Jens Lemanski

The chapters in this timely volume aim to answer the growing interest in Arthur Schopenhauer’s logic, mathematics, and philosophy of language by comprehensively exploring his work on mathematical evidence, logic diagrams, and problems of semantics. Thus, this work addresses the lack of research on these subjects in the context of Schopenhauer’s oeuvre by exposing their links to modern research areas, such as the “proof without words” movement, analytic philosophy and diagrammatic reasoning, demonstrating its continued relevance to current discourse on logic. Beginning with Schopenhauer’s philosophy of language, the chapters examine the individual aspects of his semantics, semiotics, translation theory, language criticism, and communication theory. Additionally, Schopenhauer’s anticipation of modern contextualism is analyzed. The second section then addresses his logic, examining proof theory, metalogic, system of natural deduction, conversion theory, logical geometry, and the history of logic. Special focus is given to the role of the Euler diagrams used frequently in his lectures and their significance to broader context of his logic. In the final section, chapters discuss Schopenhauer’s philosophy of mathematics while synthesizing all topics from the previous sections, emphasizing the relationship between intuition and concept. Aimed at a variety of academics, including researchers of Schopenhauer, philosophers, historians, logicians, mathematicians, and linguists, this title serves as a unique and vital resource for those interested in expanding their knowledge of Schopenhauer’s work as it relates to modern mathematical and logical study.

In early modernity, one can find many spatial logic diagrams whose geometric forms share a family resemblance with religious art and symbols. The family resemblance these diagrams bear in form is often based on a vesica piscis or on a cross: Both logic diagrams and spiritual symbols focus on the intersection or conjunction of two or more entities, e.g. subject and predicate, on the one hand, or god and man, on the other. This paper deals with the development and function of logic diagrams, their analogy to religious art and symbols, and their modern application in artificial intelligence.

In recent years the cultural pessimistic position has become known, according to which we live in an “age of post-truth.” This thesis is supported by the observation of an increasing use of argumenta ad passiones in politics. In contrast to this view, I believe that “time” and “representation” play a more decisive role in individual post-truth arguments than the appeal to passiones. By analysing typical post-truth arguments, I arrive at a much more positive view on the present age: the designation of individual arguments as “post-truth” is already an expression of a process of enlightenment.

A particularly promising trail on the search for forgotten logic diagrams leads to Upper Lusatia in the 17th century, more precisely to Christian Weise and his students. Samuel Grosser, who later became rector in Görlitz, and Johann Christian Lange, who later became professor of logic at the University of Gießen, are the most prominent to have published remarkable logic diagrams. Even more remarkable, however, is the fact that Lange's interest in these diagrams ultimately gave rise to the idea of building a logic machine.

The paper presents the diagrammatic calculus CL, which combines features of tree, Euler-type, Venn-type diagrams and squares of opposition. In its basic form, `CL' (= Cubus Logicus) organizes terms in the form of a square or cube. By applying the arrows of the square of opposition to CL, judgments and inferences can be displayed. Thus CL offers on the one hand an intuitive method to display ontologies and on the other hand a diagrammatic tool to check inferences. The paper focuses mainly on the adaptation of the square of opposition in CL and offers an algebraic notation, which corresponds to the diagrammatic representation.

Logicians have often suggested that the use of Euler-type diagrams has influenced the idea of the quantification of the predicate. This is mainly due to the fact that Euler-type diagrams display more information than is required in traditional syllogistics. The paper supports this argument and extends it by a further step: Euler-type diagrams not only illustrate the quantification of the predicate, but also solve problems of traditional proof theory, which prevented an overall quantification of the predicate. Thus, Euler-type diagrams can be called the natural basis of syllogistic reasoning and can even go beyond. In the paper, these arguments are presented in connection with the book Nucleus Logicae Weisaniae by Johann Christian Lange from 1712.

