Synthetic geometry Vs. analytic geometry

[ajb]

Let me first (loosely) define both synthetic and analytic geometry.

 

Synthetic geometry- deductive system based on postulates. The geometric objects are endowed with geometric properties from the axioms. This is includes the "high school" geometry of drawing lines and measuring angles etc. and from there making deductive statements. A geometry can be defined as a set plus a symmetric, reflexive relation. (Does it really need to be a set?)

 

 

Analytic geometry-represent geometric objects using local coordinates. This makes much use of algebra (and in the differential case, calculus). I will also include algebraic geometry here (so sheaves, affine varieties etc are included), the idea being that we use algebra to do (define) geometry.

 

The modern point of view of geometry is that "geometry" = "associative algebra", and more so we do not need to restrict ourselves to commutative algebras. One thing we generally lose is the notion of a point and hence (at least with out evoking the functor of points) we are not dealing with set theoretical objects.

 

So, can modern geometry ever be synthetic as it relies heavily on modern algebra and is not set theoretical? Or is modern geometry really geometry? Or is the distinction between synthetic and analytic geometry artificial and obtrusive in modern geometry?

 

Any thoughts?