**MATHEMATICS**. From prehistoric times until today, arguably the most fundamental duality of abstraction has been represented by the formally inverted concepts now called shape and number.

It seems that while studies of

*shape*(geometries) are essentially building-up, apparently right-brain-biased, synthetically-driven notions, so studies of

*number*(arithmetics) are basically breaking-down, apparently left-brain-biased, analytically-driven notions. Human efforts to make the two kinds of mathematics work together have had three main phases.

The Greek desire to unify the ideas necessitated a crucial choice -- between shaping number and numbering shape. They chose to

**, creating synthetic geometry, the right brain biased mathematics that was dominant in the West for 2000 years.***shape number*In the 17th century Descartes (and arguably Fermat) reversed the Greek option and chose to

*number***by inventing analytic geometry. Up to this point, human attempts at mathematical synergy seem to have reflected large scale brain forms -- at the left-right brain level.***shape*The 18th century saw the beginning of another kind of synergy -- a search for fundamentals -- in a phase called

**. In broad terms, set theory deals with elements of thought, sets of the elements and relations between elements and between sets. The ideas are basic to all mathematical abstraction and thus arguably to all thought. It is suspected that in due course set theory will be seen to reflect small-scale brain forms such as neurons and their sets just as earlier theories reflected large-scale brain forms.***set theory*
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