The concept of a point from Euclid to Hilbert

At the session of the second synthesis Terry Bollinger mentioned that Euclid's Elements begins with the definition of a point. A definition that is delightful: "A point is that which has no parts."

I would like to emphasize that 2300 years later, in Hilbert's axiomatic theory for the same entities in the same Euclidean space, the situation with the definition for a point has changed significantly, namely.

For Hilbert, a point is a primary (aka primitive) concept, along with two others: an infinite straight line and a plane.

For these three (there are no other primary) types of objects, 6 primary relations are introduced - all but one are binary, betweenness is ternary.

So what is a point for Hilbert?

This is any class of objects if it is possible to provide two more classes of objects for it, specify the geometry relations on all three, and check that the axioms of geometry are satisfied. The class of objects for which the axioms are satisfied as for a point can be considered points. I think Hilbert once joked "even beer mugs". This example illustrates how we apply our axiomatic theories: specify real or ideal objects for primary classes (sorts), specify primary relations on them, and check that the axioms are satisfied. If the axioms are satisfied (for example, theories of undirected graphs), go ahead, the theory is applicable in all its glory.