Although understanding is the object of a growing literature in epistemology and the philosophy of science, only few studies have concerned understanding in mathematics. This essay offers an account of a fundamental form of mathematical understanding: proof understanding. The account builds on a simple idea, namely that understanding a proof amounts to rationally reconstructing its underlying plan. This characterization is fleshed out by specifying the relevant notion of plan and the associated process of rational reconstruction, building in part on Bratman's (...) theory of planning agency. It is argued that the proposed account can explain a significant range of distinctive phenomena commonly associated with proof understanding by mathematicians and philosophers. It is further argued, on the basis of a case study, that the account can yield precise diagnostics of understanding failures and can suggest ways to overcome them. Reflecting on the approach developed here, the essay concludes with some remarks on how to shape a general methodology common to the study of mathematical and scientific understanding and focused on human agency. (shrink)

Contrary to most interpretations, the article argues that the scandalous dimensions of Machiavelli's thought are the outcome of his critical reflections on humanist discourse. Machiavelli drew the radical conclusions from the humanist rejection of the classical and medieval vision of man as a creature with an objective telos, within a rational order of Being. From this perspective the article accounts for Machiavelli's separation of politics from ethics, his anti-Christian stance as well as his appreciation of the social and political importance (...) of religion, his views concerning corruption and the fragility of the political body, and ultimately his anthropological premises, notably his perception of man as a creature driven by indeterminate and insatiable desire. (shrink)

A characteristic feature of the philosophy of mathematical practice is to attend to what people do when they do mathematics. But what does it mean to do mathematics? This question raises several issues regarding the nature of action, activity, and agency in mathematical practice. The present chapter reviews contributions in the field that have attempted to theorize about these notions. It begins with some motivations for taking agents seriously in the philosophical study of mathematical practice. The core of the chapter (...) discusses, in turn, what it means to carry out mathematical activities, do things with mathematical artifacts, engage with mathematical proofs, and perform mathematical actions prescribed by mathematical texts. Taken together, the various lines of work reported here provide an initial, but already sophisticated, picture of what it means to do mathematics. The chapter ends with some suggestions for future research on agency in mathematical practice. (shrink)

The Qur'an depicts fluctuating relations between Muslims and non-Muslims. While at times such relations can be conciliatory and harmonious, at others they are inimical, uneasy, or distant. Still, the Qur'an acknowledges the necessary ontological reality of the human difference. This is evidenced in many verses. Thus, I will argue that an “attentive” and “worldly” reading of the Qur'an is crucial to curb misunderstanding of the way ‘difference’ is perceived in Islam by Muslims and non-Muslims alike. A close reading is primarily (...) a multiple form of communication. It is needed to resist the networked systems of power and control dominated by images and mass media in the Arab world and Western. It is exceptionally important to free the interpretation of Qur'an from the grip of Muslim and non-Muslim extremists and Islamophobes who read some of its verses as evidence of essentialized enmity harbored by Muslims towards all non-Muslims. (shrink)

Book reviewed in this article:Thomas F. Green, Voices: The Educational Formation of Conscience.

Les situations de séparation conflictuelle entre parents peuvent provoquer une rupture du lien entre l’enfant et un de ses parents. L’espace de rencontre ( er ), en mettant en place des droits de visite encadrés, est alors le dernier recours de ces parents pour retisser un lien. Dans l’ er dont il est question dans cet article, les visites concernent principalement le père, la mère étant alors le parent gardien. Dans ce cadre ont été rencontrés des enfants qui affichaient une (...) volonté de rompre tout lien avec leur père, ce refus massif s’apparentant à un clivage de loyauté. Pour comprendre cette attitude, l’article interroge, à travers le cas de Marie, la dimension psychosociale de la fonction paternelle, la dimension intrapsychique et la dynamique relationnelle dans cette triade père-mère-enfant afin d’envisager le dispositif à mettre en place dans les situations de conflit de loyauté. (shrink)

Poincaré and Prawitz have both developed an account of how one can acquire knowledge through reasoning by mathematical induction. Surprisingly, their two accounts are very close to each other: both consider that what underlies reasoning by mathematical induction is a certain chain of inferences by modus ponens ‘moving along’, so to speak, the well-ordered structure of the natural numbers. Yet, Poincaré’s central point is that such a chain of inferences is not sufficient to account for the knowledge acquisition of the (...) universal propositions that constitute the conclusions of inferences by mathematical induction, as this process would require to draw an infinite number of inferences. In this paper, we propose to examine Poincaré’s point—that we will call the closure issue—in the context of Prawitz’s framework where inferences are represented as operations on grounds. We shall see that the closure issue is a challenge that also faces Prawitz’s own account of mathematical induction and which points out to an epistemic gap that the chain of modus ponens cannot bridge. One way to address the challenge is to introduce suitable additional inferential operations that would allow to fill the gap. We will end the paper by sketching such a possible solution. (shrink)

Machiavelli’s historical writings, notably the Discourses on Livy and the Florentine Histories, continued the tradition of the humanist historical enterprise. They were informed by th...

