Language, Logic, and Mathematics in Schopenhauer

Description

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. Contents An Introduction to Language, Logic and Mathematicsin Schopenhauer 1 Interpretations and Contradictions 2 System and World 3 Language, Logic and Mathematics 4 Intuition and Diagrams 5 Overview of the Volume 5.1 Part I: Language 5.2 Part II: Logic 5.3 Part III: Mathematics References Part I Language Language as an “Indispensable Tool and Organ” of Reason: Intuition, Concept and Word in Schopenhauer 1 Introduction 2 Real Empirical Objects 3 Concepts 4 Words 5 Conclusive Remarks References Problems in Reconstructing Schopenhauer's Theory of Meaning: With Reference to His Influence on Wittgenstein 1 Introduction 2 Schopenhauer's Theory of Language and its Links to the Representational and Use Theories 3 The Ambivalent Use of the Terms ``Concept'' and ``Word'' 4 Consequences of Ambivalence for Understanding Schopenhauer's Theory of Meaning 5 Meaning as Sense and Reference 6 Use Theory of Meaning as Unnatural Way of Language Acquisition 7 Conclusions and Prospects References Concept Diagrams and the Context Principle 1 Introduction 2 Euler-Type Diagrams and Concept Diagrams 2.1 Euler and Euler Diagrams 2.2 Concept Diagrams 3 The Primacy Question in Logic 3.1 The Kant–Frege Thesis 3.2 Criticism of the Kant–Frege Thesis 4 Schopenhauer's Concept Diagrams and the Context Principle 4.1 Representationalism and Explanation 4.1.1 Concepts and Spheres 4.1.2 Origin, Emergence, and Development of Concepts 4.1.3 Interpretation 4.2 Rationalism and Understanding 4.2.1 Translating and Understanding Languages 4.2.2 Schopenhauer's Use Theory of Meaning and His Context Principle 4.2.3 Interpretation 5 Rational Representationalism References A Comment on Lemanski's “Concept Diagrams and the Context Principle” References The World as Will and I-Language: Schopenhauer's Philosophy as Precursor of Cognitive Sciences 1 Chomsky's Conception of E- and I-Language 2 Schopenhauer's E- and I-Language 3 Language Processing According to Schopenhauer and the Primacy of I-Language 4 Results References Schopenhauer's Perceptive Invective 1 Introduction 2 Schopenhauer's Philosophy of Language 3 Learning from the Tirades 4 Conclusion References Part II Logic Schopenhauer's Eulerian Diagrams 1 Introduction 2 Eulerian Diagrams 3 Spheres and Concepts 4 Relations of Concepts 5 Complex Diagrams 6 Conclusion References Schopenhauer's Logic in Its Historical Context 1 Introduction 2 Logic as an Already-Perfected Science and Intuitive Way of Thinking 3 The History of Logic After Aristotle 3.1 The Laws of Thought at the Beginning of Logic 3.2 The Scholastic Mnemonics 3.3 Hypothetical and Disjunctive Inferences 3.4 The Separation of Concepts and Representations of Perception 3.5 The Fourth Figure 3.6 Criticisms of Aristotelian Logic 4 Left Off Schopenhauer's List: The Port-Royal Logic and Kant's Logic References Arthur Schopenhauer on Naturalness in Logic 1 Introduction 2 Unnatural and Natural Deduction in Syllogistics 2.1 Unnaturalness of FIV 2.2 Naturalness of FI 2.3 Naturalness of FII 2.4 Naturalness of FIII 3 Natural Completeness and Unnatural Redundancy 3.1 Mental-Linguistic Completeness and Redundancy 3.2 Diagrammatic Completeness and Redundancy 4 Conclusion and Outlook 4.1 Summary 4.2 Schopenhauer's Logic in the Context of Current Debates References Schopenhauer and the Equational Form of Predication 1 Introduction 2 Schopenhauer's Use of Equality Signs in Logic 3 Schopenhauer's Account of Quantity and Quality of Judgments 4 Possible Sources for Schopenhauer's Equational Symbolism 4.1 Reimarus, Kant, and Fichte 4.2 Herbart 4.3 Maimon 4.4 Schulze 5 A Tentative Assessment References From Euler Diagrams in Schopenhauer to Aristotelian Diagrams in Logical Geometry 1 Introduction 2 Some Background on Logical Geometry 2.1 Defining the Aristotelian Relations in a Boolean Algebra 2.2 The α-Structures and Their Properties 3 Two Basic Examples 3.1 From an Euler Diagram to a Classical Square of Opposition 3.2 From an Euler Diagram to a Degenerate Square of Opposition 4 Partitions, Euler Diagrams and Aristotelian Diagrams 5 Boolean Subtypes of Aristotelian Diagrams 6 Conclusion References Metalogic, Schopenhauer and Universal Logic 1 Indiosyncralogical Schopenhauer 2 Metamathematics, Metalogic, and Universal Logic 2.1 Origin and Nature of Metamathematics 2.2 From Metamathematics to Metalogic 2.3 From Metalogic to Universal Logic 3 Schopenhauer's Theory of the Metalogical 3.1 The Tricky and Crutchy Euclid 3.2 Metalogical Truths: Where They Are and What They Are 3.3 The Femininity and Triviality of Metalogical Truths 4 Formulation, Axiomatization, Interaction, Reflection 4.1 Reformulations, Semiotical Changes, and Mathematical Interaction 4.2 The Modern Axiomatic Methodology 4.3 Multi-Level Analysis and Productive Self-Reflection Acknowledgements and Memories References Part III Mathematics Schopenhauer and the Mathematical Intuition as the Foundation of Geometry 1 Introduction 2 Intuition in Mathematics 3 Role and Purpose of Mathematics 4 A British Debate 4.1 Whewell on the Study of Mathematics 4.2 Hamilton's Review 5 Intuition and the Foundations of Geometry References Schopenhauer on Intuition and Proof in Mathematics 1 Preliminary Remarks 2 Schopenhauer's Theory of Cognition 3 The Principle of Sufficient Reason 4 Geometrical Proof Procedures 5 Some Problems 6 On Disinclination for Mathematics References Schopenhauer on Diagrammatic Proof 1 Two Challenges to The Legitimacy of Picture Proofs 2 Schopenhauer's Advocacy 3 Applying Schopenhauer's Remarks 3.1 Particularity and Generality 3.2 Misleading Pictures 4 Conclusion References From Necessary Truths to Feelings: The Foundations of Mathematics in Leibniz and Schopenhauer 1 A Denied Source 2 `Necessary Truths' and Reminiscence 3 Necessary Truths in Euclid's System 4 Confused Ideas Between Images and Feeling References