Mathematicians tend to think of themselves as scientists investigating the features of real mathematical things, and the wildly successful application of mathematics in the physical sciences reinforces this picture of mathematics as an objective study. For philosophers, however, this realism about mathematics raises serious questions: What are mathematical things? Where are they? How do we know about them? Offering a scrupulously fair treatment of both mathematical and philosophical concerns, Penelope Maddy here delineates and defends a novel version of mathematical realism. (...) She answers the traditional questions and poses a challenging new one, refocusing philosophical attention on the pressing foundational issues of contemporary mathematics. (shrink)

Naturalism in Mathematics investigates how the most fundamental assumptions of mathematics can be justified. One prevalent philosophical approach to the problem--realism--is examined and rejected in favor of another approach--naturalism. Penelope Maddy defines this naturalism, explains the motivation for it, and shows how it can be successfully applied in set theory. Her clear, original treatment of this fundamental issue is informed by current work in both philosophy and mathematics, and will be accessible and enlightening to readers from both disciplines.

This paper outlines a second-philosophical account of arithmetic that places it on a distinctive ground between those of logic and set theory.

This chapter compares and contrasts Quine’s naturalism with the versions of two post-Quineans on the nature of science, logic, and mathematics. The role of indispensability in the philosophy of mathematics is treated in detail.

Mathematics depends on proofs, and proofs must begin somewhere, from some fundamental assumptions. For nearly a century, the axioms of set theory have played this role, so the question of how these axioms are properly judged takes on a central importance. Approaching the question from a broadly naturalistic or second-philosophical point of view, Defending the Axioms isolates the appropriate methods for such evaluations and investigates the ontological and epistemological backdrop that makes them appropriate. In the end, a new account of (...) the objectivity of mathematics emerges, one refreshingly free of metaphysical commitments. (shrink)

This paper explores women's accounts of prostitution in terms of the lived experience of the body, drawing on life story narratives and arts images created by women in the sex industry. These narratives show that women's experiences of prostitution constitute a spectrum of (dis)embodiment that is inflected, not determined, by settings and contexts. Theoretical approaches to embodiment were sought that acknowledged tensions between violation and a sense of empowerment. Therefore, the ontology of selling sex, and associated experiences such as violence, (...) drug use and self-harm are explored using feminist applications of Merleau-Ponty's notion of the `habit body'. A key focus is how the body is constituted by embodied experiences of abuse (e.g. the work of Parkins and of Weiss), and how women negotiate ownership of the body within commercial sex transactions. This highlights the process of repositioning the body by selling sex, in accordance with the accumulated experiences of the habit body. Thus, to borrow from Wendy Parkins, women perceive that they are acting meaningfully through the body, even when reproducing dynamics of objectification and dissociation. (shrink)

The effort to fit simple logical truths–like `if it's either red or green and it's not red, then it must be green'–into Kant's account of knowledge turns up a position more subtle and intriguing than might be expected at first glance.

Feminist scholars have long critiqued the fashion industry’s ultra-thin beauty standards as harmful to women. Combining data from three qualitative studies of women’s clothing retailers—of bras, plus-size clothing, and bridal wear—we shift the analytical focus away from glamorized media images toward the seemingly mundane realm of clothing size standards, examining how women encounter, understand, and navigate these standards in their daily lives. We conceptualize clothing size standards as “floating signifiers,” given their lack of consistency within and across brands and the (...) extent to which women engage in identity work and body work in relation to them. Our findings indicate that the instability of these unregulated standards allows some women—particularly those with bodies located closest to the boundaries between size categories—to claim conformity to body ideals and to access some of the associated psychological, social, and material privileges. However, even as individual women may benefit by distancing themselves from stigmatized size categories, this pattern renders women’s body acceptance tenuous while simultaneously reinforcing hierarchies among women based on body size and shape. (shrink)

Many philosophers these days consider themselves naturalists, but it's doubtful any two of them intend the same position by the term. In Second Philosophy, Penelope Maddy describes and practices a particularly austere form of naturalism called "Second Philosophy". Without a definitive criterion for what counts as "science" and what doesn't, Second Philosophy can't be specified directly ("trust only the methods of science" for example), so Maddy proceeds instead by illustrating the behaviors of an idealized inquirer she calls the "Second Philosopher". (...) mhis Second Philosopher begins from perceptual common sense experimentation, theory formation and testing, working all the while to asses, correct and improve her methods as she goes. Second Philosophy is then the result of the Second Philosopher's investigations. Maddy delineates the Second Philosopher's approach by tracing her reactions to various familiar skeptical and transcendental views (Descartes, Kant, Carnap, late Putnam, van Fraassen), comparing her methods to those of other self-described naturalists (especially Quine), and examining a prominent contemporary debate (between disquotationalists and correspondence theorists in the theory of truth) to extract a properly second-philosophical line of thought. She then undertakes to practice Second Philosophy in her reflections on the ground of logical truth, the methodology, ontology and epistemology of mathematics, and the general prospects for metaphysics naturalized. (shrink)

Many philosophers claim to be naturalists, but there is no common understanding of what naturalism is. Maddy proposes an austere form of naturalism called 'Second Philosophy', using the persona of an idealized inquirer, and she puts this method into practice in illuminating reflections on logical truth, philosophy of mathematics, and metaphysics.

