We investigate the expressive power of memory logics. These are modal logics extended with the possibility to store (or remove) the current node of evaluation in (or from) a memory, and to perform membership tests on the current memory. From this perspective, the hybrid logic (↓), for example, can be thought of as a particular case of a memory logic where the memory is an indexed list of elements of the domain.

Educational interventions are necessary to develop mathematical competence at early ages and prevent widespread mathematics learning failure in the education system as indicated by the results of European reports. Numerous studies agree that domain-specific predictors related to mathematics are symbolic and non-symbolic magnitude comparison, as well as, number line estimation. The goal of this study was to design 4 digital learning app games to train specific cognitive bases of mathematical learning in order to create resources and promote the use of (...) these technologies in the educational community and to promote effective scientific transfer and increase the research visibility. This study involved 193 preschoolers aged 57–79 months. A quasi-experimental design was carried out with 3 groups created after scores were obtained in a standardised mathematical competence assessment test, i.e., low-performance group, high-performance group, and control group. The results show that training with the 4 digital learning app games focusing on magnitude, subitizing, number facts, and estimation tasks improved the numerical skills of the experimental groups, compared to the control group. The implications of the study were, on the one hand, provided verified technological tools for teaching early mathematical competence. On the other hand, this study supports other studies on the importance of cognitive precursors in mathematics performance. (shrink)

Primeramente, haremos un repaso por la noción de diálogo, relación o encuentro en las artes y el pensamiento que se desarrollan durante los años sesenta del pasado siglo y fuertemente a partir del episodio de revueltas del 68. Sostenemos que en las artes, lo dialógico representó una estrategia política por partida doble: la conversación y el encuentro eran tanto un vector de ruptura con los modos de apreciar y vivir lo sensible y de entender el hecho artístico, como un procedimiento (...) para someter el individualismo y construir una presencia corporal y común considerada perdida y/o necesaria. Seguidamente veremos en qué sentido el diálogo se presenta como una de las claves de la filosofía política actual preocupada por combatir al sujeto individualista e hiperproductivo que propugna el neoliberalismo. Finalmente, trataremos de llevar a efecto esta invitación a dialogar. Así, la última sección de este escrito tomará la forma de una conversación entre nosotros, para interrogarnos juntos sobre algunas de las cuestiones anteriormente esbozadas. (shrink)

It is not yet clear which response behavior requires self-regulatory effort in the moral dilemma task. Previous research has proposed that utilitarian responses require cognitive control, but subsequent studies have found inconsistencies with the empirical predictions of that hypothesis. In this paper we treat participants’ sensitivity to utilitarian gradients as a measure of performance. We confronted participants (N = 82) with a set of five dilemmas evoking a gradient of mean utilitarian responses in a 4-point scale and collected data on (...) heart rate variability and on utilitarian responses. We found positive correlations between tonic and phasic HRV and sensitivity to the utilitarian gradient in the high tonic group, but not in the low tonic group. Moreover, the low tonic group misplaced a scenario with a selfish incentive at the high end of the gradient. Results suggest that performance is represented by sensitivity correlated with HRV and accompanied with a reasonable placement of individual scenarios within the gradient. (shrink)

. Se problematiza la comprensión de la violencia en la obra del pensador peruano José Carlos Mariátegui (1894-1930). A partir de la premisa que tal comprensión articula concepciones y concreciones de la violencia, se reconocen, primero, dos concepciones distintas: como-medio y como-reacción. En cuanto a las concreciones se argumentará que esta se encuentra permeada por una nueva agencia política, por el concepto de mito de Georges Sorel, que tendrá sobre ella unos efectos más formales que prácticos, y por las (...) divergencias del panorama social peruano respecto al clima social internacional –donde las condiciones de una situación revolucionaria propuestas por Lenin resultarán claves para entender la postura de Mariátegui–. (shrink)

