Research in History and Philosophy of Mathematics: The CSHPM 2019-2020 Volume (Annals of the Canadian Society for History and Philosophy of ... et de philosophie des mathématiques)

This volume contains eleven papers that have been collected by the Canadian Society for History and Philosophy of Mathematics/Société canadienne d’histoire et de philosophie des mathématiques. It showcases rigorously-reviewed contemporary scholarship on an interesting variety of topics in the history and philosophy of mathematics, as well as the teaching of the history of mathematics. Topics considered include The mathematics and astronomy in Nathaniel Torperly’s only published work, Diclides Coelometricae, seu valvae astronomicae universal Connections between the work of Urbain Le Verrier, Carl Gustav Jacob Jacobi, and Augustin-Louis Cauchy on the algebraic eigenvalue problem An evaluation of Ken Manders’ argument against conceiving of the diagrams in Euclid’s Elements in semantic terms The development of undergraduate modern algebra courses in the United States Ways of using the history of mathematics to teachthe foundations of mathematical analysis Written by leading scholars in the field, these papers are accessible not only to mathematicians and students of the history and philosophy of mathematics, but also to anyone with a general interest in mathematics. Preface Editorial Board Contents Contributors The Most Obscure and Inconvenient Tables Ever Constructed? 1 Introduction 2 Overview of Torporley's Astrology 3 All Circles Great and Small 3.1 Coordinate Systems 4 The Law of Tangents 4.1 Spherical Trigonometry 5 Torporley's Miter and Menelaus's Figure 5.1 Considering MT I 5.2 Considering MT II 6 The Spherical Right Triangle 7 The Quadrans or Right Gateway 8 The Quincunx or Left Gateway 9 Conclusion References Commercializing Arithmetic: The Case of Edward Hatton 1 Introduction 2 Edward Hatton and his Publishers 3 A Textbook: The Merchant's Magazine 3.1 Commercial Arithmetic 3.2 Merchant's Accompts 3.3 Book-Keeping 3.4 Later Editions 4 Ready Reckoners: Comes Commercii 5 Ready Reckoners: Index to Interest 6 Posters 7 Other Works Appendix: A Summary Hatton Bibliography References Leading to Poncelet: A Story of Collinear Points 1 Introduction 2 Pappus 3 Desargues 4 Monge 5 Carnot 6 Brianchon 7 Poncelet References Cauchy, Le Verrier et Jacobi sur le problème algébrique des valeurs propres et les inégalités séculaires des mouvementsdes planètes 1 Introduction 2 L'inégalité séculaire des planètes 3 Cauchy et le problème des valeurs propres en 1829 4 Le Verrier (1840) : Trouver les coefficients du polynôme caractéristique 5 Trouver les racines d'un polynôme 6 La méthode de Jacobi (1846) 6.1 La partie théorique 6.2 La partie pratique 7 Conclusion Références Mathematics in Astronomy at Harvard College Before 1839 as a Case Study for Teaching Historical Writing in Mathematics Courses 1 Introduction 2 John Winthrop 3 Samuel Williams 4 John Farrar 5 Conclusion References `Lectures for Women' and the Founding of Newnham College, Cambridge 1 Prologue 2 Lectures for Women 3 Newnham College 4 Conclusion References Are Euclid's Diagrams `Representations'? On an Argument by Ken Manders 1 Introduction 2 Reductio Proofs and the `Classical View' of Diagrams 3 A Partial Version of the Classical View 4 A Semantics for Diagrams 5 By Way of Conclusion: What Is a Semantics Good for? References Abstraction by Embedding and Constraint-Based Design 1 Introduction 2 Stick-Figure Abstraction 3 Abstraction by Embedding 4 Abstraction by Constraint-Based Design 4.1 Fourier Analysis and a Challenge It Posed 4.2 How to Define the Real Numbers 4.2.1 Heine 4.2.2 Cantor 4.2.3 Dedekind 4.3 Discussion 4.4 Epistemic Practices 4.4.1 Constraint-Based Design 4.4.2 Abstraction 4.4.3 Beyond Abstraction 4.4.4 Summary 5 Concluding Remarks References The Birth of Undergraduate Modern Algebra in the United States 1 Introduction 2 What Is Modern Algebra 3 Algebra in the Nineteenth Century: Examples of a Successful Start 4 Modern Algebra in the American Undergraduate Curriculum 4.1 About Cajori Two 4.2 About Table 1 4.3 About Table 2 5 The Story in a Nutshell and a Caution 6 The Environment Concerning Modern Algebra Courses 6.1 Doubts About Abstraction and Applications 6.1.1 The Double Mindset: Abstraction Versus Concrete Mathematics 6.1.2 Applications of Modern Algebra Did Not Seem Compelling 6.2 Lack of a Suitable Modern Algebra Textbook 6.3 The “Marginal Interest” and Interrupted Pace of Research 6.4 Polarizing Anxieties Related to Research 7 The Fading of Reasons to Hesitate 7.1 Advances in Research in Algebra 7.1.1 The Arrival of Emmy Noether and Abstract Ring Theory 7.1.2 The Development of Abstract Field Theory 7.1.3 The Use of Group Theory in Quantum Mechanics 7.2 The Appearance of Textbooks 7.2.1 The Appearance of Bartel van der Waerden's Moderne Algebra 7.2.2 The Appearance of Birkhoff and Mac Lane's A Survey of Modern Algebra 7.3 “Marginal and Interrupted” Give Way to a Plenitude of Research in Four Important Journals 7.4 The Fading of the Polarizing Anxieties Related to Research 7.5 Leadership from the Top 7.6 Conclusion 8 Related Wider Issues References History as a Source of Mathematical Narrative in Developing Students' Interpretations of Mathematics 1 Introduction 2 A Brief Account of What History Is 2.1 History as Literature 2.2 History as Science/Social Science 2.3 History as a Narrative 3 Mathematics and Narrative 3.1 History as a Source of Mathematical Narrative 3.2 The Context of Two History of Mathematics Courses 3.3 Changes in Students' Views 4 Conclusion References Thoughts on Using the History of Mathematics to Teach the Foundations of Mathematical Analysis 1 The Problem 2 Mathematics as Reasoning 3 Why Is There a Problem About Calculus? 4 Proofs as Sequences of Statements 5 Bridge Course: Mathematical Theories 6 Bridge Course: Set Theory and Logic 7 Analysis: Limits 8 The Real Numbers 9 Conclusion References J. S. Silverberg, The Most Obscure and Inconvenient Tables ever Constructed -- D. J. Melville, Commercializing Arithmetic: The Case of Edward Hatton -- C. Baltus, Leading to Poncelet: A Story of Collinear Points -- R. Godard, Cauchy, Le Verrier et Jacobi sur le problème algébrique des valeurs propres et les inégalités séculaires des mouvements des planètes -- A. Ackerberg-Hastings, Mathematics in Astronomy at Harvard College Before 1839 as a Case Study for Teaching Historical Writing in Mathematics Courses -- J. J. Tattersall, S. L. McMurran, "Lectures for Women" and the Founding of Newnham College, Cambridge -- D. Waszek, Are Euclid's Diagrams "Representations"? On an Argument by Ken Manders -- B. Buldt, Abstraction by Embedding and Constraint-Based Design -- W. Meyer, The Birth of Undergraduate Modern Algebra in the United States -- P. Liu, History as a Source of Mathematical Narrative in Developing Students' Interpretations of Mathematics -- F. Kamareddine, J. P. Seldin, Thoughts on Using the History of Mathematics to Teach the Foundations of Mathematical Analysis