I expand modal normativism, a theory of metaphysical modality, to give a normativist account of metaphysical explanation. According to modal normativism, basic modal claims do not have a descriptive function, but instead have the normative function of enabling language users to express semantic rules that govern the use of ordinary non-modal vocabulary. However, a worry for modal normativism is that it doesn’t keep up with all of the important and interesting metaphysics we can do by giving and evaluating metaphysical explanations. (...) So, I advance modal normativism by arguing that metaphysical explanations also have a normative rather than descriptive function. In particular, non-causal explanatory claims have formal and semantic properties that make them expressively stricter than basic modal claims and so are better suited to express fine-grained aspects of semantic rules. A major payoff of my normativist account of metaphysical explanations is that it yields a plausible story about how we come to evaluate and know metaphysical explanations—we do this primarily by conceptual analysis. I also respond to a number of objections, including the objection that the epistemic payoffs of my view are not worth the metaphysical costs. (shrink)

This paper defends the idea that there might be vagueness or indeterminacy in the world itself---as opposed to merely in our representations of the world---against the charges of incoherence and unintelligibility. First we consider the idea that the world might contain vague *properties and relations*; we show that this idea is already implied by certain well-understood views concerning the semantics of vague predicates (most notably the fuzzy view). Next we consider the idea that the world might contain vague *objects*; we (...) argue that an object is indeterminate in a certain respect (colour, size, etc.) just in case it is a borderline case of a maximally specific colour (size, etc.) property. Finally we consider the idea that the *world as a whole* might be indeterminate; we argue that the world is indeterminate just in case it lacks a determinate division into determinate objects. (shrink)

Gauss’s quadratic reciprocity theorem is among the most important results in the history of number theory. It’s also among the most mysterious: since its discovery in the late 18th century, mathematicians have regarded reciprocity as a deeply surprising fact in need of explanation. Intriguingly, though, there’s little agreement on how the theorem is best explained. Two quite different kinds of proof are most often praised as explanatory: an elementary argument that gives the theorem an intuitive geometric interpretation, due to Gauss (...) and Eisenstein, and a sophisticated proof using algebraic number theory, due to Hilbert. Philosophers have yet to look carefully at such explanatory disagreements in mathematics. I do so here. According to the view I defend, there are two important explanatory virtues—depth and transparency—which different proofs (and other potential explanations) possess to different degrees. Although not mutually exclusive in principle, the packages of features associated with the two stand in some tension with one another, so that very deep explanations are rarely transparent, and vice versa. After developing the theory of depth and transparency and applying it to the case of quadratic reciprocity, I draw some morals about the nature of mathematical explanation. (shrink)

Conceptual engineering and conceptual ethics are branches of philosophy concerned with questions about how to assess and ameliorate our representational devices (such as concepts and words). It's a part of philosophy concerned with questions about which concepts we should use (and why), how concepts can be improved, when concepts should be abandoned, and how proposals for amelioration can be implemented. Central parts of the history of philosophy have engaged with these issues, but the focus of this volume is on applications (...) to work in contemporary philosophy of language and mind, epistemology, gender and race theory, ethics, philosophy of science, and philosophical logic. This is the first volume devoted entirely to conceptual engineering and conceptual ethics. The volume explores the possibilities, benefits, problems, and applications of conceptual engineering and conceptual ethics. It consists of twenty chapters written by leading philosophers. (shrink)

Not all scientific explanations work by describing causal connections between events or the world's overall causal structure. In addition, mathematicians regard some proofs as explaining why the theorems being proved do in fact hold. This book proposes new philosophical accounts of many kinds of non-causal explanations in science and mathematics.

_''Philosophy' is a word which has been used in many ways, some wider, some narrower. I propose to use it in a very wide sense, which I will now try to explain.'_ - _ Bertrand Russell Nearly forty years since its first publication, History of Western Philosophy_ remains unchallenged as the ultimate introduction to its subject, while claiming classic status in its own right. It is the bestselling philosophy book of the twentieth century and one of the most important philosophical (...) works of all time. This compact and affordable paperback edition makes this comprehensive and brilliantly-written text readily available for a new generation of readers. As part of our commitment to Russell publishing, the delux version of this bestselling title will continue to be available. 1961: 848pp: Pb: 0-145-07854-7: £16.99. (shrink)

