One way to see them: A point is an infinitesimally small sphere.
Gravity and quantum theory imply that no length shorter than a Planck length can be measured or observed.
Given this, points are idealizations that have no basis in measurements. Space is NOT smooth and is not made of points.
But wait! How can we describe the world without space? Space MUST be smooth! How else can we write down evolution equations?
Space must be smooth and must not be smooth. What a bad situation to be in. Any theory of everything must show a way out.
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