In this note we show that the classical modal technology of Sahlqvist formulas gives quick proofs of the completeness theorems in [8] (D. Gregory, Completeness and decidability results for some propositional modal logics containing “actually” operators, Journal of Philosophical Logic 30(1): 57–78, 2001) and vastly generalizes them. Moreover, as a corollary, interpolation theorems for the logics considered in [8] are obtained. We then compare Gregory's modal language enriched with an “actually” operator with the work of Arthur Prior now known under (...) the name of hybrid logic. This analysis relates the “actually” axioms to standard hybrid axioms, yields the decidability results in [8], and provides a number of complexity results. Finally, we use a bisimulation argument to show that the hybrid language is strictly more expressive than Gregory's language. (shrink)

A modal logic is developed to deal with finite ordered binary trees a they are used in linguistics. A modal language is introduced with operators for the ‘mother of’, ‘first daughter of’ and ‘second daughter of’ relations together with their transitive reflexive closures. The relevant class of tree models is defined and three linguistic applications of this language are discussed: context free grammars, command relations, and trees decorated with feature structures. An axiomatic proof system is given for which completeness is (...) shown with respect to the class of finite ordered binary trees. A number of decidability results follow. (shrink)

Pragmatists have traditionally been enemies of representationalism but friends of naturalism, when naturalism is understood to pertain to human subjects, in the sense of Hume and Nietzsche. In this volume Huw Price presents his distinctive version of this traditional combination, as delivered in his René Descartes Lectures at Tilburg University in 2008. Price contrasts his view with other contemporary forms of philosophical naturalism, comparing it with other pragmatist and neo-pragmatist views such as those of Robert Brandom and Simon Blackburn. (...) Linking their different 'expressivist' programmes, Price argues for a radical global expressivism that combines key elements from both. With Paul Horwich and Michael Williams, Brandom and Blackburn respond to Price in new essays. Price replies in the closing essay, emphasising links between his views and those of Wilfrid Sellars. The volume will be of great interest to advanced students of philosophy of language and metaphysics. (shrink)

Pursues the theoretical level of the two‐tiered epistemology of set theoretic realism, the level at which more abstract axioms can be justified by their consequences at more intuitive levels. I outline the pre‐axiomatic development of set theory out of Cantor's researches, describe how axiomatization arose in the course of Zermelo's efforts to prove Cantor's Well‐ordering Theorem, and review the controversy over the Axiom of Choice. Cantor's Continuum Hypothesis and various questions of descriptive set theory were eventually shown to be independent (...) of the standard axioms, and new axiom candidates—Gödel's Axiom of Constructibility, on the one hand, determinacy and large cardinal axioms, on the other—offer dramatically different solutions. This situation presents a new epistemic challenge to the set theoretic realist: on what grounds can we adjudicate between new axiom candidates? (shrink)

Outlines a physicalistic version of set theoretic realism, and compares and contrasts it with Field's nominalism, with structuralism, and with modalism. I conclude that, despite metaphysical differences, versions of all these views share the new challenge raised in Ch. 4: how are new axiom candidates to be rationally evaluated?

Begins with a review of Benacerraf's metaphysical challenge to mathematical realism based on sets: how, for example, can number theory be the study of particular sets when other sets with the same structural relations would seem to do just as well? The set theoretic realist gives the straightforward response that numbers are not particular sets, but properties of sets. I close with a digression on the prospects for ‘Frege numbers’—i.e. numbers construed as proper classes.

Mathematical axioms have traditionally been thought of as obvious or self-evident truths, but current set theoretic work in the search for new axioms belies this conception. This raises epistemological questions about what other forms of justification are possible, and how they should be judged.

