Mathematics depends on proofs, and proofs must begin somewhere, from some fundamental assumptions. For nearly a century, the axioms of set theory have played this role, so the question of how these axioms are properly judged takes on a central importance. Approaching the question from a broadly naturalistic or second-philosophical point of view, Defending the Axioms isolates the appropriate methods for such evaluations and investigates the ontological and epistemological backdrop that makes them appropriate. In the end, a new account of (...) the objectivity of mathematics emerges, one refreshingly free of metaphysical commitments. (shrink)

A key to modern studies of 18th century Western philosophy, the Treatise considers numerous classic philosophical issues, including causation, existence, freedom and necessity and morality. This abridged edition has an introduction which explain's Hume's thought and places it in the context of its times.

Mathematicians tend to think of themselves as scientists investigating the features of real mathematical things, and the wildly successful application of mathematics in the physical sciences reinforces this picture of mathematics as an objective study. For philosophers, however, this realism about mathematics raises serious questions: What are mathematical things? Where are they? How do we know about them? Offering a scrupulously fair treatment of both mathematical and philosophical concerns, Penelope Maddy here delineates and defends a novel version of mathematical realism. (...) She answers the traditional questions and poses a challenging new one, refocusing philosophical attention on the pressing foundational issues of contemporary mathematics. (shrink)

What is intellectual humility? In this essay, we aim to answer this question by assessing several contemporary accounts of intellectual humility, developing our own account, offering two reasons for our account, and meeting two objections and solving one puzzle.

From the ferment of recent debates about the intellectual virtues, Roberts and Wood develop an approach they call 'regulative epistemology', exploring the connection between knowledge and intellectual virtue. In the course of their argument they analyse particular virtues of intellectual life - such as courage, generosity, and humility - in detail.

Over the past several decades, scholars of liberal and democratic theory have shown a heightened interest in the role that various virtues might play in promoting the good/free society. Yet within this recent "return" to the virtues, one quality that has been almost entirely left out of the discussion is humility. In this essay, I critically address this lacuna and offer a defense of a particular form of humility, what I call democratic humility. After considering a range of moral and (...) political objections to the concept of humility, Iprovide an account that seeks to address some of these long-standing difficulties while arguing that the idea of humility, recuperated as an ethos of civic attentiveness, may be one of the most important virtues for late-modern societies marked by ethical and cultural pluralism. (shrink)

We review different conceptions of the set-theoretic multiverse and evaluate their features and strengths. In Sect. 1, we set the stage by briefly discussing the opposition between the ‘universe view’ and the ‘multiverse view’. Furthermore, we propose to classify multiverse conceptions in terms of their adherence to some form of mathematical realism. In Sect. 2, we use this classification to review four major conceptions. Finally, in Sect. 3, we focus on the distinction between actualism and potentialism with regard to the (...) universe of sets, then we discuss the Zermelian view, featuring a ‘vertical’ multiverse, and give special attention to this multiverse conception in light of the hyperuniverse programme introduced in Arrigoni and Friedman (2013). We argue that the distinctive feature of the multiverse conception chosen for the hyperuniverse programme is its utility for finding new candidates for axioms of set theory. (shrink)

We review different conceptions of the set-theoretic multiverse and evaluate their features and strengths. In Sect. 1, we set the stage by briefly discussing the opposition between the ‘universe view’ and the ‘multiverse view’. Furthermore, we propose to classify multiverse conceptions in terms of their adherence to some form of mathematical realism. In Sect. 2, we use this classification to review four major conceptions. Finally, in Sect. 3, we focus on the distinction between actualism and potentialism with regard to the (...) universe of sets, then we discuss the Zermelian view, featuring a ‘vertical’ multiverse, and give special attention to this multiverse conception in light of the hyperuniverse programme introduced in Arrigoni-Friedman :77–96, 2013). We argue that the distinctive feature of the multiverse conception chosen for the hyperuniverse programme is its utility for finding new candidates for axioms of set theory. (shrink)

