Definability in the Foundations of Euclidean Geometry and the Product Rule for Derivations

In this talk, we discuss the results of investigations that began with a solution to an open problem posed by Schwabhäuser and Szczerba regarding definability (without parameters) in the three dimensional Euclidean geometry of lines, asking whether intersection was definable from perpendicularity (two lines intersecting at a right angle). It is not. The result is a “new” 3-dimensional geometry of lines, which we call perpendicular geometry, since it can be formalized from perpendicularity. Further investigations produce a rather complete classification of possible geometries arising from elementary Euclidean binary relations between lines in R3, modulo the determination of (metrical) projective geometries formalized by binary relations between points. The classification shows that in a sense made precise, perpendicular geometry is the only new geometry that can arise from binary geometric relations, except for possible new projective plane geometries, which we conjecture do not exist. Generalizing to geometries of s-flats in n-dimensional Euclidean geometry, we state a theorem which provides the essential first step towards a similar classification for the general case. The theorem states that parallel is definable in such a geometry no matter what binary geometric relation that one might chose to formalize geometry (with an enumerated list of trivial exceptions). To what extent this is true for ternary or higher order geometric relations is open, even for lines in R3. We conjecture that it remains true, that is, that parallel is definable from anything “except for the exceptions”. Finally, we note that perpendicular geometry, whose automorphism group is connected with derivations, sheds some rather curious light on the relationship between the product rule for derivations over a ring, and the sum rule. For example, it is a direct consequence of perpendicular geometry that the product rule for the cross product of 3dimensional vectors implies the sum rule. It is conjectured that this is also true of all finite dimensional semi-simple Lie algebras over the complex numbers. In [2], Schwabhäuser and Szczerba investigated the possibility of formulating Euclidean geometry based on lines as primitive notions, together with geometric relations between lines. (An n-ary relation between lines in R is called geometric provided that the relation is preserved under similarity transformations, that is, compositions of rotations, reflections, dilations, and translations.) They showed that for any n ≥ 2, the binary relation of perpendicularity (two lines intersecting at a right angle), together with the ternary relation of copunctuality (three lines intersecting at a single point) suffices to formalize n-dimensional Euclidean geometry. (Essentially, these relations suffice to interpret the points as equivalence classes of pairs of intersecting lines.) They went on to show that for n ≥ 4, the