Mathematical proof is the primary form of justification for mathematical knowledge, but in order to count as a proper justification for a piece of mathematical knowl- edge, a mathematical proof must be rigorous. What does it mean then for a mathematical proof to be rigorous? According to what I shall call the standard view, a mathematical proof is rigorous if and only if it can be routinely translated into a formal proof. The standard view is almost an orthodoxy among contemporary (...) mathematicians, and is endorsed by many logicians and philosophers, but it has also been heavily criticized in the philosophy of mathematics literature. Progress on the debate between the proponents and opponents of the standard view is, however, currently blocked by a major obstacle, namely the absence of a precise formulation of it. To remedy this deficiency, I undertake in this paper to provide a precise formulation and a thorough evaluation of the standard view of mathematical rigor. The upshot of this study is that the standard view is more robust to criticisms than it transpires from the various arguments advanced against it, but that it also requires a certain conception of how mathematical proofs are judged to be rigorous in mathematical practice, a conception that can be challenged on empirical grounds by exhibiting rigor judgments of mathematical proofs in mathematical practice conflicting with it. (shrink)

ABSTRACTIn this essay I agree with Nicholas Burbules that ‘Phronesis’ is an ethical and political category that grounds the possibility of intercultural communication in translation from one particular context to another rather than in the presumption of one or another account of universalism. After a brief review of the development of this idea in key milestones of Western philosophy, I argue that it requires an education in dialogue across difference that can foster hope for peaceful coexistence among diverse traditions and (...) perspectives in diverse democratic societies. (shrink)

Name der Zeitschrift: Naharaim Jahrgang: 10 Heft: 1 Seiten: 39-55.

Over the previous three decades, the area of computer networks has progressed significantly, from traditional static networks to dynamically designed architecture. The primary purpose of software-defined networking is to create an open, programmable network. Conventional network devices, such as routers and switches, may make routing decisions and forward packets; however, SDN divides these components into the Data plane and the Control plane by splitting distinct features away. As a result, switches can only forward packets and cannot make routing decisions; the (...) controller makes routing decisions. OpenFlow is the communication interface between the switches and the controller. It is a protocol that allows the controller to identify the network packet’s path across the switches. This project uses the SDN environment to implement the firefly optimization algorithm to determine the shortest path between two nodes in a network. The firefly optimization algorithm was implemented using Ryu control. The results reveal that using the firefly optimization algorithm improves the selected short path between the source and destination. (shrink)

This research suggests a comparison between two tasks to evaluate oral communication strategies (CSs) through storytelling and interview and to compare the use of CSs between high and low proficient students. The results of the storytelling and oral interview are compared to check if CSs vary with the task and what tasks are better for each of the selected CSs. A number of 60 Spanish learners of English participated in this investigation, and a total of 232 protocols were analysed to (...) obtain the results. The main results are that CSs vary to adapt to task demands and that the frequency and type of CSs used by the students are affected by the level of proficiency of the students. This highlights the importance of selecting the appropriate task for the evaluation to be significant and the necessity to adapt the syllabus to include elements that introduce some productive CSs. (shrink)

This article revisits the question of the modernity of the Renaissance by examining the political language of Florentine civic humanism and by critically analyzing the debate over Hans Baron’s interpretation of the movement. It engages two debates that are usually conducted separately: one concerning the originality of civic humanism in comparison to medieval thought, and the other concerning the political and social function of the civic humanists’ political republicanism in fifteenth-century Florence. The article’s main contention is that humanist political discourse (...) rejected the perception of social and political reality as being part of, or reflecting, a metaphysical and divine order or things, and thus undermined the traditional justifications for political hierarchies and power relations. This created the conditions of possibility for the distinctively modern aspiration for a social and political order based on liberty and equality. It also resulted in the birth of a distinctively modern form of ideology, one that legitimizes the social order by disguising its inequalities and structures of domination. Humanism, like modern political thought generally, thus simultaneously constructs and reflects the dialectic of emancipation and domination so central to modernity itself. (shrink)