Este artículo propone una lectura retrospectiva y crítica de la visita del Papa Francisco a Colombia, realizada en septiembre de 2017, y se apoya en algunas nociones provenientes del pensamiento de Bernard Lonergan S.J. No pretende analizar o recordar su coyuntura, sino captar y exponer los desafíos generados en aquella visita, que se vislumbran como posibilidad de sanación en nuestra historia. Son asuntos estructurales que apelan tanto la vida creyente como la vida social y política, y no meras declaraciones de (...) referencia de un líder religioso. (shrink)

1. El redescubrimiento del carácter social de la existencia humana constituye uno de los rasgos fundamentales de nuestra época. El hombre es un ser social en todo. Esto no excluye, sin embargo, la dimensión personal, autónoma de la existencia.2. La acentuación de la socialidad constitutiva del hombre condujo a la ideologíadel determinismo social o sociologismo. El hombre es visto sólo como un producto o resultado de las estructurasy procesos sociales. La independencia y la libertad del hombre son ilusiones. Si bien (...) nuestra existencia implica una relación de dependencia respecto de nuestros semejantes, esta relación no es de carácterdeterminista. Esto se hace evidente a través del análisis del concepto fenomenológico de "facticidad".3. La facticidad designa la clase típicamente humana de hechos que condicionan la existencia del hombre. La facticidad social presenta un carácter cósico, pero en ella está siempre presente la intencionalidad de la existencia; de ahí su contingencia esencial. La necesidad de la facticidad no es una necesidad de naturaleza sino una "necesidad en libertad".4. La función de la facticidad social es hacer que la persona se autodesarrolle y realice en plenitud sus posibilidades. Existen muchas formas de facticidad social que restringen y obstruyen la existencia dela persona, en lugar de desarrollarla. Debemos asumir una actitud crítica frente a nuestra facticidad. Sin esa actitud crítica no hay progreso posible. La facticidad social necesita de la persona para poder progresar y crear mejores condiciones para todos.5. Si la persona es el sentido de la facticidad social, la educación, como encargada de trasmitir la facticidad, debe contribuir ·eficazmente al proceso de personalización. La sociedad es una parte esencial del hombre; luego, asumir la facticidad social es una forma de alcanzar la plena autonomía. Educar es personalizar, pero para ello es necesario previamente socializar. La socialización, sin embargo, no debe entenderse como mera adaptación, sino como un proceso donde los modelos y patrones de la facticidad social son asumidos no pasiva sino activamente, en la perspectiva del cambio y la innovación. Sólo quien asume con mentalidad creadora a la sociedad tiene posibilidad de transformarla y mejorarla. (shrink)

Denaro e informazione sono entrambi segni della civilizzazione umana. La loro connessione con il linguaggio è evidente. Il loro valore si basa sulla fiducia nella creatività del lavoro umano. La minaccia odierna è di ridurre denaro e informazione a mera merce.

