Tractatus Logico-Philosophicus by Ludwig Wittgenstein: three parallel tree-structured editions. (1) Tree-structured arrangement of the German text, edited by David G. Stern, Joachim Schulte and Katia Saporiti. (2) Tree-structured arrangement of the English translation by Ogden and Ramsey, edited by David G. Stern. (3) Tree-structured arrangement of the English translation by Pears and McGuinness, edited by David G. Stern.

The article presents an anti-exceptionalist view of philosophical methodology, on which it is much closer to the methodology of other disciplines than many philosophers like to think. Like mathematics, it is a science, but not a natural science. Its methods are not primarily experimental, though it can draw on the results of natural science. Like foundational mathematics, its methods are abductive as well as deductive. As in the natural sciences, much progress in philosophy consists in the construction of better models (...) rather than in the discovery of new laws. We should not worry about whether philosophy is a priori or a posteriori, because the distinction is epistemologically superficial. (shrink)

In this paper I present an account of a distinctive form of ‘practical’ or ‘deliberative’ uncertainty that has been central in debates in both ethics and metaethics. Many writers have assumed that such uncertainty concerns a special normative question, such as what we ought to do ‘all things considered.’ I argue against this assumption and instead endorse an alternative view of such uncertainty, which combines elements of both metaethical cognitivism and non-cognitivism. A notable consequence of this view is that even (...) if there are objective and irreducible truths about how we ought to act, all things considered, the ‘central deliberative question,’ as it’s sometimes called, doesn’t concern such truths. Instead, that question doesn’t have a true answer. (shrink)

Modern empiricism has been conditioned in large part by two dogmas. One is a belief in some fundamental cleavage between truths which are analytic, or grounded in meanings independently of matters of fact, and truth which are synthetic, or grounded in fact. The other dogma is reductionism: the belief that each meaningful statement is equivalent to some logical construct upon terms which refer to immediate experience. Both dogmas, I shall argue, are ill founded. One effect of abandoning them is, as (...) we shall see, a blurring of the supposed boundary between speculative metaphysics and natural science. Another effect is a shift toward pragmatism. (shrink)

This volume consists of the first of the John Dewey Lectures delivered under the auspices of Columbia University's Philosophy Department as well as other essays by the author. Intended to clarify the meaning of the philosophical doctrines propounded by Professor Quine in 'Word and Objects', the essays included herein both support and expand those doctrines.

In providing a good foundation for mathematics, set theorists often aim to develop the strongest theories possible and avoid those theories that place undue restrictions on the capacity to possess strength. For example, adding a measurable cardinal to $ZFC$ is thought to give a stronger theory than adding $V=L$ and the latter is thought to be more restrictive than the former. The two main proponents of this style of account are Penelope Maddy and John Steel. In this paper, I’ll offer (...) a third account that is intended to provide a simple analysis of restrictiveness based on the algebraic concept of retraction in the category of theories. I will also deliver some results and arguments that suggest some plausible alternative approaches to analyzing restrictiveness do not live up to their intuitive motivation. (shrink)

Logic is usually thought to concern itself only with features that sentences and arguments possess in virtue of their logical structures or forms. The logical form of a sentence or argument is determined by its syntactic or semantic structure and by the placement of certain expressions called “logical constants.”[1] Thus, for example, the sentences Every boy loves some girl. and Some boy loves every girl. are thought to differ in logical form, even though they share a common syntactic and semantic (...) structure, because they differ in the placement of the logical constants “every” and “some”. By contrast, the sentences Every girl loves some boy. and Every boy loves some girl. are thought to have the same logical form, because “girl” and “boy” are not logical constants. Thus, in order to settle questions about logical form, and ultimately about which arguments are logically valid and which sentences logically true, we must distinguish the “logical constants” of a language from its nonlogical expressions. (shrink)

