The aim of this study is to investigate students’ conceptions about proof in mathematics and mathematics teaching. A five‐point Likert‐type questionnaire was administered in order to gather data. The sample of the study included 33 first‐year secondary school mathematics students . The data collected were analysed and interpreted using the methods of qualitative and quantitative analysis. The results have revealed that the students think that mathematical proof has an important place in mathematics and mathematics education. The students’ studying methods for (...) exams based on imitative reasoning which can be described as a type of reasoning built on copying proof, for example, by looking at a textbook or course notes proof or through remembering a proof algorithm. Moreover, they addressed to the differences between mathematics taught in high school and university as the main cause of their difficulties in proof and proving. (shrink)

This paper argues that a certain type of self-referential sentence falsifies the widespread assumption that a declarative sentence's meaning is identical to its truth condition. It then argues that this problem cannot be assimilated to certain other problems that the assumption in question is independently known to face.

Is democracy a requirement of justice or an instrument for realizing it? The correct answer to this question, I argue, depends on the background circumstances against which democracy is defended. In the presence of thin reasonable disagreement about justice, we should value democracy only instrumentally (if at all); in the presence of thick reasonable disagreement about justice, we should value it also intrinsically, as a necessary demand of justice. Since the latter type of disagreement is pervasive in real-world politics, I (...) conclude that theories of justice designed for our world should be centrally concerned with democracy. (shrink)

An anti-dispositionalist interpretation of grammatical knowledge would maintain that such knowledge exists whether or not it can be behaviourally manifested; a dispositionalist interpretation, on the other hand, would identify that knowledge with the in principle possibility of certain behavioural manifestations. The purpose of this paper is to present a preliminary case for the dispositionalist interpretation by accomplishing two complementary tasks: first, rejecting a prominent argument against the dispositionalist interpretation; second, advancing an original argument against the anti-dispositionalist interpretation. Both tasks involve (...) rebuttals of certain key theses associated with Chomskyan grammatical metatheory, and therefore, provide new opportunities for assessing its viability. -/- . (shrink)

In his recent bookThe Idea of Justice, Amartya Sen suggests that political philosophy should move beyond the dominant, Rawls-inspired, methodological paradigm – what Sen calls ‘transcendental institutionalism’ – towards a more practically oriented approach to justice: ‘realization-focused comparison’. In this article, I argue that Sen's call for a paradigm shift in thinking about justice is unwarranted. I show that his criticisms of the Rawlsian approach are either based on misunderstandings, or correct but of little consequence, and conclude that the Rawlsian (...) approach already delivers much of what Sen himself wants from a theory of justice. (shrink)

A globalized world, some argue, needs a global democracy. But there is considerable disagreement about whether global democracy is an ideal worth pursuing. One of the main grounds for scepticism is captured by the slogan: “No global demos, no global democracy.” The fact that a key precondition of democracy—a demos—is absent at the global level, some argue, speaks against the pursuit of global democracy. The paper discusses four interpretations of the skeptical slogan—each based on a specific account of the notion (...) of “the demos”—and concludes that none of them establishes that the global democratic ideal must be abandoned. In so doing, the paper (i) systematizes different types of objections against global democracy, thus bringing some clarity to an otherwise intricate debate and (ii) offers a robust but qualified defense of the global democratic ideal. (shrink)

This article provides a conceptual map of the debate on ideal and non‐ideal theory. It argues that this debate encompasses a number of different questions, which have not been kept sufficiently separate in the literature. In particular, the article distinguishes between the following three interpretations of the ‘ideal vs. non‐ideal theory’ contrast: (i) full compliance vs. partial compliance theory; (ii) utopian vs. realistic theory; (iii) end‐state vs. transitional theory. The article advances critical reflections on each of these sub‐debates, and highlights (...) areas for future research in the field. (shrink)

