After the 1930s, the research into the foundations of mathematics changed.None of its main directions (logicism, formalism and intuitionism) had any longer the pretension to be the only true mathematics.Usually, the determining factor in the change is considered to be Gödel?s work, while Heyting?s role is neglected.In contrast, in this paper I first describe how Heyting directly suggested the abandonment of the big foundational questions and the putting forward of a new kind of foundational research consisting in the isolation of (...) formal, intuitive, logical and platonistic elements within classical mathematics.Furthermore, I describe how Heyting indirectly influenced the abandon?ment of the old directions of foundational research by making out some lists of degrees of evidence that exist within intuitionism. (shrink)

The use of the three labels to denote the three foundational schools of the early twentieth century are now part of literature. Yet, neither their number nor the...

Friedrich Nietzsche and Luitzen Egbertus Jan Brouwer had strong personalities and freely expressed unconventional opinions. In particular, they dared to challenge the traditional view that considered Aristotelian logic as being absolute and intrinsic to man. Although they formed this opinion in different ways and in different contexts, they both based it on a view of life that considered it as a struggle for power in which logic was a weapon. Therefore, it is interesting to carry out an in-depth analysis on (...) the origins, the core of their opinions concerning logic and the consequences. As a matter of fact, a detailed analysis and a comparison of their contributions to the philosophy of logic have not yet been carried out. In this paper, I intend to present the results of my research.. (shrink)

Julius König is famous for his mistaken attempt to demonstrate that the continuum hypothesis was false. It is also known that the only positive result that could have survived from his proof is the paradox which bears his name. Less famous is his 1914 book Neue Grundlagen der Logik, Arithmetik und Mengenlehre. Still, it contains original contributions to logic, like the concept of metatheory and the solution of paradoxes based on the refusal of the law of bivalence. We are going (...) to discover them by analysing the content of the book. (shrink)

Paul Bernays was both a philosopher and a mathematician. He is famous for his logical-mathematical production (also in collaboration with David Hilbert), while his philosophical works have been given less consideration. The present article is an attempt to reconstruct the way that led Bernays from his early writings on ethics to his final epistemological thought.

A reproach has been done many times to post-modernism: its picking up mathematical notions or results, mostly by misrepresenting their real content, in order to strike the readers and obtaining their assent only by impressing them . In this paper I intend to point out that although Alain Badiou’s approach to philosophy starts with taking distance both from analytic philosophy and from French post-modernism, the categories that he uses for labelling logicism, formalism and intuitionism do not reflect the real content (...) of the foundational schools. Hence, a re-thinking from him would be required about them, otherwise he would risk the same reproaches as post-modernists. (shrink)

This paper presents the content of the unpublished notes that the Dutch mathematician Arend Heyting wrote in different periods of his life on solipsism and that are preserved in Heyting's archive at the University of Amsterdam. Most of the notes are quoted here and translated into English. Their study shows the originality of Heyting's reflections on a subject that was typical of his master, L. E. J. Brouwer, the father of intuitionism.

At the beginning of the twentieth century, among the foundational schools of mathematics appeared ‘intuitionism’ by Dutchman L. E. J. Brouwer, who based arithmetic on the intuition of time and all mental constructions that could be made out of it. His pupil Arend Heyting was the first populariser of intuitionism, and he repeatedly emphasised that no philosophy was required to practise intuitionism so that such mathematics could be shared by anyone. Still, stimulated by invitations to humanistic conferences, he wrote a (...) series of notes, preserved in the State Archives, Haarlem, about solipsism. In them, he operated a series of theoretical reflections consisting of the stripping away of the patterns that are part of our consciousness and their subsequent progressive re-introduction, in order to understand the formation of our Self as distinct from the natural world and other humans. In 1996, following a stroke, neuroscientist Jill Bolte Taylor experienced the deprivation of certain abilities in her brain and their successive regaining. In her experience there are remarkable similarities with what Heyting had hypothesised about the formation of the self. The purpose of this article is to highlight them and point out how Heyting's theoretical construct was found to be embodied in the brain. (shrink)

