This volume offers a selection of the most interesting and important work from recent years in the philosophy of mathematics, which has always been closely linked to, and has exerted a significant influence upon, the main stream of analytical philosophy. The issues discussed are of interest throughout philosophy, and no mathematical expertise is required of the reader. Contributors include W.V. Quine, W.D. Hart, Michael Dummett, Charles Parsons, Paul Benacerraf, Penelope Maddy, W.W. Tait, Hilary Putnam, George Boolos, Daniel Isaacson, Stewart (...) class='Hi'>Shapiro, and Hartry Field. (shrink)

Skeptical of the arguments put forth in Robert Duncan's long-awaited, post-humously published The H. D. Book, this review essay questions the elevation of Pre-Raphaelite, Aestheticist, and Decadent poetry that forms the basis of Duncan's revisionist canon—a revision in which Wallace Stevens and T. S. Eliot are dismissed as “merely rational,” while H. D. and Duncan himself are elevated to the uppermost ranks, just beneath Ezra Pound. The essay focuses on the peculiarity of “Wardour Street” diction returning to poetry in the (...) postmodern era (especially in the verse of Gjertrud Schnackenberg) and connects this development to the nostalgia evidenced in Duncan's occultism. Placing the New Critics with the poets Karl Shapiro and Richard Wilbur, Duncan dismisses them all as “academics” and “descendants of those ministers of the seventeenth and eighteenth century, holding out against the magic of poetry as once they held out against the magic-religion of witch cults.” Emphasizing that it is the witch cults with which Duncan evidently sides, the essay concludes that Duncan's reranking of poets is no more reliable than his attitude toward rationality. (shrink)

The subject of this paper is the philosophical problem of accounting for the relationship between mathematics and non-mathematical reality. The first section, devoted to the importance of the problem, suggests that many of the reasons for engaging in philosophy at all make an account of the relationship between mathematics and reality a priority, not only in philosophy of mathematics and philosophy of science, but also in general epistemology/metaphysics. This is followed by a (rather brief) survey of the major, traditional philosophies (...) of mathematics indicating how each is prepared to deal with the present problem. It is shown that (the standard formulations of) some views seem to deny outright that there is a relationship between mathematics and any non-mathematical reality; such philosophies are clearly unacceptable. Other views leave the relationship rather mysterious and, thus, are incomplete at best. The final, more speculative section provides the direction of a positive account. A structuralist philosophy of mathematics is outlined and it is proposed that mathematics applies to reality though the discovery of mathematical structures underlying the non-mathematical universe. (shrink)

We examine George Boolos's proposed abstraction principle for extensions based on the limitation-of-size conception, New V, from several perspectives. Crispin Wright once suggested that New V could serve as part of a neo-logicist development of real analysis. We show that it fails both of the conservativeness criteria for abstraction principles that Wright proposes. Thus, we support Boolos against Wright. We also show that, when combined with the axioms for Boolos's iterative notion of set, New V yields a system equivalent to (...) full Zermelo-Fraenkel set theory with a principle of global choice. This advances Boolos's longstanding interest in the foundations of set theory. (shrink)

The neo-logicist argues tliat standard mathematics can be derived by purely logical means from abstraction principles—such as Hume's Principle— which are held to lie 'epistcmically innocent'. We show that the second-order axiom of comprehension applied to non-instantiated properties and the standard first-order existential instantiation and universal elimination principles are essential for the derivation of key results, specifically a theorem of infinity, but have not been shown to be epistemically innocent. We conclude that the epistemic innocence of mathematics has not been (...) established by the neo-logicist. (shrink)

