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Utterly Fundamental Stuff (2)       — On Intervals —

Projective geometry and Euclid both measure intervals by intervals, but Euclid
  • assumes that an interval has an absolute size,
  • assumes that two or more intervals can be of equal size [video],
  • and, using sums of equal ‘units’, tacitly conflates number with geometry.

Projective Geometry and Euclid part company here, because there is no exclusively geometric way to validate these assumptions, meaning that,

for Projective Geometry, size and equality do not exist,
and, in direct consequence,
it can compare intervals only through incidence.

Intervals are matched – or mismatched – by incidence alone.

A List of Consequences :-

  1. An interval is always the incidence of no fewer than three geometric elements (see sketch on the right).

  2. Because an element is not three things, but just one thing (namely, itself),
    • and an interval is three things (elements), not one thing,
    • an element cannot be an interval,
    • and an interval cannot be an element.

  3. Because, from (2) above,
    • no geometric element is an interval,
    • and because sizes are intervals,
    it must follow that no geometric element has size.
Points haven't size (not even zero):
Lines haven't length (not even ∞):
Planes haven't area (not even ∞2).

As there are three elements – point, line and plane – there are three fundamental types of interval ...

  1. Interpunctual - between two points incident with a line
  2. Interlinear     - between two lines incident in a point and a plane
  3. Interplanar    - between two planes incident in a line

We note in passing that, for every type, whenever one interval is formed, two are in fact formed.  For example, we may trace out one interval going right from A to B on the line l, shown on the left above, and may also trace out a second interval going left from A to B.  (The latter tracing works because a line has no ends.)  This fact is of significance when investigating the validity of infinitesimals for non-Euclidean, projective geometry.

Intervals have ends, Elements do not.

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