La littérature témoigne d’une tendance croissante à soutenir l’intuitionisme par la phénoménologie. Le disciple de Brouwer Arend Heyting est considéré comme un précurseur de cette tendance, parce qu’il usait d’une terminologie phénoménologique en vue de définir la négation intuitioniste, en élaborant la première logique intuitioniste. Dans cet article, l’auteur tente d’explorer — en référence aux matériaux inédits conservés aux Archives Heyting — ce qui, dans la pensée de Heyting, est compatible avec la phénoménologie. Dans la conclusion, l’auteur suggère que Heyting (...) et Husserl, pour autant qu’ils pensent tous deux que la conscience doit être le véritable commencement de la connaissance, partagent une même attitude antipsychologiste coexistant avec une tentative de dépassement du solipsisme. Toutefois, le concept phénoménologique de degré d’évidence ne peut être appliqué à l’échelle de l’évidence de Heyting (incluant les petits nombres naturels, les grands nombres naturels, les suites infinies, le quantificateur universel), d’un côté parce qu’on ne voit pas clairement si la seconde est partagée par tous les intuitionistes, et de l’autre parce que le premier présuppose une évidence révisable qui ne cadre pas avec le point de vue de Heyting. En outre, les conceptions de la nature de la mathématique et de la logique et de leurs relations mutuelles défendues par Husserl et Heyting sont essentiellement différentes. D’un point de vue intuitioniste, la mathématique est le domaine de l’évidence, alors que la logique transcrit ses régularités. D’un point de vue phénoménologique, la mathématique demeure en dehors du domaine de l’évidence. La logique apophantique coïncide avec la mathématique (sans que l’une absorbe l’autre), mais la logique transcendantale se situe à un niveau plus élevé. (shrink)

Kant è stato il principale riferimento epistemologico di quella ricerca sui fondamenti della matematica che ha avuto in Gottlob Frege il suo massimo esponente. In seguito, tuttavia, al tramonto dei programmi logicista, intuizionista e formalista, l'esigenza di garantire la certezza della matematica passa in secondo piano rispetto al problema della natura degli enti matematici. E in tal modo che, almeno a livello terminologico, Platone subentra a Kant come essenziale referente di una diversa costruzione epistemologica. Terreno fertile della ricerca fondazionale diviene (...) infatti, a partire dagli anni '60, il dibattito fra i sostenitori e gli avversari del 'platonismo matematico': etichetta che, utilizzata per indicare la trascendenza degli enti matematici, ha non di rado banalizzato o reso marginale l'originario pensiero di Platone. Il dibattito è ancor oggi un crogiolo di idee e tendenze, tra le quali emerge, accanto alla necessità di un maggiore approfondimento dei risultati e della reale attività del matematico, l'opportunità di interrogare i testi del filosofo greco. Nella sterminata letteratura contemporanea sui fondamenti della matematica, Platone e Kant si presentano, dunque, come problematici fili d'Arianna per orientarsi, mettere ordine, cogliere una linea evolutiva o anche soltanto originali spunti teorici per nuovi orizzonti. (shrink)

Though E. W. Beth is famous for his contributions to logic aspects of his philosophical reflections and details of its development are almost unknown. In his work four periods can be distinguished: the neo-kantian, the anti-kantian, the anti-irrationalist and the logical one. Within this framework it is possible to individuate a core around which Beth developed his reflections: it is the interplay between philosophy and the sciences. His philosophy was always linked to the sciences in two ways: He steadily checked (...) his philosophical ideas against the current situation in foundational research and he occupied himself with the general question of the relationship between philosophy and the sciences. (shrink)

Though E. W. Beth is famous for his contributions to logic aspects of his philosophical reflections and details of its development are almost unknown. In his work four periods can be distinguished: the neo-kantian, the anti-kantian, the anti-irrationalist and the logical one. Within this framework it is possible to individuate a core around which Beth developed his reflections: it is the interplay between philosophy and the sciences. His philosophy was always linked to the sciences in two ways: He steadily checked (...) his philosophical ideas against the current situation in foundational research and he occupied himself with the general question of the relationship between philosophy and the sciences. (shrink)