Graham Priest's In Contradiction (Dordrecht: Martinus Nijhoff Publishers, 1987, chapter 3) contains an argument concerning the intuitive, or ‘naïve’ notion of (arithmetic) proof, or provability. He argues that the intuitively provable arithmetic sentences constitute a recursively enumerable set, which has a Gödel sentence which is itself intuitively provable. The incompleteness theorem does not apply, since the set of provable arithmetic sentences is not consistent. The purpose of this article is to sharpen Priest's argument, avoiding reference to informal notions, consensus, or (...) Church's thesis. We add Priest's dialetheic semantics to ordinary Peano arithmetic PA, to produce a recursively axiomatized formal system PA★ that contains its own truth predicate. Whether one is a dialetheist or not, PA★ is a legitimate, rigorously defined formal system, and one can explore its proof‐theoretic properties. The system is inconsistent (but presumably non‐trivial), and it proves its own Gödel sentence as well as its own soundness. Although this much is perhaps welcome to the dialetheist, it has some untoward consequences. There are purely arithmetic (indeed, Π0) sentences that are both provable and refutable in PA★. So if the dialetheist maintains that PA★ is sound, then he must hold that there are true contradictions in the most elementary language of arithmetic. Moreover, the thorough dialetheist must hold that there is a number g which both is and is not the code of a derivation of the indicated Gödel sentence of PA★. For the thorough dialetheist, it follows ordinary PA and even Robinson arithmetic are themselves inconsistent theories. I argue that this is a bitter pill for the dialetheist to swallow. (shrink)

Some philosophers have argued that moral agency is characteristic of humans alone and that its absence from other animals justifies granting higher moral status to humans. However, human beings do not have a monopoly on moral agency, which admits of varying degrees and does not require mastery of moral principles. The view that all and only humans possess moral agency indicates our underestimation of the mental lives of other animals. Since many other animals are moral agents (to varying degrees), they (...) are also subject to (limited) moral obligations, examples of which are provided in this paper. But, while moral agency is sufficient for significant moral status, it is by no means necessary. (shrink)

§1. Overview. Philosophers and mathematicians have drawn lots of conclusions from Gödel's incompleteness theorems, and related results from mathematical logic. Languages, minds, and machines figure prominently in the discussion. Gödel's theorems surely tell us something about these important matters. But what?A descriptive title for this paper would be “Gödel, Lucas, Penrose, Turing, Feferman, Dummett, mechanism, optimism, reflection, and indefinite extensibility”. Adding “God and the Devil” would probably be redundant. Despite the breath-taking, whirlwind tour, I have the modest aim of forging (...) connections between different parts of this literature and clearing up some confusions, together with the less modest aim of not introducing any more confusions.I propose to focus on three spheres within the literature on incompleteness. The first, and primary, one concerns arguments that Gödel's theorem refutes the mechanistic thesis that the human mind is, or can be accurately modeled as, a digital computer or a Turing machine. The most famous instance is the much reprinted J. R. Lucas [18]. To summarize, suppose that a mechanist provides plans for a machine,M, and claims that the output ofMconsists of all and only the arithmetic truths that a human, or the totality of human mathematicians, will ever or can ever know. We assume that the output ofMis consistent. (shrink)

For the past four decades, Anglo-American legal philosophy has been preoccupied – some might say obsessed – with something called the “Hart-Dworkin” debate. Since the appearance in 1967 of “The Model of Rules I,” Ronald Dworkin’s seminal critique of H.L.A. Hart’s theory of legal positivism, countless books and articles have been written either defending Hart against Dworkin’s objections or defending Dworkin against Hart’s defenders. My purpose in this essay is not to declare an ultimate victor; rather it is to identify (...) precisely the core issue around which the debate is organized. Is the Hart-Dworkin debate, for example, about whether the law contains principles, as well as rules? Or does it concern whether judges have discretion in hard cases? Is it about the proper way to interpret legal texts in the American legal system? Or is it about the very possibility of conceptual jurisprudence? Although trying to capture the essence of a philosophical debate can be tricky, I think that there is an important unity to the Hart-Dworkin debate that can be described in a relatively straightforward manner. I suggest that the debate is organized around one of the most profound issues in the philosophy of law, namely, the relation between legality and morality. Dworkin’s basic strategy throughout the course of the debate has been to argue that, in one form or another, legality is ultimately determined not by social facts alone, but by moral facts as well. This contention directly challenges, and threatens to undermine, the positivist picture about the nature of law, in which legality is never determined by morality, but solely by social practice. As one might expect, the response by Hart and his followers has been to argue that this dependence of legality on morality is either merely apparent or does not, in fact, undermine the social foundations of law. The Hart-Dworkin debate, I also try to show, is not a monolithic entity. In the second half of the paper, I describe how Dworkin modified his critique to circumvent the responses of Hart’s followers, thereby inaugurating a new phase in the debate. Virtually no attention, however, has been paid to this latter challenge, which is especially surprising given that none of the previous positivistic defenses are helpful against it. I then sketch out a possible response positivists might offer to this extremely powerful objection. (shrink)

