A theory of magnitudes involves criteria for their equivalence, comparison and addition. In this article we examine these aspects from an abstract viewpoint, by focusing on the so-called De Zolt’s postulate in the theory of equivalence of plane polygons (“If a polygon is divided into polygonal parts in any given way, then the union of all but one of these parts is not equivalent to the given polygon”). We formulate an abstract version of this postulate and derive it from some (...) selected principles for magnitudes. We also formulate and derive an abstract version of Euclid’s Common Notion 5 (“The whole is greater than the part”), and analyze its logical relation to the former proposition. These results prove to be relevant for the clarification of some key conceptual aspects of Hilbert’s proof of De Zolt’s postulate, in his classicalFoundations of Geometry(1899). Furthermore, our abstract treatment of this central proposition provides interesting insights for the development of a well-behaved theory ofcompatiblemagnitudes. (shrink)

The paper outlines an interpretation of one of the most important and original contributions of David Hilbert’s monograph Foundations of Geometry , namely his internal arithmetization of geometry. It is claimed that Hilbert’s profound interest in the problem of the introduction of numbers into geometry responded to certain epistemological aims and methodological concerns that were fundamental to his early axiomatic investigations into the foundations of elementary geometry. In particular, it is shown that a central concern that motivated Hilbert’s axiomatic investigations (...) from very early on was the aim of providing an independent basis for geometry. Accordingly, these concerns about an independent grounding for elementary geometry determined very clear methodological constraints in the process of embedding it into a formal axiomatic system. It is argued that Hilbert not only sought to show that geometry could be considered a pure mathematical theory, once it was presented as a formal axiomatic system; he also aimed at showing that in the construction of such an axiomatic system one could proceed purely geometrically, avoiding concept formations borrowed from other mathematical disciplines like arithmetic or analysis. (shrink)

The paper surveys different notions of implicit definition. In particular, we offer an examination of a kind of definition commonly used in formal axiomatics, which in general terms is understood as providing a definition of the primitive terminology of an axiomatic theory. We argue that such “structural definitions” can be semantically understood in two different ways, namely as specifications of the meaning of the primitive terms of a theory and as definitions of higher-order mathematical concepts or structures. We analyze these (...) two conceptions of structural definition both in the history of modern axiomatics and in contemporary philosophical debates. Based on that, we give a systematic assessment of the underlying semantics of these two ways of understanding the definiens of such definitions, by considering alternative model-theoretic and inferential accounts of meaning. (shrink)

This paper provides a detailed study of David Hilbert’s axiomatization of the theory of plane area, in the classical monograph Foundation of Geometry. On the one hand, we offer a precise contextualization of this theory by considering it against its nineteenth-century geometrical background. Specifically, we examine some crucial steps in the emergence of the modern theory of geometrical equivalence. On the other hand, we analyze from a more conceptual perspective the significance of Hilbert’s theory of area for the foundational program (...) pursued in Foundations. We argue that this theory played a fundamental role in the general attempt to provide a new independent basis for Euclidean geometry. Furthermore, we contend that our examination proves relevant for understanding the requirement of “purity of the method” in the tradition of modern synthetic geometry. (shrink)

El artículo documenta y analiza las vicisitudes en torno a la incorporación de Hilbert de su famoso axioma de completitud, en el sistema axiomático para la geometría euclídea. Esta tarea es emprendida sobre la base del material que aportan sus notas manuscritas para clases, correspondientes al período 1894–1905. Se argumenta que este análisis histórico y conceptual no sólo permite ganar claridad respecto de cómo Hilbert concibió originalmentela naturaleza y función del axioma de completitud en su versión geométrica, sino que además (...) permite disipar equívocos en cuanto a la relación de este axioma con la propiedad metalógica de completitud de un sistema axiomático, tal como fue concebida por Hilbert en esta etapa inicial.The paper reports and analyzes the vicissitudes around Hilbert’s inclusion of his famous axiom of completeness, into his axiomatic system for Euclidean geometry. This task is undertaken on the basis of his unpublished notes for lecture courses, corresponding to the period 1894–1905. It is argued that this historical and conceptual analysis not only sheds new light on how Hilbert conceived originally the nature of his geometrical axiom of completeness, but also it allows to clarify some misunderstandings regarding this axiom and the metalogical property of completeness of an axiomatic system, as it was understoodby Hilbert in this initial stage. (shrink)

