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)

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 This task is undertaken on the basis of his unpublished notes for lecture courses, corresponding to the period 1894–1905.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)

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)

The material contained in the following translation was given in substance by Professor Hilbertas a course of lectures on euclidean geometry at the University of G]ottingen during the wintersemester of 1898-1899. The results of his investigation were re-arranged and put into the formin which they appear here as a memorial address published in connection with the celebration atthe unveiling of the Gauss-Weber monument at G]ottingen, in June, 1899. In the French edition, which appeared soon after, Professor Hilbert made some additions, (...) particularly in the concludingremarks, where he gave an account of the results of a recent investigation made by Dr. Dehn.These additions have been incorporated in the following translation.Geometry, like arithmetic, requires for its logical development only a small number ofsimple, fundamental principles. These fundamental principles are called the axioms ofgeometry. The choice of the axioms and the investigation of their relations to one anotheris a problem which, since the time of Euclid, has been discussed in numerous excellentmemoirs to be found in the mathematical literature.1 This problem is tantamount to thelogical analysis of our intuition of space. (shrink)

En este artículo realizamos una reconstrucción del Programa original de Hilbert antes del surgimiento de los teoremas limitativos de la tercera década del siglo pasado. Para tal reconstrucción empezaremos por mostrar lo que Torretti llama los primeros titubeos formales de Hilbert, es decir, la defensa por el método axiomático como enfoque fundamentante. Seguidamente, mostraremos como estos titubeos formales se establecen como un verdadero programa de investigación lógico-matemático y como dentro de dicho programa la inquietud por la decidibilidad de los (...) problemas matemáticos y en específico la decidibilidad de la Lógica de primer orden cobra peso. Luego pasamos a analizar como la inquietud por la decibilidad toma lugar dentro del pensamiento filosófico-matemático de Hilbert presentándose como uno de los grandes problemas a los cuales la metamatemática debe encontrar una solución, esto lo hacemos mostrando un contraste con autores, como John von Neumann y Roberto Torretti, quienes de alguna u otra manera no interpretan el problema de la decidibilidad de la Lógica de primer orden como un problema de peso dentro del programa original de Hilbert. Finalmente argumentamos que el resultado meta-teórico de Church puede entenderse como una refutación del optimismo intelectual que permea a todo el programa original de Hilbert. (shrink)

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)

A investigação de um domínio amplo da realidade, como a evolução dosseres vivos, pode dar origem a diferentes “teorias”, cada uma consoantecom uma particular perspectiva que se considere. Para que se proceda auma análise detalhada dos pressupostos e conceitos que baseiam umadeterminada visão, o método axiomático parece ser a melhor das opções.Neste artigo, são discutidas algumas teorias da biologia evolutiva de umponto de vista axiomático, mostrando-se de que forma se pode apresentarum “predicado de Suppes” para a teoria sintética da (...) evolução, comodetalhado em outro local . Estaformulação é utilizada, dentre outras coisas, para se discutir a noção deaptidão darwiniana, bem como para explorar as relações entre a teoriadarwiniana da seleção natural e a genética de populações.The investigation of a wide field of knowledge, as evolution of living beings,may originate different “theories”, each one acting as a particularperspective we have about the domain. In order to proceed a philosophicalanalysis of the underlying assumptions of a particular view, it seems thatthe use of the axiomatic method is still the better way. Here, we discuss some theories of evolutionary biology from an axiomatic point of view, bymeans of a Suppes predicate for the synthetic theory of evolution, alreadydetailed in MAGALHÃES and KRAUSE . This formulation is hereused, among other things, to deal with the concept of Darwinian fitness, soas to explore the relationships between Darwinian theory and populationgenetics. (shrink)

We give a general account of the goals of axiomatization, introducing a variant on Detlefsen’s notion of ‘complete descriptive axiomatization’. We describe how distinctions between the Greek and modern view of number, magnitude, and proportion impact the interpretation of Hilbert’s axiomatization of geometry. We argue, as did Hilbert, that Euclid’s propositions concerning polygons, area, and similar triangles are derivable from Hilbert’s first-order axioms. We argue that Hilbert’s axioms including continuity show much more than the geometrical propositions of Euclid’s theorems and (...) thus are an immodest complete descriptive axiomatization of that subject. (shrink)