In this paper we present an overview of the current interpretations of the first volume of Arthur Schopenhauer’s main work The World as Will and Representation (W I) and discuss their problems. We discuss four issues, which in our opinion must clarify a current interpretation implicitly or explicitly, if it claims to be an interpretation of the whole book: (1) What does Schopenhauer mean by the fact that his work shares only one (single) thought? (2) How are the individual books of W I related? (3) Do we have to read Schopenhauer’s W I as a normative guide to the denial of will-to-live or as a neutral description of the world? (4) Do the often discussed contradictions and aporias within the book follow a plan or are they errors of thought?

A widely debated question in current research centres on determining the precursors to G. W. F. Hegel's theory of recognition. Until now Fichte, Rousseau and Aristotle have been discussed. However, the present paper analyses a further surprising correspondence between Marsilio Ficino's theory of love and Hegel's theory of recognition. Here it is shown that Hegel studied Ficino in 1793 and that we can discover syntactical, semantical, and structural vestiges of Ficino's De amore II 8 in Hegel's early fragments on religion (1793) and love (1797), which are closely related to the general theory of recognition found in the Phenomenology of Spirit. Not only may this thesis be relevant for Hegel or Ficino scholarship, but it could also be a further indication that social theories with normative content are an integral characteristic of (early) modern self-consciousness.

in recent years, the research on Schopenhauer has shown a change in the interpretation of his main work, «The World as Will and Presentation», from (1) a normative and linear instruction which guides the reader from idealism to mysticism, pessimism and nothingness to (2) value-free and independent descriptions of the world with all phenomena (like idealism, mysticism, nothingness etc.) in it. thus Schopenhauer’s main work has become an empirical or baconian approach—something like a «philosophical cosmography»—. this fundamental change of interpretation radically puts into question what Schopenhauer means by characterizing his main work as an «organic system». the present paper attempts to give an answer to this question, reviewing the self-reflexive, methodological and metaphilosophical hints which Schopenhauer gives in the first volume of his «World» («as Will and presentation»).

Die Frage ›Warum ist überhaupt etwas und nicht vielmehr nichts?‹ gehört zu den ebenso traditionsreichen wie umstrittenen Problemen der Philosophie. Bereits mehrmals in die Mottenkiste der Philosophiegeschichte verbannt, erlebt sie doch zuverlässig ihre Renaissancen. Der vorliegende Band nimmt sich der ›Grundfrage‹ in einer ideengeschichtlichen Perspektive an. Dabei stellt sich heraus, dass die systematisch keineswegs erst mit Leibniz auftauchende Frage in ihrer Geschichte von der Antike bis zur gegenwärtigen analytischen Philosophie nicht nur jeweils unterschiedliche Antworten provoziert hat, sondern vor allem auch ganz verschieden gestellt worden ist: Formuliert Leibniz »Pourquoi il y a plus tôt quelque chose que rien?«, heißt es bei Schelling »Warum ist nicht nichts, warum ist überhaupt etwas?«, während Schopenhauer ihr eine existentielle Wendung gibt (»Lieber nichts als etwas«). Heideggers Auseinandersetzung mit dem Nihilismus führt zu der Frage: »Warum ist überhaupt Seiendes und nicht vielmehr Nichts?«, während Arendt sie ins Politische wendet (»Warum ist überhaupt jemand und nicht niemand?«). Der Band wird durch einen Überblick über die vielschichtige Diskussion der ›letzten Warum-Frage‹ in der Tradition der Analytischen Philosophie sowie einen Antwortversuch aus Sicht der aktuellen Physik und Kosmologie abgerundet.

Inspired by various research results on the history of philosophy that have not yet been compiled, the paper pursues the thesis that the question "Why is there something rather than nothing" was formulated long before Leibniz. In reviewing this thesis, the 'fundamental question' is differentiated into two individual questions: "Why is there something at all" and 'Why isn't rather nothing?". On the basis of this systematic distinction, the paper examines the history of philosophy from early history to scholasticism with regard to the occurrence of the above mentioned individual questions and their connection to the fundamental question.

