When Form Becomes Substance: Power of Gestures, Diagrammatical Intuition and Phenomenology of Space (English and French Edition)

This interdisciplinary volume collects contributions from experts in their respective fields with as common theme diagrams. Diagrams play a fundamental role in the mathematical visualization and philosophical analysis of forms in space. Some of the most interesting and profound recent developments in contemporary sciences, whether in topology, geometry, dynamic systems theory, quantum field theory or string theory, have been made possible by the introduction of new types of diagrams, which, in addition to their essential role in the discovery of new classes of spaces and phenomena, have contributed to enriching and clarifying the meaning of the operations, structures and properties that are at the heart of these spaces and phenomena. The volume gives a closer look at the scope and the nature of diagrams as constituents of mathematical and physical thought, their function in contemporary artistic work, and appraise, in particular, the actual importance of the diagrams of knots, of braids, of fields, of interaction, of strings in topology and geometry, in quantum physics and in cosmology, but also in theory of perception, in plastic arts and in philosophy. The editors carefully curated this volume to be an inspiration to students and researchers in philosophy, phenomenology, mathematics and the sciences, as well as artists, musicians and the general interested audience. Introduction Topological Visualisation, or How to Apprehend the Invisible Feynman Diagrams: A New Way Forward in Theoretical Physics Diagrammatics and Invariants in Knot and Braid Theory Philosophical and Scientific Implications Diagrammatics and Category Theory A French Singularity in Epistemological Field: Gilles Châtelet Phenomenology of Space (and Time) and Diagrammatic Epistemology Towards a “Diagrammatic Critique of Aesthetics” Acknowledgments Contents Part I Logic, Forms and Diagrams The Semiotics of Laws of Form Introduction Finding Distinction Finding Primary Arithmetic Finding Logic Finding Mathematics The Arctic Essay Epilogue References Can We “Show” the Correctness of Reasoning? On the Role of Diagrammatic Spatialization in Logical Justification Introduction The Eulerian Thesis: The Logical Correctness of the Diagrams “Jumps to the Eyes” Determining the Logical Framework of Our Research The Late Emergence of “Analytical” Logic Diagrams in a Pedagogical Context First Section: The Cognitive Advantages of the Diagrammatic Method Second Section: Can Diagrams Be Given the Task of Validating Reasoning? Third Section: Can a Diagram Show (“donner à voir”) the Nature of a Proposition or the Correctness of a Reasoning? Back to Euler Showing the Nature of a Proposition Showing the Correctness of a Reasoning Some Remarks on the Relationship Between the Principles of Logic and Spatiality in Syllogistic (From Aristotle to Hamilton) Conclusion: Summary and Discussion References Articles and Monographs Proceedings of International Colloquia Catégorification et méthode Le polynôme de Jones La catégorification du polynôme de Jones La méthode de catégorification La théorie topologique quantique des champs Diagrammes et méthode Référenes Part II Geometrical Spaces and Topological Knots, Old and New Which Came First, the Circle or the Wheel? From Idea (δεα) to Concrete Construction Introduction Geometric Ideas and their Diagrams Digital Fabrication Geometry in Higher Dimension Bibliography Sitography The Classical Style in Contemporary Geometry: Views from a Person Working in the Field Introduction Fragments of Basic Algebraic Geometry Sterographic Projections in Dimension 1 and 2 (Figs. 4 and 5) Origins of Birational Geometry and Classification Cremona Transformations (Fig. 6) de Jonquiéres Transformations (Fig. 8) Classical Problems and Rational Parametrizations: Curves and Surfaces Classical Problems and Rational Parametrizations: Cubics The Classical Turn in Algebraic Geometry References Knots, Diagrams and Kids' Shoelaces. On Space and their Forms Introductive Remarks: Shoelaces, Knots, and the Intuition of Space Exploring and Visualizing 3-manifols and the Importance of Topology Equivalence of Images and of Forms: Manifolds, Knots and Diagrams Embeddings and Isotopies Mathematical Propaedeutic for the Understanding of Knots Knots and Links: Equivalence, Invariants and the Knot Complement The Alexander and Jones Polynomials Crossing Changes of Knots Back to Classical Invariants of Knots and Links Historical Note on Knots and their Diagrams Equivalence of Knots and Links From the Alexander Polynomial to Seifert Surfaces for Knots Reidemeister Moves and Classical Knot Invariants Dehn Surgery of Knots and his Work on Knot Theory and 3-dimensional Manifolds The Fundamental Group of Knots and Links Invariants of 3-manifolds and Hyperbolic Knots The Importance of the Linking Number in Molecular Biology Geometrical and Topological Properties of the Double Helix and Supercoiling Knots, Links, and Topological Quantum Field Theories: An Overview Knots and Dynamics Systems The Energy of Knots References Part III Diagrams, Graphs and Representation Diagrammes planaires qui représentent des objets de dimension 1, 2, 3 et 4 Problème de classification Approche combinatoire Noeuds (dimension 1) Surfaces (dimension 2) Codage planaire des surfaces Variétés de dimension 3 Polyèdres spéciaux Polyèdres spéciaux épaississable Reconstruction