Es común escuchar que el mundo Occidental debe a los árabes el descubrimiento del álgebra. No obstante, el desarrollo de esta disciplina puede interpretarse como un crisol de distintas tradiciones científicas que fue posible gracias a la clasificación, traducción y crítica tanto de los clásicos como de las obras que los árabes obtuvieron de los pueblos que conquistaron. Entre estos trabajos se encontraba Los Elementos de Euclides. Los Elementos fueron cuidadosamente traducidos durante el califato de Al-Ma’mūn por el matemático Mohammed (...) ibn-Musa Al-Khwārizmī, autor de Al-jabr wa’l muqābalah, quien sentó los fundamentos de la disciplina que más tarde sería conocida como álgebra y quien, en la primera parte de su obra, nos proporciona tres métodos para resolver tres tipos de ecuaciones que llama “ecuaciones combinadas”. A lo largo de este artículo se ofrecen argumentos para sostener que los métodos para resolver estas ecuaciones constituyen una reinterpretación, en el terreno algebraico, de los teoremas 6, 7 y 8 del segundo libro de Los Elementos de Euclides. Mi objetivo es dar respuesta a la siguiente pregunta: ¿Qué lectura de Los elementos Euclides posibilitó la emergencia del álgebra en el mundo árabe? Para responderla necesario explorar el acercamiento que los árabes tuvieron con el Libro II de Los Elementos, con el fin de proponer una interpretación de los posibles factores que los llevaron a formular el álgebra. Esto último, en particular, se encuentra en la primera parte de la obra de Al-Khwārizmī. Palabras clave: Álgebra, Al-Khwārizmī, Euclides, Ecuación, Teorema. (shrink)

Un aspecto fundamental en la tradición matemática pitagórica, es la descripción de eventos musicales o acústicos, en términos de la Teoría de la Proporción. El artículo destaca sus aspectos fundamentales, su progreso en la concepción euclidiana y permite apreciar su peso filosófico, con el análisis filológico de algunas de sus más importantes definiciones.

Descriptions of musical or acoustic events in terms of Proportional Theory, are one of the most significant features in the Pythagorean mathematical tradition. This article emphasizes its fundamental aspects, its progress in the Euclidean conception and, with the Philological analysis of some of its more important definitions, allows appreciating its philosophical weight.

Dentro de la filosofía de la práctica matemática, la cuestión de la pureza del método ha ido ganando un lugar en las agendas de investigación y publicaciones. Usualmente se asume que “pura” es un predicado de soluciones o demostraciones, el cual resulta satisfecho cuando estas son intrínsecas a los problemas o teoremas. En este artículo motivo la adopción de una concepción más hospitalaria, de acuerdo con la cual la pureza del método emerge con naturalidad en la práctica de (...) construir teorías matemáticas autónomas, y muestro cómo a partir de esta la concepción usual se ve enriquecida. (shrink)

El objetivo de este trabajo es proponer un acercamiento metafísico al problema del método filosófico como proceso de reflexión y, con ello, enunciar que toda investigación filosófica se sustenta en la noción de relación como concepto metafísico absoluto. La tesis que sostengo señala, primero, que la relación es la expresión ontológica fundamental, y segundo, la noción de porción es el concepto mínimo de sentido en toda multiplicidad ontológica y todo criterio gnoseológico. Al asumirlas como premisas deduzco de ellas la (...) proporción (analogía) que hace posible establecer un vínculo congruente entre un mundo ontológico α y un criterio gnoseológico β apoyado en una lógica de magmas, indispensable para acudir a que llamo lógica gamma. (shrink)

