© 2016

Space, Number, and Geometry from Helmholtz to Cassirer


  • Offers a new historical reconstruction of the philosophical debate on non-Euclidean geometry in neo-Kantianism

  • Brings a new approach to Helmholtz's philosophy of mathematics

  • Relevance for the current debate about the relativized a priori


Part of the Archimedes book series (ARIM, volume 46)

Table of contents

  1. Front Matter
    Pages i-xx
  2. Francesca Biagioli
    Pages 1-21
  3. Francesca Biagioli
    Pages 51-80
  4. Francesca Biagioli
    Pages 81-116
  5. Back Matter
    Pages 229-239

About this book


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.


Hermann von Helmholtz Neo-Kantianism Non-Euclidean Geometry Physical Geometry Space and Geometry Spatial Intuition

Authors and affiliations

  1. 1.ZukunftskollegUniversity of KonstanzKonstanzGermany

About the authors

Francesca Biagioli completed her PhD in philosophy and history of science at the University of Turin, Italy, in 2012. Her areas of specialization are Kant, neo-Kantianism, and the history of philosophy of science in the 19th Century. She is the author of articles about the philosophy of science of neo-Kantians such as Hermann Cohen, Alois Riehl and Ernst Cassirer, and scientists and mathematicians such as Hermann von Helmholtz and Otto Hölder.

Bibliographic information


“Francesca Biagioli’s Space, Number, and Geometry from Helmholtz to Cassirer is a substantial and pathbreaking contribution to the energetic and growing field of researchers delving into the physics, physiology, psychology, and mathematics of the nineteenth and twentieth centuries. … It is a clear, accurate, and deep account of fascinating and philosophically momentous implications of the move to relativity theory.” (Lydia Patton, Journal for General Philosophy of Science, Vol. 50, 2019)