Where does manifold collide?

A space that near every point looks like ordinary flat space, so that the tools of calculus can be carried onto it even when its overall shape is curved or twisted. A sphere and a doughnut surface are manifolds: zoom in far enough on either and each patch resembles a flat plane, and the patches are stitched together by smooth changes of coordinates. Lee's 2012 Introduction to Smooth Manifolds sets out how such spaces are built from local charts and used as the setting for differential geometry, including the curved spacetime of general relativity. The referent is an abstract geometric object assembled from patches of flat space, not a branched pipe in an engine or the raw material of perception in a mind.[1]

Unit

One manifold: an n-dimensional space covered by charts onto flat space

Recovery means

Not applicable

Measured by

Characterised by its dimension, its atlas of coordinate charts, and invariants such as curvature and topology

In Kant's philosophy, the raw and unordered multiplicity of impressions that sensibility delivers before the mind has organised it: this given variety, the manifold of intuition, is what the understanding's categories gather and bind together into one unified experience of an object. Kant's 1781 Critique of Pure Reason, here in the 1998 Guyer and Wood translation, argues that without an act of synthesis the manifold would stay a scattered plurality and no coherent experience could arise. The referent is a technical term for the unsynthesised content of perception, not a curved geometric space or a branched pipe that carries a gas.[2]

Unit

The given multiplicity of intuition awaiting synthesis by the categories

Recovery means

Not applicable

Measured by

Not measured empirically; it is a posited element in Kant's analysis of how experience is possible