The article examines Alberti’s dialogue Della famiglia as a reflection on the foundations of ethics and politics from the perspective of humanist discourse. The polyphonic work presents and critically examines several views. The authorial voice of the text rehearses the traditional philosophical view the humanists inherited, according to which humans are sociable by nature. However, some of the interlocutors reject this convenient view, implying that it cannot be squared with the humanist critique of the premises of mainstream classical and medieval (...) philosophy. Another line of argumentation in the dialogue attempts to harness the key humanist notion of gloria as the basis of human association, according to which humans self-interestedly seek fame and glory but to gain them they must promote the common good. But also this position is contested in the dialogue, with some arguing that the agonistic competition for glory produces envy and resentment rather than a civic spirit and as such threatens social cohesion. Finally, Della famiglia also explores the implications of the radical possibility that men and women are asocial creatures, by casting them as selfish economic actors who are only interested in making material gains. (shrink)

The figure of the intellectual looms large in modern history, and yet his or her social place has always been full of ambiguity and ironies. Between Utopia and Dystopia is a study of the movement that created the identity of the universal intellectual: Erasmian humanism. Focusing on the writings of Erasmus and Thomas More, Hanan Yoran argues that, in contrast to other groups of humanists, Erasmus and the circle gathered around him generated the social space—the Erasmian Republic of Letters—that (...) allowed them a considerable measure of independence. The identity of the autonomous intellectual enabled the Erasmian humanists to criticize established customs and institutions and to elaborate a reform program for Christendom. At the same time, however, the very notion of the universal intellectual presented a problem for the discourse of Erasmian humanism itself. It distanced the Erasmian humanists from concrete public activity and, as such, clashed with their commitment to the ideal of an active life. Furthermore, citizenship in the Republic of Letters threatened to lock the Erasmian humanists into a disembodied intellectual sphere, thus undermining their convictions concerning intellectual activity and the production of knowledge. Between Utopia and Dystopia will be of interest to scholars and students interested in Renaissance humanism, early modern intellectual and cultural history, and political thought. It also has much to contribute to debates over the identity, social place, and historical role of intellectuals. (shrink)