In this paper, we develop an alternative strategy, Platonized Naturalism, for reconciling naturalism and Platonism and to account for our knowledge of mathematical objects and properties. A systematic (Principled) Platonism based on a comprehension principle that asserts the existence of a plenitude of abstract objects is not just consistent with, but required (on transcendental grounds) for naturalism. Such a comprehension principle is synthetic, and it is known a priori. Its synthetic a priori character is grounded in the fact that it (...) is an essential part of the logic in which any scientific theory will be formulated and so underlies (our understanding of) the meaningfulness of any such theory (this is why it is required for naturalism). Moreover, the comprehension principle satisfies naturalist standards of reference, knowledge, and ontological parsimony! As part of our argument, we identify mathematical objects as abstract individuals in the domain governed by the comprehension principle, and we show that our knowledge of mathematical truths is linked to our knowledge of that principle. (shrink)

It is widely agreed that the intelligibility of modal metaphysics has been vindicated. Quine's arguments to the contrary supposedly confused analyticity with metaphysical necessity, and rigid with non-rigid designators.2 But even if modal metaphysics is intelligible, it could be misconceived. It could be that metaphysical necessity is not absolute necessity – the strictest real notion of necessity – and that no proposition of traditional metaphysical interest is necessary in every real sense. If there were nothing otherwise “uniquely metaphysically significant” about (...) metaphysical necessity, then paradigmatic metaphysical necessities would be necessary in one sense of “necessary”, not necessary in another, and that would be it. The question of whether they were necessary simpliciter would be like the question of whether the Parallel Postulate is true simpliciter – understood as a pure mathematical conjecture, rather than as a hypothesis about physical spacetime. In a sense, the latter question has no objective answer. In this article, I argue that paradigmatic questions of modal metaphysics are like the Parallel Postulate question. I then discuss the deflationary ramifications of this argument. I conclude with an alternative conception of the space of possibility. According to this conception, there is no objective boundary between possibility and impossibility. Along the way, I sketch an analogy between modal metaphysics and set theory. (shrink)

This paper explores the impact of quantification into predicate position on the metaphysics of properties, arguing that two familiar debates about properties are fundamentally altered by recasting them in a second-order setting. Two theories of properties are outlined, differing over whether the existence of properties is expressed using first-order or second-order quantifiers. It is argued that the second-order theory: provides good reason to regard debate about the locations of properties as contentless; resolves debate about whether properties are particulars or universals (...) in favour of universals. (shrink)

Non-naturalist realists are committed to the belief, famously voiced by Parfit, that if there are no non-natural facts then nothing matters. But it is morally objectionable to conditionalise all our moral commitments on the question of whether there are non-natural facts. Non-natural facts are causally inefficacious, and so make no difference to the world of our experience. And to be a realist about such facts is to hold that they are mind-independent. It is compatible with our experiences that there are (...) no non-natural facts, or that they are very different from what we think. As Nagel says, realism makes scepticism intelligible. So the non-naturalist must hold that you might be wrong that your partner matters, even if you are correct about every natural, causal fact about your history and relationship. But to hold that conditional attitude to your partner would be a moral betrayal. So believing non-naturalist realism involves doing something immoral. (shrink)

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

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)

Mathematics plays an inordinate role in the work of many of famous Western philosophers, from the time of Plato, through Husserl and Wittgenstein, and even to the present. Why? This paper points to the experience of learning or making mathematics, with an emphasis on proof. It distinguishes two sources of the perennial impact of mathematics on philosophy. They are classified as Ancient and Enlightenment. Plato is emblematic of the former, and Kant of the latter. The Ancient fascination arises from the (...) sense that mathematics explores something ‘out there’. This is illustrated by recent discussions by distinguished contemporary mathematicians. The Enlightenment strand often uses Kant's argot: ‘absolute necessity’, ‘apodictic certainty’ and ‘a priori’ judgement or knowledge. The experience of being compelled by proof, the sense that something must be true, that a result is certain, generates the philosophy. It also creates the illusion that mathematics is certain. Kant's leading question, ‘How is pure mathematics possible?’, is easily misunderstood because the modern distinction between pure and applied is an artefact of the 19th century. As Russell put it, the issue is to explain ‘the apparent power of anticipating facts about things of which we have no experience’. More generally the question is, how is it that pure mathematics is so rich in applications? Some six types of application are distinguished, each of which engenders its own philosophical problems which are descendants of the Enlightenment, and which differ from those descended from the Ancient strand. (shrink)

An original and elegant work of metaethics, this book brings a new clarity and rigor to the discussion of these tangled issues, and will significantly alter the ...

