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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 indeed quantifiable.

It follows that a line is not, as it is often held to be, a sum of contiguous points, because a sum is a quantity, not a quality. Moreover, because a point is, ‘just a place’, it cannot have a size as well, not even of zero. So a line cannot be a sum of sizeless points. In any case, size is an interval. If a point is to have a size, it must be an interval. It is not an interval.

So a line may extend through places, but does not consist of those places. Line and point may be incident one with the other, but neither can be part, or a component, of the other.

Similarly,
a plane is one thing, not an interval between two things, so has no size, and cannot be a component.
Likewise,
a line is one thing, not an interval between two things, so has no size, and cannot be a component.
Consequently,
A sizeless plane cannot be a sum of sizeless points or lines, though it may spread through both and be incident with them.

The corollaries are that
geometric elements do not have parts,
and that incidence is non-analytic.

It also follows that

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

The latter of these two statements may seem laughably self-evident, and the former self-evidently laughable. But they are both true; each is simply the other's complement, and as such express the fundamental “Duality” of Line and Point.

From this we may perhaps already sense that common-sense and intuition are not the reliable guides we may have supposed them to be, and that there may be need to approach the self-evident with refreshed caution.

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

It is by these axioms that we know that “parallel” lines meet. Because parallels certainly have a plane in common, by axiom 2 above, they certainly also have a point in common, and, perforce, meet.

If they do not meet, axiom number 3 in the list above applies: they have neither plane nor point in common and are in fact not parallel. They are skew.

However, Projective Geometry has no axiomatic means of “knowing” whether or not lines are in actual fact 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.

Notwithstanding, it is widely assumed that a means to measure distance exists, and it is also widely supposed that parallel lines, such as straight railway tracks, cannot meet, because they will, by such measurement, always be found to have constant separation.

Now, this is mistaken, on all counts, but let us for the moment assume that absolute and universally valid measurement - of anything - is in fact possible.

 The ‘separation’ invoked here is a lineal distance, which is reckoned point-to-point along a line. (Here, point A to point B along line a.) 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. (Here, line a to line b, around point A.) 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.

And there can be no angle, of any value whatsoever, between lines which do not meet. There is no plane angle between skews.