According to noncognitivists, when we say that stealing is wrong, what we are doing is more like venting our feelings about stealing or encouraging one another not to steal, than like stating facts about morality. These ideas challenge the core not only of much thinking about morality and metaethics, but also of much philosophical thought about language and meaning. _Noncognitivism in Ethics_ is an outstanding introduction to these theories, ranging from their early history through the latest contemporary developments. Beginning with (...) a general introduction to metaethics, Mark Schroeder introduces and assesses three principal kinds of noncognitivist theory: the speech-act theories of Ayer, Stevenson, and Hare, the expressivist theories of Blackburn and Gibbard, and hybrid theories. He pays particular attention both to the philosophical problems about what moral facts could be about or how they could matter which noncognitivism seeks to solve, and to the deep problems that it faces, including the task of explaining both the nature of moral thought and the complexity of moral attitudes, and the ‘Frege-Geach’ problem. Schroeder makes even the most difficult material accessible by offering crucial background along the way. Also included are exercises at the end of each chapter, chapter summaries, and a glossary of technical terms - making _Noncognitivism in Ethics_ essential reading for all students of ethics and metaethics. (shrink)

This book is a defense of modal realism; the thesis that our world is but one of a plurality of worlds, and that the individuals that inhabit our world are only a few out of all the inhabitants of all the worlds. Lewis argues that the philosophical utility of modal realism is a good reason for believing that it is true.

"Williams's masterful translation satisfies (at last!) a long-standing need. There are lots of good translations of Augustine's great work, but until now we have been forced to choose between those that strive to replicate in English something of the majesty and beauty of Augustine's Latin style and those that opt instead to convey the careful precision of his philosophical terminology and argumentation. Finally, Williams has succeeded in capturing both sides of Augustine's mind in a richly evocative, impeccably reliable, elegantly readable (...) presentation of one of the most impressive achievements in Western thought--Augustine's Confessions." --Scott MacDonald, Professor of Philosophy and Norma K. Regan Professor in Christian Studies, Cornell University. (shrink)

This book is Dewey's most fully developed treatment of logic as the theory of Inquiry. It is a later work which reflects, in part, Dewey's readings of C.S. Peirce during the 1930's. -/- Reprinted in Series: The collected works of John Dewey / ed. by Jo Ann Boydston, 3,12.; The later works, 1925 - 1953, Vol. 12.

Although all mathematical truths are necessary, mathematicians take certain combinations of mathematical truths to be ‘coincidental’, ‘accidental’, or ‘fortuitous’. The notion of a ‘ mathematical coincidence’ has so far failed to receive sufficient attention from philosophers. I argue that a mathematical coincidence is not merely an unforeseen or surprising mathematical result, and that being a misleading combination of mathematical facts is neither necessary nor sufficient for qualifying as a mathematical coincidence. I argue that although the components of a mathematical coincidence (...) may possess a common explainer, they have no common explanation ; that two mathematical facts have a unified explanation makes their truth non-coincidental. I suggest that any motivation we may have for thinking that there are mathematical coincidences should also motivate us to think that there are mathematical explanations, since the notion of a mathematical coincidence can be understood only in terms of the notion of a mathematical explanation. I also argue that the notion of a mathematical coincidence plays an important role in scientific explanation. When two phenomenological laws of nature are similar, despite concerning physically distinct processes, it may be that any correct scientific explanation of their similarity proceeds by revealing their similarity to be no mathematical coincidence. (shrink)

First published in 1946, History of Western Philosophy went on to become the best-selling philosophy book of the twentieth century. A dazzlingly ambitious project, it remains unchallenged to this day as the ultimate introduction to Western philosophy. Providing a sophisticated overview of the ideas that have perplexed people from time immemorial, it is 'long on wit, intelligence and curmudgeonly scepticism', as the New York Times noted, and it is this, coupled with the sheer brilliance of its scholarship, that has made (...) Russell's History of Western Philosophy one of the most important philosophical works of all time. (shrink)

This paper defends the idea that there might be vagueness or indeterminacy in the world itself--as opposed to merely in our representations of the world--against the charges of incoherence and unintelligibility. First we consider the idea that the world might contain vague properties and relations ; we show that this idea is already implied by certain well-understood views concerning the semantics of vague predicates (most notably the fuzzy view). Next we consider the idea that the world might contain vague objects (...) ; we argue that an object is indeterminate in a certain respect (colour, size, etc.) just in case it is a borderline case of a maximally specific colour (size, etc.) property. Finally we consider the idea that the world as a whole might be indeterminate; we argue that the world is indeterminate just in case it lacks a determinate division into determinate objects. (shrink)

Contains an overall account of morality in its philosophical format particularly with regard to problems of observation, evidence, and truth.

