I am reading the incredible Greenberg(2008, Euclidean and Non-Euclidean Geometries)' book.
I am not a mathematician. My doubt is the following: is there an objective difference between definitions and postulates? It seems to me that the difference is merely conventional, not strictly objective. Am I wrong?
Please, I would like to know what philosophically-minded people think about that.
Is there a stritcly objective difference between definitions and postulates?
Yes; a definition introduces a new term using previously known terms (or undefined basic ones).
A postulate, like an axiom, states a property about some basic term.
You can see in Greenberg about undefined terms: point, line, defined terms: parallelism (a relation between two lines) and postulates (or axioms) like the parallel one asserting the existence of a unique line that is parallel to a given line through a point not on that line.
The basic difference is that an axiom/postulate, like the parallel one, can assert the existence of something: a set that is empty, a line that is parallel to a given one, while a definition introduces a new term of the language. It acts as an abbreviation for a long description; a definition cannot "conjure" the existence of something out of the blue.