A través de nuestro estudio sobre el argumento de la semiótica de la historia no contada en la novela «The Crow’s Leg», pretendíamos mostrar la imagen de las prohibiciones religiosas, morales y políticas mediante la interacción de discursos con los niveles sociales y culturales destacados por el contexto semántico en el discurso. El estudio utilizó el planteamiento semiótico que señala los mecanismos del discurso narrativo que intervienen en las dimensiones semióticas y el significado de la historia no contada a través (...) de la estrategia de los temas planteados (religión–política–ética). Al finalizar nuestro estudio concluimos la importancia que tienen las dimensiones semióticas que se extienden desde el significante al significado centrándonos de manera más profunda en el tema de la historia no contada a través de la significación de la prohibición religiosa, en la que enfocamos la centralidad de la mujer y la interactuación con hombres en muchos dominios que son considerados por el principado como vetados por la Sharía, que abordamos como una prohibición religiosa. En este estudio también nos referimos a la esencia de los conceptos semióticos y a la definición de contenido mediante herramientas metodológicas que estudian el mundo semántico de la novela y sus unidades morales menores y fundamentales que solo se alcanzan en su relación con los demás elementos y que se basan en un eje semántico común en la historia no contada y sus interpretaciones en el cuerpo de la novela. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:Where's the Rhetoric? Imagining a Unified Field by S. Scott GrahamJoshua HananWhere's the Rhetoric? Imagining a Unified Field. By S. Scott Graham. Columbus: Ohio State University Press, 2020. 194 pp. Paperback $29.95. ISBN: 978-0-8142-5771-5.The proliferation of materialist perspectives in rhetorical studies has generated feelings of disciplinary crisis and fragmentation. Early materialist formulations of rhetoric, such as those put forward by Michael Calvin McGee and Raymie McKerrow, conceptualized materiality (...) discursively and, thus, maintained continuity with more traditional accounts of rhetoric as a practice of "symbolic action." However, beginning with texts such as Ronald Walter Greene's "Another Materialist Rhetoric" and Jack Selzer and Sharon Crowley's edited collection Rhetorical Bodies, scholars began emphasizing the ontological and embodied rhetoricity of physical contexts and environments over discursive and ideological conceptions of materiality. This turn toward the ontological and embodied has rapidly expanded over the past twenty years, with numerous scholars now offering new materialist, postcritical, ecological, computational, and digital perspectives on rhetoric that privilege concepts such as affect, circulation, and assemblage over more traditional rhetorical terminology.It is in response to this tension between standard rhetorical perspectives and materialist rhetorical approaches that we can appreciate the interventions of S. Scott Graham's recent book, Where's the Rhetoric? Imagining a Unified Field. Rejecting the view that the materialist turns in rhetorical studies (particularly rhetorical new materialisms [hereafter referred to as RNM] and computational rhetoric) have left the discipline more fragmented and less capable of defending a unified perspective on rhetoric, Graham believes it is possible to generate a new unified theory that can affirm the lines of scholarly influence that have given rise to RNM (what Graham calls modern rhetoric's "right branch") as well as the more [End Page 218] traditional lines of scholarly influence that have led to a formulation of rhetoric as "symbolic action" (what Graham calls modern rhetoric's "left branch"). To accomplish this task, Graham argues that we should avoid the tendency to view RNM as "other" to traditional narratives about rhetoric and, instead, consider how these latter perspectives are compatible with the former. Much like unified field theories in physics that seek to bridge older perspectives on general relativity with newer perspectives on quantum mechanics (e.g., string theory and quantum loop gravity), Graham believes it is possible to achieve similar results in rhetorical studies by approaching standard rhetorical perspectives from the ontological viewpoints enabled by RNM.What makes Graham's angle on this claim particularly unique is his premise that a proto-new materialist perspective has underpinned some of the most influential left branch approaches to rhetoric all along. While most rhetoricians start from the present when introducing concepts associated with RNM, Graham, following historically informed thinkers such as Scot Barnett, Debra Hawhee, and Thomas Rickert, demonstrates that there is a line of thinking about "symbolic action" as "situated action" that goes back to the relational approach to metaphysics put forward by philosopher Henri Bergson in the late nineteenth and early twentieth centuries. Thus, as Graham argues, if we begin with Bergson's relational ontology, rather than the postmodernist and social constructivist philosophies that (explicitly and implicitly) shaped interpretations of rhetoric's left branch during the second half of the twentieth century, we do not "need to re-engineer rhetoric" to fit the latest trends of RNM (41). From Graham's perspective, many standard rhetorical perspectives have been hospitable toward new materialism from the outset. To appreciate this fact, we simply need to recuperate the Bergsonian legacy that informs this tradition.To make a case for this Bergsonian approach to rhetoric, the first main chapter of Graham's book (chapter 2) engages in detail with the work of Kenneth Burke. As a crucial founder of the symbolic action paradigm, Graham believes that if he can show the influence of Bergson on Burke's thought, he can, in turn, demonstrate how scholarship informed by Burke is also influenced by Bergson. To trace the influence of Bergson on Burke's thinking, Graham focuses on Burke's early work, especially Permanence and Change. In contrast to Burke's later writings (e.g., A Grammar of Motives, A... (shrink)

In recent years, philosophical work directly concerned with the practice of mathematics has intensified, giving rise to a movement known as the philosophy of mathematical practice . In this paper we offer a survey of this movement aimed at mathematics educators. We first describe the core questions philosophers of mathematical practice investigate as well as the philosophical methods they use to tackle them. We then provide a selective overview of work in the philosophy of mathematical practice covering topics including the (...) distinction between formal and informal proofs, visualization and artefacts, mathematical explanation and understanding, value judgments, and mathematical design. We conclude with some remarks on the potential connections between the philosophy of mathematical practice and mathematics education. (shrink)