Begins with a presentation and elaboration of Benacerraf's epistemic challenge to realism: how can we gain knowledge of an acausal world of non‐spatio‐temporal abstracts? I then outline a theory of perception based in part on neurological theories of Hebb and developmental evidence from Piaget, and I argue in these terms that we can, in fact, perceive sets of medium‐sized physical objects. This account of perception is elaborated into an account of physical and mathematical intuition, faculties that produce various rudimentary beliefs (...) that underlie the simplest physical and set theoretic assumptions. I conclude by comparing and contrasting this epistemology with some controversial passages from Gödel's writings. (shrink)

The early sections describe the pre‐theoretic realism of the mathematician, survey the basic forms of realism in philosophy, and attempt to disentangle the issues of realism from debates over the nature of truth. The final section begins by laying out traditional Platonism, intuitionism, and formalism. Quine's famous critique of Carnap's conventionalism then leads to Quine's realism, and Putnam's developments thereof, while a different sort of mathematical realism is found in Gödel's writings. Set theoretic realism emerges as an effort to build (...) on the strengths of Quine's and Gödel's realisms, while avoiding their weaknesses and filling in their lacunae. (shrink)

Permaculture is primarily a thinking tool for designing low carbon, highly productive systems. It originated in Australia in the 1970s and was conceived by Bill Mollison and David Holmgren as a response to the devastating effects of a temperate European agriculture on the fragile soils of an ancient antipodean landscape. Like the dust bowls of the Great Plains in the USA in the 1930s, an alien agriculture has the capacity to turn a delicately balanced ecology into desert. Their initial response (...) was to design a permanent agriculture with tree crops and other perennials inhabiting all the niches from the canopy to the ground cover and below. The soil is left untilled to establish its own robust micro-ecology. Key to this is that the land must be biodiverse and stable for future generations. From perennial tree crops, permaculture has developed into an integrated system of design that encompasses everything from agriculture, horticulture, architecture, and ecology, as well as economy and legal systems for businesses and communities. (shrink)

The idea of combining logics, structures, and theories has recently been attracting interest in areas as diverse as constraint logic programming, theorem proving, verification, computational linguistics, artificial intelligence and indeed, various branches of logic itself. It would be an exaggeration to claim that these (scattered, and by-and-large independent) investigations have crystallized into an enterprise meriting the title "combined methods"; nonetheless, a number of interesting themes are emerging. This introduction notes some prominent ones and relates them to the papers in this (...) special issue. (shrink)

The Logical Must is an examination of Wittgenstein's philosophy of logic, early and late, from an austere naturalistic perspective called "Second Philosophy.".

A plea for natural philosophy --On the question of realism --Hume and Reid --Moore's hands --Wittgenstein on hinges --A note on truth and reference --The philosophy of logic --A Second Philosophy of logic --Psychology and the a priori sciences --Do numbers exist? --Enhanced if-thenism.

The article challenges conventional assumptions regarding the question of incest survival within contemporary discourses. A textual analysis of Kathryn Harrison's autobiographical novel tracing her consensual sexual relationship with her father is used to address the issue of failed or unresolved mourning as a prototypically ‘modern’ cultural phenomenon. Psychoanalytically informed feminist literary criticism is used to explore the parallels between the cultural construction of femininity and failed or postponed mourning in western historical and philosophical traditions. Following the work of Juliana Schiesari (...) and Kathleen Woodward, the article contends that melancholia is a gendered discourse that has historically privileged male theorists and film-makers, such as Barthes and Fassbinder. The article suggests that contemporary women writers, such as Harrison, are engaged in a revisionary approach to the construction of loss within their writing. By situating themselves at the heart of the contemporary family narrative, instead of the ‘melancholic’ margins, they are able to produce a counter-discourse that dispels the conventional dynamics of the traditional family romance. By using the ‘writing cure’ to overcome ideological loss, the desiring daughter is able to challenge misogynist constructions of femininity within contemporary literature. (shrink)

It’s often said that set theory provides a foundation for classical mathematics because every classical mathematical object can be modeled as a set and every classical mathematical theorem can be proved from the axioms of set theory. This is obviously a remarkable mathematical fact, but it isn’t obvious what makes it ‘foundational’. This paper begins with a taxonomy of the jobs set theory does that might reasonably be regarded as foundational. It then moves on to category-theoretic and univalent foundations, exploring (...) to what extent they do these same jobs, and to what extent they might do other jobs also reasonably regarded as foundational. (shrink)

What Do Philosophers Do? takes up the leading arguments for radical skepticism from an everyday point of view. A range of philosophical methods are examined and employed, for a revealing portrait of what philosophers do, and perhaps a quiet suggestion for what they should do, for what they do best.

Within a year of each other, three one-volume general dictionaries of philosophy have recently appeared; when our future colleagues in philosophy look back on the 1990s they may well think of it as the decade of reference works. But however productive these years may prove to be in this genre, clearly visible somewhere around the top of the heap will be this handy, useful, entertaining, and instructive contribution from Simon Blackburn. Its two immediate competitors are the Cambridge Dictionary of (...) Philosophy, edited by Robert Audi, and the Oxford Companion to Philosophy, edited by Ted Honderich. No detailed comparison will be attempted here, but two points stand out as obviously giving Blackburn’s dictionary a rather different use and flavor from the others: while containing a closely comparable number of entries, it is distinctly shorter and handier in format; and it has all been written by a single author. (shrink)