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Infinity in its Proper Place.

Projective Geometry works with just three primitive qualities (‘primitive’ in the sense of ‘prime’—i.e., ‘first’):

• The Point:           this is just “place”, “location”, “somewhere”.
• The Line:            this is just “extension”, “straightness”.
• The Plane:          this is just “spread”, “flatness”.

None of these qualities can be made “more” or “less” what it is. For example, a point cannot be made more of a place than it is, or less.

Thus these three qualities are just that—qualities.

Accordingly, they are not quantified, or quantifiable.

It follows that a line is not a sum of contiguous points, because a sum is a quantity, not a quality. Also, a point has no size—of any value whatsoever, not even zero. A line may extend through places, but does not consist of those places.

It also follows

• that a line does not have a location, as location is the quality of a point, not of a line, and
• a point does not have extension, as extension is the quality of a line, not of a point.

The latter of these two statements will probably be regarded by most as almost laughably self-evident, while the former will probably be considered self-evidently ridiculous. But the statements are both true, each being simply each other's complement. We have a first hint here of something fundamental called, “Duality”, and also a hint that common-sense and intuition are not quite the reliable guides we may have supposed them to be.

The Axioms of Incidence list the ways in which the three qualities interact. In particular—

1. If two lines have a point in common, they must also have a plane in common
2. If two lines have a plane in common, they must also have a point in common
3. If two lines have no point in common, they are skew, and have nothing in common
 If two straight lines (such as 1 & 2, on the left) meet, they have both a single point, and a plane (π) in common. Conversely, if two straight lines have a plane (π) in common, then they must either be co-incident, or meet in a single point.

Axioms 1 and 2, besides being dual, brook no exceptions.

By them, even “parallel” lines (perhaps such as those marked by ? above left), having a plane in common, quite simply must have a point in common—for, if they do not, axiom number 3 in the list above applies, and they must be skew.

However, Projective Geometry has no axiomatic means of “knowing” whether or not lines are parallel, since that is a matter of measurement, of size, of units and of quantity. None of these is ‘sponsored’ by the axioms. The whereabouts of the meeting point of parallels (said to be the ‘infinite’) is likewise a quantitative issue of measurement.

Projective Geometry does not know what parallelity and infinity are.

But, if, in the intrinsically-measureless projective-geometric context, we come nevertheless to believe that we have discovered a valid way to measure—which always means defining, assigning and counting units—and use this way to give numerical meaning to the word, “constant”,

and then go on to describe parallels as having “constant” separation,
we blunder quite fundamentally
.

 The ‘separation’ invoked here is a lineal distance, which is reckoned point-to-point along a line. Please enable Java for an interactive construction (with Cinderella). Accordingly, there is never a lineal distance between lines, simply because lines are not points. So, the ‘constant separation’ statement above is wrong. It is an idea misapplied. The proper measure of distance between lines is rotational, and this is reckoned line-to-line around a point. Please enable Java for an interactive construction (with Cinderella).

By that rotational (angular) reckoning,
lines are parallel iff ("if and only if") the rotational distance between them is zero.
That is, lines which, by some measure-system, have zero rotational separation, are parallel in that measure-system.