O artigo tem por objeto o exame de três registros de gritantes e distintos paradoxos na Doutrina do Direito de Kant. Registros feitos em tempos e contextos históricos diferentes por Friedrich Bouterwek, Marcus Willaschek e Balthazar Barbosa Filho. Bouterwek atribuiu a Kant a mais paradoxal das proposições jamais enunciadas por qualquer autor, a de que a mera ideia de soberania deve obrigar-nos a obedecer como a nosso inquestionável senhor a quem quer que se haja estabelecido como tal, sem que (...) caiba indagar quem lhe deu o direito de comandar-nos. Willaschek aponta a incompatibilidade de duas teses centrais da doutrina kantiana: a do caráter externo dos vínculos jurídicos e a da incondicionalidade obrigacional do direito positivo, pois não é possível entender como é possível termo-nos como obrigados por imperativos jurídicos e, ao mesmo, vermo-nos internamente isentados do dever de obedecê-los. O ponto crítico de Balthazar é alegar que não pode haver na filosofia kantiana uma crítica da razão político e jurídica, simplesmente porque o conceito de imputação, base da normatividade própria dessas esferas, pressupõe uma pluralidade de agentes livres que, justamente, só pode ser uma pressuposição, pois nosso acesso à normatividade prática só pode ter lugar em primeira pessoa. No exame a que o artigo submete essas alegações, o artigo argumenta, em objeção à tese de Balthazar, que o caráter universal e categórico da força que vincula o sujeito quando confrontado com a lei moral em primeira pessoa necessariamente se desvaneceria se, ao mesmo tempo, ele não fosse tomado pela evidência de que a realidade objetiva dos princípios morais é não só instanciável, mas assegurada pela múltipla instanciação. Com relação às dificuldades levantadas por Willaschek e Bouterwek, o artigo argumenta que o princípio exeundum e statu naturali, enquanto norma metapositiva, anterior à divisão do domínio prático entre doutrina do direito e doutrina da virtude, permite ao mesmo tempo compreender a exigência de obediência ao poder constituído e a restrição das obrigações jurídico-políticas exclusivamente ao foro externo.AbstractThe object of the article is to examine three claims about three distinct and allegedly blatant paradoxes in Kant's Doctrine of Right. These three critical points had been made in distinct times and contexts by Friedrich Bouterwek, Marcus Willaschek e Balthazar Barbosa Filho. Bouterwek attributed to Kant the most paradoxical of all paradoxical propositions, the statement that by the mere idea of sovereignty we are obliged to obey as our lord who has imposed himself upon us, without questioning from where he got such right. Willaschek points out the incompatibility of two main theses of Kantian doctrine of right: the claims that the legal bounds are of external character and that they are the source of unconditional obligations, since it seems impossible to understand how it would be possible to be obliged by juridical norms and decisions and at the same time to be exempted of the internal duty of compliance. The radical objection of Professor Balthazar is the claim that in the context of Kantian Philosophy it is impossible to admit a critique of the juridical and political reason because the concept of imputation, ground of the normativity in these domains, requires not only the presupposition of free agents, but a true and secure epistemic access to them, which is, according to him, impossible considering that the moral law and the other practical principles are accessible for us only in the first person. In the course of the appraisal of such claims, the article contest that objection arguing that the universal and categorical force of the normative bound experienced by the subject when confronted with the moral law in the first person would ineluctably vanish if, at the same time, he had not been taken by the evidence that the objective reality of the moral principles is secured by multiple instancing. Regarding the difficulties raised by Willaschek and Bouterwek, the article argues that the principle exeundum e statu naturali, as a norm of meta-positive character, prior to the division of practical domains between the doctrine of right and the doctrine of virtue, is the cue both to the understanding of the requirement of unquestioning obedience to the constituted power and to the restriction of the validity of this requirement only in foro externo. (shrink)

Los autores conversan sobre la distinta relación que tienen con la filosofía las lenguas española y francesa, encontrando la explicación de esa diferencia principalmente en los “espíritus” que nos separan, no obstante nuestra considerable cercanía lingüística. Mientras que la Reforma y la Contrarreforma exigieron de Francia un “humanismo del saber objetivo, del individuo y del progreso”, la cultura española dio de sí “un paradójico humanismo de la fe, de la expansión y de los juegos de la apariencia”. El “espíritu de (...) la filosofía” (Descartes, el ancestro de Hegel) se desarrolló en el norte de Europa, y en la Europa mediterránea floreció más bien “el espíritu de la ficción” (el Cervantes de Unamuno, de Ortega, de Borges y de Kundera). La inquietud de un mundo que bien podría no ser sino sueño, o incluso locura –de la que terminará por resurgir, al lado del cervantino y del calderoniano, el cogito cartesiano–, es sin embargo una inquietud española en primer lugar, y eso es algo que amerita ser estudiado, y sobre todo pensado. Una clave muy importante para dicho estudio es la constatación, subrayada por Nancy en respuesta al problema de la calificación de nuestra filosofía como “mera literatura”, de la “pérdida de confianza en las aventuras del sentido” que ha caracterizado a la Modernidad (o al Occidente en su versión nórdica, ese “espíritu de la filosofía alemana” que tras el efecto del nazismo se mudó a Francia y a Estados Unidos). Ahí donde Unamuno señalaba la necesidad, antes que de europeizar a España, de “españolizar a Europa”, Jean-Luc Nancy reconoce que Occidente necesita atreverse, una vez más, a abrirse a la aventura, y al riesgo del pensamiento. (shrink)

