While we are commonly told that the distinctive method of mathematics is rigorous proof, and that the special topic of mathematics is abstract structure, there has been no agreement among mathematicians, logicians, or philosophers as to just what either of these assertions means. John P. Burgess clarifies the nature of mathematical rigor and of mathematical structure, and above all of the relation between the two, taking into account some of the latest developments in mathematics, including the rise of experimental (...) mathematics on the one hand and computerized formal proofs on the other hand. Along the way, a great many historical developments in mathematics, philosophy, and logic are surveyed. Yet very little in the way of background knowledge on the part of the reader is presupposed. (shrink)

What is the simplest and most natural axiomatic replacement for the set-theoretic definition of the minimal fixed point on the Kleene scheme in Kripke’s theory of truth? What is the simplest and most natural set of axioms and rules for truth whose adoption by a subject who had never heard the word "true" before would give that subject an understanding of truth for which the minimal fixed point on the Kleene scheme would be a good model? Several axiomatic systems, old (...) and new, are examined and evaluated as candidate answers to these questions, with results of Harvey Friedman playing a significant role in the examination. (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)

Dummett's case against platonism rests on arguments concerning the acquisition and manifestation of knowledge of meaning. Dummett's arguments are here criticized from a viewpoint less Davidsonian than Chomskian. Dummett's case against formalism is obscure because in its prescriptive considerations are not clearly separated from descriptive. Dummett's implicit value judgments are here made explicit and questioned. ?Combat Revisionism!? Chairman Mao.

In lieu of an abstract, here is a brief excerpt of the content:Reviewed by:For More than One Voice: Toward a Philosophy of Vocal ExpressionSarah K. Burgess and Stuart J. MurrayFor More than One Voice: Toward a Philosophy of Vocal Expression. Adriana Cavarero. Trans. Paul A. Kottman. Stanford: Stanford University Press, 2005. Pp. 262. $65.00, hardcover; $24.95, paperback.Adriana Cavarero's most recent book, For More than One Voice, offers the reader a critique of Western metaphysics that challenges the hegemony of speech's (...) relation to thought within politics. Revisiting Aristotle's Politics, Cavarero examines the famous definition of man as "zoon logon echon"—usually translated as "rational animal." She is critical of the tradition that renders logos here as "reason" or "rationality," modeled on the abstract and disembodied Platonic idea. Reading the Politics alongside the Poetics, she brings politics into a distinctly rhetorical sphere, and therefore figures logos as "phone semantike." In this light, man is a "speaking animal," an animal who speaks with a voice (phone) that is meaningful (semantike—an adjectival form that qualifies the voice itself). Cavarero's argument is that voice and meaning—phonic and semantic—must be sharply distinguished. The history of Western philosophy, she contends, has effaced this vital difference and focused instead on speech as a unitary phenomenon in which the vocalic and the semantic are always already joined as meaningful signification. In this way, the singularity of the vocalic itself has been elided in favor of speech that strongly privileges meaning. She therefore wishes to deliver us from language, as it were, through a recuperation of the force of the vocalic, the acoustic, the resonant and sonorous auditory quality of phone.Absent in the history of Western philosophy, this sonorous aspect of the vocalic stands as a corrective to our logocentric metaphysical tradition that singularly values the visual over the acoustic, semantic content over vocal utterances, and an abstract, anonymous "what" over a particular, embodied "who." Cavarero claims that the vocalic is an "anti-metaphysical" force that exceeds and challenges signification. In other words, the vocalic immediately communicates the uniqueness of the speaker without recourse to the signifying content of his or her speech. The voice is essentially relational—a relation that is described as "the condition of every communication.... the communicability of the communicable, or the significance of signification" (29). Indeed, for Cavarero the voice is that which "precedes, generates, and exceeds verbal communication" [End Page 166] (29–30). In this way, remarkably, Cavarero refigures the voice as a presymbolic origin that is both temporally and ontologically prior to signification, as that which is a condition of possibility for signification, and as that which is never exhausted by any signifying speech act as such.The voice is the expression of one unique, embodied individual to another. This expression, a "reciprocal invocation" founded in the sonorous, rhythmic, and resonant materiality of the voice, constitutes a "duet," a musical exchange of voices in and through which individuals mutually invoke one another. Cavarero resists inscribing such a relation within the symbolic realm; yet for her this relation is a "wordless language" that takes place in the paradigmatic scene between mother and infant. Here, in this originary scene, there is not yet the movement of language or speech as such, but the rhythm and cadence of demand and response—the voices of mother and infant that invoke one another. Drawing on Hélène Cixous's and Julia Kristeva's well-known work on the maternal chora, Cavarero demonstrates that the cadence of the "la-la melody" exchanged between mother and infant "configure[s] a reciprocal dependence" (170) that takes a structural form in the movement between demand and response.For Cavarero, the "scene of infancy" enjoys a primacy: it is both ontologically and chronologically prior to the institution of language and law. The maternal chora is figured as that which invisibly underwrites what will be understood as the law of the father—a law that is said to mark the inauguration of language. But if the law of the father is rational and semantic, the maternal should not be conceived as simply irrational and pure sound. For Cavarero, the maternal vocalic demonstrates its own reason through its rhythmic and musical resonance... (shrink)