The purpose of this paper is to assess the prospects for a neo‐logicist development of set theory based on a restriction of Frege's Basic Law V, which we call (RV): ∀P∀Q[Ext(P) = Ext(Q) ≡ [(BAD(P) & BAD(Q)) ∨ ∀x(Px ≡ Qx)]] BAD is taken as a primitive property of properties. We explore the features it must have for (RV) to sanction the various strong axioms of Zermelo–Fraenkel set theory. The primary interpretation is where ‘BAD’ is Dummett's ‘indefinitely extensible’.1 Background: what (...) and why?2 Framework3 GOOD candidates, indefinite extensibility4 The framework of (RV) alone, or almost alone5 The axioms6 Brief closing. (shrink)

Hellman and Shapiro explore the development of the idea of the continuous, from the Aristotelian view that a true continuum cannot be composed of points to the now standard, entirely punctiform frameworks for analysis and geometry. They then investigate the underlying metaphysical issues concerning the nature of space or space-time.

Though the “internal point of view” is perhaps H.L.A. Hart’s greatestcontribution to legal theory, this concept is also often and easily misunderstood. This is unfortunate, not only because these misreadings distort Hart’s theory, but, more importantly, because they prevent us from appreciating the infirmities of sanction-centered theories of law and the compelling reasons why they ought to be rejected. In this paper, I try to address some of these confusions. What, exactly, is the internal point of view? What role (or (...) roles) does it play in Hart’s theory? And how does an adequate appreciation for the centrality of the internal point of view lead to the rejection of sanction-centered theories? (shrink)

My talk today will be about Newton’s avowed methodology, and specifically the place of experiment in his conception of science, and how his ideas changed significantly over the course of his career. I also want to look at his actual scientific practice and see how this influenced his views on the nature of the experimental sciences.

Quite unexpectedly, cognitive psychologists find their field intimately connected to a whole new intellectual landscape that had previously seemed remote, unfamiliar, and all but irrelevant. Yet the proliferating connections tying together the cognitive and evolutionary communities promise to transform both fields, with each supplying necessary principles, methods, and a species of rigor that the other lacks. (Cosmides and Tooby, 1994, p. 85).

Sections 3.16 and 3.23 of Roger Penrose's Shadows of the mind (Oxford, Oxford University Press, 1994) contain a subtle and intriguing new argument against mechanism, the thesis that the human mind can be accurately modeled by a Turing machine. The argument, based on the incompleteness theorem, is designed to meet standard objections to the original Lucas-Penrose formulations. The new argument, however, seems to invoke an unrestricted truth predicate (and an unrestricted knowability predicate). If so, its premises are inconsistent. The usual (...) ways of restricting the predicates either invalidate Penrose's reasoning or require presuppositions that the mechanist can reject. (shrink)

Zermelo’s Theorem that the axiom of choice is equivalent to the principle that every set can be well-ordered goes through in third-order logic, but in second-order logic we run into expressivity issues. In this note, we show that in a natural extension of second-order logic weaker than third-order logic, choice still implies the well-ordering principle. Moreover, this extended second-order logic with choice is conservative over ordinary second-order logic with the well-ordering principle. We also discuss a variant choice principle, due to (...) Hilbert and Ackermann, which neither implies nor is implied by the well-ordering principle. (shrink)

In previous work, Hellman and Shapiro present a regions-based account of a one-dimensional continuum. This paper produces a more Aristotelian theory, eschewing the existence of points and the use of infinite sets or pluralities. We first show how to modify the original theory. There are a number of theorems that have to be added as axioms. Building on some work by Linnebo, we then show how to take the ‘potential’ nature of the usual operations seriously, by using a modal (...) language, and we show that the two approaches are equivalent. (shrink)

This chapter provides an overview of second-order logic and higher-order logic generally. It provides the basic formal languages, deductive systems, and model-theoretic semantics, including a brief account of George Boolos’s interpretation of second-order languages in terms of the plural construction. It then goes into some of the arguments in favor of second-order logic.