In this article I present a disagreement between classical and constructive approaches to predicativity regarding the predicative status of so-called generalised inductive definitions. I begin by offering some motivation for an enquiry in the predicative foundations of constructive mathematics, by looking at contemporary work at the intersection between mathematics and computer science. I then review the background notions and spell out the above-mentioned disagreement between classical and constructive approaches to predicativity. Finally, I look at possible ways of defending the constructive (...) predicativity of inductive definitions. (shrink)

In this article, Laura Quintana elaborates on a conceptual distinction between violence and rage. Along with this distinction, she recognises that while rage may possess a destructive potential, it can also be politicised in emancipatory practices that confront conditions of injustice and structural violence. Her analysis centers on contemporary political movements in Latin America, which she views as collective manifestations of rage. Within these movements, the manifestation of rage is intertwined with forms of care and communal labor. Quintana characterises (...) such movements as embodying a “dignified rage” (digna rabia) that calls for an emancipatory transformation of the status quo. She contends that these practices can be understood in terms of the notion of “anti-violence,” as it has been coined by Étienne Balibar. In this she diverges from Judith Butler, since, according to Quintana’s analysis, such practices challenge the demand for non-violence that has often served to neutralise the disruptive force of such manifestations and movements. (shrink)

In this article I argue that characterizations of anger as a hostile emotion may be mistaken. My project is empirically informed and is partly descriptive, partly diagnostic. It is descriptive in that I am concerned with what anger is, and how it tends to manifest, rather than with what anger should be or how moral anger is manifested. The orthodox view on anger takes it to be, descriptively, an emotion that aims for retribution. This view fits well with anger being (...) a hostile emotion, as retribution is punitive. I will argue that a different view of anger deserves our attention. On this alternative view, anger aims for recognition of harms done, rather than for the punishment of those who have committed them. I argue that we have reason to favour a strong view that excludes retribution from anger’s main aims. In addition, I offer a diagnosis of the reasons that led the retributive view of anger to become, and remain, orthodoxy. This diagnosis provides indirect reason to give my descriptive proposal serious consideration, for it highlights that the orthodox view has dominated folk and philosophical conceptions of anger for reasons that do not speak in favour of the view’s veracity. The view that anger is a hostile emotion will therefore emerge as in need of serious scrutiny. (shrink)

Timothy Williamson has argued that, unless the speech act of assertion were supposed to be governed by his so-called Knowledge Rule, one could not explain why sentences of the form "A and I do not know that A" are unassertable. This paper advances three objections against that argument, of which the first two aim to show that, even assuming that Williamson's explanandum has been properly circumscribed, his explanation would not be correct, and the third aims to show that his explanandum (...) has not been properly circumscribed. (shrink)

Frank Jackson often writes as if his descriptivist account of public language meanings were just plain common sense. How else are we to explain how different speakers manage to communicate using a public language? And how else can we explain how individuals arrive at confident judgments about the reference of their words in hypothetical scenarios? Our aim in this paper is to show just how controversial the psychological assumptions behind in Jackson’s semantic theory really are. First, we explain how Jackson’s (...) theory goes well beyond the commonsense platitudes he cites in its defence. Second, we sketch an alternative explanation of those platitudes, the jazz model of meaning, which we argue is more psychologically realistic. We conclude that the psychological picture presupposed by Jackson’s semantic theory stands in need of a much more substantial defence than he has so far offered. (shrink)

Much contemporary political philosophy claims to be Kant-inspired, but its aims and method differ from Kant's own. In his recent book, Force and Freedom, Arthur Ripstein advocates a more orthodox Kantian outlook, presenting it as superior to dominant (Kant-inspired) views. The most striking feature of this outlook is its attempt to ground the whole of political morality in one right: the right to freedom, understood as the right to be independent of others’ choices. Is Ripstein's Kantian project successful? In this (...) research note I argue that it is not. First, I suggest that Ripstein's notion of freedom is viciously circular. It is meant to ground all rights, but in fact it presupposes an account of those rights. Second, I show that—independently of its inability to ground a whole political morality—such a moralized understanding of freedom is normatively unappealing. (shrink)