Sobre a continuidade metodológica em Michel Foucault: da fundamentação de uma teoria do enunciado para o cuidado de si Resumo: Uma das dificuldades em se compreender a categoria de enunciado no pensamento de Michel Foucault está relacionada com múltiplas definições tautológicas dessa categoria no livro A arqueologia do saber. O primeiro objetivo deste artigo é oferecer uma descrição do enunciado e sua íntima dependência do nível referencial no pensamento arqueológico de Michel Foucault. O segundo objetivo, interrelacionado ao primeiro objetivo, será (...) mostrar o quanto as reflexões que desembocaram na concepção “cuidado de si”, vieram à tona porque o fim da análise arqueológica é a reconstituição das práticas concretas de uma época que definiram as condições do dizer e evidenciavam os modos de saber, pois o nível referencial é o nível de maior relevância para a reconstituição histórica das práticas discursivas por fontes discursivas. Esse método trouxe à luz, a ideia de que o trabalho fundamental da arte, de que temos de cuidar, a área principal à qual se deve aplicar valores estéticos, é a si próprio, à própria existência. Consequentemente, para esse pesquisador, não temos que escolher entre o nosso mundo e o mundo grego, desde que se possa ver que alguns dos princípios mais importantes de nossa ética têm sido relacionados num certo momento a uma estética da existência. Palavras-chave: Enunciado; Referencial; Formação dos objetos; Cuidado de si; Foucault. On the methodological continuity in Michel Foucault: from the foundation of a theory of the statement to the care of the self Abstract: One of the difficulties in understanding the category of enunciation in Michel Foucault's thought is related to multiple tautological definitions of this category in the book The archeology of knowledge. The first objective of this article is to offer a description of the statement and its intimate dependence on the referential level in Michel Foucault's archaeological thought. The second objective, interrelated to the first objective, will show how the reflections that led to the concept of “self-care”, came to light because the end of the archaeological analysis is the reconstitution of the concrete practices of an era that defined the conditions of saying and evidenced the ways of knowing, because the referential level is the most relevant level for the historical reconstruction of discursive practices by discursive sources. This method brought to light, the idea that the fundamental work of art, that we have to take care of, the main area to which aesthetic values ​​must be applied, is itself, to existence itself. Consequently, for this researcher, we do not have to choose between our world and the Greek world, as long as it can be seen that some of the most important principles of our ethics have been related at a certain moment to an aesthetics of existence. Key-words: Statement; Referential; Formation of objects; Self-care; Foucault. Sobre la continuidad metodológica en Michel Foucault: de la fundación de una teoría del enunciado al cuidado del yo Resumen: Una de las dificultades para comprender la categoría de enunciaciado en el pensamiento de Michel Foucault está relacionada con las múltiples definiciones tautológicas de esta categoría en el libro La arqueología del saber. El primer objetivo de este artículo es ofrecer una descripción del enunciado y su íntima dependencia del nivel referencial en el pensamiento arqueológico de Michel Foucault. El segundo objetivo, interrelacionado con el primero, será mostrar cómo las reflexiones que llevaron al concepto de “cuidado de sií”, salieron a la luz porque el fin del análisis arqueológico es la reconstitución de las prácticas concretas de una época que definió las condiciones del decir y evidenció las formas de conocer, porque el nivel referencial es el nivel más relevante para la reconstrucción histórica de prácticas discursivas por fuentes discursivas. Este método sacó a la luz la idea de que la obra de arte fundamental, que tenemos que cuidar, el ámbito principal al que se deben aplicar los valores estéticos, es ella misma, la existencia misma. En consecuencia, para este investigador, no tenemos que elegir entre nuestro mundo y el mundo griego, siempre y cuando se pueda constatar que algunos de los principios más importantes de nuestra ética se han relacionado en un momento determinado con una estética de la existencia. Palabra clave: Enunciado; Referencial; Formación de objetos; Cuidado de sí; Foucault. Data de registro: 24/02/2021 Data de aceite: 01/12/2021. (shrink)

It is commonplace to view the rigor of the mathematics in Euclid's Elements in the way an experienced teacher views the work of an earnest beginner: respectable relative to an early stage of development, but ultimately flawed. Given the close connection in content between Euclid's Elements and high-school geometry classes, this is understandable. Euclid, it seems, never realized what everyone who moves beyond elementary geometry into more advanced mathematics is now customarily taught: a fully rigorous proof cannot rely on geometric (...) intuition. In his arguments he seems to call illicitly upon our understanding of how objects like triangles and circles behave rather than grounding everything rigorously in axioms.Though widespread, the attitude is in a historical sense puzzling. For over two millenia, mathematicians of all levels studied the arguments in Elements and found nothing substantial missing. The book, on the contrary, represented the limit of mathematical explicitness. It served as the paradigm for careful and exact reasoning. How it could enjoy this reputation for so long is mysterious if careful and exact reasoning demands that all inferences be grounded in a modern axiomatic theory in the way Hilbert did in his famous Foundations of Geometry. By these standards, Euclid's work is deeply flawed. The holes in his arguments are not minor and excusable, but massive and cryptic.With his book Euclid and His Twentieth Century Rivals, Nathaniel Miller makes substantial progress in clearing this mystery up. The book is an explication of FG , a formal system of proof developed by Miller which reconstructs Euclid's deductions as essentially diagrammatic. The holes in Euclid's arguments are taken to appear precisely at those steps which are unintelligible without an accompanying geometric diagram. Interpreting the reasoning in the Elements in terms of a modern axiomatization , …. (shrink)

