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)

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)

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)

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)

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 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)

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)

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)

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.

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)

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)

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 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 ...

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)

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)

Formalizing Euclid’s first axiom. Bulletin of Symbolic Logic. 20 (2014) 404–5. (Coauthor: Daniel Novotný) -/- Euclid [fl. 300 BCE] divides his basic principles into what came to be called ‘postulates’ and ‘axioms’—two words that are synonyms today but which are commonly used to translate Greek words meant by Euclid as contrasting terms. -/- Euclid’s postulates are specifically geometric: they concern geometric magnitudes, shapes, figures, etc.—nothing else. The first: “to draw a line from any point to any (...) point”; the last: the parallel postulate. -/- Euclid’s axioms are general principles of magnitude: they concern geometric magnitudes and magnitudes of other kinds as well even numbers. The first is often translated “Things that equal the same thing equal one another”. -/- There are other differences that are or might become important. -/- Aristotle [fl. 350 BCE] meticulously separated his basic principles [archai, singular archê] according to subject matter: geometrical, arithmetic, astronomical, etc. However, he made no distinction that can be assimilated to Euclid’s postulate/axiom distinction. -/- Today we divide basic principles into non-logical [topic-specific] and logical [topic-neutral] but this too is not the same as Euclid’s. In this regard it is important to be cognizant of the difference between equality and identity—a distinction often crudely ignored by modern logicians. Tarski is a rare exception. The four angles of a rectangle are equal to—not identical to—one another; the size of one angle of a rectangle is identical to the size of any other of its angles. No two angles are identical to each other. -/- The sentence ‘Things that equal the same thing equal one another’ contains no occurrence of the word ‘magnitude’. This paper considers the problem of formalizing the proposition Euclid intended as a principle of magnitudes while being faithful to the logical form and to its information content. (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.

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)

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)

There are styles in thinking just as there are in dress. Their justification is often as slight, and once a style in thought has been established, it is followed with the same unconscious readiness. The possibility of styles lies, of course, in a lack of fixed criterions. In dress, the choice to a large extent is a matter of taste, and because taste varies from individual to individual and from time to time, styles also change. Although the foundations of science (...) are far more stable than those of taste, they yet have some touch of uncertainty. We live in a relative world, not an absolute one, and thus there is always room for the point of view to shift. We like to feel that we ourselves and the world in general are growing wiser, and thus that while there were errors in the thinking of the past, they are now being rapidly eliminated. The discovery that some philosopher of several thousand years ago held almost our identical views is always disconcerting. (shrink)

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)

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.

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)

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.