In the nineteenth and twentieth centuries many mathematicians referred to intuition as the indispensable research tool for obtaining new results. In this essay we will analyse a group of mathematicians who interacted with Luitzen Egbertus Jan Brouwer in order to compare their conceptions of intuition. We will see how to the same word “intuition” very different meanings corresponded: they varied from geometrical vision, to a unitary view of a demonstration, to the perception of time, to the faculty of considering concepts (...) that habitually occur in our thinking separately. Furthermore, we will discover that these different meanings had a philosophical, very relevant counterside: they passed from a racial characterization of mathematics to a pluralistic view of it. (shrink)

This book summarizes the intense research that the author performed for his Ph.D. thesis , revised and with the addition of an intuitionistic critique of Husserl's concept of number. His starting point consisted of a double conviction: 1) Brouwerian intuitionism is a valid way of doing mathematics but is grounded on a weak philosophy; 2) Husserlian phenomenology can provide a suitable philosophical ground for intuitionism. In order to let intuitionism and phenomenology match, he had to solve in general two problems: (...) 1) the question of the reciprocal indifference that the authors had toward each other's theorizing which indeed they knew; 2) Husserl's general attitude of accepting classical mathematics, which contrasts with the critical attitude of the intuitionists.Moreover, in the specific case focused on by this book—concerning choice sequences, that is, sequences that can be completely lawless and that were admitted as mathematical entities by Brouwer—van Atten had to handle a further problem: such sequences do not fit the characteristic of omnitemporality that the late Husserl seemed to consider a necessary attribute of entities if they were to be considered mathematical. In as far as they are lawless we cannot predict at any moment how they will develop in the future, hence they are not determined at any particular time—they will be determined only in the future.Let us begin with van Atten's first conviction, i.e., that Brouwerian intuitionism is grounded on a weak philosophy. We have to recall that Brouwer provided a mystical attitude for proposing his intuitionism: man can obtain happiness only by keeping himself closed inside the inner Self. Any attempted movement towards the exterior world is a source of pain: in particular language and the sciences. Mathematics, in order to be morally acceptable …. (shrink)

This paper critically examines Coeckelbergh’s (2023) performative view of moral status. Drawing parallels to Searle’s social ontology, two key claims of the performative view are identified: (1) Making a moral status claim is equivalent to making a moral status declaration. (2) A successful declaration establishes the institutional fact that the entity has moral status. Closer examination, however, reveals flaws in both claims. The second claim faces a dilemma: individual instances of moral status declaration are likely to fail because they do (...) not conform to established moral discourse conventions, and reliance on declarations becomes both unnecessary and implausible for explaining widespread collective recognition of moral status. As for the first claim, accepting it undermines the potential for meaningful moral disagreement. As a remedy, this paper proposed a shift in perspective: interpreting moral status claims as assertions rather than declarations. This refined perspective provides a more plausible framework for understanding moral status and avoids the pitfalls associated with the performative view. (shrink)

Improving human characteristics goes beyond compensating for an impairment. This book explores the rich and complex relationship between enhancement and impairment, showing that the study of disability offers new ways of thinking about the social and ethical implications of improving the human condition.

Proclus, successor of the School of Athens in the fifth century AD, is one of the last great voices of a pagan polytheistic world in crisis in the face of the gradual rise of Christianity. His writing bears witness not only to one of the most influential metaphysical representations for the constitution of the idea of a philosophical "system" based on a complex "language game" based on the conceptual sphere of "order" and "unity", but also of an attempt to reconcile (...) rhetorics with philosophy. In the three chapters in which the book is structured (style, system, unity), then, a conscious intention is evident in Proclus, to adapt the content of his systematic metaphysics to literary form. Hence there is a consistency traced in Proclus' stylistic choices, with the aesthetic norms of a rhetorical canon, probably established in the cycle of readings of the Neoplatonic school. His writing bears witness not only to one of the most influential metaphysical representations for the constitution of the idea of a philosophical "system" based on a complex "language game" based on the conceptual sphere of "order" and "unity", but also of an attempt to reconcile rhetorics with philosophy. In the three chapters in which the book is structured (style, system, unity), then, a conscious intention is evident in Proclus, to adapt the content of his systematic metaphysics to literary form. Hence there is a consistency traced in Proclus' stylistic choices, with the aesthetic norms of a rhetorical canon, probably established in the cycle of readings of the Neoplatonic school. The way in which expressive "art" is subordinated in Proclus to philosophy is one of the most emblematic examples of a development in late antiquity, that of the enkuklios paideia - the union of all the arts necessary for the formation of man - that would later constitute in the Latin world the cycle of liberal arts. (shrink)