We will examine and discuss the nature and content of thedefinitions of the first part of Spinoza’s Ethics in relation to Euclid’s Elements.Our main purpose is to point out that the Spinozistic definitionsare distinct and discrepant from Euclidean definitions in some significantaspects. From this, we will point out and consider the differences andsimilarities that exist between the Euclidean geometric method and theSpinozistic geometric method, in order to problematize the theme of thepresence of the geometric method in Spinoza’s Ethics.

The present article provides a mereological analysis of Euclid’s planar geometry as presented in the first two books of his Elements. As a standard of comparison, a brief survey of the basic concepts of planar geometry formulated in a set-theoretic framework is given in Section 2. Section 3.2, then, develops the theories of incidence and order using a blend of mereology and convex geometry. Section 3.3 explains Euclid’s “megethology”, i.e., his theory of magnitudes. In Euclid’s system of geometry, megethology takes (...) over the role played by the theory of congruence in modern accounts of geometry. Mereology and megethology are connected by Euclid’s Axiom 5: “The whole is greater than the part.” Section 4 compares Euclid’s theory of polygonal area, based on his “Whole-Greater-Than-Part” principle, to the account provided by Hilbert in his Grundlagen der Geometrie. An hypothesis is set forth why modern treatments of geometry abandon Euclid’s Axiom 5. Finally, in Section 5, the adequacy of atomistic mereology as a framework for a formal reconstruction of Euclid’s system of geometry is discussed. (shrink)

Girolamo Saccheri (1667--1733) was an Italian Jesuit priest, scholastic philosopher, and mathematician. He earned a permanent place in the history of mathematics by discovering and rigorously deducing an elaborate chain of consequences of an axiom-set for what is now known as hyperbolic (or Lobachevskian) plane geometry. Reviewer's remarks: (1) On two pages of this book Saccheri refers to his previous and equally original book Logica demonstrativa (Turin, 1697) to which 14 of the 16 pages of the editor's "Introduction" are devoted. (...) At the time of the first edition, 1920, the editor was apparently not acquainted with the secondary literature on Logica demonstrativa which continued to grow in the period preceding the second edition \ref[see D. J. Struik, in Dictionary of scientific biography, Vol. 12, 55--57, Scribner's, New York, 1975]. Of special interest in this connection is a series of three articles by A. F. Emch [Scripta Math. 3 (1935), 51--60; Zbl 10, 386; ibid. 3 (1935), 143--152; Zbl 11, 193; ibid. 3 (1935), 221--333; Zbl 12, 98]. (2) It seems curious that modern writers believe that demonstration of the "nondeducibility" of the parallel postulate vindicates Euclid whereas at first Saccheri seems to have thought that demonstration of its "deducibility" is what would vindicate Euclid. Saccheri is perfectly clear in his commitment to the ancient (and now discredited) view that it is wrong to take as an "axiom" a proposition which is not a "primal verity", which is not "known through itself". So it would seem that Saccheri should think that he was convicting Euclid of error by deducing the parallel postulate. The resolution of this confusion is that Saccheri thought that he had proved, not merely that the parallel postulate was true, but that it was a "primal verity" and, thus, that Euclid was correct in taking it as an "axiom". As implausible as this claim about Saccheri may seem, the passage on p. 237, lines 3--15, seems to admit of no other interpretation. Indeed, Emch takes it this way. (3) As has been noted by many others, Saccheri was fascinated, if not obsessed, by what may be called "reflexive indirect deductions", indirect deductions which show that a conclusion follows from given premises by a chain of reasoning beginning with the given premises augmented by the denial of the desired conclusion and ending with the conclusion itself. It is obvious, of course, that this is simply a species of ordinary indirect deduction; a conclusion follows from given premises if a contradiction is deducible from those given premises augmented by the denial of the conclusion---and it is immaterial whether the contradiction involves one of the premises, the denial of the conclusion, or even, as often happens, intermediate propositions distinct from the given premises and the denial of the conclusion. Saccheri seemed to think that a proposition proved in this way was deduced from its own denial and, thus, that its denial was self-contradictory (p. 207). Inference from this mistake to the idea that propositions proved in this way are "primal verities" would involve yet another confusion. The reviewer gratefully acknowledges extensive communication with his former doctoral students J. Gasser and M. Scanlan. ADDED 14 March 14, 2015: (1) Wikipedia reports that many of Saccheri's ideas have a precedent in the 11th Century Persian polymath Omar Khayyám's Discussion of Difficulties in Euclid, a fact ignored in most Western sources until recently. It is unclear whether Saccheri had access to this work in translation, or developed his ideas independently. (2) This book is another exemplification of the huge difference between indirect deduction and indirect reduction. Indirect deduction requires making an assumption that is inconsistent with the premises previously adopted. This means that the reasoner must perform a certain mental act of assuming a certain proposition. It case the premises are all known truths, indirect deduction—which would then be indirect proof—requires the reasoner to assume a falsehood. This fact has been noted by several prominent mathematicians including Hardy, Hilbert, and Tarski. Indirect reduction requires no new assumption. Indirect reduction is simply a transformation of an argument in one form into another argument in a different form. In an indirect reduction one proposition in the old premise set is replaced by the contradictory opposite of the old conclusion and the new conclusion becomes the contradictory opposite of the replaced premise. Roughly and schematically, P,Q/R becomes P,~R/~Q or ~R, Q/~P. Saccheri’s work involved indirect deduction not indirect reduction. (3) The distinction between indirect deduction and indirect reduction has largely slipped through the cracks, the cracks between medieval-oriented logic and modern-oriented logic. The medievalists have a heavy investment in reduction and, though they have heard of deduction, they think that deduction is a form of reduction, or vice versa, or in some cases they think that the word ‘deduction’ is the modern way of referring to reduction. The modernists have no interest in reduction, i.e. in the process of transforming one argument into another having exactly the same number of premises. Modern logicians, like Aristotle, are concerned with deducing a single proposition from a set of propositions. Some focus on deducing a single proposition from the null set—something difficult to relate to reduction. (shrink)