Codage graphique Mouvements locaux Calcul graphique Variétés de dimension 4 Codage graphique des ombres References From Singularities to Graphs Introduction What Is the Meaning of Such Graphs? What Does it Mean to Resolve the Singularities of an Algebraic Surface? Representations of Surface Singularities Around 1900 Du Val's Singularities, Coxeter's Diagrams and the Birth of Dual Graphs Mumford's Paper on the Links of Surface Singularities Waldhausen's Graph Manifolds and Neumann's Calculus with Graphs Conclusion References Part IV Diagrams, Physical Forces and Path Integrals Mathematical Aspects of Feynman Path Integrals, Divergences, Quantum Fields and Diagrams, and Some More General Reflections Introduction The Case of Quantum Fields Divergences and Diagrams Some Conclusions, Philosophical Remarks and Reflections References Some Remarks on Penrose Diagrams Introduction Reflexions on the Different Notions of Dimensions The Notion of a Function A Second Kind of Diagram The Introduction of Infinity Some Complements to Conformal Geometry and General Relativity Cosmology and Conformal Diagrams Philosophical Comments Final Complements References Part V Phenomenology in and of Mathematical Diagrams Phénoménologie, représentations, combinatoire Représentations géométriques: une approche classique Représentations géométriques: l'approche mathématique Représentations géométriques: l'approche phénoménologique Partitions non croisées Représentations graphiques De l'usage des représentations diagrammatiques. De la portée des représentations diagrammatiques Références Husserl, Intentionality and Mathematics: Geometry and Category Theory Intentionality and Space The Idea of a Mannigfaltigkeitslehre Space and Time in Phenomenology Issues of Foundation in Phenomenology through Category Theory References Diagrams of Time and Syntaxes of Consciousness: A Contribution to the Phenomenology of Visualization The Time Diagram Has Been Touched Upon (Varela, Weyl) Phenomenological Elucidation of the Subjective Resources of the Mathematical Construction of Linear Time The Specious Time of Neurophenomenology Phenomenology of the Use of Diagrams in Science Including Phenomenology Diagrammatic Underpinnings of Scientific Knowledge The Use of Diagrams in Phenomenology A Second Example: Intersubjective Constitution and Relativisation of the Original Coordinate System Symbolisation and Formal Writing in Phenomenology Symbolisation and Diagrammatisation of Intentional Analysis Intentionality as a System of Modification and Its Symbolism Analysis of the System of Continuous, i.e. Temporal Changes in 1913 An Example of Moving from an Analysis to a Mathematical Model Phenomenology and Diagrammatic of Lived Time The Dialectic of Phenomenological Reflection, Symbolisation and Diagram Construction Between 1905 and 1918 Diagrams of Retentions and Drafts of a Chronometry From Infinitesimal Analysis to Complex Analysis of Phenomenological Time Gaps in the 1905 Diagrams An Important Gap in the Initial Retention Diagram: The Meaning of the Zero Diagrams of the on-in-the-Other (or Nesting) and Intertwining of Retentions and Protentions Diagrams in Reflection Conclusion: Perspectives for a General Theory of Possible Times References Part VI Diagrams, Gestures and Subjectivity A Topological Analysis of Space-Time-Consciousness: Self, Self-Self, Self-Other Knot Logic: Linking as Mutuality Belongingness: Not-I, Knot-I Self-Mutuality as Mutuality: Mutuality as Self-Mutuality Plurality of Now's Music Kauffman's Universe and HYK's Self References Gestes, diagrammes et subjectivité Un exemple de diagramme social (et ses gestes) Prendre sur soi, le 0-milieu Rapport à soi, la verticale Tourner autour de soi, le trou Le virtuel-Temps événementiel Some Prolegomena for a Contemporary “Critique of Imagination” Six Guiding Theses Imagination: Common Sense Notions Imagination: Philosophic Insights from Leibniz, Hume, Kant and Husserl Imagination: Hints from Neuroscience Bibliography Le langage diagrammatique au-delà de la différencephénoménologique Introduction La première forme canonique de notre rapport au monde : la signifiance De la signifiance au calcul : brève histoire de la différence phénoménologique Conclusion Part VII Diagrams: from Mathematics to Aesthetics Ars diagrammaticae Le phénomène de compactification Du diagramme comme preuve par l'image Quelques précisions maintenant concernant la désintrication des concepts d'Image, de Figure & de Diagramme. Grid Diagram: Deleuze's Aesthetics Applied to Maggs's Photographs Grid as Diagram in Portraits Faciality of Portraits Grid and Virtuality Dada Diagrams/Dada Portraits Routes of Reference: Technical Diagrams as Art Nomenclature as Poetic Potential Conclusion Les jeux de l'unilatère L'invariant, Persée (8 = 8). 8, par exemple A Plat Le diagramme à l'œuvre Part VIII Poetics and Politics of Diagrams Diagrammes du possible : de l'espace des phases au sujet politique L'espace des phases: exemple des diagrammes qui ne sont pas représentation Faute de programmes on trace des diagrammes La crise du sujet politique La dimension diagrammatique de l'écriture littéraire : un formalisme dynamique inscrit dans la sensorialité du langage DE DIAGRAMME EN DIAGRAMME LA DIMENSION DIAGRAMMATIQUE DE L'ECRITURE LITTERAIRE SPÉCIFICITÉ DE L'ÉCRITURE LITTÉRAIRE Une autre langue dans la langue Iconicité et abstraction Diagrammatisation de l'écriture ; « sensorialisation » d'une nouvelle logique Avec Charles Baudelaire et Emily Dickinson