When in May 1930, the Council of Trinity College, Cambridge, had to decide whether to renew Wittgenstein's research grant, it turned to Bertrand Russell for an assessment of the work Wittgenstein had been doing over the past year. His verdict: "The theories contained in this new work . . . are novel, very original and indubitably important. Whether they are true, I do not know. As a logician who likes simplicity, I should like to think that they are not, but (...) from what I have read of them I am quite sure that he ought to have an opportunity to work them out, since, when completed, they may easily prove to constitute a whole new philosophy." "[ Philosophical Remarks ] contains the seeds of Wittgenstein's later philosophy of mind and of mathematics. Principally, he here discusses the role of indispensable in language, criticizing Russell's The Analysis of Mind . He modifies the Tractatus 's picture theory of meaning by stressing that the connection between the proposition and reality is not found in the picture itself. He analyzes generality in and out of mathematics, and the notions of proof and experiment. He formulates a pain/private-language argument and discusses both behaviorism and the verifiability principle. The work is difficult but important, and it belongs in every philosophy collection."--Robert Hoffman, Philosophy "Any serious student of Wittgenstein's work will want to study his Philosophical Remarks as a transitional book between his two great masterpieces. The Remarks is thus indispensible for anyone who seeks a complete understanding of Wittgenstein's philosophy."--Leonard Linsky, American Philosophical Association. (shrink)

This classic and controversial book examines the roots of the idea of virtue, diagnoses the reasons for its absence in modern life, and proposes a path for its recovery.

What is the Moral Problem? NORMATIVE ETHICS VS. META-ETHICS It is a common fact of everyday life that we appraise each others' behaviour and attitudes from ...

Recasting important questions about truth and objectivity in new and helpful terms, his book will become a focus in the contemporary debates over realism, and ...

It is my aim in this paper to show that the contemporary assimilation of essence to modality is fundamentally misguided and that, as a consequence, the corresponding conception of metaphysics should be given up. It is not my view that the modal account fails to capture anything which might reasonably be called a concept of essence. My point, rather, is that the notion of essence which is of central importance to the metaphysics of identity is not to be understood in (...) modal terms or even to be regarded as extensionally equivalent to a modal notion. The one notion is, if I am right, a highly refined version of the other; it is like a sieve which performs a similar function but with a much finer mesh. (shrink)

If the world itself is metaphysically indeterminate in a specified respect, what follows? In this paper, we develop a theory of metaphysical indeterminacy answering this question.

We shall argue that the attempt carried out by certain philosophers in this century to parrot the language, the method, and the results of mathematics has harmed philosophy. Such an attempt results from a misunderstanding of both mathematics and philosophy, and has harmed both subjects.

The critics in this volume raise several important challenges to the modal normativist position developed in Norms and Necessity, including whether the relation I claim holds between semantic rules and necessity claims generates spurious claims of metaphysical necessity, whether the view is circular (implicitly relying on a more 'robust' form of modal realism), and whether it conflicts with truth-conditional semantics. They also raise probing questions about how it compares to other views of modality, including a Lewisian view and an essentialist (...) view. In these replies, I respond to these challenges in ways that precisify the relevant understanding of ‘semantic rules' and the forms they can take, that make clearer the direction of explanation in modal normativism in ways that show the view doesn't rely on a more ‘robust’ form of modal realism, and that give reason for thinking that there is actually no conflict between modal normativism and truth-conditional semantics. I also aim to give a fuller assessment of how it compares to other approaches to modality, including an essentialist approach. (shrink)

The purpose of this article is to explore the extent to which mathematics is subject to open texture and the extent to which mathematics resists open texture. The resistance is tied to the importance of proof and, in particular, rigor, in mathematics.

Philosophical theories often hinge on claims about what is necessary or possible. But what are possibilities and necessities, and how could we come to know about them? This book aims to help demystify the methodology of philosophy, by treating such claims not as attempted descriptions of strange facts or distant 'possible worlds', but rather as ways of expressing rules or norms.

ABSTRACTIn this paper I investigate how conceptual engineering applies to mathematical concepts in particular. I begin with a discussion of Waismann’s notion of open texture, and compare it to Shapiro’s modern usage of the term. Next I set out the position taken by Lakatos which sees mathematical concepts as dynamic and open to improvement and development, arguing that Waismann’s open texture applies to mathematical concepts too. With the perspective of mathematics as open-textured, I make the case that this allows us (...) to deploy the tools of conceptual engineering in mathematics. I will examine Cappelen’s recent argument that there are no conceptual safe spaces and consider whether mathematics constitutes a counterexample. I argue that it does not, drawing on Haslanger’s distinction between manifest and operative concepts, and applying this in a novel way to set-theoretic foundations. I then set out some of the questions that need to be engaged with to establish mathematics as involving a kind of conceptual engineering. I finish with a case study of how the tools of conceptual engineering will give us a way to progress in the debate between advocates of the Universe view and the Multiverse view in set theory. (shrink)

In this ambitious study, David Corfield attacks the widely held view that it is the nature of mathematical knowledge which has shaped the way in which mathematics is treated philosophically and claims that contingent factors have brought us to the present thematically limited discipline. Illustrating his discussion with a wealth of examples, he sets out a variety of approaches to new thinking about the philosophy of mathematics, ranging from an exploration of whether computers producing mathematical proofs or conjectures are doing (...) real mathematics, to the use of analogy, the prospects for a Bayesian confirmation theory, the notion of a mathematical research programme and the ways in which new concepts are justified. His inspiring book challenges both philosophers and mathematicians to develop the broadest and richest philosophical resources for work in their disciplines and points clearly to the ways in which this can be done. (shrink)

In this collection of essays, Bromberger explores the centrality of questions and predicaments they create in scientific research. He discusses the nature of explanation, theory, and the foundations of linguistics.

