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Domain extension in mathematics occurs whenever a given mathematical domain is augmented so as to include new elements. Manders argues that the advantages of important cases of domain extension are captured by the model-theoretic notions of existential closure and model completion. In the specific case of domain extension via ideal elements, I argue, Manders’s proposed explanation does not suffice. I then develop and formalize a different approach to domain extension based on Dedekind’s Habilitationsrede, to which Manders’s account is compared. I (...) conclude with an examination of three possible stances towards extensions via ideal elements. (shrink)

This essay offers a working analysis of the trait of intellectual perseverance. It argues that intellectual perseverance is a disposition to overcome obstacles, so as to continue to perform intellectual actions, in pursuit of one’s intellectual goals. The trait of intellectual perseverance is not always an intellectual virtue. This essay provides a pluralist analysis of what makes it an intellectual virtue, when it is one. Along the way, it argues that the virtue of intellectual perseverance can be contrasted with both (...) a vice of deficiency and a vice of excess. It also suggests that the virtues of intellectual courage and intellectual self-control are types of intellectual perseverance. The essay ends with several open questions about the virtue of intellectual perseverance. My hope is that this essay will stimulate further interest in, and analysis of, this important intellectual trait. (shrink)

In this paper, we review the history of quasicrystals from their sensational discovery in 1982, initially “forbidden” by the rules of classical crystallography, to 2011 when Dan Shechtman was awarded the Nobel Prize in Chemistry. We then discuss the discovery of quasicrystals in philosophical terms of anomalies behavior that led to a paradigm shift as offered by philosopher and historian of science Thomas Kuhn in ‘The Structure of Scientific Revolutions’. This discovery, which found expression in the redefinition of the concept (...) crystal from being periodically arranged to producing sharp peaks in the Bragg diffraction pattern, is analyzed according to the Kuhn Cycle. We relate the quasicrystal revolution to the non-Euclidean geometry revolution and argues that since “great minds think alike” there is a diffusion of ideas between scientific revolutions, or a resonance between different disciplines at different times. The story behind quasicrystals is an excellent example of a paradigm shift, demonstrating the nature of scientific discoveries and breakthroughs. (shrink)

Mathematicians often intentionally leave gaps in their proofs. Based on interviews with mathematicians about their refereeing practices, this paper examines the character of intentional gaps in published proofs. We observe that mathematicians’ refereeing practices limit the number of certain intentional gaps in published proofs. The results provide some new perspectives on the traditional philosophical questions of the nature of proof and of what grounds mathematical knowledge.

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 chapter focuses on alternative logics. It discusses a hierarchy of logical reform. It presents case studies that illustrate particular aspects of the logical revisionism discussed in the chapter. The first case study is of intuitionistic logic. The second case study turns to quantum logic, a system proposed on empirical grounds as a resolution of the antinomies of quantum mechanics. The third case study is concerned with systems of relevance logic, which have been the subject of an especially detailed reform (...) program. Finally, the fourth case study is paraconsistent logic, perhaps the most controversial of serious proposals. (shrink)

Deep disagreements are characteristically resistant to rational resolution. This paper explores the contribution a virtue theoretic approach to argumentation can make towards settling the practical matter of what to do when confronted with apparent deep disagreement, with particular attention to the virtue of courage.

A panoramic survey of the vast spectrum of modern and contemporary mathematics and the new philosophical possibilities they suggest. A panoramic survey of the vast spectrum of modern and contemporary mathematics and the new philosophical possibilities they suggest, this book gives the inquisitive non-specialist an insight into the conceptual transformations and intellectual orientations of modern and contemporary mathematics. The predominant analytic approach, with its focus on the formal, the elementary and the foundational, has effectively divorced philosophy from the real practice (...) of mathematics and the profound conceptual shifts in the discipline over the last century. The first part discusses the specificity of modern (1830–1950) and contemporary (1950 to the present) mathematics, and reviews the failure of mainstream philosophy of mathematics to address this specificity. Building on the work of the few exceptional thinkers to have engaged with the “real mathematics” of their era (including Lautman, Deleuze, Badiou, de Lorenzo and Châtelet), Zalamea challenges philosophy's self-imposed ignorance of the “making of mathematics.” In the second part, thirteen detailed case studies examine the greatest creators in the field, mapping the central advances accomplished in mathematics over the last half-century, exploring in vivid detail the characteristic creative gestures of modern master Grothendieck and contemporary creators including Lawvere, Shelah, Connes, and Freyd. Drawing on these concrete examples, and oriented by a unique philosophical constellation (Peirce, Lautman, Merleau-Ponty), in the third part Zalamea sets out the program for a sophisticated new epistemology, one that will avail itself of the powerful conceptual instruments forged by the mathematical mind, but which have until now remained largely neglected by philosophers. (shrink)

