1NUNwLXhkvD6uUDRyvbnQZQy5D1ABP2iDE (7453) ·
When constructions in Space are extended into the immeasurably large, unboundedness is to be distinguished from infinitude; one belongs to relations of extension, the other to metric relations
Replies
1NUNwLXhkvD6uUDRyvbnQZQy5D1ABP2iDE (7453) ·
That space is an unbounded triply extended manifold is an assumption which is employed for every apprehension of the external world, ...
1NUNwLXhkvD6uUDRyvbnQZQy5D1ABP2iDE (7453) ·
... by which at every moment the domain of actual perception is supplemented, and by which the possible locations of a sought for object are constructed; and in these applications it is continually confirmed.
1NUNwLXhkvD6uUDRyvbnQZQy5D1ABP2iDE (7453) ·
The unboundedness of space consequently has a greater empirical certainty than any experience of the external world.
1NUNwLXhkvD6uUDRyvbnQZQy5D1ABP2iDE (7453) ·
But its infinitude does not in any way follow from this; quite the contrary, Space would necessarily be finite if one assumed independence of bodies from position, ...
1NUNwLXhkvD6uUDRyvbnQZQy5D1ABP2iDE (7453) ·
... and thus ascribed to it a constant curvature, as long as this curvature had ever so small a positive value.
1NUNwLXhkvD6uUDRyvbnQZQy5D1ABP2iDE (7453) ·
If one prolonged the initial directions lying in a surface direction into the shortest lines, one would obtain an unbounded surface with constant positive curvature, ...
1NUNwLXhkvD6uUDRyvbnQZQy5D1ABP2iDE (7453) ·
... and thus a surface which in a flat triply extended manifold would take the form of a sphere, and consequently be infinite.