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)

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)

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.

Economic explanations of inequality in liberal economies often appeal to social norms. This article argues that recent economic policies and institutional interventions, in favor of deregulation and away from participatory practices in both the political and economic spheres, have elicited norms that increasingly resemble those of the economic theory that has informed these interventions. This has altered the social ethos in an inegalitarian direction—individuals are more likely to conceive of themselves as acquisitive maximizers in their daily life, and there has (...) been a reduction in the number of institutional settings for people to deliberate on their values, commitments, and other priorities. (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.

The work of Ludwig Wittgenstein called Tractatus Logico-Philosophicus, at the same time that represents a great contribution to a change of conception concerning the language and it’s capacities to deal with the reality and a comprehension of the external world, brings within itself a series of argumentative/conceptual problems as well, which seem that have got out of author’s control, revealing a polemic work with a strong presence in the debates about Philosophy Of Language and Ontology. The dogmatic aspect, compounding a (...) work grounding on aphorisms, brings a reflection almost entirely subjective and internal to a new system peculiar to Wittgenstein, generating certainly many doubts about the power of it to establish a public consensus. Putting this aside, it is also convenient to notice that the author himself, when decides to deal with the definition of sensible concepts of the Philosophy, turns out entering in contradictions with his own words many times, revealing, maybe, why he had had abandoned these text’s ideas to assume a more pragmatical posture in his later works. We will see how and where some of these contradictions occur, making as well a dialog with another authors in order to have a more diverse comprehension of the terms treated. (shrink)

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)

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)

O artigo visa discutir no campo educacional a presença das tecnologias e suas influencias quanto à crise de desempenho do indivíduo dentro da sociedade, analisar como a escola e a educação passaram a lidar com a propagação de informações por parte dos alunos, e compreender os impactos advindos desse isolamento social provocado pela Pandemia da COVID-19. Com base em autores como Foucault (2009), Han (2018), França (2018) e Serres (2013). Concluindo que se deve ter criticidade ao utilizar esses recursos, ponderando (...) o que se busca a partir deles, sendo que os mesmos podem ampliar como também contribuir para a limitação e submissão do sujeito. (shrink)

Artificial intelligence is a current tool that is used in different areas of the human life with the purpose of facilitating and directing the processes in distinct sceneries of society. This research aimed to use AI as an educational resource in teaching and learning process in the area of biology for students at the Adventist School CEA Cambulos in the city of Cali. A qualitative research methodology was used, with data collection techniques such as observation and interviews, presenting an action (...) research approach. The main results were the potential of AI to enhance learning, the disposition of students to learn through it, and the need for training in the educational sector for its inclusive use and achieving responsible and effective results. That is how it is considered the challenge of reflecting the imminent arrival of AI in the student context and being prepared to be part of the transformation that can be generated in pedagogical processes. (shrink)

Relying upon a very close reading of all of the definitions given in Euclid’s Elements, I argue that this mathematical treatise contains a philosophical treatment of mathematical objects. Specifically, I show that Euclid draws elaborate metaphysical distinctions between substances and non-substantial attributes of substances, different kinds of substance, and different kinds of non-substance. While the general metaphysical theory adopted in the Elements resembles that of Aristotle in many respects, Euclid does not employ Aristotle’s terminology, or indeed, any (...) philosophical terminology at all. Instead, Euclid systematically uses different types of definition to distinguish between metaphysically different kinds of mathematical object. (shrink)

El objetivo de este trabajo es mostrar la influencia socrática sobre la filosofía de Euclides de Mégara, en contra de la interpretación que señala la influencia de los eleáticos sobre su teoría. Para ello indicaré que la doctrina de Euclides exhibe una fuerte impronta socrática, al menos en lo que concierne a: su labor como escritor de diálogos socráticos, el uso de la dialéctica erística, el desarrollo de algunos postulados éticos sobre la conducta frente a la muerte, el auto-dominio y (...) la unidad de la virtud, y la presentación de una metafísica que coloca al bien en un lugar privilegiado. Aunque esta metafísica excede el socratismo, no puede ser identificada con los postulados de la filosofía eleática. (shrink)

In this system, the properties of space were believed to be in accord with the geometry of Euclid ; and one might have expected that the correctness of the ...