In this ambitious study, David Corfield attacks the widely held view that it is the nature of mathematical knowledge which has shaped the way in which mathematics is treated philosophically, and claims that contingent factors have brought us to the present thematically limited discipline. Illustrating his discussion with a wealth of examples, he sets out a variety of new ways to think philosophically about mathematics, ranging from an exploration of whether computers producing mathematical proofs or conjectures are doing real mathematics, (...) to the use of analogy, the prospects for a Bayesian confirmation theory, the notion of a mathematical research programme, and the ways in which new concepts are justified. His highly original book challenges both philosophers and mathematicians to develop the broadest and richest philosophical resources for work in their disciplines, and points clearly to the ways in which this can be done. (shrink)

What do mathematicians mean when they use terms such as ‘deep’, ‘elegant’, and ‘beautiful’? By applying empirical methods developed by social psychologists, we demonstrate that mathematicians' appraisals of proofs vary on four dimensions: aesthetics, intricacy, utility, and precision. We pay particular attention to mathematical beauty and show that, contrary to the classical view, beauty and simplicity are almost entirely unrelated in mathematics.

I develop a non-representationalist account of mathematical thought, on which the point of mathematical theorizing is to provide us with the conceptual capacity to structure and articulate information about the physical world in an epistemically useful way. On my view, accepting a mathematical theory is not a matter of having a belief about some subject matter; it is rather a matter of structuring logical space, in a sense to be made precise. This provides an elegant account of the cognitive utility (...) of mathematics. Further, it makes explicit how the brand of non-representationalism I develop is compatible with there being substantive rationality constraints on our mathematical theorizing. (shrink)

Modality presents notorious philosophical problems, including the epistemic problem of how we could come to know modal facts and metaphysical problems about how to place modal facts in the natural world. These problems arise from thinking of modal claims as attempts to describe modal features of this world that explain what makes them true. Here I propose a different view of modal discourse in which talk about what is “metaphysically necessary” does not aim to describe modal features of the world, (...) but, rather, provides a particularly useful way of expressing constitutive semantic and conceptual rules in the object language. The result is a “modal normativist” view that enables us to avoid the epistemic problems of modality and mitigate the metaphysical worries, while also leaving open the possibility of a unified account of the function of modal language. Finally, I address a serious challenge: we have the norms we do in order to track the modal facts of the world, so that the order of explanation must go in the opposite direction. I close by showing how the normativist may answer that challenge. (shrink)

This paper argues that new light may be shed on mathematical reasoning in its non-pathological forms by careful observation of its pathologies. The first section explores the application to mathematics of recent work on fallacy theory, specifically the concept of an ‘argumentation scheme’: a characteristic pattern under which many similar inferential steps may be subsumed. Fallacies may then be understood as argumentation schemes used inappropriately. The next section demonstrates how some specific mathematical fallacies may be characterized in terms of argumentation (...) schemes. The third section considers the phenomenon of correct answers which result from incorrect methods. This turns out to pose some deep questions concerning the nature of mathematical knowledge. In particular, it is argued that a satisfactory epistemology for mathematical practice must address the role of luck. (shrink)

Needs, Values, Truth brings together of some of the most important and influential writings by a leading contemporary philosopher, drawn from twenty-five years of his work in the broad area of the philosophy of value. The author ranges between problems of ethics, meta-ethics, philosophy of mind, and philosophy of logic and language, looking at questions relating to meaning, truth and objectivity in judgements of value. For this third edition he has added a new essay on incommensurability, in addition to making (...) minor revisions to the existing text. The volume will stand as a definitive summation of his work in this area. (shrink)

Explanation in mathematics has recently attracted increased attention from philosophers. The central issue is taken to be how to distinguish between two types of mathematical proofs: those that explain why what they prove is true and those that merely prove theorems without explaining why they are true. This way of framing the issue neglects the possibility of mathematical explanations that are not proofs at all. This paper addresses what it would take for a non-proof to explain. The paper focuses on (...) a particular example of an explanatory non-proof: an argument that mathematicians regard as explaining why a given theorem holds regarding the derivative of an infinite sum of differentiable functions. The paper contrasts this explanatory non-proof with various non-explanatory proofs of the same theorem. The paper offers an account of what makes the given non-proof explanatory. This account is motivated by investigating the difficulties that arise when we try to extend Mark Steiner’s influential account of explanatory proofs to cover this explanatory non-proof. (shrink)