Background Despite the great benefits of intelligent assistive technology (IAT) for dementia care – for example, the enhanced safety and increased independence of people with dementia and their caregivers – its practical adoption is still limited. The social and ethical issues pertaining to IAT in dementia care, shaped by factors such as culture, may explain these limitations. However, most studies have focused on understanding these issues within one cultural setting only. Therefore, the aim of this study was to explore and (...) compare the attitudes of Israeli and German dementia experts toward IAT in dementia care, to contribute to a more cultural-comparative perspective. Methods Semi-structured interviews were conducted with 35 experts (15 Israelis and 20 Germans) in key roles in health and community services for people with dementia as well as in the fields of dementia and IAT (e.g., computer science, electrical/biomedical engineering, ethics, nursing, and gerontology). Thematic content analysis was used to analyze the data. Findings Israeli and German experts identified the same social accelerators in the development and implementation of IAT in dementia care (i.e., changes in family structure and social digitization) and benefits of adopting IAT (e.g., enhancing the safety of people with dementia and increasing their independence). However, there were differences in inhibitor/risk assessments between the two groups. Namely, economic considerations and the cognitive capacity of people with dementia were identified by both groups as inhibitors, while Israeli experts additionally reported stigma and ageism. Whereas both groups agreed that IAT might reduce human connection, and that the technology is not yet reliable enough, German experts highlighted concerns regarding privacy; in contrast, Israeli experts prioritized safety over privacy. Conclusions Our research findings allow for the identification of relevant similarities but also important differences between German and Israeli experts’ perspectives. As such, an important basis has been provided for a more in-depth discussion regarding where, why, and how culturally-sensitive technology development is needed. (shrink)

New materialism is one of the most important emerging trends in the humanities and social sciences, but it is also one of the least understood. This is because, as a term of ongoing contest...

In practice, mathematical proofs are most often the result of careful planning by the agents who produced them. As a consequence, each mathematical proof inherits a plan in virtue of the way it is produced, a plan which underlies its “architecture” or “unity”. This paper provides an account of plans and planning in the context of mathematical proofs. The approach adopted here consists in looking for these notions not in mathematical proofs themselves, but in the agents who produced them. The (...) starting point is to recognize that to each mathematical proof corresponds a proof activity which consists of a sequence of deductive inferences—i.e., a sequence of epistemic actions—and that any written mathematical proof is only a report of its corresponding proof activity. The main idea to be developed is that the plan of a mathematical proof is to be conceived and analyzed as the plan of the agent(s) who carried out the corresponding proof activity. The core of the paper is thus devoted to the development of an account of plans and planning in the context of proof activities. The account is based on the theory of planning agency developed by Michael Bratman in the philosophy of action. It is fleshed out by providing an analysis of the notions of intention—the elementary components of plans—and practical reasoning—the process by which plans are constructed—in the context of proof activities. These two notions are then used to offer a precise characterization of the desired notion of plan for proof activities. A fruitful connection can then be established between the resulting framework and the recent theme of modularity in mathematics introduced by Jeremy Avigad. This connection is exploited to yield the concept of modular presentations of mathematical proofs which has direct implications for how to write and present mathematical proofs so as to deliver various epistemic benefits. The account is finally compared to the technique of proof planning developed by Alan Bundy and colleagues in the field of automated theorem proving. The paper concludes with some remarks on how the framework can be used to provide an analysis of understanding and explanation in the context of mathematical proofs. (shrink)

This chapter contains sections titled: Sources of Jewish Tradition Education in Biblical and Rabbinic Thought The Study of Sacred Texts.

In this short but pointed “Open Letter,” Ashrawi lists a number of aggressions perpetrated by Israeli troops and armed settlers against Palestinian civilians in the Occupied Territories, to redress the ”converse version of reality” promoted by U.S. Secretary of State Madeleine Albright, who claims that the U.S. seeks to be “an even-handed peace broker,” yet presents Israel as a victimized, besieged country, rather than an occupying force guilty of grievous UN-recognized crimes against humanity.

We propose a dynamic-epistemic analysis of the different epistemic operations constitutive of the process of interrogative inquiry, as described by Hintikka’s Interrogative Model of Inquiry (IMI). We develop a dynamic logic of questions for representing interrogative steps, based on Hintikka’s treatment of questions in the IMI, along with a dynamic logic of inferences for representing deductive steps, based on the tableau method. We then merge these two systems into a dynamic logic of interrogative inquiry which articulates a joint treatment of (...) questions and inferences, providing thereby a unified framework representing the informational dynamics of interrogative inquiry. We provide sound and complete axiomatic systems for the three dynamic logics that we introduce, we compare our framework with existing approaches, and we finally propose several directions for further work. (shrink)