Twentieth-century developments in logic and mathematics have led many people to view Euclid’s proofs as inherently informal, especially due to the use of diagrams in proofs. In _Euclid and His Twentieth-Century Rivals_, Nathaniel Miller discusses the history of diagrams in Euclidean Geometry, develops a formal system for working with them, and concludes that they can indeed be used rigorously. Miller also introduces a diagrammatic computer proof system, based on this formal system. This volume will be of interest to mathematicians, (...) computer scientists, and anyone interested in the use of diagrams in geometry. (shrink)

The lack of specific arithmetical axioms in Book VII has puzzled historians of mathematics. It is hardly possible in our view to ascribe to the Greeks a conscious undertaking to axiomatize arithmetic. The view that associates the beginnings of the axiomatization of arithmetic with the works of Grassman [1861], Dedekind [1888] and Peano [1889] seems to be more plausible. In this connection a number of interesting historical problems have been raised, for instance, why arithmetic was axiomatized so late. This question (...) was first posed and quite conclusively answered by Yanovskaja [1956]. Her major and quite conclusive argument was that “algorithms in arithmetic have absolute character, while in geometry we have to do with reducibility algorithms”. In this paper we are going to draw attention on certain peculiarities of the construction of the arithmetical Books of Euclid’s "Elements" and show that Euclidean arithmetic is constructed by effective procedures. In spite of the use of inference by reductio ad absurdum and methods equivalent to mathematical induction, Euclidean arithmetic retains its finitary character. It is not full arithmetic, but a finitary fragment of classical arithmetic, and thereby there was not any internal reason for its axiomatization. (shrink)

ArgumentThe Fourth Postulate of Euclid’s Elements states that all right angles are equal. This principle has always been considered problematic in the deductive economy of the treatise, and even the ancient interpreters were confused about its mathematical role and its epistemological status. The present essay reconsiders the ancient testimonies on the Fourth Postulate, showing that there is no certain evidence for its authenticity, nor for its spuriousness. The paper also considers modern mathematical interpretations of this postulate, pointing out various (...) anachronisms. It further discusses the validity of the ancient proof by superposition of the Fourth Postulate. Finally, the article proposes an interpretation of the history of the concept of angle in Greek geometry between Euclid and Apollonius, and puts forward a conjecture on the interpolation of the Fourth Postulate in the Hellenistic age. The essay contributes to a general reassessment of the axiomatic foundations of ancient mathematics. (shrink)

The “common notions” prefacing the Elements of Euclid are a very peculiar set of axioms, and their authenticity, as well as their actual role in the demonstrations, have been object of debate. In the first part of this essay, I offer a survey of the evidence for the authenticity of the common notions, and conclude that only three of them are likely to have been in place at the times of Euclid, whereas others were added in Late Antiquity. (...) In the second part of the essay, I consider the meaning and uses of the common notions in Greek mathematics, and argue that they were originally conceived in order to axiomatize a theory of equivalence in geometry. I also claim that two interpolated common notions responded to different epistemic needs and regulated diagrammatic inferences. (shrink)

In his well-known paper on Euclid’s geometry, Ken Manders sketches an argument against conceiving the diagrams of the Elements in ‘semantic’ terms, that is, against treating them as representations—resting his case on Euclid’s striking use of ‘impossible’ diagrams in some proofs by contradiction. This paper spells out, clarifies and assesses Manders’s argument, showing that it only succeeds against a particular semantic view of diagrams and can be evaded by adopting others, but arguing that Manders nevertheless makes a compelling (...) case that semantic analyses ought to be relegated to a secondary role for the study of mathematical practices. (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)

The work of Euclid is situated in the Aristotelian rather than the Platonic perspective, although it incorporates the methodological results of both. The Alexandrian mathematician stamped with an Aristotelian seal the whole system of Greek mathematics.

Among the writings devoted to the reflection of “visual” and solar rays on various mirrors, there are two that preceded many others and that occupy a central position in the history of catoptrics: one is attributed to Euclid, the other to Ptolemy. To these two names, we add a third, hitherto unknown, Thiasos. In this article, we take up the textual and conceptual history of this catoptric research tradition.