In Jordan, research ethics committees exist in most health settings. However, little is known about Jordanian public views regarding the ethics of clinical research. This study aimed to evaluate Jordanian public understanding and attitudes about ethics in clinical research. A questionnaire was used to collect information that included demographics, public knowledge, and attitudes towards ethics in clinical research. It was administered via face-to-face interviews in two major cities in Jordan from 1st June to 15th August 2017. Of the 2315 respondents, (...) 2.33% were found to have poor knowledge, 22.16% had fair knowledge, and 75.51% had good knowledge of ethics in clinical research. Furthermore, 75.81% of respondents had positive attitudes towards ethics in research. However, only 45.23% reported that they trust clinical researchers in Jordan. Even though a large majority of respondents were aware of key features of research ethics, efforts are needed to address negative perceptions and knowledge deficits. (shrink)

In this paper, I show that Polleit and Mariano (2011) are right in concluding that Credit Default Swaps (CDS) are per se unobjectionable from Rothbard’s libertarian perspective on property rights and contract theory, but that they fail to derive this conclusion properly. I therefore outline the proper explanation. In addition, though Polleit and Mariano are correct in pointing out that speculation with CDS can conceivably hurt the borrowers’ interests, they fail to grasp that this can be the case only in (...) some peculiar circumstances that I identify. In other words, they miss the bigger picture, the one outside special circumstances, in which CDS trading has the opposite effect. That is, CDS facilitate debt accumulation, including government debt accumulation. Finally, I point out how this can precipitate the collapse of fiat money regimes. An incidental goal of the analysis is to provide a better account than Polleit and Mariano of recent government interventions in and around CDS markets. (shrink)

Minding Time: A Philosophical and Theoretical Approach to the Psychology of Time offers an innovative philosophical account of the most fundamental kinds of time representation.

This monograph addresses the question of the increasing irrelevance of philosophy, which has seen scientists as well as philosophers concluding that philosophy is dead and has dissolved into the sciences. It seeks to answer the question of whether or not philosophy can still be fruitful and what kind of philosophy can be such. The author argues that from its very beginning philosophy has focused on knowledge and methods for acquiring knowledge. This view, however, has generally been abandoned in the last (...) century with the belief that, unlike the sciences, philosophy makes no observations or experiments and requires only thought. Thus, in order for philosophy to once again be relevant, it needs to return to its roots and focus on knowledge as well as methods for acquiring knowledge. Accordingly, this book deals with several questions about knowledge that are essential to this view of philosophy, including mathematical knowledge. Coverage examines such issues as the nature of knowledge; plausibility and common sense; knowledge as problem solving; modeling scientific knowledge; mathematical objects, definitions, diagrams; mathematics and reality; and more. This monograph presents a new approach to philosophy, epistemology, and the philosophy of mathematics. It will appeal to graduate students and researchers with interests in the role of knowledge, the analytic method, models of science, and mathematics and reality. (shrink)

Many of the computing systems programmed using Machine Learning are opaque: it is difficult to know why they do what they do or how they work. Explainable Artificial Intelligence aims to develop analytic techniques that render opaque computing systems transparent, but lacks a normative framework with which to evaluate these techniques’ explanatory successes. The aim of the present discussion is to develop such a framework, paying particular attention to different stakeholders’ distinct explanatory requirements. Building on an analysis of “opacity” from (...) philosophy of science, this framework is modeled after accounts of explanation in cognitive science. The framework distinguishes between the explanation-seeking questions that are likely to be asked by different stakeholders, and specifies the general ways in which these questions should be answered so as to allow these stakeholders to perform their roles in the Machine Learning ecosystem. By applying the normative framework to recently developed techniques such as input heatmapping, feature-detector visualization, and diagnostic classification, it is possible to determine whether and to what extent techniques from Explainable Artificial Intelligence can be used to render opaque computing systems transparent and, thus, whether they can be used to solve the Black Box Problem. (shrink)

