Model And Mathematics: From The 19th To The 21st Century

Contents
1 How to Grasp an Abstraction: Mathematical Models and Their Vicissitudes Between 1850 and 1950. Introduction
	I. Models at the End of the Nineteenth Century: Between Maxwell’s ‘Fictitious Substances’ and Boltzmann’s ‘Tangible Representation’
	II. 1850s/1870s: ‘Analogy’ and ‘Model’ in Maxwell
	III. 1880–1900: ‘Anschauung’ and ‘Bild’ (Klein and Brill)
	IV. 1900s–1930s: From Material Analogies and ‘Geometric Models’ to Formal Analogies and Language-Oriented Models
		(1) 1891/1899/1936: Mathematics and the New Definition of ‘Model’
		(2) 1931/1925–6: The ‘Pencil and Paper Models’ of Biology and the Precursors of Modeling
		V. 1940s: Lévi-Strauss and Mathematical Models in Anthropology
		VI. Conclusion: The Model in the Twentieth Century: Fictitious, Fragmentary, Temporary
Part I Historical Perspectives and Case Studies
2 Knowing by Drawing: Geometric Material Models in Nineteenth Century France
	Introduction
	Geometry and Model Drawing
		Drawing, Models, and Analysis
		Geometric Drawing in the Royal Engineering Schools
		The Foundation of École Polytechnique
		Mutual Instruction Versus Academic Pedantry
		Monge’s “Cabinet Des Modèles”
		A Polytechnic Culture of Drawing
	The Canons of Geometric Drawing: Models and the Artillery School
		The Alliance Between Practice and Theory
		Learning by Drawing at the Conservatoire and Beyond
		Olivier’s String Models
		Bardin’s Plaster Models
		Model Drawing in Superior Primary Education
	The Models of Higher Geometry
		Naturalistic Mathematics
		The Darboux-Caron Wooden Models
		Models and the 1902 Educational Reform in France
		The Golden Age of Mathematical Models in View of the Decline of Model Drawing
		Open Questions: Models, Mathematical Modelization, and the Graphical Method
	Conclusions
3 Wilhelm Fiedler and His Models—The Polytechnic Side
	Wilhelm Fiedler
	Some Remarks on Teaching and Early Models
	Models in Fiedler’s Correspondence
	Models in Fiedler’s Teaching and Publishing
	Conclusions
4 Models from the Nineteenth Century Used for Visualizing Optical Phenomena and Line Geometry
	Introduction
	Optics Stimulating Mathematics Simulating Optics
	Constructing Fresnel’s Wave Surface
	Constructing Infinitely Thin Pencils of Rays
	Kummer Surfaces
	Plücker’s Complex Surfaces
	On Deforming Quartics
5 Modeling Parallel Transport
	Introduction
	Historical Context: Localization of the Models in Space and Time
	The Notion of Parallel Transport
	The Context of the History of Mathematics
	The Levi-Civita Connection
	A Mechanical Model of Parallel Transport
	Later History
	Concluding Remarks
6 The Great Yogurt Project: Models and Symmetry Principles in Early Particle Physics
	Introduction: The Coral Gables Conferences on “Symmetry Principles at High Energy” and the Yogurt Project
	‘Models’ and ‘Theories’ as Actors’ Categories in Early Theoretical Particle Physics
	Mathematical Practices of Rotations and the Emergence of the Gell-Mann-Nishijima Model of Particle Classification
	The Search for a Theory of Isospin and Strangeness in the 1950s
	The Path from SU(2) to SU(3), or: Did Particle Physicist Know Group Theory?
	Beyond SU(3)—The Mathematical Marriage of Space-Time and Internal Symmetries
	The Rise and Fall of SU(6)
	Conclusion: The End of the Yogurt Project?
7 Interview with Myfanwy E. Evans: Entanglements On and Models of Periodic Minimal Surfaces
8 The Dialectics Archetypes/Types (Universal Categorical Constructions/Concrete Models) in the Work of Alexander Grothendieck
	Archetypes and Types in the Tôhoku and the Rapport
	Types and Archetypes in Pursuing Stacks and Dérivateurs
	Models in Récoltes et Semailles
	Conclusion
Part II Epistemological and Conceptual Perspectives
9 ‘Analogies,’ ‘Interpretations,’ ‘Images,’ ‘Systems,’ and ‘Models’: Some Remarks on the History of Abstract Representation in the Sciences Since the Nineteenth Century
	Dynamical Analogies, Physical/Mechanical Analogies, Mathematical Analogies
	Interpretations of Non-Euclidean Geometry
	Systems, Spielräume, Euklidische Modelle: Some Remarks by Felix Hausdorff, Ca. 1900
	Images and Dynamical Models: Heinrich Hertz Once Again
	Epilogue: The Rise of (Modern) Mathematical Models
10 Mappings, Models, Abstraction, and Imaging: Mathematical Contributions to Modern Thinking Circa 1900
	Generalities
	The Riemann Inflexion
	Reflections in Science and Mathematics … and New Flashes
	Helmholtz and Hertz
	Longue Durée
	Other Reflections
11 Thinking with Notations: Epistemic Actions and Epistemic Activities in Mathematical Practice
	The Applicability ‘Problem’
	Philosophies of Mathematical Practice
	Notations, Formalisms, Models
	Practices, Agents, Actions
	Epistemic Actions and Their Limits
	What ‘Epistemic Actions’ in Mathematics Might Be
		The Use of Gestures and Symbolic Operations in Instructional Settings
		Applying Material Models to Mathematics
		Re-proving Theorems
		Notations as ‘Institutionalized’ (Long-Term) Epistemic Actions?
12 Matrices—Compensating the Loss of Anschauung
	Introduction
	Immanuel Kant’s Philosophy of Applied Mathematics
	The Loss of Anschauung in the Nineteenth Century and the Declaration of Anschaulichkeit as a Model in Geometry
	Matrices as New Tools for Compensating the Loss of Anschauung in Physics
	Early Twentieth Century Debate on Anschauung and Anschaulichkeit in Physics
	Surreality of the New Physics
	Conclusion
Part III From Production Processes to Exhibition Practices
13 Interview with Anja Sattelmacher: Between Viewing and Touching—Models and Their Materiality
14 Interview with Ulf Hashagen: Exhibitions and Mathematical Models in the Nineteenth and Twentieth Centuries
15 Interview with Andreas Daniel Matt: Real-Time Mathematics