Analysis
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A linear measure, as on the ‘ruler’ illustrated to the left, is generally made by calibrating a strip of "dimensionally stable" material, such as seasoned wood, or a plastic like Perspex (a.k.a. Lucite, or Plexiglas).
Dividers, such as the pair illustrated on the right, might be used to mark off contiguous, non-overlapping intervals along the strip. If the setting of the arms of the dividers is not altered during this operation, the resulting intervals are assumed to be...
- of absolute size and
- equal in size to each other.
If they are the latter, they are called “units”.
The concept of absoluteness of size arises from the usually tacit, and usually unquestioned, assumption that the interval between the divider-points is independent of...
- the general location, and
- the orientation
...of the pair of dividers.
Thus, the interval between the divider-points is expected to be the same whether the (unaltered) dividers are lying East/West somewhere in London, or lying Zenith/Nadir somewhere in Beijing. or indeed lying along the Vernal Equinox at the Jovian North Pole.
The second assumption (of equality) follows from the first.
In other words, Distance is taken to be a primitive,
that is, a “given” thing, which may be metered as a count of “units”, as defined above.
There are no geometric axioms for either Distance or Equality,
so neither has geometric meaning.
Indeed, absolute (i.e., immutable) distance contradicts the axioms—
Assertion of absolute distance leads to the corresponding (but false) assertion that parallels cannot meet because they preserve a constant separation (i.e., a linear distance).
But only skew lines fail to meet, having no plane–indeed, having nothing at all–in common. Parallels, having a plane in common, are not skew. Furthermore, there is never linear distance between lines.
It follows that distance is neither primitive, nor absolute.
And, directly from this, there is no geometric way to detect absolute equality.
But the truly underlying assumptions here are that a line may be...
- broken into parts (that is, analysed)—specifically, into equal, unit parts—and
- assembled from such parts to any desired length.
Now, physical items can be broken, and assembled,
but geometric elements can not.
Analysis and Assembly are not “sponsored”
by the axioms of Synthetic Geometry
If we wish to measure in strict accordance with geometric axioms,
and those only,
then we must discover how these axioms may be used
to define an interval.