Traditional geometry concerns itself with planimetric and stereometric considerations, which are at the root of the division between plane and solid geometry. To raise the issue of the relation between these two areas brings with it a host of different problems that pertain to mathematical practice, epistemology, semantics, ontology, methodology, and logic. In addition, issues of psychology and pedagogy are also important here. To our knowledge there is no single contribution that studies in detail even one of the aforementioned areas.

This paper provides a definitive answer, based on considerations derived from first-order logic, to the question regarding the status of elementary geometry, whether elementary geometry can be reduced to algebra. The answer we arrive at is negative, and is based on a series of structural questions that can be asked only inside the geometric formal theory, as well as the consideration of reverse geometry, which is the art of finding minimal axiom systems strong enough to prove certain geometrical theorems, given (...) that there are no algebraic structures that one could associate with those minimal axiom systems. (shrink)