Este escrito da cuenta de un recorrido de lecturas. Se han seleccionado algunos textos de Arturo Roig con la preocupación de responder interrogantes vinculados con las herramientas hermenéuticas y metodológicas propias del autor en su labor como filósofo e historiador de las ideas. Si se ha determinado la posición de Roig frente al análisis del discurso, sus estudios y su crítica al estructuralismo, es para comprender una metodología que no se queda en el análisis formal de los discursos, sino que (...) atiende al carácter histórico y situado de los mismos. También permite identificar e interrogar sobre las relaciones entre lenguaje y sujeto, discurso e ideología, y la historicidad en el pensamiento latinoamericano mediante el abordaje de una historia episódica, antípodas de una idea de continuidad y despliegue lineal. Esto permite comprender el vínculo con una antropología de la emergencia, cuyo carácter final es el de ponerse a sí mismo como valioso.This article presents an overview of several readings. Some of Arturo Roig's writings have been selected with the aim of answering questions related to the hermeneutical and methodological tools proper to the author in his work as philosopher and historian of ideas. If Roig's position facing the analysis of discourse, his studies and his criticism of structuralism has been determined in this article, it is simply to understand a methodology that does not end with the formal analysis of discourses but that rather considers the historical settings of such discourses. The approach also allows for an identification and interrogation of the relationship between language and subject, discourse and ideology, and historicity in Latin American thought, through an episodical history, the antipodes of an idea of continuity and lineal course. This makes it possible to understand the bond with an anthropology of the emergency, whose final character is to place itself as valuable. (shrink)

Este ensaio introdutório faz uma breve apresentação do tratado de óptica atribuído a Euclides de Alexandria, inserindo-o no contexto das teorias sobre a visão formuladas pelas doutrinas filosóficas antigas. Ressalta-se o antagonismo entre a análise geométrica da visão, empreendida por Euclides, e as considerações filosóficas acerca dos processos físicos subjacentes à sensação visual. Pretende-se mostrar que o objeto da óptica euclidiana é a percepção visual daquilo que Aristóteles denomina "sensível comum". This introductory essay provides an abridged presentation of the optical (...) treatise attributed to Euclid of Alexandria, placing it in the context of theories about vision formulated by the ancient philosophical doctrines. I emphasize the antagonism between the geometric analysis of vision, undertaken by Euclid, and the philosophical considerations about the physical processes underlying visual sensation. In addition, I aim to show that the object of Euclidean optics is the visual perception of what Aristotle calls "common sensible". (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)

