The Oxford Handbook of Generality in Mathematics and the Sciences (Oxford Handbooks)

Generality is a key value in scientific discourses and practices. Throughout history, it has received a variety of meanings and of uses. This collection of original essays aims to inquire into this diversity. Through case studies taken from the history of mathematics, physics and the life sciences, the book provides evidence of different ways of understanding the general in various contexts. It aims at showing how collectives have valued generality and how they have worked with specific types of «general» entities, procedures, and arguments. The books connects history and philosophy of mathematics and the sciences at the intersection of two of the most fruitful contemporary lines of research: historical epistemology, in which values (e.g. «objectivity», «accuracy») are studied from a historical viewpoint and the philosophy of scientific practice, in which conceptual developments are seen as embedded in networks of social, instrumental, and textual practices. Each chapter provides a self-contained case-study, with a clear exposition of the scientific content at stake. The collection covers a wide range of scientific domains - with an emphasis on mathematics - and historical periods. It thus allows a comparative perspective which suggests a non-linear pattern for a history of generality. The introductory chapter spells out the key issues and points to the connections between the chapters. Cover 1 The Oxford Handbook of Generality in Mathematics and the Sciences 4 Copyright 5 Dedication 6 Contents 8 List of Contributors 12 1 Prologue: generality as a component of an epistemological culture 16 Part I The meaning and value of generality 58 Section I.1 Epistemic and epistemological values 60 2 The value of generality in Michel Chasles’s historiography of geometry 62 3 Generality in Leibniz’s mathematics 105 Section I.2 Actors’ reflections on generality in science 126 4 The problem of a “general” theory in mathematics: Aristotle and Euclid 128 5 Generality, generalization, and induction in Poincaré’s philosophy 150 Part II Statements and concepts: the formulation of the general 180 Section II.1 Developing a new kind of statement 182 6 Elaboration of a statement on the degree of generality of a property: Poincaré’s work on the recurrence theorem 184 7 Generality and structures in functional analysis: the influence of Stefan Banach 238 Section II.2 A diachronic approach: continuity and reinterpretation 270 8 How general are genera? The genus in systematic zoology 272 9 Homology: an expression of generality in the life sciences 301 Section II.3 Circulation between epistemological cultures 312 10 The role of genericity in the history of dynamical systems theory 314 Part III Practices of generality 340 Section III.1 Scientists at work 342 11 Leibnizian analysis, canonical objects, and generalization 344 12 Models, structure, and generality in Clerk Maxwell’s theory of electromagnetism 360 Section III.2 A diachronic approach: continuity and contrasts 372 13 Biological generality: general anatomy from Xavier Bichat to Louis Ranvier 374 14 Questions of generality as probes into nineteenth-​century mathematical analysis 400 Section III.3 A synchronic approach: controversies 426 15 Universality versus generality: an interpretation of the dispute over tangents between Descartes and Fermat 428 16 Algebraic generality versus arithmetic generality in the 1874 448 17 Practices of generalizationin mathematical physics, in biology,and in evolutionary strategies 483 Section III.4 Circulation between epistemological cultures 496 18 A process of generalization:Kummer’s creation of ideal numbers 498 Index 516 This handbook examines how actors have valued generality in mathematics and the sciences and how they worked with specific types of “general” entities, procedures, and arguments. It argues that actors have shaped these various types of generality, mainly by introducing specific terminologies to distinguish between different levels or forms of generality, as well as designing means to work with them, or to work in relation to them. The book is organized into three parts. Part I deals with the meaning and value of generality, and more specifically the value of generality in Michel Chasles’s historiography of geometry and generality in Gottfried Leibniz’s mathematics. Part II focuses on statements and concepts that make up the general, covering topics such as Henri Poincaré’s work on the recurrence theorem and the role of genericity in the history of dynamical systems theory. Part III explores the practices of generality, including the dispute over tangents between René Descartes and Pierre de Fermat, generality in James Clerk Maxwell’s theory of electromagnetism, and practices of generalization in mathematical physics, biology, and evolutionary strategies. This Collection Of Original Essays Aims To Inquire Into The Diversity Of Generality. Through Case Studies Taken From The History Of Mathematics, Physics And The Life Sciences, The Book Provides Evidence Of Different Ways Of Understanding The General In Various Contexts.