The Islamic education curriculum in Indonesia tends to be partial and dichotomous. However, Muhammadiyah has reformed the holistic–integrative curriculum as a solution for the sustainability of education. This study aims to reveal a special curriculum reform in the holistic–integrative Muhammadiyah education system to solve the dichotomous problems and the inadequacy in the Islamic education curriculum in Indonesia. Furthermore, the study describes a specific curriculum model that includes all aspects of students’ personality and integrates science and technology with Islamic values. Data (...) were collected from previous studies and special curriculum documents of Al-Islam, Kemuhammadiyahan and Arabic Language (ISMUBA). Curriculum documents were obtained from elementary, junior and Muhammadiyah senior high school, published by Primary and Secondary Education Assembly of Muhammadiyah Central Board in 2017. Data analysis used a content model with a systematic and rational framework. The results showed that the special reform curriculum based on the holistic–integrative model develops students’ potentials, including spiritual, emotional, intellectual and transcendental intelligence, in an integrated manner. Contribution: The holistic–integrative curriculum is a future Islamic education model for the sustainability of private Islamic education system in Indonesia that can form a good, faithful, pious and a noble community. (shrink)

The deviation of mathematical proof—proof in mathematical practice—from the ideal of formal proof—proof in formal logic—has led many philosophers of mathematics to reconsider the commonly accepted view according to which the notion of formal proof provides an accurate descriptive account of mathematical proof. This, in turn, has motivated a search for alternative accounts of mathematical proof purporting to be more faithful to the reality of mathematical practice. Yet, in order to develop and evaluate such alternative accounts, it appears as a (...) necessary prerequisite to first possess a clear picture of what the deviation of mathematical proof from formal proof consists in. The present work aims to contribute building such a picture by investigating the relation between the elementary steps of deduction constituting the two types of proofs—mathematical inference and logical inference. Many claims have been made in the literature regarding the relation between mathematical inference and logical inference, most of them stating that the former is lacking properties that are constitutive of the latter. Such differentiating claims are, however, usually put forward without a clear conception of the properties occurring in them, and are generally considered to be immediately justified by our direct acquaintance, or phenomenological experience, with the two types of inferences. The present study purports to advance our understanding of the relation between mathematical inference and logical inference by developing a detailed philosophical analysis of the differentiating claims, that is, an analysis of the meaning of the differentiating claims—through the properties that occur in them—as well as the reasons that support them. To this end, we provide at the outset a representative list of the different properties of logical inference that have occurred in the differentiating claims, and we notice that they all boil down to the three properties of formality, generality, and mechanicality. For each one of these properties, our analysis proceeds in two steps: we first provide precise conceptual characterizations of the different ways logical inference has been said to be formal, general, and mechanical, in the philosophical and logical literature on formal proof; we then examine why mathematical inference does not appear to be formal, general, and mechanical, for the different variations of these notions identified. Our study results in a precise conceptual apparatus for expressing and discussing the properties differentiating mathematical inference from logical inference, and provides a first inventory of the various reasons supporting the observations of those differences. The differentiating claims constitute thus a set of data that any philosophical account of mathematical inference and proof purporting to be more faithful to mathematical practice ought to be able to accommodate and explain. (shrink)

Topological relations such as inside, outside, or intersection are ubiquitous to our spatial thinking. Here, we examined how people reason deductively with topological relations between points, lines, and circles in geometric diagrams. We hypothesized in particular that a counterexample search generally underlies this type of reasoning. We first verified that educated adults without specific math training were able to produce correct diagrammatic representations contained in the premisses of an inference. Our first experiment then revealed that subjects who correctly judged an (...) inference as invalid almost always produced a counterexample to support their answer. Noticeably, even if the counterexample always bore a certain level of similarity to the initial diagram, we observed that an object was more likely to be varied between the two drawings if it was present in the conclusion of the inference. Experiments 2 and 3 then directly probed counterexample search. While participants were asked to evaluate a conclusion on the basis of a given diagram and some premisses, we modulated the difficulty of reaching a counterexample from the diagram. Our results indicate that both decreasing the counterexample density and increasing the counterexample distance impaired reasoning performance. Taken together, our results suggest that a search procedure for counterexamples, which proceeds object‐wise, could underlie diagram‐based geometric reasoning. Transposing points, lines, and circles to our spatial environment, the present study may ultimately provide insights on how humans reason about topological relations between positions, paths, and regions. (shrink)