In this paper, I begin by defending permissivism: the claim that, sometimes, there is more than one way to rationally respond to a given body of evidence. Then I argue that, if we accept permissivism, certain worries that arise as a result of learning that our beliefs were caused by the communities we grew up in, the schools we went to, or other irrelevant influences dissipate. The basic strategy is as follows: First, I try to pinpoint what makes irrelevant influences (...) worrying and I come up with two candidate principles. I then argue that one principle should be rejected because it is inconsistent with permissivism. The principle we should accept implies that it is sometimes rational to maintain our beliefs, even upon learning that they were caused by irrelevant influences. (shrink)

The question of whether it is ever permissible to believe on insufficient evidence has once again become a live question. Greater attention is now being paid to practical dimensions of belief, namely issues related to epistemic virtue, doxastic responsibility, and voluntarism. In this book, McCormick argues that the standards used to evaluate beliefs are not isolated from other evaluative domains. The ultimate criteria for assessing beliefs are the same as those for assessing action because beliefs and actions are both products (...) of agency. Two important implications of this thesis, both of which deviate from the dominant view in contemporary philosophy, are 1) it can be permissible to believe for non-evidential reasons, and 2) we have a robust control over many of our beliefs, a control sufficient to ground attributions of responsibility for belief. (shrink)

This chapter contains section titled: I II III Acknowledgement.

Daniel Weiskopf has recently raised an apparently powerful objection against the so-called “extended mind thesis” with regard to beliefs. His argument is that since alleged cases of “extended beliefs” lack a characteristic feature of beliefs properly so called (newly acquired beliefs are usually integrated with already existing beliefs rapidly, automatically and unconsciously), they do not count as genuine beliefs properly so called. We defend the extended mind thesis by arguing that Weiskopf is wrong. First, we suggest an alternative account of (...) informational integration that is compatible with externally stored beliefs’ being beliefs properly so called, emphasizes the crucial role action plays for cognition, and stresses the embodied and situationally embedded nature of human cognizers. Second, we argue that even if informational integration were usually rapid, automatic, and unconscious, this would not be an essential feature of beliefs. Third, we argue that even if rapid, automatic, and unconscious informational integration were characteristic of our commonsense conception of beliefs, externally stored “beliefs” would still be sufficiently similar to beliefs properly so called for them to be grouped together for all practical and scientific purposes. (shrink)

This paper is about the connection between rationality and accuracy. I show that one natural picture about how rationality and accuracy are connected emerges if we assume that rational agents are rationally omniscient. I then develop an alternative picture that allows us to relax this assumption, in order to accommodate certain views about higher order evidence.

Greaves and Wallace argue that conditionalization maximizes expected accuracy. In this paper I show that their result only applies to a restricted range of cases. I then show that the update procedure that maximizes expected accuracy in general is one in which, upon learning P, we conditionalize, not on P, but on the proposition that we learned P. After proving this result, I provide further generalizations and show that much of the accuracy-first epistemology program is committed to KK-like iteration principles (...) and to the existence of a class of propositions that rational agents will be certain of if and only if they are true. (shrink)

In recent years, permissivism—the claim that a body of evidence can rationalize more than one response—has enjoyed somewhat of a revival. But it is once again being threatened, this time by a host of new and interesting arguments that, at their core, are challenging the permissivist to explain why rationality matters. A version of the challenge that I am especially interested in is this: if permissivism is true, why should we expect the rational credences to be more accurate than the (...) irrational ones? My aim is to turn this challenge on its head and argue that, actually, those who deny permissivism will have a harder time responding to such a challenge than those who accept it. (shrink)