This title is part of UC Press's Voices Revived program, which commemorates University of California Press’s mission to seek out and cultivate the brightest minds and give them voice, reach, and impact. Drawing on a backlist dating to 1893, Voices Revived makes high-quality, peer-reviewed scholarship accessible once again using print-on-demand technology. This title was originally published in 1986.

In this paper I explore how intellectual humility manifests in mathematical practices. To do this I employ accounts of this virtue as developed by virtue epistemologists in three case studies of mathematical activity. As a contribution to a Topical Collection on virtue theory of mathematical practices this paper explores in how far existing virtue-theoretic frameworks can be applied to a philosophical analysis of mathematical practices. I argue that the individual accounts of intellectual humility are successful at tracking some manifestations of (...) this virtue in mathematical practices and fail to track others. There are two upshots to this. First, the accounts of the intellectual virtues provided by virtue epistemologists are insightful for the development of a virtue theory of mathematical practices but require adjustments in some cases. Second, the case studies reveal dimensions of intellectual humility virtue epistemologists have thus far overlooked in their theoretical reflections. (shrink)

In a recent article, Azzouni has argued in favor of a version of formalism according to which ordinary mathematical proofs indicate mechanically checkable derivations. This is taken to account for the quasi-universal agreement among mathematicians on the validity of their proofs. Here, the author subjects these claims to a critical examination, recalls the technical details about formalization and mechanical checking of proofs, and illustrates the main argument with aanalysis of examples. In the author's view, much of mathematical reasoning presents genuine (...) meaning-dependent mathematical characteristics that cannot be captured by formal calculi. ‘…there is a conflict between mathematical practice and the formalist doctrine.’ [Kreisel, 1969, p. 39]. (shrink)

What is the nature of deep disagreement? In this paper, I consider two similar albeit seemingly rival answers to this question: the Wittgensteinian theory, according to which deep disagreements are disagreements over hinge propositions, and the fundamental epistemic principle theory, according to which deep disagreements are disagreements over fundamental epistemic principles. I assess these theories against a set of desiderata for a satisfactory theory of deep disagreement, and argue that while the fundamental epistemic principle theory does better than the Wittgensteinian (...) theory on this score, the fundamental epistemic principle theory nevertheless struggles to explain the variety of deep disagreement. (shrink)

Deep disagreements concern our most basic and fundamental commitments. Such disagreements seem to be problematic because they appear to manifest epistemic incommensurability in our epistemic systems, and thereby lead to epistemic relativism. This problem is confronted via consideration of a Wittgensteinian hinge epistemology. On the face of it, this proposal exacerbates the problem of deep disagreements by granting that our most fundamental commitments are essentially arationally held. It is argued, however, that a hinge epistemology, properly understood, does not licence epistemic (...) incommensurability or epistemic relativism at all. On the contrary, such an epistemology in fact shows us how to rationally respond to deep disagreements. It is claimed that if we can resist these consequences even from the perspective of a hinge epistemology, then we should be very suspicious of the idea that deep disagreements in general are as epistemologically problematic as has been widely supposed. (shrink)

This article provides a spatial analysis of the conceptual framework of fluid dynamics during the nineteenth century, focusing on the transition from the Euler equation to the Navier–Stokes equation. A dynamic version of Peter Gärdenfors's theory of conceptual spaces is applied which distinguishes changes of five types: addition and deletion of special laws; change of metric; change in importance; change in separability; addition and deletion of dimensions. The case instantiates all types but the deletion of dimensions. We also provide a (...) new view upon limiting case reduction at the conceptual level that clarifies the relation between the predecessor and successor conceptual framework. The nineteenth-century development of fluid dynamics is argued to be an instance of normal science development. (shrink)