The author reviews and summarizes, in as jargon-free way as he is capable of, the form of anti-platonist anti-nominalism he has previously developed in works since the 1980s, and considers what additions and amendments are called for in the light of such recently much-discussed views on the existence and nature of mathematical objects as those known as hyperintensional metaphysics, natural language ontology, and mathematical structuralism.

As we move around in our environment, and interact with it, many of the most important problems we face involve the processing of spatial information. We have to be able to navigate by perceiving and remembering the locations and orientations of the objects around us relative to ourself; we have to sense and act upon these objects; and we need to move through space to position ourselves in favourable locations or to avoid dangerous ones. While this appears so simple that (...) we don't even think about it, the difficulty of solving these problems has been shown in the repeated failure of artificial systems to perform these kinds of tasks efficiently. In contrast, humans and other animals routinely overcome these problems every single day. This book examines some of the neural substrates and mechanisms that support these remarkable abilities. The hippocampus and the parietal cortex have been implicated in various core spatial behaviours, such as the ability to localise an object and navigate to it. Damage to these areas in humans and animals leads to impairment of these spatial functions. This collection of papers, written by internationally recognized experts in the field, reviews the evidence that each area is involved in spatial cognition, examines the mechanisms underlying the generation of spatial behaviours, and considers the relative roles of the parietal and hippocampal areas, including how each interacts with the other. The papers integrate a wide range of theoretical and experimental approaches, and touch on broader issues relating to memory and imagery. As such, this book represents the state of the art of current research into the neural basis of spatial cognition. It should be of interest to anyone - researchers or graduate students - working in the areas of cognitive neuroscience, neuroanatomy, neuropsychology, and cognition generally. (shrink)

The question, "Which modal logic is the right one for logical necessity?," divides into two questions, one about model-theoretic validity, the other about proof-theoretic demonstrability. The arguments of Halldén and others that the right validity argument is S5, and the right demonstrability logic includes S4, are reviewed, and certain common objections are argued to be fallacious. A new argument, based on work of Supecki and Bryll, is presented for the claim that the right demonstrability logic must be contained in S5, (...) and a more speculative argument for the claim that it does not include S4.2 is also presented. (shrink)

Metasemantics comprises new work on the philosophical foundations of linguistic semantics, by a diverse group of established and emerging experts in the philosophy of language, metaphysics, and the theory of content. The science of semantics aspires to systematically specify the meanings of linguistic expressions in context. The paradigmatic metasemantic question is accordingly: what more basic or fundamental features of the world metaphysically determine these semantic facts? Efforts to answer this question inevitably raise others, including: where are the boundaries of semantics?; (...) what is the essence of the meaning relation?; which framework should we use for semantic theorizing?; and what are the intrinsic natures of semantic values? Metasemantic inquiry has long been recognized as a central part of the philosophy of language, but recent developments in metaphysics and semantics itself now allow us to approach these classic questions with an unprecedented degree of precision. The essays collected here provide promising new perspectives on old problems, pose questions that suggest novel research projects, and taken together, greatly sharpen our understanding of linguistic representation. (shrink)

Rigorous proof is supposed to guarantee that the premises invoked imply the conclusion reached, and the problem of rigor may be described as that of bringing together the perspectives of formal logic and mathematical practice on how this is to be achieved. This problem has recently raised a lot of discussion among philosophers of mathematics. We survey some possible solutions and argue that failure to understand its terms properly has led to misunderstandings in the literature.