In his new book, Carlos Alberto Torres, an internationally renowned critical theorist of education, explores the early writings of Paulo Freire whose ideas have had a tremendous and long-lasting impact on the world of pedagogy and politics. Torres analyzes Freire's works, from the 1960s and 1970s, before Freire gained worldwide recognition for his Pedagogy of the Oppressed. Offering an in-depth look into the formative thinking of Freire, Torres identifies how his ideas produced frameworks for educating global citizens, building community (...) and mutual respect, creating social responsibility, instilling an appreciation for diversity, promoting multiple literacies, and social justice education. This volume is the result of more than 3 decades of research with access to Freire's personal library and the archives of the Paulo Freire Institute, as well as the author's extensive conversations with Paulo Freire over two decades--Dr. Torres was Freire's adviser during his tenure as Secretary of Education in the Municipality of São Paulo, Brazil, 1989-1991. (shrink)

I am a theoretical physicist, and, following Aristotles' injunction (Aristotle, Physics III, 202b 34), I do consider it my responsibility to discuss the problem of the notion of infinity in the world--in particular, to "inquire whether there is such a thing or not." I will do so here by illustrating some aspects of the notion of infinity in the natural sciences.

Why do we remember the past and not the future? What does it mean for time to "flow"? Do we exist in time or does time exist in us? In lyric, accessible prose, Carlo Rovelli invites us to consider questions about the nature of time that continue to puzzle physicists and philosophers alike. For most readers this is unfamiliar terrain. We all experience time, but the more scientists learn about it, the more mysterious it remains. We think of it as (...) uniform and universal, moving steadily from past to future, measured by clocks. Rovelli tears down these assumptions one by one, revealing a strange universe where at the most fundamental level time disappears. He explains how the theory of quantum gravity attempts to understand and give meaning to the resulting extreme landscape of this timeless world. Weaving together ideas from philosophy, science and literature, he suggests that our perception of the flow of time depends on our perspective, better understood starting from the structure of our brain and emotions than from the physical universe. (shrink)

En el marco del desafío que implica abordar el fenómeno de la informalidad urbana a escala metropolitana en términos cuantitativos, el presente trabajo se propone recuperar fuentes de datos secundarias que han georreferenciado las urbanizaciones informales en el territorio y ponerlas en diálogo con las posibilidades que brindan los datos censales para trabajar a nivel microespacial. De esta manera, se busca dar cuenta de las características sociodemográficas y habitacionales de las urbanizaciones informales de la aglomeración Gran Buenos Aires en 2010 (...) y las especificidades que adquieren dos formas que condensan aspectos socio-urbanos distintos: las villas y los asentamientos. Se evalúan posibilidades y limitaciones de esta metodología y, como resultado, se analizan las particularidades que atraviesan a estos tipos de hábitat en términos de estructura etaria, origen migratorio, nivel educativo, composición y tamaño de los hogares, condiciones habitacionales, y los contrastes que emergen entre ellos y respecto del conjunto urbano. (shrink)

Latin-American immigration has transformed Chilean schools into new multicultural scenarios. Studies about intergroup dynamics among students from different cultural backgrounds and their psychological consequences are still limited in south–south migration contexts. Literature has suggested that intergroup relations influence students’ satisfaction with school, and they could be improved by the development of competences to cope with cultural differences. This study aims to verify if cultural self-efficacy and its dimensions mediated the influence of prejudice on satisfaction with school, in a sample composed (...) byN= 690 Chilean and Latin-American immigrant secondary students. Results showed that cultural self-efficacy reduced the effect of prejudice in satisfaction with school, in the cases of both immigrant and Chilean students. The dimensions of cultural self-efficacy in processing information from other cultures and mixing with different others make the difference. Findings’ contributions for the understanding of adolescents’ intergroup relations and psychosocial interventions at school are discussed. (shrink)

In this paper, I show that Polleit and Mariano (2011) are right in concluding that Credit Default Swaps (CDS) are per se unobjectionable from Rothbard's libertarian perspec..