Françios Viète was a geometer in search of better techniques for astronomical calculation. Through his theorem on angular sections he found a use for higher-dimensional geometric magnitudes which allowed him to create an algebra for geometry. We show that unlike traditional numerical algebra, the knowns and unknowns in Viète’s logistice speciosa are the relative sizes of non-arithmetized magnitudes in which the “calculations” must respect dimension. Along with this foundational shift Viète adopted a radically new notation based in Greek geometric equalities. (...) His letters stand for values rather than types, and his given values are undetermined. Where previously algebra was founded in polynomials as aggregations, Viète became the first modern algebraist in working with polynomials built from operations, and the notations reflect these conceptions. Viète’s innovations are situated in the context of sixteenth-century practice, and we examine the interpretation of Jacob Klein, the only historian to have conducted a serious inquiry into the ontology of Viète’s “species”. (shrink)

In this paper, I argue that there is neither valid deductive support nor strong inductive support for Kuhn’s incommensurability thesis. There is no valid deductive support for Kuhn’s incommensurability thesis because, from the fact that the reference of the same kind terms changes or discontinues from one theoretical framework to another, it does not necessarily follow that these two theoretical frameworks are taxonomically incommensurable. There is no strong inductive support for Kuhn’s incommensurability thesis, since there are rebutting defeaters against it (...) in the form of episodes from the history of science that do not exhibit discontinuity and replacement, as Kuhn’s incommensurability thesis predicts, but rather continuity and supplementation. If this is correct, then there are no compelling epistemic reasons to believe that Kuhn’s incommensurability thesis is true or probable. (shrink)

This paper offers an analysis of the structure of epistemic vice-charging, the critical practice of charging other persons with epistemic vice. Several desiderata for a robust vice-charge are offered and two deep obstacles to the practice of epistemic vice-charging are then identified and discussed. The problem of responsibility is that few of us enjoy conditions that are required for effective socialisation as responsible epistemic agents. The problem of consensus is that the efficacy of a vice-charge is contingent upon a degree (...) of consensus between critic and target that is unlikely or impossible where vice-charging is most likely to be provoked. It emerges that a robust critical practice of vice-charging is possible in principle, but very difficult in practice. (shrink)

The existence of fundamental moral disagreements is a central problem for moral realism and has often been contrasted with an alleged absence of disagreement in mathematics. However, mathematicians do in fact disagree on fundamental questions, for example on which set-theoretic axioms are true, and some philosophers have argued that this increases the plausibility of moral vis-à-vis mathematical realism. I argue that the analogy between mathematical and moral disagreement is not as straightforward as those arguments present it. In particular, I argue (...) that pluralist accounts of mathematics render fundamental mathematical disagreements compatible with mathematical realism in a way in which moral disagreements and moral realism are not. 11. (shrink)

Social revolutions--that is critical periods of decisive, qualitative change--are a commonly acknowledged historical fact. But can the idea of revolutionary upheaval be extended to the world of ideas and theoretical debate? The publication of Kuhn's The Structure of Scientific Revolutions in 1962 led to an exciting discussion of revolutions in the natural sciences. A fascinating, but little known, off-shoot of this was a debate which began in the United States in the mid-1970's as to whether the concept of revolution could (...) be applied to mathematics as well as science. Michael Grove declared that revolutions never occur in mathematics, while Joseph Dauben argued that there have been mathematical revolutions and gave some examples. This book is the first comprehensive examination of the question. It reprints the original papers of Grove, Dauben, and Mehrtens, together with additional chapters giving their current views. To this are added new contributions from nine further experts in the history of mathematics, who each discuss an important episode and consider whether it was a revolution. The whole question of mathematical revolutions is thus examined comprehensively and from a variety of perspectives. This thought-provoking volume will interest mathematicians, philosophers, and historians alike. (shrink)

Develops a logical analysis of dialogue in which two or more parties attempt to advance their own interests. It includes a classification of the major types of dialogues and a discussion of several important informal fallacies.