This paper analyzes the theory of area developed by Euclid in the Elements and its modern reinterpretation in Hilbert’s influential monograph Foundations of Geometry. Particular attention is bestowed upon the role that two specific principles play in these theories, namely the famous common notion 5 and the geometrical proposition known as De Zolt’s postulate. On the one hand, we argue that an adequate elucidation of how these two principles are conceptually related in the theories of Euclid and Hilbert is highly (...) relevant for a better understanding of the respective geometrical practices. On the other hand, we claim that these conceptual relations unveil interesting issues between the two main contemporary approaches to the study of area of plane rectilinear figures, i.e., the geometrical approach consisting in the geometrical theory of equivalence and the metrical approach based on the notion of measure of area. Finally, in an appendix logical relations among equivalence, comparison and addition of magnitudes are examined schematically in an abstract setting. (shrink)

El artículo presenta una interpretación del abordaje axiomático temprano a la geometría de David Hilbert, i.e., el desarrollado entre 1891 y 1905. Se sostiene que diversos aspectos de este abordaje, a primera vista problemáticos, se pueden comprender mejor si se contrastan con una de sus influencias más importantes en este periodo: la teoría pictórica [Bildtheorie] de Heinrich Hertz. En particular se argumenta que un análisis de la concepción axiomática de Hilbert a la luz de la teoría de Hertz permite aliviar (...) ciertas tensiones en la concepción hilbertiana; más precisamente, las tensiones que surgen de mantener, al mismo tiempo, una posición axiomática abstracta o formal y una concepción empirista de la geometría, que la considera una ciencia natural. (shrink)

Theoria, Volume 88, Issue 3, Page 629-652, June 2022.

Sobre la base que aportan las notas manuscritas de David Hilbert para cursos sobre geometría, el artículo procura contextualizar y analizar una de las contribuciones más importantes y novedosas de su célebre monografía Fundamentos de la geometría, a saber: el cálculo de segmentos lineales. Se argumenta que, además de ser un resultado matemático importante, Hilbert depositó en su aritmética de segmentos un destacado significado epistemológico y metodológico. En particular, se afirma que para Hilbert este resultado representaba un claro ejemplo de (...) uno de los rasgos más fructíferos y atractivos de su nuevo método axiomático formal, o sea, la capacidad de descubrir y exhibir conexiones estructurales o internas entre diferentes teorías matemáticas. On the basis of a set of unpublished notes for lecture courses on geometry, the paper seeks to contextualize and analyze one of the most important and original contributions of David Hilbert's celebrated monograph Foundations of Geometry, namely its arithmetic of line segments. It is argued that Hilbert attributed to his arithmetic of segments an important epistemological and methodological meaning, in addition to its relevance as an original mathematical result. In particular, it is claimed that for Hilbert his arithmetic of segments represented a clear example of one of the most fruitful and attractive traits of his new formal axiomatic method, i.e., the power to discover and exhibit inner or structural connections among different mathematical theories. (shrink)