It is shown how a diagram on the reverse of a Greek coin of Aegina of the fifth century b.c.e., is simply constructed with the help of Proposition 36 of Euclid’s Book on Divisions [of Figures], and it is conjectured in the absence of contemporary evidence that, since Euclid expressly designated this proposition to be the last in the book, he may have had in mind the diagram, which, some 200 years after its appearance on the coinage, may (...) still have played a significant role in geometry. (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)

Our aim is to propose several itineraries which follow the scenario having as a first step Euclid’s Fifth Postulate; as a second step the Bolyai–Lobachevsky’s non-Euclidean geometries and as a third step Einstein’s relativity theory. The role of Euclid’s fifth postulate is successively assumed by Archimedes’ axiom; Zermelo’s choice axiom; Cantor’s continuum hypothesis; von Neumann’s foundation axiom for set theory; Church–Turing thesis and Turing’s computability; the validity of classical logic under the form of the principles of identity, non-contradiction (...) and excluded middle; the macroscopic universe assumed as form of existence, including the fact that humans are macroscopic beings; human language and semiosis assumed as the unique way to approach understanding and communication; the old Greek ideal of order, symmetry and harmony, adopted for representation of the world in Art and Poetry. The common denominator of these starting steps is the fact that they all correspond to the sensorial-empirical perception of the world. We point out the strong interaction among the steps corresponding, in these different itineraries, to Einstein’s relativity theory. We infer in this way the need of a transdisciplinary approach. (shrink)

Philosophy in the period immediately after Aristotle is sometimes thought to be marked by the decline of natural philosophy and philosophical disinterest in contemporary achievements in the sciences. But in one area at least, the early third century B.C.E. was a time of productive interaction between such disparate fields as epistemology, physics and geometry. Debates between the sceptics and the dogmatic philosophical schools focus on epistemological problems about the possibility of self-evident appearances, but there is evidence from Euclid's day (...) of a quite different response. The sceptical challenge provoked the development of theories explaining error formation, showing how illusions can be studied systematically and are subject to prediction. Such theories do not legitimate claims about the nature of the underlying entities perceived, but provide justification for forming expectations about future perceptions. While it overtly focuses on purely geometrical considerations, the Euclidean model of optics nonetheless provides support for certain views about the nature of vision and the physics of light. Moreover, by offering a model in which the image received is not thought to be a perspicuous mirroring of the object seen, Euclid may have helped promote a view of perception as something reconstructed from information received, not as a mere form transferred into the eye. The ancient sceptic may indeed have fulfilled his promise to promote inquiry by focusing attention on problems that escape the attention of a hasty theorist. (shrink)

J. H. Lambert proved important results of what we now think of as non-Euclidean geometries, and gave examples of surfaces satisfying their theorems. I use his philosophical views to explain why he did not think the certainty of Euclidean geometry was threatened by the development of what we regard as alternatives to it. Lambert holds that theories other than Euclid's fall prey to skeptical doubt. So despite their satisfiability, for him these theories are not equal to Euclid's in (...) justification. Contrary to recent interpretations, then, Lambert does not conceive of mathematical justification as semantic. According to Lambert, Euclid overcomes doubt by means of postulates. Euclid's theory thus owes its justification not to the existence of the surfaces that satisfy it, but to the postulates according to which these "models" are constructed. To understand Lambert's view of postulates and the doubt they answer, I examine his criticism of Christian Wolff's views. I argue that Lambert's view reflects insight into traditional mathematical practice and has value as a foil for contemporary, model-theoretic, views of justification. (shrink)

Frege’s argument against the ancient Greek conception of numbers as 'multitudes of units’ has been hailed as one of the most successful in his "Grundlagen". The aim of this paper is to show that despite Frege’s best efforts, the Euclidean conception remains a viable alternative to the Fregean conception of numbers by arguing that neither a dilemma argument Frege brings against the Euclidean conception nor a possible argument against it based on the truth of what is known as "Hume’s Principle" (...) succeeds. (shrink)