In this article, I sketch a Kant-inspired liberal account of human rights: the freedom-centred view. This account conceptualizes human rights as entitlements that any political authority—any state in the first instance—must secure to qualify as a guarantor of its subjects' innate right to freedom. On this picture, when a state (or state-like institution) protects human rights, it reasonably qualifies as a moral agent to be treated with respect. By contrast, when a state (or state-like institution) fails to protect human rights, (...) it loses its moral status and becomes liable to both internal and external interference. I argue that this account not only steers a middle course between so-called natural-law and political approaches to human rights but also satisfies three important theoretical desiderata— explanatory power, functional specificity, and critical capacity. (shrink)

Reference to the state is ubiquitous in debates about global justice. Some authors see the state as central to the justification of principles of justice, and thereby reject their extension to the international realm. Others emphasize its role in the implementation of those principles. This chapter scrutinizes the variety of ways in which the state figures in the global-justice debate. Our discussion suggests that, although the state should have a prominent role in theorizing about global justice, contrary to what is (...) commonly thought, acknowledging this role does not lead to anti-cosmopolitan conclusions, but to the defense of an “intermediate” position about global justice. From a justificatory perspective, we argue, the state remains a key locus for the application of egalitarian principles of justice, but is not the only one. From the perspective of implementation, we suggest that state institutions are increasingly fragile in a heavily interdependent world, and need to be supplemented—though not supplanted—with supranational authorities. (shrink)

Predicativity is a notable example of fruitful interaction between philosophy and mathematical logic. It originated at the beginning of the 20th century from methodological and philosophical reflections on a changing concept of set. A clarification of this notion has prompted the development of fundamental new technical instruments, from Russell's type theory to an important chapter in proof theory, which saw the decisive involvement of Kreisel, Feferman and Schütte. The technical outcomes of predica-tivity have since taken a life of their own, (...) but have also produced a deeper understanding of the notion of predicativity, therefore witnessing the " light logic throws on problems in the foundations of mathematics. " (Feferman 1998, p. vii) Predicativity has been at the center of a considerable part of Feferman's work: over the years he has explored alternative ways of explicating and analyzing this notion and has shown that predicative mathematics extends much further than expected within ordinary mathematics. The aim of this note is to outline the principal features of predicativity, from its original motivations at the start of the past century to its logical analysis in the 1950-60's. The hope is to convey why predicativity is a fascinating subject, which has attracted Feferman's attention over the years. (shrink)

The aim of this chapter is to discuss the relevance that emotions can play in our epistemic life considering the state of the art of the philosophical debate on emotions. The strategy is the one of focusing on the three main models on emotions as evaluative judgements, bodily feelings, and perceptions, following the fil rouge of emotion intentionality for rising questions about their epistemic functions. From this examination, a major challenge to mainstream epistemology arises, the one that asks to provide (...) prominence to the epistemic agent and to her affects. This chapter discusses these implications, also providing an overview of the many alternatives available nowadays in epistemology, arguing for an open, comprehensive, and interdisciplinary approach to emotions in knowledge. (shrink)

Recent work at the junction of epistemology and political theory focuses on the notion of epistemic injustice, the injustice of being wronged as a knower. Miranda Fricker (2007) identifies two kinds of epistemic injustice. I focus here on hermeneutical injustice in an attempt to identify a difficulty for Fricker's account. In particular, I consider the significance of background social conditions and suggest that an epistemic injustice should not rely on other forms of disadvantage to achieve its status as an injustice. (...) Thus reformulated, the notion of epistemic injustice can help us to achieve an even deeper understanding of the relationship between knowledge and privilege. (shrink)

A life without liberty and material resources is not a good life. Equally, a life devoid of meaningful social relationships—such as friendships, family attachments, and romances—is not a good life. From this it is tempting to conclude that just as individuals have rights to liberty and material resources, they also have rights to access meaningful social relationships. I argue that this conclusion can be defended only in a narrow set of cases. ‘Pure’ social relationship deprivation—that is, deprivation that is not (...) caused, or accompanied, by deficits in liberties and material resources— mostly generates demands of private beneficence. I suggest that social relationship deprivation is unjust, hence a rights violation, only when it is due to factors—such as one’s race—that are irrelevant to one’s being a good participant in social relationships. I thus conclude that access to meaningful social relationships is not a fundamental concern for theories of justice. (shrink)

