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

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.

Les enjeux de la controverse Frege-Hilbert sur les fondements de la géométrie : une étude philosophique sur la logique et les mathématiques

L’auteur entreprend dans ce mémoire de faire une présentation des débats axiologiques de philosophie de la logique sous-jacents à la controverse opposant Frege et Hilbert sur les fondements de la géométrie. Contre le parti pris philosophique selon lequel la logique est une discipline achevée, l’auteur entreprend une mise en contexte des positions de Frege et Hilbert afin de montrer que dans leur conception de la logique se trouvent des paradigmes incommensurables, résultant de l’influence de traditions philosophiques et scientifiques diverses. Dans cette perspective, Frege est le défenseur de la vision traditionnelle de la logique comme medium universel de la science, tel qu’incarnée dans la géométrie euclidienne. La logique symbolique de Frege est ainsi vue comme la mise en oeuvre de moyens raffinés pour lutter contre la (( perversion des sciences )) ayant lieu au 19ième siècle et pour la défense de la vision traditionnelle de la science. à l’opposé, l’approche métathéorique de Hilbert représente la conception moderne dite algébrique de la logique telle que développée au 19ième sous l’influence des métamathématiques, et certains rapprochements avec les conceptions (( model-theoretic )) et catégorielles de la logique viennent appuyer cette thèse.This memoir presents some axiological debates of philosophy of logic underlying the Frege-Hilbert controversy on the foundations of geometry. Against the philosophical bias according to which logic is an achieved discipline, a contextualized presentation of the respective positions of Frege and Hilbert is done in order to show that incommensurable paradigms are found in their view of logic, that is due to the influence of various philosophical and scientific traditions. From this standpoint, Frege is the defender of the traditionalist view of logic as the universal medium of science, as embodied in Euclidean geometry. In this perspective, Frege’s symbolic logic is seen as the achievement of a refined means to counter the 19th-century perversion of science with the purpose of defending the traditional conception of the role of science. On the other hand, Hilbert’s metatheoretical approach represents the so-called algebraic modern conception of logic as developed in the 19th century under the influence of metamathematics. Following this, parallels between Hilbert’s approach and the model-theoretical and categorical conceptions of logic are drawn to show their proximity