The Rise and Fall of the German Combinatorial Analysis (Frontiers in the History of Science)

This text presents the ideas of a particular group of mathematicians of the late 18th century known as “the German combinatorial school” and its influence. The book tackles several questions concerning the emergence and historical development of the German combinatorial analysis, which was the unfinished scientific research project of that group of mathematicians. The historical survey covers the three main episodes in the evolution of that research project: its theoretical antecedents (which go back to the innovative ideas on mathematical analysis of the late 17th century) and first formulation, its consolidation as a foundationalist project of mathematical analysis, and its dissolution at the beginning of the 19th century. In addition, the book analyzes the influence of the ideas of the combinatorial school on German mathematics throughout the 19th century. Acknowledgements Contents 1 Introduction 1.1 Historiography on the Combinatorial Analysis 1.2 An Alternative Historical Interpretation of the German Combinatorial Analysis 1.3 Content of the Book 2 A History of the Binomial and Multinomial Theorems 2.1 The Binomial Theorem 2.1.1 Newton's Theorem 2.1.2 Newton's Theorem and the Method of Fluxions 2.1.3 Newton's Theorem in Germany 2.1.3.1 Differential Proofs of the Binomial Theorem for Rational Exponents 2.1.3.2 Functional Proofs of the Binomial Theorem for Rational Exponents 2.1.3.3 Proofs of the Binomial Theorem for Rational Exponents by the Method of Series 2.2 The Multinomial Theorem 2.2.1 Moivre's Multinomial Theorem 2.2.2 The Reception of the Multinomial Theorem 2.2.2.1 Proving the Multinomial Theorem by the Binomial Theorem 2.2.2.2 Fluxional Proofs of the Multinomial Theorem 2.2.2.3 A Combinatorial Proof of the Multinomial Theorem 2.2.3 The Reception of the Multinomial Theorem in Germany 2.2.3.1 Proving the Multinomial Theorem by the Binomial Theorem in Germany 2.2.3.2 Differential Proofs of the Multinomial Theorem 2.2.3.3 A Combinatorial Proof of the Multinomial Theorem in Germany 3 The Emergence of the German Combinatorial Analysis 3.1 Hindenburg on Mathematical Tables 3.2 Hindenburg and the Theory of Series 3.2.1 Hindenburg's First Approach to the Multinomial Theorem 3.2.1.1 The First Infinitinomii 3.2.1.2 The New Method of Hindenburg 3.2.2 The Second Infinitinomii, and the Sudden Apparition of Leibniz 3.3 The German Combinatorial Analysis as a Research Project 3.3.1 In Search of a Conceptual Background for the New System 3.3.2 The Combinatorial Operations of the New System 3.3.3 Creation of a New Mathematical Symbolism 3.3.4 Hindenburg's Scientific Work Program 4 The Consolidation of the German Combinatorial Analysis 4.1 The Reversion of Series and the Rise of the German Combinatorial School 4.1.1 Hindenburg, Science Editor 4.1.2 Lagrange's Inversion Formula and the Reversion of Series 4.1.2.1 Lagrange's Inversion Formula 4.1.2.2 Lagrange's Proof of Newton's Theorem on Reversion of Series 4.1.3 The Rise of the German Combinatorial School: An Interpretation 4.1.3.1 A Combinatorial and non-recursive Formula for the Reversion of Series 4.1.3.2 The Theory of Dimension Symbols 4.1.3.3 Unmasking the Usurper 4.1.3.4 A Proof of Eschenbach's Formula 4.1.3.5 Eschenbach-Rothe's Formula and Lagrange's Inversion Theorem 4.1.3.6 Hindenburg's Silence 4.2 The Most Important Theorem in the Whole of Analysis 4.2.1 Back to Moivre 4.2.1.1 The Method of Combinatorial Involutions 4.2.1.2 A Combinatorial Law of Nature 4.2.1.3 Corroborating Evidence for the Combinatorial Law of Nature 4.2.2 Independence, Primacy, and the Multinomial Theorem 4.2.2.1 The Independence of the Multinomial Theorem 4.2.2.2 A Foundational Combinatorial Proof of the Binomial Theorem 4.2.3 Hindenburg's Project Implementation 4.2.3.1 Toward an Algebra of Involutions 4.2.3.2 Involutions and Reversion of Series 4.2.3.3 Product, Power, and Division of Series 4.2.3.4 A New Analytic-Combinatorial Formula 4.2.3.5 Other Contributions to the Combinatorial Analysis 5 The Decline of the German Combinatorial Analysis 5.1 Second Thoughts About the Foundation of Analysis 5.1.1 The Foundation of Analysis Crumbles 5.1.2 First Textbooks 5.1.2.1 Stahl's Grundriss 5.1.2.2 Weingärtner's Lehrbuch 5.1.3 A Last Combinatorial Proof of the Binomial Theorem 5.2 Against the Current 5.2.1 A Non-Combinatorial Approach to the Multinomial Theorem 5.2.2 The Exponent Calculus 5.2.3 The Combinatorial Characteristic 5.2.3.1 The Analytic Tachygraphy 5.2.3.2 Ducit in Vitium Culpae Fuga 5.2.3.3 A Universal Notation System 6 A Combinatorial Current of Thought 6.1 Isolated Research 6.2 Combinatorics and Mathematics 6.2.1 Foundationalist Current 6.2.2 Countercurrent: Algebraic Purity 6.2.3 Syntactic Current 6.2.4 Non-Foundationalist Current 6.3 Vicissitudes Abroad 7 Epilogue References