schematization, which strongly resembles key features of Proclus’ philosophy of mathematics, is in full accordance with Descartes’ mathematical practice in later works such as La Géométrie from 1637. Although due to its reliance on a form of geometric intuition, it may sound obsolete, I would like to show that (...) this has strong echoes in contemporary philosophy of mathematics, in particular in the trend of the so called “philosophy of mathematical practice”. Indeed Ken Manders’ study on the Euclidean practice, along with Reviel Netz’s historical studies on ancient Greek Geometry, indicate that mathematical imagination can play a central role in mathematical knowledge as bearing specific forms of inference. Moreover, this role can be formalized into sound logical systems. One question of general epistemology is thus to understand this mysterious role of the imagination in reasoning and to assess its relevance for other mathematical practices. Drawing from Edwin Hutchins’ study of “material anchors” in human reasoning, I would like to show that Descartes’ epistemology of mathematics may prove to be a helpful resource in the analysis of mathematical knowledge. (shrink)

If Wolff from the very beginning was strongly influenced by Cartesianism, and his concept of system was guided by the geometric model of scientific method, Leibniz relies on the German tradition of “methodological thinking”, originated by Joachim Jung. Leibniz’s conception of sys-tem focuses mainly on algebraic analysis. The application of the term “system” mainly to meta-physics can be explained by the fact that it is this science that deals with synthetic principles and is designed to develop a rational view of (...) the world. Based on the idea of “real metaphysics”, Leibniz justifies these principles as such that make possible the laws of physics and rational discourse in general. Leibniz does not provide a system of philosophy in the sense of traditional school philos-ophy. He considers metaphysics as the most consistent expression of a systematic mind and indi-cates heuristic principles for the construction and improvement of reasonable hypotheses. (shrink)

In the present paper we propose to approach the Spinoza`s methodological project from a philosophical and historical perspective broad enough to adequately understand the reasons that led him to adopt geometric method to expose his philosophy. Even if the topic has been widely discussed by Spinoza´s commentators in the four centuries since the Ethics was published, we believe that the approaches are either inadequate or suffer from some fragmentation, in the sense that they address this or that aspect, but don`t (...) put together all pieces that make up the puzzle. Likewise, we consider that a full understanding of such application of geometric method transcends the mere question of whether it was successful or not in terms of logical rigor. So that, the fundamental task in the present work is unite various aspects that help us understand Spinoza`s methodological choice in a synoptic vision, in which both historical and systematic aspects can be included. To this end, we start from a hypothesis which allow us glimpse a certain unity behind the reasons that lead Spinoza to choose the geometric method, namely: our philosopher tries to reach an ideal of science, which combines some conceptual elements of Aristotelian tradition with others from Cartesian philosophy. (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)

For Kant as for Hegel method is not a structure or procedure imported into philosophy from without, as, e.g. a mathematical demonstration in modern physics or in the proof-structure of philosophies such as Spinoza’s or Wolff’s. For both Hegel and Kant method is the arrangement that reason gives its contents and cognitions; for both, that is, method and object do not fall asunder, unlike in all disciplines other than philosophy. For Kant method is the design and plan of the whole, (...) the scientific form that guides the organization of cognitions (KrV A 707/B 736, Ak 24, 780). Likewise, Hegel writes that method is the consciousness of the form of its inner movement (WL 1, 49, SL 53, W 3 47, PhS 28). Unfortunately, Hegel never considers Kant an example or a precursor or a positive role model. It is important to ask why Hegel never takes seriously Kant’s Doctrine of Method. Why, if he shares so many central points with the Architectonic of the first Critique, does he never acknowledge Kant as a possible ally? Why does he misunderstand Kant on analysis and synthesis as he does? These are some of the questions I plan to discuss in this paper. (shrink)

TheGenerales Inquisitiones de Analysi Notionum et Veritatumis Leibniz’s most substantive work in the area of logic. Leibniz’s central aim in this treatise is to develop a symbolic calculus of terms that is capable of underwriting all valid modes of syllogistic and propositional reasoning. The present paper provides a systematic reconstruction of the calculus developed by Leibniz in theGenerales Inquisitiones. We investigate the most significant logical features of this calculus and prove that it is both sound and complete with respect to (...) a simple class of enriched Boolean algebras which we call auto-Boolean algebras. Moreover, we show that Leibniz’s calculus can reproduce all the laws of classical propositional logic, thus allowing Leibniz to achieve his goal of reducing propositional reasoning to algebraic reasoning about terms. (shrink)

ZusammenfassungDer Beitrag befasst sich mit der Mathematisierung in der praktisch-technischen Ausbildung im letzten Drittel des 18. und ersten Drittel des 19. Jahrhunderts. Anhand von zwei Fällen werden Variantenreichtum und Problemlagen des praktisch-mathematischen Wissens in der Zeit um 1800 herausgearbeitet. Nach einer Einführung zu institutionellem Rahmen und sozialem Status der praktischen Mathematiker (eigentlich mathematischen Praktiker) in den deutschen Gebieten, beschäftigt sich der erste Teil mit dem Bergmeister Johann Andreas Scheidhauer (1718–1784). Die Biographie dieses bislang wenig beachteten Praktikers führt zu einer Analyse (...) seiner Handschriften und deren Rezeption, in erster Linie durch Johann Friedrich Lempe (1757–1801), Professor für Mathematik an der Bergakademie Freiberg. Ein Praktiker der nachfolgenden Generation steht im Fokus des zweiten Teils. Carl Friedrich Steiner (1774–1840) war ein Baumeister im Weimar der Goethezeit. Gewissermaßen reflektiert sein Lehrbuch zur Geometry descriptive die Folgen der fortlaufenden Mathematisierung und den Stellenwert mathematischen Wissens in der Ausbildung junger Praktiker und Künstler. Beide Teilstudien geben Aufschluss zum historischen Verständnis praktischer Mathematik. (shrink)