This article compares the research literature published by Goethe and Schopenhauer on the subject of evolutionary theory. The metatheoretical thesis is supported that the results of any comparison between Schopenhauer or Goethe and a concrete theory of evolution always depend on three dimensions: 1. the scientific point of view of the comparative, 2. the meaning and definition of the central terms and 3. the selection of the analyzed primary literature. It is also argued that no previously published study was able to present convincing arguments for the fact that Schopenhauer or Goethe represented a 'theory of evolution'.

It is little known that Schopenhauer (1788–1860) made thorough use of Euler diagrams in his works. One specific diagram depicts a high number of concepts in relation to Good and Evil. It is, hence, uncharacteristic as logicians of that time seldom used diagrams for more than three terms (the number demanded by syllogisms). The objective of this paper is to make sense of this diagram by explaining its function and inquiring whether it could be viewed as an early serious attempt to construct complex diagrams.

From the beginning of the 16th century to the end of the 18th century, there were not less than ten philosophers who focused extensively on Venn’s ostensible analytical diagrams, as noted by modern historians of logic (Venn, Gardner, Baron, Coumet et al.). But what was the reason for early modern philosophers to use logic or analytical diagrams? Among modern historians of logic one can find two theses which are closely connected to each other: M. Gardner states that since the Middle Ages certain logic diagrams were used just in order to teach “dull-witted students”. Therefore, logic diagrams were just a means to an end. According to P. Bernhard, the appreciation of logic diagrams had not started prior to the 1960s, therefore the fact that logic diagrams become an end the point of research arose very late. The paper will focus on the question whether logic resp. analytical diagrams were just means in the history of (early) modern logic or not. In contrast to Gardner, I will argue that logic diagrams were not only used as a tool for “dull-witted students”, but rather as a tool used by didactic reformers in early modern logic. In predating Bernhard’s thesis, I will argue that in the 1820s logic diagrams had already become a value in themselves in Arthur Schopenhauer’s lectures on logic, especially in proof theory.

Das folgende Essay beschäftigt sich mit Zeitmetaphern im Erkenntnisakt. Durch eine historische Untersuchung von dialektischen Denkern der Antike wie Platon, Plotin und Porphyrius und der Neuzeit wie Fichte, Schopenhauer und Wittgenstein versucht es, die These Karl Poppers zu widerlegen, dass die Nachkonstruktion der Vorgänge bei der Auslösung eines Einfalls kein philosophisches Thema sei. Im Unterschied zu großen Teilen der modernen Philosophie zeigt die Philosophiegeschichte, dass Denken und Sprechen nicht unbedingt als identisch angesehen werden. Vielmehr zeigt sich eine Diskrepanz zwischen dem qualitativen Denkakt und der quantitativen Begrifflichkeit, die dadurch zustande kommt, dass das prä-dikative Denken erst in eine Subjekt-Objekt-Struktur übertragen werden muss, um in einer natürlichen Sprache mittgeteilt werden zu können. Gedanken und Begriffe weisen somit eine eigene Geschwindigkeit auf, die nur im Akt des Philosophierens überwunden werden soll. The following essay is concerned with time metaphors in the act of cognition. On the basis of a historical study regarding dialectical philosophers of the antiquity like Plato, Plotinus and Porphyry and the modern age like Fichte, Schopenhauer and Wittgenstein it attempts to disprove the assumption of Karl Popper that the reconstruction of the process involved in the stimulation and release of an inspiration is not a philosophical theme. In contrast to the many groups of modern philosophy the history of philosophy shows that thinking and speaking are not necessarily identical. Actually the discrepancy is between the qualitative act of speaking and the quantitative concept arriving from the fact that the pre-dictive thinking has to be conveyed to a structure of ‘subject-object’, which is to communicate thinking through the form of a natural language. Thinking and concept have thus their own velocity which is only overcome by the act of philosophizing

Logicians commonly speak in a relatively undifferentiated way about pre-euler diagrams. The thesis of this paper, however, is that there were three periods in the early modern era in which euler-type diagrams (line diagrams as well as circle diagrams) were expansively used. Expansive periods are characterized by continuity, and regressive periods by discontinuity: While on the one hand an ongoing awareness of the use of euler-type diagrams occurred within an expansive period, after a subsequent phase of regression the entire knowledge about the systematic application and the history of euler-type diagrams was lost. I will argue that the first expansive period lasted from Vives (1531) to Alsted (1614). The second period began around 1660 with Weigel and ended in 1712 with lange. The third period of expansion started around 1760 with the works of Ploucquet, euler and lambert. Finally, it is shown that euler-type diagrams became popular in the debate about intuition which took place in the 1790s between leibnizians and Kantians. The article is thus limited to the historical periodization between 1530 and 1800.