Introduction -- Youth, a contradictory formation: 1910-18 -- Workers' democracy and factory-councils: 1919-20 -- Passage to maturity: 1921-6 -- Methodological observations on the prison notebooks -- The 'extended' theory of the state -- The party as 'collective intellectual' -- The current relevance and universality of Gramsci.

All human civilizations have thought that the world was made of sky above and the Earth below. All except one. For the Greeks, the Earth was a rock floating in space, and under the earth there was no ground, no turtles, nor the gigantic columns of which the Bible speaks. How did the Greeks understand that the Earth is suspended in nothingness? Who understood this and how? It is this unique "scientific revolution" of Anaximander of which the author speaks, which (...) Karl Popper called "one of the most daring ideas, revolutionary and portentous in the entire history of human thought" The concern is with the conflict that was opened, still burning, with the nature of scientific thought, its critical and rebellious character, the force with which it subverts the order of things and our image of the world of scientific knowledge, and the highly effective "error" at the same time --as we teach physics in the twentieth century. Speaking of Anaximander, the author invites us to reflect on what his ideas mean to the scientific revolution opened by Einstein. (shrink)

v. 1. Ensino secundário e médio -- v. 2. Estudos sobre o ensino primário -- Planejamento educacional e contribuições à educação do futuro.

This article aims to clarify the sharp contrast between the “short path” of Heideggerian ontology and the “long path” adopted by Ricoeur as a representation of the many necessary mediations in the constitution of his philosophical anthropology. Heidegger breaks into contemporary thought with the publication of his treatise “Being and Time” (1927) as a kind of direct settlement in the field of ontology. In contrast, Ricoeur is seen - and sees himself - as a thinker of conceptual mediations in his (...) long itinerary, since adherence to Husserlian phenomenology and subsequent hermeneutic inflection, passing through confrontations with Freudian psychoanalysis, structuralism and analytical philosophy, to the creation of the book “Oneself as Another” (1990). However, the two philosophers built their works on the common horizon of hermeneutics; therefore, are there possible points of convergence between them? And how could we interpret the relevance of this possible interlocution? These questions cannot be answered without a thorough comparative reading. This text does not intend to answer them, but only to suggest elements for further investigations. (shrink)

The received view of dynamical explanation is that dynamical cognitive science seeks to provide covering law explanations of cognitive phenomena. By analyzing three prominent examples of dynamicist research, I show that the received view is misleading: some dynamical explanations are mechanistic explanations, and in this way resemble computational and connectionist explanations. Interestingly, these dynamical explanations invoke the mathematical framework of dynamical systems theory to describe mechanisms far more complex and distributed than the ones typically considered by philosophers. Therefore, contemporary dynamicist (...) research reveals the need for a more sophisticated account of mechanistic explanation. (shrink)

Models developed using machine learning are increasingly prevalent in scientific research. At the same time, these models are notoriously opaque. Explainable AI aims to mitigate the impact of opacity by rendering opaque models transparent. More than being just the solution to a problem, however, Explainable AI can also play an invaluable role in scientific exploration. This paper describes how post-hoc analytic techniques from Explainable AI can be used to refine target phenomena in medical science, to identify starting points for future (...) investigations of causal relationships, and to generate possible explanations of target phenomena in cognitive science. In this way, this paper describes how Explainable AI—over and above machine learning itself—contributes to the efficiency and scope of data-driven scientific research. (shrink)