Institutional investors are increasingly focusing on firms that prioritise Corporate Social Responsibility. In the absence of any objective measure of a firm's CSR Performance, their investment choices are largely guided by independent rating indices that rank firms according to their social performance metrics. As a result, firms looking to increase their attractiveness as targets of social investment focus their CSR efforts on increasing the visibility of activities that are recognised by such indices. However, the validity of these indices as accurate (...) measures of firms' actual social performance has repeatedly been called into question. This means that the ability of these indices to measure and report on firms' actual social impact cannot be ascertained with any degree of accuracy. The result is that firms are incentivised to engage in activities that cannot be said to improve social responsibility, and may even ultimately harm society. Thus, another method of measuring CSP must be found that enables firms to measure their true impact on society. We propose a new approach to measuring CSP that is integrated with stakeholder theory. Such an approach provides managers of firms with an interest in engaging in real social development for the purposes of ensuring firm survival with the ability to understand their social obligations, and the ability to measure the resulting benefit to society. (shrink)

According to Gupta and Belnap, the “extensional behavior” of ‘true’ matches that of a circularly defined predicate. Besides promising to explain semantic paradoxicality, their general theory of circular predicates significantly liberalizes the framework of truth-conditional semantics. The authors’ discussions of the rationale behind that liberalization invoke two distinct senses in which a circular predicate’s semantic behavior is explained by a “revision rule” carrying hypothetical information about its extension. Neither attempted explanation succeeds. Their theory may however be modified to employ a (...) relativized notion of extension. The resulting contextualist semantics for ‘true’ construes circularity as a pragmatic phenomenon. (shrink)

The purpose of this note is to examine the relationship between the practice of mathematics and the philosophy of mathematics, ontology in particular. One conclusion is that the enterprises are (or should be) closely related, with neither one dominating the other. One cannot 'read off' the correct way to do mathematics from the true ontology, for example, nor can one ‘read off’ the true ontology from mathematics as practiced.

The paradox of the Unexpected Hanging, related prediction paradoxes, and the Sorites paradoxes all involve reasoning about ordered collections of entities: days ordered by date in the case of the Unexpected Hanging; men ordered by the number of hairs on their heads the case of the bald man version of the Sorites. The reasoning then assigns each entity a value that depends on the previously assigned value of one of the neighboring entities. The final result is paradoxical because it conflicts (...) with the obviously correct, commonsensical value. The paradox is due to the serial procedure of assigning a value based on the newly assigned value of the neighbor. An alternative procedure is to assign each value based only on the original values of neighbors - a parallel procedure. That procedure does not give paradoxical answers. (shrink)

The idea that logic and reasoning are somehow related goes back to antiquity. It clearly underlies much of the work in logic, as witnessed by the development of computability, and formal and mechanical deductive systems, for example. On the other hand, a platitude is that logic is the study of correct reasoning; and reasoning is cognitive if anything Is. Thus, the relationship between logic, computation, and correct reasoning makes an interesting and historically central case study for mechanism. The purpose of (...) this article is to begin the articulation of this relationship, pointing out its sources and its limitations. (shrink)

Since virtually every mathematical theory can be interpreted in Zermelo-Fraenkel set theory, it is a foundation for mathematics. There are other foundations, such as alternate set theories, higher-order logic, ramified type theory, and category theory. Whether set theory is the right foundation for mathematics depends on what a foundation is for. One purpose is to provide the ultimate metaphysical basis for mathematics. A second is to assure the basic epistemological coherence of all mathematical knowledge. A third is to serve mathematics, (...) by lending insight into the various fields and suggesting fruitful techniques of research. A fourth purpose of a foundation is to provide an arena for exploring relations and interactions between mathematical fields. While set theory does better with regard to some of these and worse with regard to others, it has become the de facto arena for deciding questions of existence, something one might expect of a foundation. Given the different goals, there is little point to determining a single foundation for all of mathematics. (shrink)

