Friedman M., Krauthausen K. (eds.) Model And Mathematics: From The 19th To The 21st Century

Basel: Birkhäuser, 2022. — 441 p. — (Trends In The History Of Science). — ISBN 9783030978327, 303097832X.This open access book collects the historical and medial perspectives of a systematic and epistemological analysis of the complicated, multifaceted relationship between model and mathematics, ranging from, for example, the physical mathematical models of the 19th century to the simulation and digital modelling of the 21st century. The aim of this anthology is to showcase the status of the mathematical model between abstraction and realization, presentation and representation, what is modeled and what models.How to Grasp an Abstraction: Mathematical Models and Their Vicissitudes Between 1850 and 1950. Introduction.
Models at the End of the Nineteenth Century: Between Maxwell’s ‘Fictitious Substances’ and Boltzmann’s ‘Tangible Representation’.
1850s/1870s: ‘Analogy’ and ‘Model’ in Maxwell.
1880–1900: ‘Anschauung’ and ‘Bild’ (Klein and Brill).
1900s–1930s: From Material Analogies and ‘Geometric Models’ to Formal Analogies and Language-Oriented Models.
1891/1899/1936: Mathematics and the New Definition of ‘Model’.
1931/1925–6: The ‘Pencil and Paper Models’ of Biology and the Precursors of Modeling.
1940s: Lévi-Strauss and Mathematical Models in Anthropology.
Conclusion: The Model in the Twentieth Century: Fictitious, Fragmentary, Temporary.
Historical Perspectives and Case Studies.
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.
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.
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.
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.
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?
Interview with Myfanwy E. Evans: Entanglements On and Models of Periodic Minimal Surfaces.
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
Epistemological and Conceptual Perspectives.
‘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.
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.
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?
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
From Production Processes to Exhibition Practices.
Interview with Anja Sattelmacher: Between Viewing and Touching—Models and Their Materiality.
Interview with Ulf Hashagen: Exhibitions and Mathematical Models in the Nineteenth and Twentieth Centuries.
Interview with Andreas Daniel Matt: Real-Time Mathematics.