Combining in-depth analysis with strikingly apt examples of the role that courage plays in the life of human beings, this major contribution to moral philosophy argues that courage is necessary to personal achievement as well as to the common good of a civic community. Bauhn insists that courage is necessary for reinforcing people's understanding of themselves as autonomous agents, which is in turn necessary for countering widespread feelings of alienation and depression. He defines courage as the ability to confront fear, (...) but crucially distinguishes a variety of fears that give rise to different types of courage. (shrink)

In their account of theory change in logic, Aberdein and Read distinguish 'glorious' from 'inglorious' revolutions--only the former preserves all 'the key components of a theory' [1]. A widespread view, expressed in these terms, is that empirical science characteristically exhibits inglorious revolutions but that revolutions in mathematics are at most glorious [2]. Here are three possible responses: 0. Accept that empirical science and mathematics are methodologically discontinuous; 1. Argue that mathematics can exhibit inglorious revolutions; 2. Deny that inglorious revolutions are (...) characteristic of science. Where Aberdein and Read take option 1, option 2 is preferred by Mizrahi [3]. This paper seeks to resolve this disagreement through consideration of some putative mathematical revolutions. [1] Andrew Aberdein and Stephen Read, The philosophy of alternative logics, The Development of Modern Logic (Leila Haaparanta, ed.), Oxford University Press, Oxford, 2009, pp. 613-723. [2] Donald Gillies (ed.), Revolutions in Mathematics, Oxford University Press, Oxford, 1992. [3] Moti Mizrahi, Kuhn's incommensurability thesis: What's the argument?, Social Epistemology 29 (2015), no. 4, 361-378. (shrink)

This article examines the central epistemological issues tied to the recognition of disagreement.

I intend to bring recent work applying virtue theory to the study of argument to bear on a much older problem, that of disagreements that resist rational resolution, sometimes termed "deep disagreements". Just as some virtue epistemologists have lately shifted focus onto epistemic vices, I shall argue that a renewed focus on the vices of argument can help to illuminate deep disagreements. In particular, I address the role of arrogance, both as a factor in the diagnosis of deep disagreements and (...) as an obstacle to their mutually acceptable resolution. Arrogant arguers are likely to make any disagreements to which they are party seem deeper than they really are and arrogance impedes the strategies that we might adopt to resolve deep disagreements. As a case in point, since arrogant or otherwise vicious arguers cannot be trusted not to exploit such strategies for untoward ends, any policy for deep disagreement amelioration must require particularly close attention to the vices of argument, lest they be exploited by the unscrupulous. (shrink)

Douglas Walton’s multitudinous contributions to the study of argumentation seldom, if ever, directly engage with argumentation in mathematics. Nonetheless, several of the innovations with which he is most closely associated lend themselves to improving our understanding of mathematical arguments. I concentrate on two such innovations: dialogue types (§1) and argumentation schemes (§2). I argue that both devices are much more applicable to mathematical reasoning than may be commonly supposed.

While puzzles concerning the epistemic significance of disagreement are typically motivated by looking at the widespread and persistent disagreements we are aware of, almost all of the literature on the epistemic significance of disagreement has focused on cases idealized peer disagreement. This fact might itself be puzzling since it doesn’t seem that we ever encounter disagreements that meet the relevant idealized conditions. In this paper I hope to somewhat rectify this matter. I begin by closely examining what an idealized case (...) of peer disagreement looks like and what the Equal Weight View (EWV) of disagreement claims about the epistemic significance of such disagreements. After briefly defending the verdicts of the EWV in idealized disagreements I proceed to unpack the implications of stripping away the idealized conditions. In doing I show both why it is important to focus on idealized cases of peer disagreement and what we can learn from such cases that applies to the everyday cases of disagreement of which we are very familiar. (shrink)

One result of successful argumentation – able arguers presenting cogent arguments to competent audiences – is a transfer of credibility from premises to conclusions. From a purely logical perspective, neither dubious premises nor fallacious inference should lower the credibility of the target conclusion. Nevertheless, some arguments do backfire this way. Dialectical and rhetorical considerations come into play. Three inter-related conclusions emerge from a catalogue of hapless arguers and backfiring arguments. First, there are advantages to paying attention to arguers and their (...) contexts, rather than focusing narrowly on their arguments, in order to understand what can go wrong in argumentation. Traditional fallacy identification, with its exclusive attention to faulty inferences, is inadequate to explain the full range of argumentative failures. Second, the notion of an Ideal Arguer can be defined by contrast with her less than ideal peers to serve as a useful tool in argument evaluation. And third, not all of the ways that arguers raise doubts about their conclusions are pathological. On the contrary, some ways that doubts are raised concerning our intended conclusions are an integral part of ideal argumentative practice. (shrink)