Do people think that algebraic geometry might be thought of as the root of all geometry? (With differential geometry being another root). You've got Euclidean,

Do people think that algebraic geometry might be thought of as the root of all geometry? (With differential geometry being another root). You've got Euclidean, generalised to affine, generalised to projective, generalised to algebraic geometry. The objects of study in projective geometry are all algebraic sets of some kind; in fact, I'd more accurately say they're subschemes of the scheme RP^n. I think this view has been put forward by Norman Wildberger* (who admittedly has some controversial views on mathematics, but I think this suggestion is helpful for motivation). * - Without mentioning scheme theory.