In On Female Body Experience: “Throwing Like A Girl” and Other Essays, Iris Marion Young examines the feminine modalities representing women in contemporary industrial and commercial society. She reveals the “root” of these modalities, which she refers to as the state of “self-reference.” The main task of the present paper is to extend Young’s understanding of femininity by arguing that this state of “self-reference” is rooted in the state of “reversibility.” Such rootedness is essential to Young’s concept of femininity, as (...) it explains the constitution of the objectified state that represents feminine experience. It also justifies the three aspects of her definition of femininity: the non-essentiality of femininity, the structural conditions that constitute women’s situations in a certain society, and women’s lived experiences according to these situations. This paper is mainly inspired by Maurice Merleau-Ponty’s existential and phenomenological approach, as represented in his Phenomenology of Perception and The Visible and the Invisible. It is also inspired by Edmund Husserl’s concept of “double senses” as presented in his Thing and Space: Lectures of 1907. An exploration of Young’s definition of “femininity” in relation to arguments related to essentialism is followed by a discussion of her argument that the modality of self-reference is the root of the other feminine modalities. The rest of the paper looks at how her understanding of femininity can be extended by presenting the concept of reversibility as the root of the formation of the state of self-reference and the other feminine modalities. (shrink)

In this essay I critique two influential accounts of rational autonomy in common schooling that conceive liberalism as an ideal form of life, and I offer an alternative approach to democratic education that views liberal theory as concerned with coexistence among rival ways of living. This view places moral agency, not rational autonomy, at the heart of schooling in liberal societies—a moral agency grounded in initiation into dynamic traditions that enable self-definition and are accompanied by exposure to life-paths other than (...) one’s own. This alternative challenges the tendency in large diverse democracies (such as those of the US and the UK) to prefer common to particularistic schools, thereby placing many types of faith and secular schools on a more equal footing and providing moral justification for education in the national cultures of small liberal republics (such as Denmark, Israel and Lithuania) that maintain special relationships to particular groups while acknowledging the rights of all citizens. I call this approach the pedagogy of difference. (shrink)

The Interrogative Model of Inquiry and Dynamic Epistemic Logics are two central paradigms in formal epistemology. This paper is motivated by the observation of a significant complementarity between them: on the one hand, the IMI provides a framework for investigating inquiry represented as an idealized game between an Inquirer and Nature, along with an account of the interaction between questions and inferences in information-seeking processes, but is lacking a formulation in the multi-agent case; on the other hand, DELs model various (...) operations of information change in multi-agent systems, but the field is lacking a proper integration of question and inference dynamics, along with an application to the investigation of inquiry processes. The goal of this paper is to integrate the two paradigms in such a way as to combine their respective insights. To this end, we develop a formal system called DEL $$_\mathrm{IMI }$$ IMI which aims to represent the interaction between question and inference dynamics in inquiry—as described by the IMI—in a multi-agent setting, and this in such a way as to enable an investigation of inquiry games with multi-agent dimensions. The DEL $$_\mathrm{IMI }$$ IMI system is designed to represent the possible moves of such inquiry games through three types of epistemic actions: agents addressing questions to Nature, agents addressing questions to other agents, agents drawing logical inferences. We then show how the resulting framework can be used to formally define multi-agent inquiry games. We conclude by evaluating the interest of the DEL $$_\mathrm{IMI }$$ IMI system for the IMI and DELs paradigms. (shrink)

How should we assess programs dedicated to education in virtue? One influential answer draws on quantitative research designs. By measuring the inputs and processes that produce the highest levels of virtue among participants according to some reasonable criterion, in this view, we can determine which programs engender the most desired results. Although many outcomes of character education can undoubtedly be assessed in this way, taken on its own, this approach may support favorable judgments about programs that indoctrinate rather than educate, (...) because education in character entails teleological thinking that is volitional not merely determined. I argue instead that proper assessment of virtue requires an expansive view of character education in both particular and common goods that avoids the tendency to indoctrinate and an inclusive conception of measurement that takes into account qualitative in addition to quantitative methodologies. (shrink)

In this paper I ask how educational researchers can believe the subjective perceptions of qualitative participant-observers given the concern for objectivity and generalisability of experimental research in the behavioural and social sciences. I critique the most common answer to this question within the educational research community, which posits the existence of two (or more) equally legitimate epistemological paradigms—positivism and constructivism—and offer an alternative that places a priority in educational research on understanding the purposes and meanings humans attribute to educational practices. (...) Only within the context of what I call a transcendent view from somewhere—higher ideals that govern human activities—can we make sense of quantitative as well as qualitative research findings. (shrink)

This chapter contains sections titled: Religion Spirituality Education.

