Did Lobachevsky Have A Model Of His "imaginary Geometry"?

The invention of non-Euclidean geometries is often seen through the optics of Hilbertian formal axiomatic method developed later in the 19th century. However such an anachronistic approach fails to provide a sound reading of Lobachevsky's geometrical works. Although the modern notion of model of a given theory has a counterpart in Lobachevsky's writings its role in Lobachevsky's geometrical theory turns to be very unusual. Lobachevsky doesn't consider various models of Hyperbolic geometry, as the modern reader would expect, but uses a non-standard model of Euclidean plane (as a particular surface in the Hyperbolic 3-space). In this paper I consider this Lobachevsky's construction, and show how it can be better analyzed within an alternative non-Hilbertian foundational framework, which relates the history of geometry of the 19th century to some recent developments in the field.Comment: 31 pages, 8 figure

Constructive Geometry and the Parallel Postulate

Euclidean geometry consists of straightedge-and-compass constructions and reasoning about the results of those constructions. We show that Euclidean geometry can be developed using only intuitionistic logic. We consider three versions of Euclid's parallel postulate: Euclid's own formulation in his Postulate 5; Playfair's 1795 version, and a new version we call the strong parallel postulate. These differ in that Euclid's version and the new version both assert the existence of a point where two lines meet, while Playfair's version makes no existence assertion. Classically, the models of Euclidean (straightedge-and-compass) geometry are planes over Euclidean fields. We prove a similar theorem for constructive Euclidean geometry, by showing how to define addition and multiplication without a case distinction about the sign of the arguments. With intuitionistic logic, there are two possible definitions of Euclidean fields, which turn out to correspond to the different versions of the parallel axiom. In this paper, we completely settle the questions about implications between the three versions of the parallel postulate: the strong parallel postulate easily implies Euclid 5, and in fact Euclid 5 also implies the strong parallel postulate, although the proof is lengthy, depending on the verification that Euclid 5 suffices to define multiplication geometrically. We show that Playfair does not imply Euclid 5, and we also give some other independence results. Our independence proofs are given without discussing the exact choice of the other axioms of geometry; all we need is that one can interpret the geometric axioms in Euclidean field theory. The proofs use Kripke models of Euclidean field theories based on carefully constructed rings of real-valued functions.Comment: 114 pages, 39 figure

Kant's Views on Non-Euclidean Geometry

Kant's arguments for the synthetic a priori status of geometry are generally taken to have been refuted by the development of non-Euclidean geometries. Recently, however, some philosophers have argued that, on the contrary, the development of non-Euclidean geometry has confirmed Kant's views, for since a demonstration of the consistency of non-Euclidean geometry depends on a demonstration of its equi-consistency with Euclidean geometry, one need only show that the axioms of Euclidean geometry have 'intuitive content' in order to show that both Euclidean and non-Euclidean geometry are bodies of synthetic a priori truths. Michael Friedman has argued that this defence presumes a polyadic conception of logic that was foreign to Kant. According to Friedman, Kant held that geometrical reasoning itself relies essentially on intuition, and that this precludes the very possibility of non-Euclidean geometry. While Friedman's characterization of Kant's views on geometrical reasoning is correct, I argue that Friedman's conclusion that non-Euclidean geometries are logically impossible for Kant is not. I argue that Kant is best understood as a proto-constructivist and that modern constructive axiomatizations (unlike Hilbert-style axiomatizations) of both Euclidean and non-Euclidean geometry capture Kant's views on the essentially constructive nature of geometrical reasoning well

REFLECTING ON THE BASES OF GEOMETRY: CONSTRUCTION WITH THE TRACTRIX

Construction is, historically, the first form of geometry, and from early days the subtleties of how constructions are done have been considered important. In this paper, we examine geometric construction using a straightedge and a device for drawing tractrices. The tractrix was the first curve traced by the mechanical solution of an inverse tangent problem, the geometrical issue at the basis of Leibniz's conception of infinitesimal analysis. This nonalgebraic curve cannot be axiomatized simply, as the circle can. We show that some important constructions can be done based on a weak axiomatization that does not fully specify the curve, and that more may be done using its Cartesian representation

Las imágenes y la lógica del cono de luz: rastreando el giro postulacional de Robb en la física geométrica

Previous discussions of Robb’s work on space and time have offered a philosophical focus on causal interpretations of relativity theory or a historical focus on his use of non-Euclidean geometry, or else ignored altogether in discussions of relativity at Cambridge. In this paper I focus on how Robb’s work made contact with those same foundational developments in mathematics and with their applications. This contact with applications of new mathematical logic at Göttingen and Cambridge explains the transition from his electron research to his treatment of relativity in 1911 and finally to the axiomatic presentation in 1914 in terms of postulates. At the heart of Robb’s physical optics was the model of the light cone. The model underwent a transition from a working mechanical model in the Maxwellian Cambridge sense of a pedagogical and research tool to the semantic model, in the logical, model-theoretic sense. Robb tracked this transition from the 19th- to the 20th-century conception with the earliest use of the term ‘model’ in the new sense. I place his cone models in a genealogy of similar models and use their evolution to track how Robb’s physical researches were informed by his interest in geometry, logic and the foundations of mathematics. Las discusiones anteriores de la obra de Robb acerca del espacio y el tiempo han ofrecido un enfoque filosófico de las interpretaciones de la teoría de la relatividad o un enfoque histórico de su empleo de la geometría no-euclidiana, o han ignorado enteramente las discusiones de la relatividad en Cambridge. En este artículo centro mi atención en la forma cómo la obra de Robb tomó contacto con esos mismos desarrollos fundacionales en la matemática y con sus aplicaciones. El contacto con las aplicaciones de la nueva lógica matemática en Göttingen y en Cambridge explica la transición de las investigaciones de Robb sobre los electrones a su tratamiento de la relatividad en 1911 y finalmente a su presentación axiomática de 1914. En el corazón de la óptica física de Robb estaba el modelo del cono de luz. Este modelo pasó de ser un modelo mecánico operante en el sentido cantabrigense maxwelliano de herramienta didáctica y heurística a ser un modelo semántico en el sentido lógico de la teoría de modelos. Robb marcó esta transición de la concepción del siglo XIX a la del siglo XX con el uso más temprano del término “modelo” en el nuevo sentido. Sitúo sus modelos de conos en una genealogía de modelos similares y uso su evolución para seguir la pista de cómo las investigaciones físicas de Robb dependían de su interés en la geometría, la lógica y los fundamentos de las matemáticas.