nLab On the hypotheses which underlie geometry

A classical text regarded as envisioning the modern notion of (differentiable, smooth) manifolds and of what came to be called Riemannian geometry.

• Bernhard Riemann:

  
  

  Über die Hypothesen, welche der Geometrie zu Grunde liegen

  talk before the Göttingen Faculty (including Gauss, Dedekind & Weber)

  June 10, 1854

  
  

  Reprinted with historical commentary as:

  Jürgen Jost (ed.)

  Klassische Texte der Wissenschaft

  Springer (2013)

  doi:10.1007/978-3-642-35121-1

  
  

  English translation by William Clifford:

  On the hypotheses which underlie geometry

  Nature VIII (1873) 183-184

  pdf, pdf pdf

  
  

  Reprinted with historical commentary in:

  Jürgen Jost (ed.)

  On the Hypotheses Which Lie at the Bases of Geometry

  Classic Texts in the Sciences

  Springer (2016)

  doi:10.1007/978-3-319-26042-6

Quotes:

• On the potential breakdown of differential-geometric space at microscopic distances, reconsidered much later in the context of quantum gravity:

  [§III.3] Now it seems that the empirical notions on which the metric determinations of Space are based, the concept of a solid body and a light ray, lose their validity in the infinitely small; it is therefore quite definitely conceivable that the metric relations of Space in the infinitely small do not conform to the hypotheses of geometry; and in fact, one ought to assume this as soon as it permits a simpler way of explaining phenomena.