The article deals with larger stretches of talk-in-interaction and argues in favor of a descriptive approach, which integrates the structural requirements of global organization, the special type of sequential orderliness within larger units as well as the genre-orientation of these units. Drawing on previous work in conversation analysis, discourse analysis and the sociological genre analysis, the article introduces GLOBE as an analytical tool which functionally links discourse units to conventionalized communicative purposes. GLOBE reconstructs the interactive achievement of genre-oriented discourse units (...) in a three-branch analysis of jobs, devices and forms. The analytical potential of GLOBE is demonstrated in the exemplary genre-contrastive analysis of a narrative, an explanative and an argumentative excerpt. On the basis of the analyses of the three genres, the overarching constitution of genre-oriented global units is then explicated. (shrink)

The aim of this paper is to apply the accuracy based approach to epistemology to the case of higher order evidence: evidence that bears on the rationality of one's beliefs. I proceed in two stages. First, I show that the accuracy based framework that is standardly used to motivate rational requirements supports steadfastness—a position according to which higher order evidence should have no impact on one's doxastic attitudes towards first order propositions. The argument for this will require a generalization of (...) an important result by Greaves and Wallace for the claim that conditionalization maximizes expected accuracy. The generalization I provide will, among other things, allow us to apply the result to cases of self-locating evidence. In the second stage, I develop an alternative framework. Very roughly, what distinguishes the traditional approach from the alternative one is that, on the traditional picture, we're interested in evaluating the expected accuracy of conforming to an update procedure. On the alternative picture that I develop, instead of considering how good an update procedure is as a plan to conform to, we consider how good it is as a plan to make. I show how, given the use of strictly proper scoring rules, the alternative picture vindicates calibrationism: a view according to which higher order evidence should have a significant impact on our beliefs. I conclude with some thoughts about why higher order evidence poses a serious challenge for standard ways of thinking about rationality. (shrink)

David Hunter’s On Believing is an ambitious, extremely carefully argued, discussion of what it means to believe. He urges readers to re-think the way to categorize beliefs (or more precisely believ...

I argue that in many cases, there are good reasons to engage with people who hold fringe beliefs such as debunked conspiracy theories. I (1) discuss reasons for engaging with fringe beliefs; (2) discuss the conditions that need to be met for engagement to be worthwhile; (3) consider the question of how to engage with such beliefs, and defend what Jeremy Fantl has called “closed-minded engagement” and (4) address worries that such closed-minded engagement involves problematic deception or manipulation. Thinking about (...) how we engage with irrational emotions offers a way of responding to these concerns. Reflection on engagement with fringe beliefs has wider implications for two distinct philosophical discussions. First, it can help illuminate the nature of beliefs, lending support to the view that not all states which are deeply resistant to evidence thereby fail to be beliefs. Second, an implication of the view I put forth is that it need not constitute a lack of respect to adopt what Peter Strawson called “the objective stance” in relationships. (shrink)

It has been claimed that, in response to certain kinds of evidence, agents ought to adopt imprecise credences: doxastic states that are represented by sets of credence functions rather than single ones. In this paper I argue that, given some plausible constraints on accuracy measures, accuracy-centered epistemologists must reject the requirement to adopt imprecise credences. I then show that even the claim that imprecise credences are permitted is problematic for accuracy-centered epistemology. It follows that if imprecise credal states are permitted (...) or required in the cases that their defenders appeal to, then the requirements of rationality can outstrip what would be warranted by an interest in accuracy. (shrink)

The aim of this paper is to describe a problem for calibrationism: a view about higher order evidence according to which one's credences should be calibrated to one's expected degree of reliability. Calibrationism is attractive, in part, because it explains our intuitive judgments, and provides a strong motivation for certain theories about higher order evidence and peer disagreement. However, I will argue that calibrationism faces a dilemma: There are two versions of the view one might adopt. The first version, I (...) argue, has the implausible consequence that, in a wide range of cases, calibrationism is the only constraint on rational belief. The second version, in addition to having some puzzling consequences, is unmotivated. At the end of the paper I sketch a possible solution. (shrink)

The aim of this essay is to argue that, if a robust form of moral realism is true, then moral vagueness is ontic vagueness. The argument is by elimination: I show that neither semantic nor epistemic approaches to moral vagueness are satisfactory.

