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Utterly Fundamental Stuff (1)       — On Elements —

  • An element is a single, irreducible property or quality.
  • An element is not a composite. It is not made of anything. It is something.
  • An element is an essence, unique and ultimately simple.

For example, a line does not consist of – i.e., is not made of – points. It is not even made of line: it is line – line-as-such.

The corollary is that composites cannot be elements.

Thus, geometric elements do not consist of each other, but do differ in quality.  That is, the differences are qualitative, not quantitative.

Geometric elements are not numbers, and vice versa.

  1. A point is place-as-such. ‘Place’ is its single, elementary quality.
  2. A line is extension-as-such. ‘Extension’ is its only, elementary quality.
  3. A plane is spread-as-such. ‘Spread’ is its unique, elementary quality.

By (2), A curve is not a line. It is a composite of a point stepping along a line stepping around that point. The point and line remain incident.

By (3), A surface is not a plane. It is a composite of a line stepping around a point, while that point steps along that line, and as a plane steps about that line. The point, line and plane remain incident.

Thus, curves and surfaces cannot be elements.

Elementary Incidence is incidence of geometric elements, each with the others.

A Construction is some arrangement of such incidences.

An Absolute is immutable.  Nothing can change an absolute.

It is shown Next that –

Points haven't size (not even zero):
Lines haven't length (not even ∞):
Planes haven't area (not even ∞2),

And we note that

Lines and Planes haven't thickness, width or sides,

and also that,

if Elements had ends, they would be Composites, not Elements.