ABSTRACT Humor, especially in jokes, involves breaking rules of reason and social norms, such as moral, cultural, and religious standards. This transgression can produce pejoratives that insult or slander individuals or groups by exposing aspects of human life usually hidden to maintain social harmony. When the joke teller is unaffected, they and their audience share the pleasure of laughter and ridicule directed at the joke’s targets. Thus, jokes can express a form of superiority over others based on physical, cultural, or (...) behavioral traits. This study aims to connect the main philosophical theory of humor with the pragmatic theory of pejoratives, showing that the speech acts of joking and insulting overlap in certain pragmatic contexts, as it is not the words themselves that offend, but the agents who use them. (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)

RESUMO O humor, especialmente presente nas piadas, envolve a quebra de regras da razão e das normas sociais, como as morais, culturais e religiosas. Essa transgressão pode resultar em pejorativos que insultam ou caluniam indivíduos ou grupos, ao revelar aspectos da vida humana geralmente ocultos para preservar a convivência social. Quando o narrador da piada não se sente afetado, ele e seus ouvintes compartilham o prazer do riso e do escárnio contra os alvos da piada. Assim, as piadas podem expressar (...) uma relação de superioridade sobre outros, baseada em características físicas, culturais ou comportamentais. Este estudo busca articular a principal teoria filosófica do humor com a teoria pragmática dos pejorativos, evidenciando que os atos de discurso de contar piadas e de insultar se confundem em determinados contextos pragmáticos, pois o que ofende não são as palavras, mas os agentes que as empregam. (shrink)

Joseph Ratzinger desarrolla su teología sobre la Iglesia a partir de las imágenes “Cuerpo de Cristo” y “Pueblo de Dios”, resaltadas por el Concilio Vaticano II. La complementariedad entre ambas nociones es un aspecto clave de su eclesiología, en la que destacan las dimensiones eucarística y teológica. La Iglesia tiene como tarea asemejarse a Cristo y ser signo de la presencia de Dios en el mundo. En este artículo se busca presentar la eclesiología del autor alemán y precisar si es (...) posible aplicar sus planteamientos a la realidad de Latinoamérica. Se concluye que la sacramentalidad es un rasgo esencial de la Iglesia como signo de la presencia de la Trinidad en el mundo, que la misión bautismal conduce también a la fraternidad, con énfasis en la opción preferencial por los pobres y que sus propuestas son pertinentes para la misión de la Iglesia como testimonio de la esperanza en Cristo para el continente americano. (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 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)

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)

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)

_This paper argues for a novel thesis in philosophy of language and philosophy of religion: that language itself renders impossible any genuine encounter with, description of, or reference to infinite, perfect, or omnipotent beings. Unlike traditional arguments against divine attributes that focus on logical contradictions between properties (omnipotence paradoxes, problem of evil), this paper demonstrates that the very act of linguistic engagement with the concept of the divine necessarily corrupts and limits any absolute being. Drawing on insights from Wittgenstein's philosophy (...) of language, Derrida's deconstruction, and original analysis of the performative contradictions inherent in theological discourse, we establish that the linguistic prerequisite for conflict makes any "speakable paradise" logically impossible. This constitutes an original contribution to both negative theology and philosophy of language, providing a new foundation for understanding why mystical traditions consistently emphasize the ineffability of the divine._. (shrink)

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 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)

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.

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)

El objetivo de este artículo es desarrollar el concepto de «racismo del anfitrión», con el fin de comprender cómo los niños blancos aprenden a ser racistas. Esta categoría conceptual pretende esclarecer la dinámica racial en el ámbito de la infancia blanca, a menudo ignorada por los estudios étnico-raciales, lo que contribuye a la reproducción de la idea de que la blancura representa una condición neutra, universal y exenta de análisis. Este trabajo resulta necesario porque persiste la creencia generalizada de que (...) no existe racismo entre los niños y de que los niños blancos no pueden ser racistas, lo que contradice las investigaciones que demuestran la reproducción de prácticas racistas desde la infancia. Se trata de una investigación bibliográfica, basada en el análisis temático de la literatura relativa a los conceptos de racismo en la infancia. Se concluye que el niño blanco no nace racista. Sin embargo, al nacer en una sociedad marcada por el racismo, el cuerpo blanco se convierte en huésped de ese virus. De este modo, el concepto de «racismo huésped» ha demostrado ser adecuado para aclarar que es una sociedad estructuralmente racista la que desarrolla el potencial de ser racista que lleva consigo el niño blanco. (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)