The Book I of Euclid’s Elements begins with three propositiones that ask for the solution of three problems. Unlike other propositiones, these do not assert properties or relationships between geometric objects. In them, some of these objects are assumed as given, and actions are demanded in order to obtain other objects. The solution to this type of problem is a construction, and its foundations can be found in the definitions and postulates of the Geometry of Euclid. Furthermore, each (...) construction is normally accompanied by the resolution of a second problem. This consists in demonstrating that the previously offered solution is correct, i.e., that the presented construction indeed produces what was demanded in the formulation of the initial problem. The first theorem of the Elements is exactly the fourth proposition. We will consider, in a propaedeutic manner, the implications of adopting an approach to geometry where all propositions are interpreted as problems and the solution to any specific construction problem consists in offering a procedure such that, globally, the execution of this procedure on a workspace by a geometer will be simulated by an Euclid machine—which is the central object of analysis of the present article. From this perspective, the Geometry of Euclid would admit an interpretation in which it would be, in addition to a foundational milestone in the history of mathematics, also a foundational milestone in the development of the notion of effective procedure that would only fully come to light in the twentieth century. (shrink)

Euclid in the Rainforest combines the literary with the mathematical to explore logic--the one indispensable tool in man's quest to understand the world. Mazur argues that logical reasoning is not purely robotic. At its most basic level, it is a creative process guided by our intuitions and beliefs about the world.

Book 1 of Euclid's Elements opens with a set of unproved assumptions: definitions, postulates, and ‘common notions’. The common notions are general rules validating deductions that involve the relations of equality and congruence. The attested postulates are five in number, even if a part of the manuscript tradition adds a sixth, almost surely spurious, that in some manuscripts features as the ninth, and last, common notion. The postulates are called αἰτήματα both in the manuscripts of the Elements and in (...) the ancient exegetic tradition. It is not said, however, that this denomination is original, as it coincides with the nomen rei actae associated with the verbal form introducing the postulates themselves. Since antiquity it has been recognized that the postulates naturally split into two groups of quite different character: the first three are rules licensing basic constructions, the fourth and the fifth are assertions stating properties of particular geometric objects. I transcribe postulates 1–3 in the text established by Heiberg : ᾐτήσθω ἀπὸ παντὸς σημείου ἐπὶ πᾶν σημεῖον εὐθεῖαν γραμμὴν ἀγαγεῖνκαὶ πεπερασμένην εὐθεῖαν κατὰ τὸ συνεχὲς ἐπ᾽ εὐθείας ἐκβαλεῖνκαὶ παντὶ κέντρῳ καὶ διαστήματι κύκλον γράϕεσθαι. (shrink)

The issue is obscured by the fact that the word `space' can be used in four different ways. It can be used, first, as a term of pure mathematics, as when mathematicians talk of an `n-dimensional phase-space', an `n-dimensional vector-space', a `three-dimensional projective space' or a `twodimensional Riemannian space'. In this sense the word `space' means the totality of the abstract entities-the `points'-implicitly defined by the axioms. There is no doubt that there exist, iii this sense, non-Euclidean spaces, because all (...) that is claimed by such an assertion is that sets of non-Euclidean axioms constitute possible implicit definitions of abstract entities, that is to say that some sets of non-Euclidean axioms are consistent. If Kant or any other philosopher had denied this, he would have been wrong; but Kant himself took care not to deny it, 2 and there is little reason to suppose that any philosopher concerned about space has been using the word in this, the pure mathematician's, sense. (shrink)

This paper seeks to demonstrate that propositions 23–27 of the Euclidian Optics originated in the context of geometrical astronomy. These entries, which deal with the geometry of spheres and rays, present material that overlaps considerably with propositions 1–3 of Aristarchus of Samos’ On the Sizes and Distances of the Sun and the Moon. While all these theorems deal with material that could conceivably be native to celestial illumination, the proofs do not work for binocular vision. It therefore seems probable that (...) the proofs were borrowed from Aristarchus or, more likely, some common astronomical predecessor. As an extension of this observation, this paper argues that the Optics displays a far less unified purpose than has typically been assumed. Instead, it reflects a conglomerate of concerns, goals and dependencies, insofar as it presents multiple proofs that emerge from a set of geometrical concerns not directly related to vision as a physical phenomenon. These entries complicate the idea that the Optics is about explaining light, sight or false appearances in any straightforward sense. Instead, the text is oriented toward a somewhat more diffuse goal: simply to articulate the geometry of vision and rays. Optics as a discipline should should therefore be understood not as a purely mathematical explanation of vision, but as a target-field of explanation constituted by features and boundaries derived from extra-visual geometric concerns. (shrink)

We present a formal system, E, which provides a faithful model of the proofs in Euclid's Elements, including the use of diagrammatic reasoning.