El artículo ofrece una interpretación de las contribuciones de David Hilbert a la teoría de las magnitudes poligonales, desarrollada en su influyente monografía Fundamentos de la geometría, publicada en 1899. Se argumenta que la construcción de esta parte central de la geometría euclidiana represento para Hilbert un desafío muy significativo, en razón de su objetivo general de proporcionar una axiomatización estrictamente sintética de esta teoría geométrica; es decir, en virtud de sus conocidos requerimientos metodológicos y epistemológicos de construir la geometría (...) elemental evitando toda apelación esencial a presupuestos numéricos. Se sostiene además que dicho desafío consistió en ofrecer un nuevo fundamento lógicamente sólido para la teoría del área poligonal, pero que al mismo tiempo pueda ser considerado como puramente geométrico. The paper offers an interpretation of David Hilbert's contributions to the theory of polygonal magnitudes, developed in his influential monograph Foundations of Geometry, published in 1899. It is argued that the construction of this central part of Euclidean geometry presented an appealing challenge to Hilbert's general aim of providing a purely synthetic axiomatization of this geometrical theory; in other words, to his epistemological and methodological concerns of constructing elementary geometry without resorting to any kind of numerical assumption. Moreover, it is claimed this challenge consisted in offering a new foundation for the theory of polygonal area, which could be considered at the same time as logically sound and purely geometrical. (shrink)

El artículo presenta y analiza un conjunto de notas manuscritas de clases para cursos sobre geometría, dictados por David Hilbert entre 1891 y 1905. Se argumenta que en estos cursos el autor elabora la concepción de la geometría que subyace a sus investigaciones axiomáticas en Fundamentos de la geometría . Por un lado, afirmo que lo que caracteriza esta concepción de la geometría es: i) una posición axiomática abstracta o formal; ii) una posición empirista respecto del origen de la geometría (...) y de su lugar dentro de las distintas teorías matemáticas. Por otro lado, sostengo que el papel que Hilbert le confiere a la intuición geométrica en el proceso de axiomatización de esta teoría, permite apreciar claramente su oposición respecto de las posiciones formalistas con las que habitualmente es identificado. (shrink)

This paper aims to offer an interpretation of the Transcendental Deduction of the Categories which puts together two of its most distinctive and fundamental traits: the constant reference to the temporal character of the human consciousness and the use of the analytic method of exposition. We will defend the thesis that the connection between both traits is essential, i. e., it will be argued that it is precisely the usage of the analytic method what confers to the pure consciousness of (...) time a significant role in the structure of the argument. With this thesis an alternative reading of the A-Deduction is pursued, one which moves away from the predominant one which claims that Kant's persistent allusion in it to the pure time representation - and to the synthesis which make it possible - is just a direct consequence of its psychological character.El objetivo del artículo es ofrecer una interpretación de la Deducción Transcendental de las Categorías que ponga en relación dos de sus aspectos más distintivos y fundamentales: la referencia permanente al carácter temporal de la conciencia humana, y la utilización del método analítico de exposición. Se defenderá la tesis de que la conexión entre ambos aspectos es esencial, en tanto que es precisamente la adopción del método analítico lo que determina que se le confiera a la conciencia pura del tiempo un rol fundamental en la estructura del argumento. Con esta tesis se intenta ofrecer una lectura alternativa de la Deducción A, que se diferencie de la habitual según la cual la apelación constante de Kant a la representación pura del tiempo - y a las síntesis que lo hacen posible - es una consecuencia directa del carácter psicologista que la distingue. (shrink)

El artículo examina una de las contribuciones más importantes a los fundamentos de la geometría euclídea elemental lograda por David Hilbert en su obra Fundamentos de la geometría, a saber: la reconstrucción de la teoría euclídea de las proporciones y de los triángulos semejantes. Se argumenta que dicha reconstrucción no sólo estuvo motivada por la identificación de Hilbert de suposiciones implícitas en la teoría de Euclides, sino que además estuvo esencialmente ligada a la preocupación por la 'pureza del método'. Más (...) aún, se afirma que, en este caso específico, el requerimiento de Hilbert por la pureza del método posee un carácter general o fundacional, esto es, no se refiere a la demostración de un teorema en particular, sino más bien es planteado respecto de la construcción axiomática de la teoría misma. (shrink)

A theory of magnitudes involves criteria for their comparison, equivalence and addition. We examine these aspects from an abstract viewpoint, stressing independence and definability. These considerations are triggered by the so-called De Zolt’s principle in the theory of equivalence of plane polygons.