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 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)

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)

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)

Euclides administra dos modalidades comunicativas en la exposición de sus demostraciones: una es detallada, minuciosa, la otra más resumida o esbozada o sumaria. Existen pues en los Elementos demostraciones que combinan (en su comunicación) ambas modalidades expresivas, y las hay que solo recurren a la primera, careciendo de pasajes que ejemplifiquen la segunda. En Autor se identifican estos dos estilos expresivos, denominándosele al primero polimodal y al segundo unimodal. Por otra parte, algunos filósofos han resaltado el valor de distinguir entre (...) expresión y estructura de una demostración. La polimodalidad apunta, ciertamente, al plano expresivo, pero posee calado estructural. Dado que existen formas diversas de emerger la polimodalidad en la demostración euclidiana, puede resultar de interés examinarlas atendiendo a la interacción entre ambos planos y su contexto. (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 ...

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.

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)

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)

It is often claimed that the Greeks had a single, straightforward conception of number, based on units that are counted. This conception views numbers as collections of units, a perspective clearly expressed in the second definition of Euclid’s Elements, Book VII. However, we argue that there are a few traces in the Elements of a different conception of number, which we may call ‘divisive’. This conception is grounded in division and multiplication, rather than in the addition of units, and (...) it appears to have deep roots in Plato’s Dialogues. It is probably within the framework of this divisive understanding that we can fully grasp the use of the adjective πρῶτοι to describe prime numbers. (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)

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)

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.

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)

This is a scholarly contribution to an area -- the history of Greek geometrical analysis -- that is still insufficiently understood. At the time of Zeuthen, and even up to the middle of the last century, it was fashionable to treat the Data algebraically. Taisbak has abandoned this approach completely, arguing that it does nothing to help us to understand either the development of the work or the reasons for its having been copied, studied, and quoted for more than two (...) millennia. We must bear a queer sort of frustration that affects us everywhere in the Data: we get very little information, hardly any 'knowledge' of the givens. And why not? Probably because 'knowing' geometrical objects was problematic in those days when the concept of 'given' came into being, and the consequences of incommensurability was just being understood. Next to nothing is known of these items, and very little that is worth knowing: length, size, distance -- any of the attributes that can be spoken of by means of numbers. Although there have been two recent translations of the Data, this one is unique in providing, as well, an extensive commentary, which provides the insights gained from three decades of studying the work. The book is meant as a coherent and understandable account of what could have been going on in Euclid's mind, and some reasons for believing that that is what actually was going on in his mind. (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)

Though pictures are often used to present mathematical arguments, they are not typically thought to be an acceptable means for presenting mathematical arguments rigorously. With respect to the proofs in the Elements in particular, the received view is that Euclid's reliance on geometric diagrams undermines his efforts to develop a gap-free deductive theory. The central difficulty concerns the generality of the theory. How can inferences made from a particular diagrams license general mathematical results? After surveying the history behind the (...) received view, this essay provides a contrary analysis by introducing a formal account of Euclid's proofs, termed Eu. Eu solves the puzzle of generality surrounding Euclid's arguments. It specifies what diagrams Euclid's diagrams are, in a precise formal sense, and defines generality-preserving proof rules in terms of them. After the central principles behind the formalization are laid out, its implications with respect to the question of what does and does not constitute a genuine picture proof are explored. (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)

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)

Euclid's definition of proportional magnitudes in the Fifth Book of the Elements gave rise to many commentaries. We examine closely two of these commentaries, one by al-Jayy and the other by Saunderson. Both al-Jayy and Saunderson attempted to defend Euclid's definition by making explicit what Euclid had only implied. We show that the two authors explain Euclid's position in a virtually identical manner.

Cognitive Science, Volume 46, Issue 1, January 2022.