My main aim in this paper is to specify conditions that distinguish rational, or justified, hope from irrational, or unjustified hope. I begin by giving a brief characterization of hope and then turn to offering some criteria of rational hope. On my view both theoretical and practical norms are significant when assessing hope’s rationality. While others have recognized that there are theoretical and practical components to the state itself, when it comes to assessing its rationality, depending on the account, only (...) one of these dimensions matters. Either hope’s rationality is taken to be entirely derivative on the rationality of some belief that the hope entails, or the focus is entirely on the practical dimensions. If we take seriously the idea that both kinds of norms are at play, an account which treats these both as salient and as intertwining in important ways is preferable. I argue that as the practical importance of hope increases, the demand for the level of evidential support lessens and as the stakes get lower, the evidence for the likeliness of the hoped-for outcome coming to be must go up for the hope to be rational. This is so because a hope is rational insofar as it contributes to agential flourishing, but the different dimensions of hope each pick out different aspects of agency; to hope well requires agential competence in both the epistemic and practical realms. (shrink)

It seems like we care about at least two features of our credence function: gradational-accuracy and verisimilitude. Accuracy-first epistemology requires that we care about one feature of our credence function: gradational-accuracy. So if you want to be a verisimilitude-valuing accuracy-firster, you must be able to think of the value of verisimilitude as somehow built into the value of gradational-accuracy. Can this be done? In a recent article, Oddie has argued that it cannot, at least if we want the accuracy measure (...) to be proper. I argue that it can. 1Introduction2Some Nuts and Bolts3First Attempts4Oddie’s Constraint5The Good5.1Proximity over the disagreement metric 5.2Proximity over the magnitude metric 6The Bad and the Ugly 7Some More Good: The Role of Evenness of Distribution 8Some More Bad: Which Propositions to Privilege? 9Concluding Thoughts: Accuracy and Practical Value. (shrink)

A defense of imprecise credences (and other imprecise doxastic attitudes).

I investigate what we mean when we hold people responsible for beliefs. I begin by outlining a puzzle concerning our ordinary judgments about beliefs and briefly survey and critique some common responses to the puzzle. I then present my response where I argue a sense needs to be articulated in which we do have a kind of control over our beliefs if our practice of attributing responsibility for beliefs is appropriate. In developing this notion of doxastic control, I draw from (...) John Fischer's discussions of “guidance control”. A central feature of this kind of control is the idea of “ownership”. I argue that we can own our beliefs and that we expect each other to do so. We take responsibility for our beliefs and taking responsibility includes taking control of them. I end by considering objections to my view as well as some implications of it. (shrink)

Abstract: This paper defends a constraint that any satisfactory decision theory must satisfy. I show how this constraint is violated by all of the decision theories that have been endorsed in the literature that are designed to deal with cases in which opinions or values are represented by a set of functions rather than a single one. Such a decision theory is necessary to account for the existence of what Ruth Chang has called “parity” (as well as for cases in (...) which agents have incomplete preferences or imprecise credences). The problem with the all of the decision theories that have been defended to account for parity is that they are committed to a claim I call unanimity: when all of the functions in the set agree that an agent ought to do A, then an agent ought to do A. A decision theory committed to unanimity violates the constraint I defend in this paper. Thus, if parity exists, a new approach to decision theory is necessary. (shrink)

My main aim is to argue that most conceptions of doxastic agency do not respond to the skeptic’s challenge. I begin by considering some reasons for thinking that we are not doxastic agents. I then turn to a discussion of those who try to make sense of doxastic agency by appeal to belief’s reasons-responsive nature. What they end up calling agency is not robust enough to satisfy the challenge posed by the skeptics. To satisfy the skeptic, one needs to make (...) sense of the possibility of believing for nonevidential reasons. While this has been seen as an untenable view for both skeptics and anti-skeptics, I conclude by suggesting it is a position that has been too hastily dismissed. (shrink)

Go beyond reading about early learning theories and see what they look like in action in modern programs and teacher practices. With classroom vignettes and colorful photographs, this book makes the works of Jean Piaget, Erik Erikson, Lev Vygotsky, Abraham Maslow, John Dewey, Howard Gardner, and Louise Derman-Sparks visible, accessible, and easier to understand. Each theory is defined-through engaging stories and rich visuals-in relation to cognitive, social-emotional, and physical developmental domains. Use this book to build a stronger comprehension of the (...) foundations of early learning theories and become more reflective and intentional in your work with young children. (shrink)