In this paper, I focus on the concept of embeddedness as the background against which eros is a force and a power in and through interactions. To go beyond an internalist account of eros, I engage in a dialogue with some philosophical accounts of desire from an enactive perspective.This enables me to shed light on the location of the embodied tension as “in-between” lovers and “all-around” them. Crucial to this tensional account of embedded eros is the intertwining between self and (...) others’ becomings in processes of participatory sense-making. Through participatory sense-making lovers make their worlds, creating new ways of being and knowing in the ensemble. I advance some steps towards an enactive ethics of eros where, I claim, the cultivation of the space in-between and all-around lovers is the key to avoid the traps of a degenerated form of eros. (shrink)

Le blockbuster, c'est de l'explosif. A l'origine du mot, une bombe utilisée pendant la Seconde Guerre mondiale, capable de détruire tout un pâté de maisons. Et il n'y a sans doute pas d'image plus typique du blockbuster que celle d'une gigantesque explosion. Les déflagrations dont il s'agit dans ce livre ne sont pas uniquement celles que racontent les films. Produit grand public, confectionné grâce à d'immenses investissements financiers, le blockbuster s'apparente à une bombe à fragmentation, pulvérisant l'objet filmique dans des (...) contextes de consommation qui n'ont plus rien à voir avec le cinéma, se diffractant sur le marché mondial sous forme de produits dérivés, T-shirts, figurines et autres gadgets. Sept auteurs, philosophes et critiques, se sont penchés sur quelques blockbusters de leur choix. A chaque fois, les films dynamitent la pensée et l'entraînent dans des directions inattendues : les gremlins sabotent les écrans et contaminent les images, Godzilla et King Kong se font machines à échanger les regards, Batman devient le théâtre d'une déconstruction du politique, Alien met en scène les métamorphoses d'un récit qui ne change jamais, et Titanic coule comme cette industrie lourde qu'est le cinéma lui-même... (shrink)

Various philosophers authors have argued—on the basis of powerful examples—that we can have compelling moral or practical reasons to believe, even when the evidence suggests otherwise. This paper explores an alternative story, which still aims to respect widely shared intuitions about the motivating examples. Specifically, the paper proposes that what is at stake in these cases is not belief, but rather acceptance—an attitude classically characterized as taking a proposition as a premise in practical deliberation and action. I suggest that acceptance’s (...) theoretical usefulness in the ethics of belief has been hidden by its psychological obscurity. I thus aim to develop an empirically adequate and mechanistically specific psychological profile of acceptance. I characterize acceptance as centrally involving a cognitive gating function, in which we prevent a target belief state from having its characteristic downstream effects on reasoning, cognition, and action, and restructure those downstream processes. I then argue that there is substantial empirical support for the existence of the cognitive mechanisms needed to instantiate this view, coming from the science of emotion regulation. I argue that acceptance involves deploying the same mechanisms used in emotional response modulation to belief states: acceptance is doxastic response modulation. I then propose that having a better understanding of the psychological profile of acceptance leaves us better positioned to appreciate its potential usefulness for making progress on various puzzles within the ethics of belief. (shrink)

When and why do socially constructed norms—including the laws of the land, norms of etiquette, and informal customs—generate moral obligations? I argue that the answer lies in the duty to respect others, specifically to give them what I call “agency respect.” This is the kind of respect that people are owed in light of how they exercise their agency. My central thesis is this: To the extent that (i) existing norms are underpinned by people’s commitments as agents and (ii) they (...) do not conflict with morality, they place moral demands on us on agency-respect grounds. This view of the moral force of socially constructed norms, I suggest, is superior to views that deny the moral force of such norms, and it elegantly explains certain instances of wrongdoing that would otherwise remain unaccounted for. (shrink)