Some authors have claimed that ante rem structuralism has problems with structures that have indiscernible places. In response, I argue that there is no requirement that mathematical objects be individuated in a non-trivial way. Metaphysical principles and intuitions to the contrary do not stand up to ordinary mathematical practice, which presupposes an identity relation that, in a sense, cannot be defined. In complex analysis, the two square roots of –1 are indiscernible: anything true of one of them is true of (...) the other. I suggest that ‘i’ functions like a parameter in natural deduction systems. I gave an early version of this paper at a workshop on structuralism in mathematics and science, held in the Autumn of 2006, at Bristol University. Thanks to the organizers, particularly Hannes Leitgeb, James Ladyman, and Øystein Linnebo, to my commentator Richard Pettigrew, and to the audience there. The paper also benefited considerably from a preliminary session at the Arché Research Centre at the University of St Andrews. I am indebted to my colleagues Craige Roberts, for help with the linguistics literature, and Ben Caplan and Gabriel Uzquiano, for help with the metaphysics. Thanks also to Hannes Leitgeb and Jeffrey Ketland for reading an earlier version of the manuscript and making helpful suggestions. I also benefited from conversations with Richard Heck, John Mayberry, Kevin Scharp, and Jason Stanley. CiteULike Connotea Del.icio.us What's this? (shrink)

A statement of the form ‘C caused E’ obeys the requirement of proportionality precisely when C says no more than what is necessary to bring about E. The thesis that causal statements must obey this requirement might be given a semantic or a pragmatic justification. We use the idea that causal claims are contrastive to criticize both.

When conceived as an empirical claim, it is natural to wonder how one might test the hypothesis of multiple realization. I consider general issues of testability, show how they apply specifically to the hypothesis of multiple realization, and propose an auxiliary assumption that, I argue, must be conjoined to the hypothesis of multiple realization to ensure its testability. I argue further that Bechtel and Mundale go astray because they fail to appreciate the need for this auxiliary assumption. †To contact the (...) author, please write to: Department of Philosophy, University of Wisconsin–Madison, 5185 Helen C. White Hall, 600 North Park Street, Madison, WI 53706; e‐mail: [email protected]. (shrink)

There is an interesting logical/semantic issue with some mathematical languages and theories. In the language of (pure) complex analysis, the two square roots of i’ manage to pick out a unique object? This is perhaps the most prominent example of the phenomenon, but there are some others. The issue is related to matters concerning the use of definite descriptions and singular pronouns, such as donkey anaphora and the problem of indistinguishable participants. Taking a cue from some work in linguistics and (...) the philosophy of language, I suggest that i functions like a parameter in natural deduction systems. This may require some rethinking of the role of singular terms, at least in mathematical languages. (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)

Fodor (1990) argues that the theory of evolution by natural selection will not help to save naturalistic accounts of representation from the disjunction problem. This is because, he claims, the context 'was selected for representing things as F' is transparent to the substitution of predicates coextensive with F. But, I respond, from an evolutionary perspective representational contexts cannot be transparent: only under particular descriptions will a representational state appear as a "solution" to a selection "problem" and so be adaptive. Only (...) when we construe representational states as opaque in this manner are the generalizations of branches of evolutionary theory, like foraging theory, possible. (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)

Frances Egan argues that the states of computational theories of vision are individuated individualistically and, as far as the theory is concerned, are not intentional. Her argument depends on equating the goals and explanatory strategies of computational psychology with those of its algorithmic level. However, closer inspection of computational psychology reveals that the computational level plays an essential role in explaining visual processes and that explanations at this level are nonindividualistic and intentional. In conclusion, I sketch an account of content (...) in which content does the sort of explanatory work that Egan denies is possible. (shrink)

The authors developed this textbook in response to an increasing interest in ethics, and a growing number of courses on this topic that are now being offered in educational leadership programs. It is designed to fill a gap in instructional materials for teaching the ethics component of the knowledge base that has been established for the profession. The text has several purposes: First, it demonstrates the application of different ethical paradigms (the ethics of justice, care, critique, and the profession) through (...) discussion and analysis of real-life moral dilemmas that educational leaders face in their schools and communities. Second, it addresses some of the practical, pedagogical, and curricular issues related to the teaching of ethics for educational leaders. Third, it emphasizes the importance of ethics instruction from a variety of theoretical approaches. Finally, it provides a process that instructors might follow to develop their own ethics unit or course. * Part I provides an overview of why ethics is so important, especially for today's educational leaders, and describes a multiparadigm approach essential to practitioners as they grapple with ethical dilemmas. * Part II deals with the dilemmas themselves. Ethical dilemmas written by the authors' graduate students bring readers face-to-face with the kinds of dilemmas faced by practicing administrators in urban, suburban, and rural settings in an era full of complexities and contradictions. * Part III focuses on pedagogy and provides teaching notes for the instructor. The authors discuss the importance of self-reflection on the part of both instructors and students, and model how they thought through their own personal and professional ethical codes as well as reflected upon the critical incidents in their lives that shaped their teaching and frequently determined what they privileged in class. (shrink)