Philosophers have long suspected that thought and discourse about what we ought to do differ in some fundamental way from statements about what is. But the difference has proved elusive, in part because the two kinds of statement look alike. Focusing on judgments that express decisions--judgments about what is to be done, all things considered--Allan Gibbard offers a compelling argument for reconsidering, and reconfiguring, the distinctions between normative and descriptive discourse--between questions of "ought" and "is." Gibbard considers how our actions, (...) and our realities, emerge from the thousands of questions and decisions we form for ourselves. The result is a book that investigates the very nature of the questions we ask ourselves when we ask how we should live, and that clarifies the concept of "ought" by understanding the patterns of normative concepts involved in beliefs and decisions. An original and elegant work of metaethics, this book brings a new clarity and rigor to the discussion of these tangled issues, and will significantly alter the long-standing debate over "objectivity" and "factuality" in ethics. Table of Contents: I. Preliminaries 1. Introduction: A Possibility Proof 2. Intuitionism as Template: Emending Moore II. The Thing to Do 3. Planning and Ruling Out: The "Frege-Geach" Problem 4. Judgment, Disagreement, Negation 5. Supervenience and Constitution 6. Character and Import III. Normative Concepts 7. Ordinary Oughts: Meaning and Motivation 8. Normative Kinds: Patterns of Engagement 9. What to Say about the Thing to Do: The Expressivistic Turn and What it Gains Us IV. Knowing What to Do 10. Explaining with Plans 11. Knowing What to Do 12. Ideal Response Concepts 13. Deep Vindication and Practical Confidence 14. Impasse and Dissent References Index This is a remarkable book. It takes up a central and much-discussed problem - the difference between normative thought and "descriptive" thought. It develops a compelling response to that problem with ramifications for much else in philosophy. But perhaps most importantly, it brings new clarity and rigor to the discussion of these tangled issues. It will take some time to come to terms with the details of Gibbard's discussion. It is absolutely clear, however, that the book will reconfigure the debate over objectivity and "factuality" in ethics. --Gideon Rosen, Professor of Philosophy, Princeton University Gibbard,/author> writes elegantly, and the theory he develops is innovative, philosophically sophisticated, and challenging. Gibbard defends his theory vigorously and with admirable intellectual honesty. --David Copp, Professor of Philosophy, Bowling Green State University. (shrink)

§ i. After deserting for a time the old Euclidean standards of rigour, mathematics is now returning to them, and even making efforts to go beyond them. ...

Radically permissive ontologies like mereological universalism and material plenitude are typically motivated by concerns about arbitrariness or anthropocentrism: it would be objectionably arbitrary, the thought goes, to countenance only those objects that we ordinarily take there to be. Despite the prevalence of this idea, it isn't at all clear what it is for a theory to be “objectionably arbitrary,” or what follows from a commitment to avoiding arbitrariness in metaphysics. This paper aims to clarify both questions, and examines whether arguments (...) from arbitrariness really are the proper foundations for one or both varieties of ontological permissivism. I argue that these considerations (even when made more precise) are far less successful at motivating radical forms of permissivism than we often take them to be. To do better, permissivists must either supply a much more developed metaphysics of material objects, or a controversial (but tempting) conception of what we're doing when we do metaphysics. (shrink)

According to a certain pluralist view in philosophy of mathematics, there are as many mathematical objects as there can coherently be. Recently, Justin Clarke-Doane has explored what consequences the analogous view on normative properties would have. What if there is a normative pluriverse? Here I address this same question. The challenge is best seen as a challenge to an important form of normative realism. I criticize the way Clarke-Doane presents the challenge. An improved challenge is presented, and the role of (...) pluralism in this challenge is assessed. (shrink)