Beginning with a research review, the present paper shows that Hans Slugaʼs and esp. Robert Brandomʼs thesis, according to which Frege has adopted the context-principle and the priority of propositional from Kant, can solve problems in current Frege scholarship, on the one hand, but is itself fraught with further problems, on the other hand. In contrast, this paper maintains that the context-principle and the priority of the propositional are implicitly present in Fregeʼs Begriffsschrift since both have not been taken over from Kant, but rather from neo-Aristotelianism: The context-principle and the priority of the propositional were first drawn up in 1834 by O. F. Gruppe. As a consequence, A. F. Trendelenburg has published a re-arranged edition of Aristotleʼs Organon in 1836 which became one of the most influential schoolbooks on logic in the 19th century and of which Frege ought to have been aware. In the beginning of this schoolbook, using the original ancient Greek sentences of Aristotle, the priority of the propositional (De int. I 1) is deduced from the context-principle (De anima III 6).

From the mid-1600s to the beginning of the eighteenth century, there were two main circles of German scholars which focused extensively on diagrammatic reasoning and representation in logic. The first circle was formed around Erhard Weigel in Jena and consists primarily of Johann Christoph Sturm and Gottfried Wilhelm Leibniz; the second circle developed around Christian Weise in Zittau, with the support of his students, particularly Samuel Grosser and Johann Christian Lange. Each of these scholars developed an original form of using geometric diagrams in logic. In this paper, I will trace the historical notes of John Venn and other modern logicians back to the original works published in the Weigel and Weise circles and describe the development of using geometric figures for logical reasoning and representation in that period of time.

The paper distinguishes three interpretations of Kant’s so called ‘Copernican Revolution’: an epistemological, a hermeneutical and a scientific-theoretical or methodological one. It is argued that the ‘scientific-theoretical reading’ can be based on new historical evidence. Kant borrowed the metaphors ‘army of stars’ (‘Sternenheer’) and ‘spectator’ (‘Zuschauer’) from Johann Heinrich Lambert and used them in a context similar to Lambert’s. This suggests that Kant’s formula “first thoughts of Copernicus” (“den ersten Gedanken des Copernicus”) refers, again following Lambert, to the first 9 chapters of Copernicus’ De revolutionibus, which contain a change from inductive geocentrism to deductive heliocentrism. This interpretation is itself no revolution: Johann Baptist Schad claimed in 1800 that metaphysics must be regarded as a deductive rather than an inductive science. Kant explicitly agreed.

In the second preface to the Critique of Pure Reason, Kant claims that Galileo Galilei, Evangelista Torricelli and Georg Ernst Stahl caused a scientific revolution in experimental physics (B xii). In this paper, I advance the historical thesis that Kantʼs claim refers precisely to three passages from Discursus et demonstrationes mathematicae (Galilei), Lettera a Filaleti Di Timauro Antiate (Torricelli), and Beweiß von den Saltzen (Stahl). This historical thesis provides evidence for a newer systematic interpretation, which says that the topic of the second preface is not epistemological, but rather methodological.

Normally in nowadays philosophical research the term 'Neoplatonism' is coined and it was used the first time by Jacob Brucker in the first half of the 18th century. But there are signs that the concept is much older. So this essay follows the trace of the term 'Neoplatonism' from german philosophical historians, like Büsching and Brucker, back to the Cambridge Platonists and tries to demonstrate that the origin of the concept is based on some texts of the late antiquity which act first on the research of the early modern philosophy.