This very interesting and extremely useful study raises the question, by virtue of its title and what it does not do, of what is, or ought to be, meant by the philosophy of mathematics. The author begins his study of Euclid with a brief discussion of Hilbert's axiomatization of geometry. The two main points in this discussion are: "Hilbertian geometry and many other parts of modern mathematics are the study of structure", i.e., of the interpretations of axiom-systems; and intuition of (...) geometrical form becomes irrelevant to the content of mathematics. Greek mathematics, and specifically Euclid, cannot be understood in these terms; hence, Euclid's use of the axiomatic method differs from ours. One may also say that logic, "the theory of structure-preserving inference", becomes central to modern mathematics, as it was not for Euclid. But, the author continues, it does not follow that one cannot exhibit the logical structure employed by Euclid. The bulk of Mueller's book is concerned to do just this, thanks to a formalization of Euclid which attempts to be faithful to his principal concepts, while at the same time employing contemporary logical techniques. (shrink)

This very interesting and extremely useful study raises the question, by virtue of its title and what it does not do, of what is, or ought to be, meant by the philosophy of mathematics. The author begins his study of Euclid with a brief discussion of Hilbert's axiomatization of geometry. The two main points in this discussion are: "Hilbertian geometry and many other parts of modern mathematics are the study of structure", i.e., of the interpretations of axiom-systems; and intuition of (...) geometrical form becomes irrelevant to the content of mathematics. Greek mathematics, and specifically Euclid, cannot be understood in these terms; hence, Euclid's use of the axiomatic method differs from ours. One may also say that logic, "the theory of structure-preserving inference", becomes central to modern mathematics, as it was not for Euclid. But, the author continues, it does not follow that one cannot exhibit the logical structure employed by Euclid. The bulk of Mueller's book is concerned to do just this, thanks to a formalization of Euclid which attempts to be faithful to his principal concepts, while at the same time employing contemporary logical techniques. (shrink)

Colour has often been supposed to be a subjective property, a property to be analysed orretly in terms of the phenomenological aspects of human expereince. In contrast with subjectivism, an objectivist analysis of color takes color to be a property objects possess in themselves, independently of the character of human perceptual expereince. David Hilbert defends a form of objectivism that identifies color with a physical property of surfaces - their spectral reflectance. This analysis of color is shown to provide a (...) more adequate account of the features of human color vision than its subjectivist rivals. The author's account of colro also recognises that the human perceptual system provides a limited and idiosyncratic picture of the world. These limitations are shown to be consistent with a realist account of colour and to provide the necessary tools for giving an analysis of common sense knowledge of color phenomena. (shrink)

Die Leitgedanken meiner Untersuchungen über die Grundlagen der Mathematik, die ich - anknüpfend an frühere Ansätze - seit 1917 in Besprechungen mit P. BERNAYS wieder aufgenommen habe, sind von mir an verschiedenen Stellen eingehend dargelegt worden. Diesen Untersuchungen, an denen auch W. ACKERMANN beteiligt ist, haben sich seither noch verschiedene Mathematiker angeschlossen. Der hier in seinem ersten Teil vorliegende, von BERNAYS abgefaßte und noch fortzusetzende Lehrgang bezweckt eine Darstellung der Theorie nach ihren heutigen Ergebnissen. Dieser Ergebnisstand weist zugleich die Richtung (...) für die weitere Forschung in der Beweistheorie auf das Endziel hin, unsere üblichen Methoden der Mathematik samt und sonders als widerspruchsfrei zu erkennen. Im Hinblick auf dieses Ziel möchte ich hervorheben, daß die zeit weilig aufgekommene Meinung, aus gewissen neueren Ergebnissen von GÖDEL folge die Undurchführbarkeit meiner Beweistheorie, als irrtüm lich erwiesen ist. Jenes Ergebnis zeigt in der Tat auch nur, daß man für die weitergehenden Widerspruchsfreiheitsbeweise den finiten Stand punkt in einer schärferen Weise ausnutzen muß, als dieses bei der Be trachtung der elementaren Formallsmen erforderlich ist. Göttingen, im März 1934 HILBERT Vorwort zur ersten Auflage Eine Darstellung der Beweistheorie, welche aus dem HILBERTschen Ansatz zur Behandlung der mathematisch-logischen Grundlagenpro bleme erwachsen ist, wurde schon seit längerem von HILBERT ange kündigt. (shrink)

Dieser Buchtitel ist Teil des Digitalisierungsprojekts Springer Book Archives mit Publikationen, die seit den Anfängen des Verlags von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv Quellen für die historische wie auch die disziplingeschichtliche Forschung zur Verfügung, die jeweils im historischen Kontext betrachtet werden müssen. Dieser Titel erschien in der Zeit vor 1945 und wird daher in seiner zeittypischen politisch-ideologischen Ausrichtung vom Verlag nicht beworben.

