Metrics
Pure Geometry does not define absolute size,
either linear or angular.
There is no inch or metre, no radian or degree.
there are no square units of area,
or cubic units of volume,
or the steradians of solid angle.
either linear or angular.
There is no inch or metre, no radian or degree.
there are no square units of area,
or cubic units of volume,
or the steradians of solid angle.
But it does define intervals which are exactly constant
with regard to how they are devised.
Such intervals are geometrically indistinguishable,
meaning that they cannot be told apart,
and if this is so,
then they must be the same.
They are in every geometric respect equal to each other.
We adopt this as
our definition of equality, and say that
two, distinct intervals on the same element,
devised in identically the same, projective manner,
are equal.
Some may prefer to speak instead of equivalence, or identity.
The important things to note are that the comparison is not quantitative,
involves elementary incidence alone,
and does not involve absolutes.
our definition of equality, and say that
two, distinct intervals on the same element,
devised in identically the same, projective manner,
are equal.
Some may prefer to speak instead of equivalence, or identity.
The important things to note are that the comparison is not quantitative,
involves elementary incidence alone,
and does not involve absolutes.
The Standard Reference Measure(SRM)
Caveat:
There is no way to establish
absolute numerical equality
of geometric intervals, intuition
(and appearances) notwithstanding.
All of Euclidean Geometry,
and the physics that depends upon it, rely on that intuition, and take
it as a given.
This is a serious error.