Until recently, discussion of virtues in the philosophy of mathematics has been fleeting and fragmentary at best. But in the last few years this has begun to change. As virtue theory has grown ever more influential, not just in ethics where virtues may seem most at home, but particularly in epistemology and the philosophy of science, some philosophers have sought to push virtues out into unexpected areas, including mathematics and its philosophy. But there are some mathematicians already there, ready to (...) meet them, who have explicitly invoked virtues in discussing what is necessary for a mathematician to succeed. In both ethics and epistemology, virtue theory tends to emphasize character virtues, the acquired excellences of people. But people are not the only sort of thing whose excellences may be identified as virtues. Theoretical virtues have attracted attention in the philosophy of science as components of an account of theory choice. Within the philosophy of mathematics, and mathematics itself, attention to virtues has emerged from a variety of disparate sources. Theoretical virtues have been put forward both to analyse the practice of proof and to justify axioms; intellectual virtues have found multiple applications in the epistemology of mathematics; and ethical virtues have been offered as a basis for understanding the social utility of mathematical practice. Indeed, some authors have advocated virtue epistemology as the correct epistemology for mathematics (and perhaps even as the basis for progress in the metaphysics of mathematics). This topical collection brings together several of the researchers who have begun to study mathematical practices from a virtue perspective with the intention of consolidating and encouraging this trend. (shrink)

This edited work presents contemporary mathematical practice in the foundational mathematical theories, in particular set theory and the univalent foundations. It shares the work of significant scholars across the disciplines of mathematics, philosophy and computer science. Readers will discover systematic thought on criteria for a suitable foundation in mathematics and philosophical reflections around the mathematical perspectives. The first two sections focus on the two most prominent candidate theories for a foundation of mathematics. Readers may trace current research in set theory, (...) which has widely been assumed to serve as a framework for foundational issues, as well as new material elaborating on the univalent foundations, considering an approach based on homotopy type theory (HoTT). The further sections then build on this and are centred on philosophical questions connected to the foundations of mathematics. Here, the authors contribute to discussions on foundational criteria with more general thoughts on the foundations of mathematics which are not connected to particular theories. This book shares the work of some of the most important scholars in the fields of set theory (S. Friedman), non-classical logic (G. Priest) and the philosophy of mathematics (P. Maddy). The reader will become aware of the advantages of each theory and objections to it as a foundation, following the latest and best work across the disciplines and it is therefore a valuable read for anyone working on the foundations of mathematics or in the philosophy of mathematics. (shrink)

Penelope Maddy has recently addressed the set-theoretic multiverseset-theoretic multiverse, and expressed reservations on its status and merits Foundations of mathematics. Essays in honor of W. Hugh Woodin’s 60th birthday. Contemporary mathematics. American Mathematical Society, Providence, pp. 289–322, 2017). The purpose of the paper is to examine her concerns, by using the interpretative framework of set-theoretic naturalismset-theoretic naturalism. I first distinguish three main forms of ‘multiversism’multiversism, and then I proceed to analyse MaddyMaddy’s concerns. Among other things, I take into account salient (...) aspects of multiverse-related mathematics, in particular, research programmes in set theory for which the use of the multiverse seems to be crucial, and show how one may provide responses to MaddyMaddy’s concerns based on a careful analysis of ‘multiverse practice’. (shrink)

What sorts of epistemic virtues are required for effective mathematical practice? Should these be virtues of individual or collective agents? What sorts of corresponding epistemic vices might interfere with mathematical practice? How do these virtues and vices of mathematics relate to the virtue-theoretic terminology used by philosophers? We engage in these foundational questions, and explore how the richness of mathematical practices is enhanced by thinking in terms of virtues and vices, and how the philosophical picture is challenged by the complexity (...) of the case of mathematics. For example, within different social and interpersonal conditions, a trait often classified as a vice might be epistemically productive and vice versa. We illustrate that this occurs in mathematics by discussing Gerovitch’s historical study of the aggressive adversarialism of the Gelfand seminar in post-war Moscow. From this we conclude that virtue epistemologies of mathematics should avoid pre-emptive judgments about the sorts of epistemic character traits that ought to be promoted and criticised. (shrink)

Intellectual humility, I argue in this paper, is a cluster of strong attitudes directed toward one's cognitive make-up and its components, together with the cognitive and affective states that constitute their contents or bases, which serve knowledge and value-expressive functions. In order to defend this new account of humility I first examine two simpler traits: intellectual self-acceptance of epistemic limitations and intellectual modesty about epistemic successes. The position defended here addresses the shortcomings of both ignorance and accuracy based accounts of (...) humility. (shrink)