Successful athletic performance requires precision in many respects. A batter stands behind home plate awaiting the arrival of a ball that is less than three inches in diameter and moving close to 100 mph. His goal is to hit it with a ba­­t that is also less than three inches in diameter. This impressive feat requires extraordinary temporal and spatial coordination. The sweet spot of the bat must be at the same place, at the same time, as the ball. A (...) basketball player must keep a ball bouncing as she speeds from one end of the court to another, evading defensive players. She may never break pace as she lifts from the ground, throwing the ball fifteen feet toward a hoop that is eighteen inches in diameter. One task facing a psychologist involves explaining how the body does such things within the sometimes very demanding spatial and temporal constraints that a given task imposes. Part of the goal of this chapter is to sketch the commitments of an embodied approach to such an explanation. We shall see that an embodied account of motor skills draws concepts that depart radically from more traditional cognitivist theories of motor activity. Similarly, because an embodied approach to cognition introduces new ways to understand the human capacity for social interaction, it also promises to shed new light on how athletes coordinate their actions with each other. (shrink)

Over the last few decades Michael Dummett developed a rich program for assessing logic and the meaning of the terms of a language. He is also a major exponent of Frege's version of logicism in the philosophy of mathematics. Over the last decade, Neil Tennant developed an extensive version of logicism in Dummettian terms, and Dummett influenced other contemporary logicists such as Crispin Wright and Bob Hale. The purpose of this paper is to explore the prospects for Fregean logicism within (...) a broadly Dummettian framework. The conclusions are mostly negative: Dummett's views on analyticity and the logical/non-logical boundary leave little room for logicism. Dummett's considerations concerning manifestation and separability lead to a conservative extension requirement: if a sentence S is logically true, then there is a proof of S which uses only the introduction and elimination rules of the logical terms that occur in S. If basic arithmetic propositions are logically true-as the logicist contends-then there is tension between this conservation requirement and the ontological commitments of arithmetic. It follows from Dummett's manifestation requirements that if a sentence S is composed entirely of logical terminology, then there is a formal deductive system D such that S is analytic, or logically true, if and only if S is a theorem of D. There is a deep conflict between this result and the essential incompleteness, or as Dummett puts it, the indefinite extensibility, of arithmetic truth. (shrink)

I begin my discussion by considering how to relate Descartes’s more general concern with the conduct of life to the metaphysics and epistemology in the foreground of his philosophical project. I then turn to the texts in which Descartes offers his developed ethical thought and present the case for Descartes as a virtue ethicist. My argument emerges from seeing that Descartes’s conception of virtue and the good owes much to Stoic ethics, a school of thought which saw a significant revival (...) in the seventeenth century. It does, however, deviate from classical Stoicism in critical ways. Towards the end of my discussion, I return to the question of the relation between Descartes’s ethics and his metaphysics and epistemology, and I suggest that the Discourse on the Method for Rightly Conducting Reason and the Meditations on First Philosophy are invested with the virtue ethical considerations of moral education and the regulation of the passions, respectively. (shrink)

The purpose of this article is show that second-order logic, as understood through standard semantics, is intimately bound up with set theory, or some other general theory of interpretations, structures, or whatever. Contra Quine, this does not disqualify second-order logic from its role in foundational studies. To wax Quinean, why should there be a sharp border separating mathematics from logic, especially the logic of mathematics?

Three essays discuss aspects of Nietzsche's Thus Spake Zarathustra: the place of giftgiving in the portrayed economy, the meaning of feasting and parasitism, and references to the classical myth of Alcyone.

Among the aims of this book are: - The discussion of some important philosophical issues using the precision of mathematics. - The development of formal systems that contain both classical and constructive components. This allows the study of constructivity in otherwise classical contexts and represents the formalization of important intensional aspects of mathematical practice. - The direct formalization of intensional concepts (such as computability) in a mixed constructive/classical context.

According to relativist accounts of discourse about, e.g., epistemic possibility and matters of taste, the truth of propositions must be relativized to nonstandard parameters. This paper argues that the central thrust of such accounts should be understood independently of relative truth, in terms of a perspectival account of assertoric force. My point of departure is a stripped-down version of Brandom’s analysis of the normative structure of discursive practice. By generalizing that structure, I make room for an analogue of the “assessment (...) sensitivity” MacFarlane characterizes in terms of relative truth. I argue that my reformulation supplies a stronger rationale for the most distinctive feature of MacFarlane’s brand of relativism, its account of when speakers ought to retract assertions. Furthermore, I show that the view usually regarded as a “moderate” alternative to MacFarlane’s “radical” relativism requires the more radical deviation from an absolutist account of assertoric force. (shrink)

Huw Price holds that a recognizable version of expressivism about normative and modal language can be “globalized” so as to apply to all areas of discourse. He focuses on globalizing the anti-representationalism of expressivist theories. By contrast, this paper’s topic is the seldom-discussed way Price seeks to globalize the expressivist view that “non-descriptive” discourse exhibits subjectivity. I argue that Price’s own argument against the possibility of a purely objective domain conflicts with his anti-representationalism and is self-undermining. I then defend a (...) different strategy for arguing that a kind of subjectivity can emerge even in domains traditional expressivists have regarded as purely objective. I do so by using an account of assertoric practice to offer a new solution to a puzzle concerning how to describe contingent histories in the use of natural kind terms. (shrink)