Recent high-profile lawsuits have supported competing narratives that alternately depict civil litigation as an essential instrument of the rule of law and a threat to the ideal. This essay argues that each narrative captures an important element of truth and that Gerald Postema’s account of the rule of law in his book Law’s Rule helps us (albeit unwittingly) to see why. While Postema presents recourse for alleged abuses of power as a universal and enduring facet of the rule of law, (...) his conception of recourse turns out to resemble core features of the kind of adversarial litigation process exemplified by the U.S. federal civil justice system. Yet such a system both promises to promote and threatens to undermine each of the three principles that Postema claims are entailed by his understanding of the rule of law—namely, sovereignty, equality, and fidelity. Realizing recourse thus requires confronting difficult tradeoffs within each of those principles, as well as within the overarching rule-of-law ideal. And if the rule of law can’t be instantiated unequivocally in any particular set of institutions, then perhaps we should be more willing to treat the ideal as a subject of politics rather than just a constraint on it. (shrink)

According to the Categories, predicates can be ‘said of’ their subjects or they can be ‘present in’ their subjects. The said-of relation has received relatively little scholarly attention, and scholars disagree on the answers to four foundational questions about the relation. (i) What is it? (ii) Is it an essential relation? (iii) How is it related to predication? (iv) Is it primitive? I argue that A is said-of B just in case A is a formal part of B. On this (...) account, said-of is an essential relation and it is definable, but it is not defined in terms of predication. (shrink)

For nearly a century, the French have entertained an unshakable conviction that their ability to recognize themselves-to know and transmit the essence of Frenchness-depended on the teaching of the history of France. In effect, history was a discourse on France, and the teaching of history-"la pédagogie centrale du citoyen"-the means by which children were constituted as heirs and carriers of a common collective memory that made them not only citizens, but family. In this essay, I examine the rhetorical and conceptual (...) effects on history writing that emerge out of this preoccupation with the elaboration of a continuous, coherent national identity. Focusing on schoolbooks, I begin by looking at the dominant, nearly hegemonic model of French history created by Ernest Lavisse in the 1890s-a model informed by the dream of a unified, unitary French nation, embodied in and articulated through the history of France-and at the disruption of this paradigm in the aftermath of the Great War. I then consider a text written in the 1990s specifically to repudiate the kind of nationalist narratives that prevailed for most of this century-a new supranational history of Europe. I argue that, in their different experiments with fixing history, both Lavisse and the contemporary textbook authors did not so much repair a deficient history as produce a historical fixation, creating mythicized histories that are complete, closed, predictable, and at bottom ahistorical. Finally, I turn to a recent World War I novel, A Very Long Engagement by Sébastien Japrisot, in order to suggest ways in which the narrative strategies of a fiction writer may be useful to historians in thinking about a different kind of historical project. (shrink)

Critics of attempts to explain meaning in terms of truth-conditions have tended to charge their opponents with misconceptions regarding truth. I shall argue that the 'naïve' version of the truth-conditional theory which best accounts for its resilience fails for a different and more basic reason, namely, circularity arising from the contingency of meaning. One reason why this problem has been overlooked is a tendency (noted by Dummett in a different connection) to assimilate the naïve truth-conditional theory to an idealized verificationism.

Motivating my discussion is a puzzle in Spinoza’s account of the primary affects – his shift away from adopting Descartes’s list of six primitive passions in the Short Treatise to the three primary affects in the Ethics. I lay out this puzzle in Section 1. In Section 2, I approach this puzzle by considering the taxonomy offered by Descartes of the basic or primitive passions. In considering Descartes, I will also briefly consider Aquinas’s view since Descartes positions himself as rejecting (...) the Thomist account. Doing so brings out that the basic passions highlight structures of cognition for these thinkers. In Section 3, I return to Spinoza and consider what light his pared down account of the primary affects can shed on his account of the human mind. (shrink)

Much discourse on school shootings tends to imply a binary separation between what is considered normal and exceptional, between an expected course of human events and sociohistorical aberrations. In this article Harvey Shapiro suggests the need for new directions in our responses: First, he shows how responses to school shootings tend to expropriate and dismiss certain kinds of violence in order to articulate a vision of the self as sovereign, exerting power over bodily life, exercising a self-removal from community (...) conversations, and thus claiming what Giorgio Agamben and others call the “sovereign exception.” Second, he suggests how Agamben's development of Walter Benjamin's concept of “divine violence” can unmask prevailing states of exception and can inform education's efforts to challenge binaries of good and evil, urban and suburban, individual and community, justice and law, normal and exceptional, that confound our deliberations and long-term responses to mass shootings. (shrink)