The Steiner-Lehmus theorem and "triangles with congruent medians are isosceles" hold in weak geometries

We prove that (i) a generalization of the Steiner-Lehmus theorem due to A. Henderson holds in Bachmann's standard ordered metric planes, (ii) that a variant of Steiner-Lehmus holds in all metric planes, and (iii) that the fact that a triangle with two congruent medians is isosceles holds in Hjelmslev planes without double incidences of characteristic $\neq 3$

Three-dimensional hyperbolic geometry with planes and plane parallelism as only primitive notions

We show that Euclidean Möbius planes can be axiomatized in terms of circles and circle-tangency, and that 3-dimensional hyperbolic geometry can be axiomatized in terms of planes and plane-parallelism

Comparing theories: the dynamics of changing vocabulary. A case-study in relativity theory

There are several first-order logic (FOL) axiomatizations of special relativity theory in the literature, all looking essentially different but claiming to axiomatize the same physical theory. In this paper, we elaborate a comparison, in the framework of mathematical logic, between these FOL theories for special relativity. For this comparison, we use a version of mathematical definability theory in which new entities can also be defined besides new relations over already available entities. In particular, we build an interpretation of the reference-frame oriented theory SpecRel into the observationally oriented Signalling theory of James Ax. This interpretation provides SpecRel with an operational/experimental semantics. Then we make precise, "quantitative" comparisons between these two theories via using the notion of definitional equivalence. This is an application of logic to the philosophy of science and physics in the spirit of Johan van Benthem's work.Comment: 27 pages, 8 figures. To appear in Springer Book series Trends in Logi

Note di Matematica 26

Abstract. We point out the geometric significance of a part of the theorem regarding the maximality of the orthogonal group in the equiaffine group proved in Keywords: Erlanger Programm, definability, Lω 1 ω -logic MSC 2000 classification: 03C40, 14L35, 51F25, 51A99 A. Schleiermacher and K. Strambach [12] proved a very interesting result regarding the maximaility of the group of orthogonal transformations and of that of Euclidean similarities inside certain groups of affine transformations. Although similar results have been proved earlier, this is the first time that the base field for the groups in question was not the field of real numbers, but an arbitrary Pythagorean field which admits only Archimedean orderings. They also state, as geometric significance of the result regarding the maximality of the group of Euclidean motions in the unimodular group over the reals, that there is "no geometry between the classical Euclidean and the affine geometry". The aim of this note is to point out the exact geometric meaning of the positive part of the 2-dimensional part their theorem, in the case in which the underlying field is an Archimedean ordered Euclidean field. In this case their theorem states that: (1) the group G 1 of Euclidean isometries is maximal in the group H 1 of equiaffinities (affine transformations that preserve non-directed area), and that (2) the group G 2 of Euclidean similarities is maximal in the group H 2 of affine transformations. The restriction to the 2-dimensional case is not essential but simplifies the presentation. The geometric counterpart of group-theoretic results in the spirit of the Erlanger Programm is given by Beth's theorem, as was emphasized by Büch