This new anthology of early modern philosophy enriches the possibilities for teaching this period by highlighting not only metaphysics and epistemology, but also new themes such as virtue, equality and difference, education, the passions, and love. It contains the works of forty-three philosophers, including traditionally taught figures such as Descartes, Spinoza, Leibniz, Locke, Berkeley, Hume, and Kant, as well as less familiar writers such as Lord Shaftesbury, Anton Amo, Julien Offray de La Mettrie, and Denis Diderot. It also highlights the (...) contributions of women philosophers, including Margaret Cavendish, Anne Conway, Gabrielle Suchon, Sor Juana Inéz de la Cruz, and Emilie Du Châtelet. (shrink)

: McDowell opposes the view that the intentionality of language and thought remains mysterious unless it can be understood ‘from outside the conceptual order’. While he thinks the demand for such a ‘sideways-on’ understanding can be the result of ‘scientistic prejudice’, he points to Sellars's thought as exhibiting a different source: a distortion of our perspective ‘from within the conceptual order’. The distortion involves a failure on Sellars's part to see how descriptions from within the conceptual order can present expressions (...) and mental acts as related to extra-conceptual objects (a failure in turn explained by his failure to see how such relations could have normative import). In this paper, I argue that Sellars's thought suffers from no such distortion. If that is right, McDowell's examination of Sellars has not uncovered a disorder whose treatment might help relieve the desire for a sideways-on view. (shrink)

Archaeologies of Vision will be a landmark work for all scholars of visual culture as well as for those engaged with continental philosophy.

Graham Priest introduces an informal but presumably rigorous and sharp ‘provability predicate’. He argues that this predicate yields inconsistencies, along the lines of the paradox of the Knower. One long-standing claim of Priest’s is that a dialetheist can have a complete, decidable, and yet sufficiently rich mathematical theory. After all, the incompleteness theorem is, in effect, that for any recursive theory A, if A is consistent, then A is incomplete. If the antecedent fails, as it might for a dialetheist, then (...) the consequent may also fail to hold. One somewhat friendly purpose of my ‘Incompleteness and inconsistency’ was to improve the technical situation for the dialetheist, eschewing reliance on an informal provability predicate. Another, less friendly purpose was to bring out what I took to be some untoward consequences of the situation. It seems that Priest accepted at least some of the improvements that I attempted. In the second edition of In contradiction, he responded to the alleged untoward consequences. One purpose of this note is to revisit the technical and philosophical situation. There were some errors in my original presentation, brought out by discussion with Priest and by Hartry Field’s analysis of the second incompleteness theorem in such contexts. A second task here is to present a sort of Curry version of the Gödel incompleteness situation. I tentatively conclude that even for a dialetheist, an interesting and complete theory is not as easy to come by as it may look—at least not for theories of arithmetic that are plausible for a dialetheist. (shrink)

In developing a new theory of vision in Ad Vitellionem paralipomena Kepler introduced a new optical concept, pictura, which is an image projected on to a screen by a camera obscura. He distinguished this pictura from an imago, the traditional image of medieval optics that existed only in the imagination. By the 1670s a new theory of optical imagery had been developed, and Kepler's pictura and imago became real and virtual images, two aspects of a unified concept of image. The (...) new concept of image developed out of a synthesis of Kepler's determination of the geometrical location of a pictura as the limit, or focus, of refracted pencils of rays and the triangulation used by a single eye to determine the perceived location of an imago. The distinction between real and imaginary images was largely developed by Gilles Personne de Roberval and the Jesuits Francesco Eschinardi and Claude François Milliet Dechales. (shrink)

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)

Within Western political philosophy, the rights of groups has often been neglected or addressed in only the narrowest fashion. Focusing solely on whether rights are exercised by individuals or groups misses what lies at the heart of ethnocultural conflict, leaving the crucial question unanswered: can the familiar system of common citizenship rights within liberal democracies sufficiently accommodate the legitimate interests of ethnic citizens? Specifically, how does membership in an ethnic group differ from other groups, such as professional, lifestyle, or advocacy (...) groups? How important is ethnicity to personal identity and self-respect, and does accommodating these interests require more than standard citizenship rights? Crucially, what forms of ethnocultural accommodations are consistent with democratic equality, individual freedom, and political stability? Invoking numerous cases studies and addressing the issue of ethnicity from a range of perspectives, Ethnicity and Group Rights seeks to answer these questions. (shrink)

Fodor argues that the theory of evolution by natural selection will not help to save naturalistic accounts of representation from the disjunction problem. This is because, he claims, the context 'was selected for representing things as F' is transparent to the substitution of predicates coextensive with F. But, I respond, from an evolutionary perspective representational contexts cannot be transparent: only under particular descriptions will a representational state appear as a "solution" to a selection "problem" and so be adaptive. Only when (...) we construe representational states as opaque in this manner are the generalizations of branches of evolutionary theory, like foraging theory, possible. (shrink)