Thursday, March 29, 2012

Axiomatic Consistency

The pre-eminent exemplar of diagrammatic thinking is, of course, Euclid's geometry.  Unlike those diagrams relying on a constant conjunction with some particular object, the object of Euclidean geometry is axiomatically defined.
  1. "To draw a straight line from any point to any point.
  2. To produce a finite straight line continuously in a straight line.
  3. To describe a circle with any centre and distance.
  4. That all right angles are equal to one another.
  5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two, if produced indefinitely, meet on that side on which are the angles less than the two right angles." [Heath's translation]
Euclid's axioms set the constraints for what can and cannot be done in geometry, but they also specify the planar and three-dimensional space where the geometry can be applied.  The result is an implicative universality.  An axiomatically defined diagram can go anywhere, not because it applies everywhere, but because it specifies just where and when it will apply.  The axioms underpin an internally consistent unity, cut off from any particular object, that can also self-direct its applications throughout the universe.  It is a serious, if common, short-changing of axiomatic thought to limit the axioms to maintaining an internal consistency.

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