En este artículo se esboza la relación entre la libertad y la moral política en el pensamiento de Joseph Ratzinger. Para ello, se exponen los fundamentos teológicos y antropológicos que justifican el concepto de libertad y el significado de la política como ámbito de la razón moral. Por último, se propone que un adecuado concepto de libertad, cuyo presupuesto es la verdad, facilita la comprensión y el desarrollo de la actividad política a partir de su naturaleza moral.

En este artículo se estudian los escritos de Joseph Ratzinger en los que busca la mutua imbricación de la vida política con otros saberes y hace propuestas de solución para las aporías actuales. En el primer apartado se analiza cómo el planteamiento cristiano presupone que el componente social y político forma parte de la esencia humana y cómo introdujo el cristianismo un sano dualismo en las relaciones entre Estado e Iglesia. En el segundo, se investiga cuáles fueron los motivos que (...) llevaron a rechazar el fundamento sacro de la historia y del Estado. En el tercer punto se exponen las divisiones que siguieron al giro moderno: el laicismo, las Iglesias estatales, y los diversos socialismos, democráticos o totalitarios. Más adelante se estudia cómo se pasó de las guerras del siglo XX a lo que Ratzinger denomina “la dictadura global”. Finalmente se justifica por qué razón la justicia y la ética son factores claves en la búsqueda de la paz. (shrink)

O objetivo deste artigo é diluir o excesso de obscuridade presente nas análises filosóficas contemporâneas da música, e o faremos com base em uma abordagem linguísticoanalítica, a qual, como será defendido, promove uma explicação do fenômeno musical, isto é, da sua constituição, valor à sociedade e prazer sensível/mental, de uma maneira clara e objetiva enquanto um fenômeno já muito conhecido e difundido, que é o da comunicação.

RESUMO Apesar de terem sido estudos realizados por autores diferentes e de certa forma independente um do outro, as teorias das Implicaturas de Paul Grice e dos Atos de Discurso Indiretos de John Searle não apenas estão intimamente ligadas como, na verdade, são complementares entre si, de modo que se mostra de crucial importância a compreensão dos dois juntos, a fim de se obter um maior domínio da noção de “não literalidade” das enunciações. Este trabalho tratará de mostrar as principais (...) características de ambas as teorias e como elas se inter-relacionam orgânica e simbioticamente. (shrink)

The world as perceived by human beings is full of colour. The world as described by physical scientists is composed of colourless particles and fields. Philosophical theories of colour since the scientific revolution have been primarily driven by a desire to harmonize these two apparently conflicting pictures of the world. Any adequate theory of colour has to be consistent with the characteristics of colour as perceived without contradicting the deliverances of the physical sciences. Given this conception of the aim of (...) a theory of colour, there are three possibilities for resolving the apparent conflict between the scientific and perceptual facts. The first possibility is to deny that physical objects have colours. Theories of this kind admit that objects appear coloured but maintain that these appearances are misleading. The conflict is resolved by removing colour from the external world. Second, it might be that colour is a relational property. For an object to possess a particular colour it must be related in the right way to a perceiver. One common version of this view analyzes colour as a disposition to cause particular kinds of perceptual experiences in a human being. Since the physical sciences deal only with the intrinsic properties of physical objects and their relations to other physical objects and not their relations to perceiving subjects, the possibility of conflict is removed. A third possible response to the conflict is to maintain that colour really is a property of external objects and that the conflict is merely apparent. Some theories of this form maintain that colour is identical to a physical property of objects. Others maintain that colour is a property that physical objects possess over and above all their physical <blockquote> 1 </blockquote> properties. Philosophical discussions of colour typically take the form of either elaborating on one of these three possibilities or attempting to show more generally that one of these three types of responses is to be preferred to the others.. (shrink)