The definitive account of Aristotle's life and school This definitive biography shows that Aristotle's philosophy is best understood on the basis of a firm knowledge of his life and of the school he founded. First published in Italian, and now translated, updated, and expanded for English readers, this concise chronological narrative is the most authoritative account of Aristotle's life and his Lyceum available in any language. Gathering, distilling, and analyzing all the evidence and previous scholarship, Carlo Natali, one of the (...) world's leading Aristotle scholars, provides a masterful synthesis that is accessible to students yet filled with evidence and original interpretations that specialists will find informative and provocative. Cutting through the controversy and confusion that have surrounded Aristotle's biography, Natali tells the story of Aristotle's eventful life and sheds new light on his role in the foundation of the Lyceum. Natali offers the most detailed and persuasive argument yet for the view that the school, an important institution of higher learning and scientific research, was designed to foster a new intellectual way of life among Aristotle's followers, helping them fulfill an aristocratic ideal of the best way to use the leisure they enjoyed. Drawing a wealth of connections between Aristotle's life and thinking, Natali demonstrates how the two are mutually illuminating. For this edition, ancient texts have been freshly translated on the basis of the most recent critical editions; indexes have been added, including a comprehensive index of sources and an index to previous scholarship; and scholarship that has appeared since the book's original publication has been incorporated. (shrink)

Franz Brentano is not usually associated with mathematics. Generally, only Brentano’s discussion of the continuum and his critique of the mathematical accounts of it is treated in the literature. It is this detailed critique which suggests that Brentano had more than a superficial familiarity with mathematics. Indeed, considering the authors and works quoted in his lectures, Brentano appears well-informed and quite interested in the mathematical research of his time. I specifically address his lectures here as there is much less to (...) be found about mathematics in his published works. Besides Brentano’s own work, it is quite remarkable to see that practically all of his better-known students sooner or later produced a work on the philosophy of mathematics. This also encourages the supposition not just of a common interest in the matter, but of a common theoretical core. All this prompts the question: Can we speak of a Brentanist philosophy of mathematics? (shrink)

In a context of pervasive digitalization of the social world, both before and during the COVID-19 pandemic, the field of education has undergone major changes with the development of digital practices and settings. However, the physical presence of the subjects and the body remain something primordial and irreplaceable in traditional educational processes. Thus, it is often assumed that virtuality is opposed to the corporeal reality of the subjects involved in teaching, learning and studying. In this paper we aim to critically (...) challenge this assumption by addressing the phenomenon of virtuality in a more original sense: as a fundamental dimension of corporeality itself. Drawing on Merleau-Ponty’s phenomenological insights, we argue that the experience of the virtual is inherently embodied and fully real. We understand corporeality and virtuality as entangled and multidimensional phenomena. At the same time, this leads us to rethink the digital as only one possible medium in which virtuality can appear as inherent to embodied experience. We develop our phenomenological approach in three instances. First, we question and bracket two common assumptions about corporeality. From our critical perspective, assumptions about a disembodied digitality and a devirtualized corporeality represent extremes within which educational discourses currently move. Second, we address the inherent virtuality of embodied experience. We outline the heuristic concept of the virtual body to describe corporeality and virtuality as entangled and multidimensional phenomena that encompass extension, intercorporeality and intermediality. Finally, we make some remarks on the relationships of corporeality and virtuality considering the challenges of digital education. (shrink)

In the last few decades there has been a revival of interest in diagrams in mathematics. But the revival, at least at its origin, has been motivated by adherence to the view that the method of mathematics is the axiomatic method, and specifically by the attempt to fit diagrams into the axiomatic method, translating particular diagrams into statements and inference rules of a formal system. This approach does not deal with diagrams qua diagrams, and is incapable of accounting for the (...) role diagrams play as means of discovery and understanding. Alternatively, this paper purports to show that the view that the method of mathematics is the analytic method is capable of dealing with diagrams qua diagrams, and of accounting for such role. (shrink)