Since the publication of Edmund Gettier's challenge to the traditional epistemological doctrine of knowledge as justified true belief, Roberts and Wood claim that epistemologists lapsed into despondency and are currently open to novel approaches. One such approach is virtue epistemology, which can be divided into virtues as proper functions or epistemic character traits. The authors propose a notion of regulative epistemology, as opposed to a strict analytic epistemology, based on intellectual virtues that function not as rules or even as skills (...) but as habits of the heart. To that end, they divide the task of clarifying and expounding their notion in the book's two parts.In the first part, Roberts and Wood examine various components that constitute their notion of regulative epistemology. The first are the epistemic goods or goals that drive the epistemic process. What is needed, claim Roberts and Wood, is an enriched notion of these goods rather than the restricted notion of justified true belief. Epistemic agents are more than calculating devices in that …. (shrink)

This book explores how physicists, astronomers, chemists, and historians in the late nineteenth and early twentieth centuries employed ‘epistemic virtues’ such as accuracy, objectivity, and intellectual courage. In doing so, it takes the first step in providing an integrated history of the sciences and humanities. It assists in addressing such questions as: What kind of perspective would enable us to compare organic chemists in their labs with paleographers in the Vatican Archives, or anthropologists on a field trip with mathematicians poring (...) over their formulas? While the concept of epistemic virtues has previously been discussed, primarily in the contexts of the history and philosophy of science, this volume is the first to enlist the concept in bridging the gap between the histories of the sciences and the humanities. Chapters research whether epistemic virtues can serve as a tool to transcend the institutional disciplinary boundaries and thus help to attain a ‘post-disciplinary’ historiography of modern knowledge. Readers will gain a contextualization of epistemic virtues in time and space as the book shows that scholars themselves often spoke in terms of virtue and vice about their tasks and accomplishments. This collection of essays opens up new perspectives on questions, discourses, and practices shared across the disciplines, even at a time when the neo-Kantian distinction between sciences and humanities enjoyed its greatest authority. Scholars including historians of science and of the humanities, intellectual historians, virtue epistemologists, and philosophers of science will all find this book of particular interest and value. (shrink)

In this paper, I explore the relationship of virtue, argumentation, and philosophical conduct by considering the role of the specific virtue of intellectual humility in the practice of philosophical argumentation. I have three aims: first, to sketch an account of this virtue; second, to argue that it can be cultivated by engaging in argumentation with others; and third, to problematize this claim by drawing upon recent data from social psychology. My claim is that philosophical argumentation can be conducive to the (...) cultivation of virtues, including humility, but only if it is conceived and practiced in appropriately ‘edifying’ ways. (shrink)

This paper concerns would-be necessary connections between doxastic attitudes about the epistemic statuses of your doxastic attitudes, or ‘higher-order epistemic attitudes’, and the epistemic statuses of those doxastic attitudes. I will argue that, in some situations, it can be reasonable for a person to believe p and to suspend judgment about whether believing p is reasonable for her. This will set the stage for an account of the virtue of intellectual humility, on which humility is a matter of your higher-order (...) epistemic attitudes. Recent discussions in the epistemology of disagreement have assumed that the question of the proper response to disagreement about p concerns whether you ought to change your doxastic attitude towards p. My conclusion here suggests an alternative approach, on which the question of the proper response to disagreement about p concerns the proper doxastic attitude to adopt concerning the epistemic status of your doxastic attitude towards p.Send article to KindleTo send this article to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle. Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply. Find out more about the Kindle Personal Document Service.HIGHER-ORDER EPISTEMIC ATTITUDES AND INTELLECTUAL HUMILITYVolume 9, Issue 3Allan HazlettDOI: https://doi.org/10.1017/epi.2012.11Your Kindle email address Please provide your Kindle [email protected]@kindle.com Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Send article to Dropbox To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Dropbox. HIGHER-ORDER EPISTEMIC ATTITUDES AND INTELLECTUAL HUMILITYVolume 9, Issue 3Allan HazlettDOI: https://doi.org/10.1017/epi.2012.11Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Send article to Google Drive To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Google Drive. HIGHER-ORDER EPISTEMIC ATTITUDES AND INTELLECTUAL HUMILITYVolume 9, Issue 3Allan HazlettDOI: https://doi.org/10.1017/epi.2012.11Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Export citation Request permission. (shrink)

