Infinity

It is established here that parallel lines meet,
just as all line-pairs with a common plane meet.

Where is the meeting place?

For parallels, it is often said to be, “at infinity”.

But, because geometry cannot detect parallelity,
for geometry, infinity does not exist.

Geometry, as has been noted, is without units,
so does not define sizes—of anything at all—and hence cannot measure.

In particular, it cannot determine the size of a rotation.
To detect the “parallel condition”, geometry would be required to evaluate angle,
but this it simply cannot do.

• Geometry does not “know” what parallelity is.
• Infinity is not geometrically specified.
• All meeting-points are just that—meeting-points.
• There are no exceptional or special points.

Hence, if Infinity is after all to specified, it must either be done in extra-geometric ways,
or we must discover how to unitise (that is, analyse), and hence metricise, geometry......