Space, number, and geometry from Helmholtz to Cassirer
By: Francesca Biagioli
Series: Archimedes : new studies in the history and philosophy of science and technology -- volume 46; Archimedes (Dordrecht, Netherlands) -- v. 46.
Publisher: Springer
Published: 2016
Place of publication: Cham, Switzerland
Language: eng
Number of pages: 239
ISBN-13: 9783319317793; 9783319317779
Copyright date: 2016
Subjects: Geometry, Non-Euclidean; Neo-Kantianism; Science; Philosophy; History of science; Geometry; History of Western philosophy; MATHEMATICS; General; Nichteuklidische Geometrie
Description: This book offers a reconstruction of the debate on non-Euclidean geometry in neo-Kantianism between the second half of the nineteenth century and the first decades of the twentieth century. Kant famously characterized space and time as a priori forms of intuitions, which lie at the foundation of mathematical knowledge. The success of his philosophical account of space was due not least to the fact that Euclidean geometry was widely considered to be a model of certainty at his time. However, such later scientific developments as non-Euclidean geometries and Einstein's general theory of relativity called into question the certainty of Euclidean geometry and posed the problem of reconsidering space as an open question for empirical research. The transformation of the concept of space from a source of knowledge to an object of research can be traced back to a tradition, which includes such mathematicians as Carl Friedrich Gauss, Bernhard Riemann, Richard Dedekind, Felix Klein, and Henri Poincaré, and which finds one of its clearest expressions in Hermann von Helmholtz's epistemological works. Although Helmholtz formulated compelling objections to Kant, the author reconsiders different strategies for a philosophical account of the same transformation from a neo-Kantian perspective, and especially Hermann Cohen's account of the aprioricity of mathematics in terms of applicability and Ernst Cassirer's reformulation of the a priori of space in terms of a system of hypotheses. This book is ideal for students, scholars and researchers who wish to broaden their knowledge of non-Euclidean geometry or neo-Kantianism.
Notes: Includes bibliographical references and index.
Table of contents: title: Acknowledgments; Contents; Introduction; Chapter 1: Helmholtz's Relationship to Kant; 1.1 Introduction; 1.2 The Law of Causality and the Comprehensibility of Nature; 1.3 The Physiology of Vision and the Theory of Spatial Perception; 1.4 Space, Time, and Motion; References; Chapter 2: The Discussion of Kant's Transcendental Aesthetic; 2.1 Introduction; 2.2 Preliminary Remarks on Kant's Metaphysical Exposition of the Concept of Space; 2.3 The Trendelenburg-Fischer Controversy; 2.4 Cohen's Theory of the A Priori; 2.4.1 Cohen's Remarks on the Trendelenburg-Fischer Controversy.; title: 2.4.2 Experience as Scientific Knowledge and the A Priori2.5 Cohen and Cassirer; 2.5.1 Space and Time in the Development of Kant's Thought: A Reconstruction by Ernst Cassirer; 2.5.2 Substance and Function; References; Chapter 3: Axioms, Hypotheses, and Definitions; 3.1 Introduction; 3.2 Geometry and Mechanics in Nineteenth-Century Inquiries into the Foundations of Geometry; 3.2.1 Gauss's Considerations about Non-Euclidean Geometry; 3.2.2 Riemann and Helmholtz; 3.2.3 Helmholtz's World in a Convex Mirror and His Objections to Kant.; title: 3.3 Neo-Kantian Strategies for Defending the Aprioricity of Geometrical Axioms3.3.1 Riehl on Cohen's Theory of the A Priori; 3.3.2 Riehl's Arguments for the Homogeneity of Space; 3.3.3 Cohen's Discussion of Geometrical Empiricism in the Second Edition of Kant's Theory of Experience; 3.4 Cohen and Helmholtz on the Use of Analytic Method in Physical Geometry; References; Chapter 4: Number and Magnitude; 4.1 Introduction; 4.2 Helmholtz's Argument for the Objectivity of Measurement; 4.2.1 Reality and Objectivity in Helmholtz's Discussion with Jan Pieter Nicolaas Land.; title: 4.2.2 Helmholtz's Argument against Albrecht Krause: "Space Can Be Transcendental without the Axioms Being So"4.2.3 The Premises of Helmholtz's Argument: The Psychological Origin of the Number Series and the Ordinal Conception of Number; 4.2.4 The Composition of Physical Magnitudes; 4.3 Some Objections to Helmholtz; 4.3.1 Cohen, Husserl, and Frege; 4.3.2 Dedekind's Definition of Number; 4.3.3 An Internal Objection to Helmholtz: Cassirer; References; Chapter 5: Metrical Projective Geometry and the Concept of Space; 5.1 Introduction; 5.2 Metrical Projective Geometry before Klein.; title: 5.2.1 Christian von Staudt's Autonomous Foundation of Projective Geometry5.2.2 Arthur Cayley's Sixth Memoir upon Quantics; 5.3 Felix Klein's Classification of Geometries; 5.3.1 A Gap in von Staudt's Considerations: The Continuity of Real Numbers; 5.3.2 Klein's Interpretation of the Notion of Distance and the Classification of Geometries; 5.3.3 A Critical Remark by Bertrand Russell; 5.4 The Arithmetization of Mathematics: Dedekind, Klein, and Cassirer; 5.4.1 Dedekind's Logicism in the Definition of Irrational Numbers.