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Parallelity

  
 

At the outset we issue the caveat that
size is a count of equal units, possibly coupled with a further count of equal sub-divisions of a unit (fractions), and hence is quantity, not quality, and again stress that
numbers do not automatically attach to the qualitative elements of geometry.

All that follows below assumes that a properly-valid, working connection can be established between the elements of geometry and the processes of counting.

Later, we will attempt to ascertain to what extent this might be the case.

It is rather often asserted that parallel, straight lines maintain a constant distance from each other,
and thus do not - indeed, cannot - meet.
This is a version of Euclid's famous “5th Postulate”.

It is mistaken.

'Distance' of the kind referred to in such a statement (namely, linear translation) is reckoned
point-to-point along a line,
and cannot apply to line-pairs, quite simply because lines are not points.
It follows that there is never linear distance between lines.

 
 

Distance between lines is properly reckoned
line-to-line around a point,
and is rotation, not translation.
In fact, it is precisely dual to translation.

Thus,

  1. if, as above, we can assert that there is never translational distance between lines,
  2. we can at once assert the dual - that there is never rotational distance between points.
Intuition,
though a much-employed and well-trusted faculty,
is here fetched quite sharply into question:
  • probably most would regard statement 2, above, to be self-evidently so,
  • and almost certainly also consider statement 1 to be self-evidently not so.

It follows that lines are parallel if the rotational distance between them is zero.

Note the necessary participation here of a point, common to the lines.
It implies that the lines in question must have a plane in common for the rotation to exist at all—and parallels certainly have such a plane. In fact, only skew lines (with no common plane) do not meet.

So parallels meet,

and, clearly, our next question must be, “where do they meet?”

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