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Weyl and the Problem of Space

From Science to Philosophy

  • Deepens readers’ foundations of geometry, physics and the theory of continuum with an approach which is inextricably philosophical, historical and scientific

  • Presents the first historical reconstitution of Weyl’s conferences on the problem of space

  • Proposes an extensive philosophical approach to the different aspects of the problem of space

  • Provides new insights and clarifications on complicated questions

Part of the book series: Studies in History and Philosophy of Science (AUST, volume 49)

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  • ISBN: 978-3-030-11527-2
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Table of contents (15 chapters)

  1. Front Matter

    Pages i-xxvi
  2. Weyl’s Intellectual Neighborhoods and the Theory of Subjectivity

    1. Front Matter

      Pages 1-1
    2. Hermann Weyl chez Gaston Bachelard

      • Charles Alunni
      Pages 25-33
    3. Neighbourhoods and Intersubjectivity

      • Norman Sieroka
      Pages 99-122
  3. Weyl’s Theory of the Continuum: Intuitionism and Dimensionnality of Space

    1. Front Matter

      Pages 123-123
    2. Weyl and Intuitionistic Infinitesimals

      • Mark van Atten
      Pages 125-159
  4. From Aprioristic to Physical Foundations of the Metric

    1. Front Matter

      Pages 211-211
    2. Logic of Gauge

      • Alexander Afriat
      Pages 265-293
    3. The Plasticine Ball Argument

      • Julien Bernard
      Pages 295-345
  5. Weyl’s Methodological Issues: Intuition, Symbolic Thought and Manifolds of Possibilities

    1. Front Matter

      Pages 369-369
    2. Husserl and Weyl on the Constitution of Space

      • Jairo José da Silva
      Pages 389-402

About this book

This book investigates Hermann Weyl’s work on the problem of space from the early 1920s onwards. It presents new material and opens the philosophical problem of space anew, crossing the disciplines of mathematics, history of science and philosophy. With a Kantian starting point Weyl asks: among all the infinitely many conceivable metrical spaces, which one applies to the physical world? In agreement with general relativity, Weyl acknowledges that the metric can quantitatively vary with the physical situation. Despite this freedom, Weyl “deduces”, with group-theoretical technicalities, that there is only one “kind” of legitimate metric. This construction was then decisive for the development of gauge theories. Nevertheless, the question of the foundations of the metric of physical theories is only a piece of a wider epistemological problem.

Contributing authors mark out the double trajectory that goes through Weyl’s texts, from natural science to philosophy and conversely, always through the mediation of mathematics. Readers may trace the philosophical tradition to which Weyl refers and by which he is inspired (Kant, Husserl, Fichte, Leibniz, Becker etc.), and explore the mathematical tradition (Riemann, Helmholtz, Lie, Klein) that permitted Weyl to elaborate and solve his mathematical problem of space. Furthermore, this volume analyzes the role of the interlocutors with whom Weyl discussed the nature of physical space (Einstein, Cartan, De Sitter, Schrödinger, Eddington).

This volume features the work of top specialists and will appeal to postgraduates and scholars in philosophy, the history of science, mathematics, or physics.


  • Foundation of Physics
  • Group Theory
  • History of Mathematics and Physics of the 20th
  • Lie groups
  • Non-Euclidean Geometry
  • Phenomenology of Space
  • Relativity Theory
  • Relativity Cosmology
  • Inflationary Cosmology
  • Kantian tradition
  • Transcendental Phenomenological Theories of Subjectivity
  • Conceptual Construction in Weyl's Analysis
  • Intuitionistic Vision of Space
  • Metrical Spaces
  • Wesenanalyse
  • Gauge Theories
  • Problem of Space

Editors and Affiliations

  • Assistant Professor in Philosophy, Centre Gilles Gaston Granger (CGGG) UMR 7304, Aix-Marseille-University, Marseille, France

    Julien Bernard

  • Collège International de Philosophie, Paris, France, Centro de Filosofia das Ciencias Universidade de Lisboa, Lisboa, Portugal

    Carlos Lobo

About the editors

Julien Bernard is currently assistant professor at Aix-Marseille University. His laboratory is the Centre Gilles Gaston Granger, UMR 7304. Before this, he was postdoctorate researcher at Zukunftskolleg (University of Constance), and also worked for the ANR Gödel on Kurt Gödel’s unpublished papers. His research interest is primarily in philosophy of physics, mathematics and logic, focussing on the relationships between the great traditional philosophical systems (Leibnizian metaphysics, neo-Kantianism, Husserlian phenomenology) and the scientific revolutions of the beginning of the twentieth century.

He has published two monographs – L’idéalisme dans l’infinitésimal and L’Analyse mathématique du problème de l’espace (French-German commented edition of Weyl’s text) – and several specialized articles on Hermann Weyl, on the problem of space, on Gödel’s Leibnizian interpretation of quantum mechanics, and on the foundations of mathematics. He serves as reviewer for the journals Lato Sensu, Journal of Symbolic Logic, and for the GDR 3719 “Philosophie des mathématiques”. He has also been awarded by the “Prix Paul Ricoeur 2013” of University Paris-Nanterre, and was winner of several important funding programs for his Art-Philosophy-Science project “Biomorphism” (notably: the “Interdisciplinarite 2016” AMIDEX funding program at Aix-Marseille and the funding program of the Foundation Daniel and Nina Carasso, Paris).

Carlos Lobo is currently member of the Collège international de philosophie and of the Center of Philosophy of Sciences (University of Lisbon). For over 30 years, while preparing students to the Ecole Normal Supérieure competitive exam, he pursued his research in phenomenology and epistemology. After a first monograph on the methodology of transcendental phenomenology (Le phénoménologue et ses exemples, in 2000), he published several contributions on phenomenological issues such as intersubjectivity, temporality, individuation, and axiology, showing their epistemological import for the understanding and critical clarification of formal logic, probability theory, and physics. Those contributions are particularly illustrated by a series of contributions such as “Self-variation and self-modification” in The Phenomenology of Embodied Subjectivity, Springer, 2013, 263–283, “Husserl’s Reform of Logic. An introduction” in New Yearbook for Phenomenology and Phenomenological Philosophy, 2017; “Some Reasons to reopen the question of foundations of probability theory following the Rota way” in The Philosophers and Mathematicians, ed. H. Tahiri, Springer, 2018; “Le maniérisme épistémologique de Gilles Châtelet. Relativité et exploration de l’a priori esthétique chez Châtelet, Weyl et Husserl”, Revue de Synthèse, Brill, n° 137, 2018. As a translator (from German, English, and Spanish), Lobo has published Husserl’s Les leçons sur l’éthique et la théorie de la valeur, PUF, 1999, and more recently Weyl’s Philosophie des mathématiques et des sciences de la nature, Métis Presses, Geneva, 2017.

Bibliographic Information

Buying options

eBook USD 109.00
Price excludes VAT (USA)
  • ISBN: 978-3-030-11527-2
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Hardcover Book USD 149.99
Price excludes VAT (USA)