Mathematical proofs are not sequences of arbitrary deductive steps—each deductive step is, to some extent, rational. This paper aims to identify and characterize the particular form of rationality at play in mathematical proofs. The approach adopted consists in viewing mathematical proofs as reports of proof activities—that is, sequences of deductive inferences—and in characterizing the rationality of the former in terms of that of the latter. It is argued that proof activities are governed by specific norms of rational planning agency, and (...) that a deductive step in a mathematical proof qualifies as rational whenever the corresponding deductive inference in the associated proof activity figures in a plan that has been constructed rationally. It is then shown that mathematical proofs whose associated proof activities violate these norms are likely to be judged as defective by mathematical agents, thereby providing evidence that these norms are indeed present in mathematical practice. We conclude that, if mathematical proofs are not mere sequences of deductive steps, if they possess a rational structure, it is because they are the product of rational planning agents. (shrink)

Although there exist today a variety of non-deductive reliable processes able to determine the truth of certain mathematical propositions, proof remains the only form of justification accepted in mathematical practice. Some philosophers and mathematicians have contested this commonly accepted epistemic superiority of proof on the ground that mathematicians are fallible: when the deductive method is carried out by a fallible agent, then it comes with its own level of reliability, and so might happen to be equally or even less reliable (...) than existing non-deductive reliable processes—I will refer to this as the reliability argument. The aim of this article is to examine whether the reliability argument forces us to reconsider the commonly accepted epistemic superiority of the deductive method over non-deductive reliable processes. I will argue that the reliability argument is fundamentally correct, but that there is another epistemic property differentiating the deductive method from non-deductive reliable processes. This property is based on the observation that although mathematicians are fallible agents, they are also self-correcting agents. This means that when a proof is produced that only contains repairable mistakes, given enough time and energy, a mathematician or a group thereof should be able to converge towards a correct proof through a finite number of verification and correction rounds, thus providing a guarantee that the considered proposition is true, something that non-deductive reliable processes will never be able to produce. From this perspective, the standard of justification adopted in mathematical practice should be read in a diachronic way: the demand is not that any proof that is ever produced be correct—which would amount to requiring that mathematicians are infallible—but rather that, over time, proofs that contain repairable mistakes be corrected, and proofs that cannot be repaired be rejected. (shrink)

In mathematics, any form of probabilistic proof obtained through the application of a probabilistic method is not considered as a legitimate way of gaining mathematical knowledge. In a series of papers, Don Fallis has defended the thesis that there are no epistemic reasons justifying mathematicians’ rejection of probabilistic proofs. This paper identifies such an epistemic reason. More specifically, it is argued here that if one adopts a conception of mathematical knowledge in which an epistemic subject can know a mathematical proposition (...) based solely on a probabilistic proof, one is then forced to admit that such an epistemic subject can know several lottery propositions based solely on probabilistic evidence. Insofar as knowledge of lottery propositions on the basis of probabilistic evidence alone is denied by the vast majority of epistemologists, it is concluded that this constitutes an epistemic reason for rejecting probabilistic proofs as a means of acquiring mathematical knowledge. (shrink)

Euclidean diagrammatic reasoning refers to the diagrammatic inferential practice that originated in the geometrical proofs of Euclid’s Elements. A seminal philosophical analysis of this practice by Manders (‘The Euclidean diagram’, 2008) has revealed that a systematic method of reasoning underlies the use of diagrams in Euclid’s proofs, leading in turn to a logical analysis aiming to capture this method formally via proof systems. The central premise of this paper is that our understanding of Euclidean diagrammatic reasoning can be fruitfully advanced (...) by confronting these logical and philosophical analyses with the field of cognitive science. Surprisingly, central aspects of the philosophical and logical analyses resonate in very natural ways with research topics in mathematical cognition, spatial cognition and the psychology of reasoning. The paper develops these connections, concentrating on four issues: (1) the cognitive origins of Euclidean diagrammatic reasoning, (2) the cognitive representations of spatial relations in Euclidean diagrams, (3) the nature of the cognitive processes and cognitive representations involved in Euclidean diagrammatic reasoning seen as a form of visuospatial relational reasoning and (4) the complexity of Euclidean diagrammatic reasoning for the human cognitive system. For each of these issues, our analysis generates concrete experiment proposals, opening thereby the way for further empirical investigations. The paper is thus a prolegomenon to a research program on Euclidean diagrammatic reasoning at the crossroads of logic, philosophy and cognitive science. (shrink)