To answer the question of whether mathematics needs new axioms, it seems necessary to say what role axioms actually play in mathematics. A first guess is that they are inherently obvious statements that are used to guarantee the truth of theorems proved from them. However, this may neither be possible nor necessary, and it doesn’t seem to fit the historical facts. Instead, I argue that the role of axioms is to systematize uncontroversial facts that mathematicians can accept from a wide (...) variety of philosophical positions. Once the axioms are generally accepted, mathematicians can expend their energies on proving theorems instead of arguing philosophy. Given this account of the role of axioms, I give four criteria that axioms must meet in order to be accepted. Penelope Maddy has proposed a similar view in Naturalism in Mathematics, but she suggests that the philosophical questions bracketed by adopting the axioms can in fact be ignored forever. I contend that these philosophical arguments are in fact important, and should ideally be resolved at some point, but I concede that their resolution is unlikely to affect the ordinary practice of mathematics. However, they may have effects in the margins of mathematics, including with regards to the controversial “large cardinal axioms” Maddy would like to support. (shrink)

In 1932 Ludwig Wittgenstein accused Rudolf Carnap of plagiarism and seems to have gone so far as to scrawl the word ‘Plagiarism’ on one of Carnap’s offprints and initial that note as well. Priority disputes are inherently distasteful and usually sterile. And they are often impossible to adjudicate fully. I make no such attempt here. But these disputes can also be revealing about what the participants thought they were doing and what they thought they had achieved. It is in this (...) latter vein that I revisit the 1932 dispute. My primary focus will be on Carnap, and I begin by examining the accounts of the dispute by three distinguished philosopher/historians, Jaakko Hintikka, Thomas Uebel, and David Stern. The aim is not a verdict on Wittgenstein’s charge of plagiarism, but to see what the dispute and surrounding documents show about how Carnap’s views were developing in the early thirties, what antecedents those ideas may have had (including in Wittgenstein), and how Carnap saw the changes in his views. (shrink)

It is often supposed that, unlike typical axioms of mathematics, the Continuum Hypothesis (CH) is indeterminate. This position is normally defended on the ground that the CH is undecidable in a way that typical axioms are not. Call this kind of undecidability “absolute undecidability”. In this paper, I seek to understand what absolute undecidability could be such that one might hope to establish that (a) CH is absolutely undecidable, (b) typical axioms are not absolutely undecidable, and (c) if a mathematical (...) hypothesis is absolutely undecidable, then it is indeterminate. I shall argue that on no understanding of absolute undecidability could one hope to establish all of (a)–(c). However, I will identify one understanding of absolute undecidability on which one might hope to establish both (a) and (c) to the exclusion of (b). This suggests that a new style of mathematical antirealism deserves attention—one that does not depend on familiar epistemological or ontological concerns. The key idea behind this view is that typical mathematical hypotheses are indeterminate because they are relevantly similar to CH. (shrink)

It is widely agreed that the intelligibility of modal metaphysics has been vindicated. Quine's arguments to the contrary supposedly confused analyticity with metaphysical necessity, and rigid with non-rigid designators.2 But even if modal metaphysics is intelligible, it could be misconceived. It could be that metaphysical necessity is not absolute necessity – the strictest real notion of necessity – and that no proposition of traditional metaphysical interest is necessary in every real sense. If there were nothing otherwise “uniquely metaphysically significant” about (...) metaphysical necessity, then paradigmatic metaphysical necessities would be necessary in one sense of “necessary”, not necessary in another, and that would be it. The question of whether they were necessary simpliciter would be like the question of whether the Parallel Postulate is true simpliciter – understood as a pure mathematical conjecture, rather than as a hypothesis about physical spacetime. In a sense, the latter question has no objective answer. In this article, I argue that paradigmatic questions of modal metaphysics are like the Parallel Postulate question. I then discuss the deflationary ramifications of this argument. I conclude with an alternative conception of the space of possibility. According to this conception, there is no objective boundary between possibility and impossibility. Along the way, I sketch an analogy between modal metaphysics and set theory. (shrink)

