# nLab On the hypotheses which underlie geometry

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

#### Riemannian geometry

Riemannian geometry

## Applications

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

• 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

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.

Last revised on January 19, 2024 at 09:38:13. See the history of this page for a list of all contributions to it.