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)

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)

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)

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)

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...

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)

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)

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.

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)

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)

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)

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.

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)

The present paper analyzes the self‐generated explanations (from talk‐aloud protocols) that “Good” and “Poor” students produce while studying worked‐out examples of mechanics problems, and their subsequent reliance on examples during problem solving. We find that “Good” students learn with understanding: They generate many explanations which refine and expand the conditions for the action parts of the example solutions, and relate these actions to principles in the text. These self‐explanations are guided by accurate monitoring of their own understanding and misunderstanding. Such (...) learning results in example‐independent knowledge and in a better understanding of the principles presented in the text. “Poor” students do not generate sufficient self‐explanations, monitor their learning inaccurately, and subsequently rely heavily on examples. We then discuss the role of self‐explanations in facilitating problem solving, as well as the adequacy of current AI models of explanation‐based learning to account for these psychological findings. (shrink)

The present paper analyzes the self‐generated explanations (from talk‐aloud protocols) that “Good” and “Poor” students produce while studying worked‐out examples of mechanics problems, and their subsequent reliance on examples during problem solving. We find that “Good” students learn with understanding: They generate many explanations which refine and expand the conditions for the action parts of the example solutions, and relate these actions to principles in the text. These self‐explanations are guided by accurate monitoring of their own understanding and misunderstanding. Such (...) learning results in example‐independent knowledge and in a better understanding of the principles presented in the text. “Poor” students do not generate sufficient self‐explanations, monitor their learning inaccurately, and subsequently rely heavily on examples. We then discuss the role of self‐explanations in facilitating problem solving, as well as the adequacy of current AI models of explanation‐based learning to account for these psychological findings. (shrink)

Infinite time register machines (ITRMs) are register machines which act on natural numbers and which are allowed to run for arbitrarily many ordinal steps. Successor steps are determined by standard register machine commands. At limit times register contents are defined by appropriate limit operations. In this paper, we examine the ITRMs introduced by the third and fourth author (Koepke and Miller in Logic and Theory of Algorithms LNCS, pp. 306–315, 2008), where a register content at a limit time is set (...) to the lim inf of previous register contents if that limit is finite; otherwise the register is reset to 0. The theory of these machines has several similarities to the infinite time Turing machines (ITTMs) of Hamkins and Lewis. The machines can decide all ${\Pi^1_1}$ sets, yet are strictly weaker than ITTMs. As in the ITTM situation, we introduce a notion of ITRM-clockable ordinals corresponding to the running times of computations. These form a transitive initial segment of the ordinals. Furthermore we prove a Lost Melody theorem: there is a real r such that there is a program P that halts on the empty input for all oracle contents and outputs 1 iff the oracle number is r, but no program can decide for every natural number n whether or not ${n \in r}$ with the empty oracle. In an earlier paper, the third author considered another type of machines where registers were not reset at infinite lim inf’s and he called them infinite time register machines. Because the resetting machines correspond much better to ITTMs we hold that in future the resetting register machines should be called ITRMs. (shrink)

ABSTRACT What does it mean to practice a theory of recognition within the discipline of philosophy? Across an initially acrimonious French-German divide, Axel Honneth’s effort to recognise the value of contemporary French philosophy and social theory suggests that philosophy is a self-critical, outwardly oriented, and cooperative discipline. First, mobilising the idea of recognition in his own philosophical practise has permitted Honneth to notice non-deliberative aspects of social interaction that Habermas had overlooked, including the need for self-confidence and the need for (...) self-esteem. Second, although Honneth’s ascription of value to contemporary French theory also involves mis-recognition, his practice of recognition has nonetheless prompted philosophical cooperation among theorists who would not otherwise have engaged, encouraging French-influenced critical theorists to extend Honneth’s theory in ways that he had not anticipated. What emerges is an interpretation of philosophy as an ongoing intersubjective pursuit that attempts to respond to its own limitations through cooperative critique. (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)

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)

Had we grown up elsewhere or been educated differently, our view of the world would likely be radically different. What to make of this? This paper takes an accuracy-centered first-personal approach to the question of how to respond to the arbitrary nature in which many of our beliefs are formed. I show how considerations of accuracy motivate different responses to this sort of information depending on the type of attitude we take towards the belief in question upon subjecting the belief (...) to doubt. (shrink)

This article summarizes a virtual live‐streamed panel event that occurred in August 2020 and was cosponsored by the International Philosophy of Nursing Society (IPONS) and the University of California, Irvine's Center for Nursing Philosophy. The event consisted of a series of three self‐contained panel discussions focusing on the past, present and future of IPONS and was moderated by the current Chair of IPONS, Catherine Green. The first panel discussion explored the history of IPONS and the journal Nursing Philosophy. The second (...) panel involved a reflection on the challenges of doing nursing philosophy in a research‐intensive context of a Canadian university and the history and current movements in nursing philosophy in the Nordic countries. The final panel involved presentations on the future potential for philosophy in/and for nursing, the critical connections between nursing philosophy and nursing theory, dismantling racism in nursing and the potential for process philosophy to help explore nursing's unique efficacy in creating possibilities for health. The panels were followed by a lively Q&A session with participants, of which there were 252 registrants from across the globe. The event underscored the wide and diverse interests of nurses in philosophical discussion and the need for more virtual events and other connective modalities bringing nurses together to discuss and analyze the value and potential of philosophy to better understand and advance nursing theory and practice. (shrink)

