The modern point of view of geometry is highly based on algebraic constructions. Mainly the coordinate or structure ring. Aspacecan be identified with a not necessarily commutingalgebra. The mantra is

Geometry $\displaystyle \Leftrightarrow$ Algebra

Thus, can modern geometry ever be synthetic?

To me the distinction between geometry and algebra is almost non-existent, viz algebraic geometry and generalisations.

Any thoughts?