The article draws on critiques in political theory and morality to argue that womanism, a worldview rooted in Black women's lives and history, provides an alternative conceptual framework to prevailing Eurocentric thinking, for promoting socially just institutions of higher education. Presupposing a positioned, encultured, and embodied account of identity, womanism’s social change perspective holds transformative promise. It foregrounds Black women’s penchant for reaching solutions that promote communal balance, affirm one’s humanity and attend to the spiritual dimension (Phillips, 2006 Phillips, L. (...) (Ed.). (2006). The womanist reader. Routledge.[Crossref], [Google Scholar]). Directed first towards scholars of color, fostering inclusivity, communalist values and acknowledged intersectionality offers an ethic of the embodied self. As a corollary, it argues for a universal that recommends at least two guiding principles for a pedagogical philosophy. It is, first oriented towards a love of self and second towards placing all disciplines within a cultural-historical context. In the first claim, there is potential for nothing less than suspending the symbolic and ontological violence to one’s sense of belonging in academia. In the second claim, such an understanding can position scholars of color to actively re-narrate their fields. As illustrations, we focus on three academic fields of education, theology in higher education, and criminal justice. (shrink)

To enact democracy, which is to live in communication with difference, requires a formative process that involves an education of the whole person for and through civic life. Drawing on Charles Mills's theory of Herrenvolk ethics and Jonathan Lear's analysis of psychosocial lapses that ail us, Sheron Fraser-Burgess and Chris Higgins pursue a critical, historiographical, and psychosocial reading of our failures to live up to this aspiration, offering (1) a critique of our tendency to saddle ourselves with a false (...) choice between a homogenizing unity and a differentiated but fractured republic; (2) a demonstration of why we must eschew a thin universalism of principles and confront difference as embodied; (3) an argument from the ethics of risk against the urge to reify and compartmentalize difference; and (4) an evocation of how deep pluralism itself might serve as a unifying creed. Civic education is not a matter of informing but of forming and cultivating vision and values. In pursuing the credal deep pluralism that is required to do justice to the prospects and perils of our democracy-in-the-making, the task of the formative educator may be more difficult; but by embracing this creed, teachers may inspire their students to do the same. (shrink)

Parsons has given a (nonconstructive) proof that the first-order fragment of the system of Frege's Grundgesetze is consistent. Here a constructive proof of the same result is presented.

This fourth edition of one of the classic logic textbooks has been thoroughly revised by John Burgess. The aim is to increase the pedagogical value of the book for the core market of students of philosophy and for students of mathematics and computer science as well. This book has become a classic because of its accessibility to students without a mathematical background, and because it covers not simply the staple topics of an intermediate logic course such as Godel's Incompleteness (...) Theorems, but also a large number of optional topics from Turing's theory of computability to Ramsey's theorem. John Burgess has now enhanced the book by adding a selection of problems at the end of each chapter, and by reorganising and rewriting chapters to make them more independent of each other and thus to increase the range of options available to instructors as to what to cover and what to defer. (shrink)

John Burgess is the author of a rich and creative body of work which seeks to defend classical logic and mathematics through counter-criticism of their nominalist, intuitionist, relevantist, and other critics. This selection of his essays, which spans twenty-five years, addresses key topics including nominalism, neo-logicism, intuitionism, modal logic, analyticity, and translation. An introduction sets the essays in context and offers a retrospective appraisal of their aims. The volume will be of interest to a wide range of readers across (...) philosophy of mathematics, logic, and philosophy of language. (shrink)

In placing education at the centre, as The Main Enterprise of the World, Philip Kitcher has undertaken a monumental task. He has come to the field of philosophy of education captivated by the importance of its substantive preoccupations for the advancement of democratic aims. Accordingly, his book argues that the most salient obstruction to preparing citizens who will contribute to society is the seeming irreconcilability of the demands of industry, on the one hand, and of students’ personal growth, on the (...) other. In spite of his desire to accommodate diverse accounts of the human good, and his recognition of the formative role of culture, broadly conceived, there are strains in his account of human fulfilment deriving from the disjunction of the self and others. It is not evident, on his Deweyan onto-epistemology, that there is adequate attention to the imprint on an individual, and on the beliefs they come to form, of proximal social groups. The nature of the balance between ‘in-group’ and ‘out-group’ influences, between those that are near and those that are far, can profoundly affect the plausibility of Kitcher's account of a socially based sense of fulfilment. (shrink)

My contribution to the symposium on Goedel’s philosophy of mathematics at the spring 2006 Association for Symbolic Logic meeting in Montreal. Provisional version: references remain to be added. To appear in an ASL volume of proceedings of the Goedel sessions at that meeting.

Venn diagrams, which are widely used in introductory logic courses, provide a convenient and illuminating way of presenting the various theories concerning the nature of law. When combined with the Aristotelian square of opposition, these diagrams show not only how the theories are related to one another, logically, which is essential to understanding them, but also which theories are compossible. One surprising result of this approach is that it shows the substantive compatibility of the theories of law set forth by (...) H. L. A. Hart and Ronald Dworkin, who are usually pitted against one another. I show, through quotation and visual representation, that there is no essential disagreement between these jurisprudential stalwarts concerning the relation of law and morality. (shrink)