For the last forty years, two claims have been at the core of disputes about scientific change: that scientists reason rationally and that science is progressive. For most of this time discussions were polarized between philosophers, who defended traditional Enlightenment ideas about rationality and progress, and sociologists, who espoused relativism and constructivism. Recently, creative new ideas going beyond the polarized positions have come from the history of science, feminist criticism of science, psychology of science, and anthropology of science. Addressing the (...) traditional arguments as well as building on these new ideas, Miriam Solomon constructs a new epistemology of science. After discussions of the nature of empirical success and its relation to truth, Solomon offers a new, social account of scientific rationality. She shows that the pursuit of empirical success and truth can be consistent with both dissent and consensus, and that the distinction between dissent and consensus is of little epistemic significance. In building this social epistemology of science, she shows that scientific communities are not merely the locus of distributed expert knowledge and a resource for criticism but also the site of distributed decision making. Throughout, she illustrates her ideas with case studies from late-nineteenth- and twentieth-century physical and life sciences. Replacing the traditional focus on methods and heuristics to be applied by individual scientists, Solomon emphasizes science funding, administration, and policy. One of her goals is to have a positive influence on scientific decision making through practical social recommendations. (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.

After the discovery of the anthropoid ape in Asia and in Africa, eighteenth-century Holland became the crossroads of Enlightenment debates about the human species. Material evidence about human diversity reached Petrus Camper, comparative anatomist in the Netherlands, who engaged, among many other interests, in menschkunde. Could only religious doctrine support the belief of human demarcation from animals? Camper resolved the challenges raised by overseas discoveries with his thesis of the facial angle, a theory which succeeding generations distorted and misused in (...) order to justify slavery, racism, antisemitism, and genocide. Thanks to his abundant papers in Dutch archives, Camper's ideas are restored to their original state. Eighteenth-century issues differed from those of other centuries: Did orang-utans talk like humans, walk like humans; even rape humans? What was the skin pigmentation of Adam and Eve? Did the spectrum of human physiognomies around the globe reflect the Fall of Man, the Creator's bounty, or merely bizarre beauty practices? Why did the ideal beauty of the Greeks appear to be the reverse of the Hottentots? The book contains some 50 illustrations, including apes with hiking sticks or tea cups, metamorphoses of living forms, and Apollo or Venus icons which titillated the science of man. (shrink)

How is medical knowledge made? There have been radical changes in recent decades, through new methods such as consensus conferences, evidence-based medicine, translational medicine, and narrative medicine. Miriam Solomon explores their origins, aims, and epistemic strengths and weaknesses; and she offers a pluralistic approach for the future.

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)

It is commonly held that (i) beliefs are revisable in the face of counter‐evidence and (ii) beliefs are connected to actions in reliable and predictable ways. Given such a view, many argue that if a mental state fails to respond to evidence or doesn't result in the kind of behavior typical or expected of belief, it is not a belief after all, but a different state. Yet, one finds seeming counter examples of resilient beliefs that fail to respond to evidence, (...) or that do not connect to action in the way that is expected. I offer a view of belief that does not force us to exclude states as realbeliefs that we pre‐reflectively think of as beliefs, and that does not require us to “outsource” the work belief seems to do to other mental states. I propose that we view belief as a type of emotion where emotions are understood a irreducibly blended states, containing representational, motivational and phenomenological elements. (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 usage of imprecise probabilities has been advocated in many domains: A number of philosophers have argued that our belief states should be “imprecise” in response to certain sorts of evidence, and imprecise probabilities have been thought to play an important role in disciplines such as artificial intelligence, climate science, and engineering. In this paper I’m interested in the question of whether the usage of imprecise probabilities can be given a practical motivation (a motivation based on practical rather than epistemic, (...) or alethic concerns). My aim is to challenge the central motivation for using imprecise probabilities in decision-making that has been offered in the literature: the idea that, in at least some contexts, it’s desirable to be ambiguity averse. If I succeed, this will show that we need to reconsider whether there are good reasons to use imprecise probabilities in contexts in which making good decisions is what's of primary concern. (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)

The concept of a “validator” as a unit of evidence for the validity of a psychiatric category has been important for more than fifty years. Validator evidence is aggregated by expert committees (for the Diagnostic and Statistical Manual of Mental Disorders (DSM), these are referred to as “workgroups”), which use the results to make nosological decisions. Through an examination of the recent history of psychiatric research, this paper argues that it is time to reassess this traditional practice. It concludes with (...) specific suggestions for going forward. (shrink)