Holly Smith has done more than anyone to explore and defend the importance of usability for moral theories. In Making Morality Work, she develops a moral theory that is almost universally usable. But not quite. In this article, I argue that no moral theory is universally usable, in the sense that is most immediately relevant to action, even by agents who know all the normative facts. There is no moral theory knowledge of which suffices to settle deliberation about what to (...) do. However, this is unsurprising if the question of what to do is not a question of fact. One upshot of the discussion is that the search for a universally usable moral theory is misconceived. Another is that, contra Smith (341), agents who are radically uncertain need not lack autonomy. (shrink)

Philosophers have ever declared that their problems lie at a different level from the problems of the empirical sciences. Perhaps one may agree with this assertion; the question is, however, where should one seek this level. The metaphysicians wish to seek their object behind the objects of empirical science; they wish to enquire after the essence, the ultimate cause of things. But the logical analysis of the pretended propositions of metaphysics has shown that they are not propositions at all, but (...) empty word arrays, which on account of notional and emotional connections arouse the false appearance of being propositions. This conception that the “propositions” of metaphysics, including those of ethics, have no theoretical content, is to be sure still disputed. We shall not, however, enter here on its demonstration, but, under its guidance, will limit ourselves to non-metaphysical and non-ethical philosophical problems. (shrink)

A response is given here to Benacerraf's 1973 argument that mathematical platonism is incompatible with a naturalistic epistemology. Unlike almost all previous platonist responses to Benacerraf, the response given here is positive rather than negative; that is, rather than trying to find a problem with Benacerraf's argument, I accept his challenge and meet it head on by constructing an epistemology of abstract (i.e., aspatial and atemporal) mathematical objects. Thus, I show that spatio-temporal creatures like ourselves can attain knowledge about mathematical (...) objects by simply explaininghow they can do this. My argument is based upon the adoption of a particular version of platonism — full-blooded platonism — which asserts that any mathematical object which possiblycould exist actuallydoes exist. (shrink)

Bazzocchi disposes the text of the Tractatus in a user-friendly manner, exactly as Wittgenstein's decimals advise. This discloses the logical form of the book by distinct reading units, linked into a fashioned hierarchical tree. The text becomes much clearer and every reader can enjoy, finally, its formal and literary qualities.

Friedrich Nietzsche is one of the most influential thinkers of the past 150 years and On the Genealogy of Morality (1887) is his most important work on ethics and politics. A polemical contribution to moral and political theory, it offers a critique of moral values and traces the historical evolution of concepts such as guilt, conscience, responsibility, law and justice. This is a revised and updated edition of one of the most successful volumes to appear in Cambridge Texts in the (...) History of Political Thought. Keith Ansell-Pearson has modified his introduction to Nietzsche's classic text, and Carol Diethe has incorporated a number of changes to the translation itself, reflecting the considerable advances in our understanding of Nietzsche in the twelve years since this edition first appeared. In this new guise the Cambridge Texts edition of Nietzsche's Genealogy should continue to enjoy widespread adoption, at both undergraduate and graduate level. (shrink)

Logical monism is the view that there is ‘One True Logic’. This is the default position, against which pluralists react. If there were not ‘One True Logic’, it is hard to see how there could be one true theory of anything. A theory is closed under a logic! But what is logical monism? In this article, I consider semantic, logical, modal, scientific, and metaphysical proposals. I argue that, on no ‘factualist’ analysis (according to which ‘there is One True Logic’ expresses (...) a factual claim, rather than an attitude like approval), does the doctrine have both metaphysical and methodological import. Metaphysically, logics abound. Methodologically, what to infer from what is not settled by the facts, even the normative ones. I conclude that the only interesting sense in which there could be One True Logic is noncognitive. The same may be true of monism about normative areas, like moral, epistemic, and prudential ones, generally. (shrink)

Science Without Numbers caused a stir in 1980, with its bold nominalist approach to the philosophy of mathematics and science. It has been unavailable for twenty years and is now reissued in a revised edition with a substantial new preface presenting the author's current views and responses to the issues raised in subsequent debate.