The multiverse view in set theory, introduced and argued for in this article, is the view that there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe. The universe view, in contrast, asserts that there is an absolute background set concept, with a corresponding absolute set-theoretic universe in which every set-theoretic question has a definite answer. The multiverse position, I argue, explains our experience with the enormous range of set-theoretic possibilities, a phenomenon that challenges the universe (...) view. In particular, I argue that the continuum hypothesis is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse, and as a result it can no longer be settled in the manner formerly hoped for. (shrink)

In the era of information and communication, issues of misinformation and miscommunication are more pressing than ever. _Epistemic injustice - _one of the most important and ground-breaking subjects to have emerged in philosophy in recent years - refers to those forms of unfair treatment that relate to issues of knowledge, understanding, and participation in communicative practices. The Routledge Handbook of Epistemic Injustice is an outstanding reference source to the key topics, problems and debates in this exciting subject. The first collection (...) of its kind, it comprises over thirty chapters by a team of international contributors, divided into five parts: Core Concepts Liberatory Epistemologies and Axes of Oppression Schools of Thought and Subfields within Epistemology Socio-political, Ethical, and Psychological Dimensions of Knowing Case Studies of Epistemic Injustice. As well as fundamental topics such as testimonial and hermeneutic injustice and epistemic trust, the Handbook includes chapters on important issues such as social and virtue epistemology, objectivity and objectification, implicit bias, and gender and race. Also included are chapters on areas in applied ethics and philosophy, such as law, education, and healthcare. The Routledge Handbook of Epistemic Injustice is essential reading for students and researchers in ethics, epistemology, political philosophy, feminist theory, and philosophy of race. It will also be very useful for those in related fields, such as cultural studies, sociology, education and law. (shrink)

The logician Kurt Gödel published a paper in 1931 formulating what have come to be known as his 'incompleteness theorems', which prove, among other things, that within any formal system with resources sufficient to code arithmetic, questions exist which are neither provable nor disprovable on the basis of the axioms which define the system. These are among the most celebrated results in logic today. In this volume, leading philosophers and mathematicians assess important aspects of Gödel's work on the foundations and (...) philosophy of mathematics. Their essays explore almost every aspect of Godel's intellectual legacy including his concepts of intuition and analyticity, the Completeness Theorem, the set-theoretic multiverse, and the state of mathematical logic today. This groundbreaking volume will be invaluable to students, historians, logicians and philosophers of mathematics who wish to understand the current thinking on these issues. (shrink)

In The Virtues of Ignorance the author demonstrates that classical theories of virtue are flawed and developes a consequentialist theory of virtue. ;Virtues are excellences of character. They are traits which are considered to be valuable in some way. A person who is virtuous is one who has a tendency to act well. Classical philosophers, such as Plato and Aristotle, believed that virtues, as human excellences, could not involve ignorance in any way. On their view, the virtuous agent, when acting (...) well, knows what she is doing under the description which makes the action conform to virtue. For example, the generous person must know that she is, in fact, helping others in need. Yet, there is a class of virtues, the "virtues of ignorance," which essentially involve ignorance. For example, modesty requires that the agent underestimate self worth. In the first portion of the dissertation, this class of virtues is discussed, and problems raised for the traditional theories of virtue. ;In the latter portion of the dissertation, the author outlines a consequentialist theory of virtue. It is argued that the only hope of developing a unified theory of virtue is along consequentialist lines, because attempts at identifying the psychological state which is necessary for virtue have failed. On a consequentialist theory, a virtue is defined in terms of its consequences. Thus, the moral quality of the character trait is determined by how much good it brings about . The virtues of ignorance can easily be accommodated in such a theory because these traits do bring about good, even if the agent possessing the trait acts out of ignorance. (shrink)

Penelope Maddy has recently addressed the set-theoretic multiverse, and expressed reservations on its status and merits ([Maddy, 2017]). The purpose of the paper is to examine her concerns, by using the interpretative framework of set-theoretic naturalism. I first distinguish three main forms of 'multiversism', and then I proceed to analyse Maddy's concerns. Among other things, I take into account salient aspects of multiverse-related mathematics , in particular, research programmes in set theory for which the use of the multiverse seems to (...) be crucial, and show how one may provide responses to Maddy's concerns based on a careful analysis of 'multiverse practice'. (shrink)