This book discusses the problem of mathematical knowledge, and its broader philosophical ramifications. It argues that the problem of explaining the (defeasible) justification of our mathematical beliefs (‘the justificatory challenge’), arises insofar as disagreement over axioms bottoms out in disagreement over intuitions. And it argues that the problem of explaining their reliability (‘the reliability challenge’), arises to the extent that we could have easily had different beliefs. The book shows that mathematical facts are not, in general, empirically accessible, contra Quine, (...) and that they cannot be dispensed with, contra Field. However, it argues that they might be so plentiful that our knowledge of them is intelligible. The book concludes with a complementary ‘pluralism’ about modality, logic and normative theory, highlighting its revisionary implications. Metaphysically, pluralism engenders a kind of perspectivalism and indeterminacy. Methodologically, it vindicates Carnap’s pragmatism, transposed to the key of realism. (shrink)

This paper takes up an important epistemological challenge to the naturalistic moral realist: that her metaphysical commitments are difficult to square with a plausible rationalist view about the epistemology of morality. The paper begins by clarifying and generalizing this challenge. It then illustrates how the generalized challenge can be answered by a form of naturalistic moral realism that I dub joint-carving moral realism. Both my framing of this challenge and my answer advertise the methodological significance of non-fundamental epistemological theorizing, which (...) defends and deploys epistemological claims without adverting to the most fundamental epistemological facts. (shrink)

This truly philosophical book takes us back to fundamentals - the sheer experience of proof, and the enigmatic relation of mathematics to nature. It asks unexpected questions, such as 'what makes mathematics mathematics?', 'where did proof come from and how did it evolve?', and 'how did the distinction between pure and applied mathematics come into being?' In a wide-ranging discussion that is both immersed in the past and unusually attuned to the competing philosophical ideas of contemporary mathematicians, it shows that (...) proof and other forms of mathematical exploration continue to be living, evolving practices - responsive to new technologies, yet embedded in permanent facts about human beings. It distinguishes several distinct types of application of mathematics, and shows how each leads to a different philosophical conundrum. Here is a remarkable body of new philosophical thinking about proofs, applications, and other mathematical activities. (shrink)

A History of Mathematics: From Mesopotamia to Modernity covers the evolution of mathematics through time and across the major Eastern and Western civilizations. It begins in Babylon, then describes the trials and tribulations of the Greek mathematicians. The important, and often neglected, influence of both Chinese and Islamic mathematics is covered in detail, placing the description of early Western mathematics in a global context. The book concludes with modern mathematics, covering recent developments such as the advent of the computer, chaos (...) theory, topology, mathematical physics, and the solution of Fermat's Last Theorem. Containing more than 100 illustrations and figures, this text, aimed at advanced undergraduates and postgraduates, addresses the methods and challenges associated with studying the history of mathematics. The reader is introduced to the leading figures in the history of mathematics and their fields. An extensive bibliography with cross-references to key texts will provide invaluable resource to students and exercises will stretch the more advanced reader. (shrink)

A History of Mathematics: From Mesopotamia to Modernity covers the evolution of mathematics through time and across the major Eastern and Western civilizations. It begins in Babylon, then describes the trials and tribulations of the Greek mathematicians. The important, and often neglected, influence of both Chinese and Islamic mathematics is covered in detail, placing the description of early Western mathematics in a global context. The book concludes with modern mathematics, covering recent developments such as the advent of the computer, chaos (...) theory, topology, mathematical physics, and the solution of Fermat's Last Theorem.Containing more than 100 illustrations and figures, this text, aimed at advanced undergraduates and postgraduates, addresses the methods and challenges associated with studying the history of mathematics. The reader is introduced to the leading figures in the history of mathematics and their fields. An extensive bibliography with cross-references to key texts will provide invaluable resource to students and exercises will stretch the more advanced reader. (shrink)