Philosophy of mathematics is moving in a new direction: away from a foundationalism in terms of formal logic and traditional ontology, and towards a broader range of approaches that are united by a focus on mathematical practice. The scientific research network PhiMSAMP (Philosophy of Mathematics: Sociological Aspects and Mathematical Practice) consisted of researchers from a variety of backgrounds and fields, brought together by their common interest in the shift of philosophy of mathematics towards mathematical practice. Hosted by the Rheinische Friedrich- (...) Wilhelms-Universitat Bonn and funded by the Deutsche Forschungsgemeinschaft (DFG) from 2006-2010, the network organized and contributed to a number of workshops and conferences on the topic of mathematical practice. The refereed contributions in this volume represent the research results of the network and consists of contributions of the network members as well as selected paper versions of presentations at the network's mid-term conference, "Is mathematics special?" (PhiMSAMP-3) held in Vienna 2008. (shrink)

The present volume provides an introduction to the major themes of work in experimental philosophy, bringing together some of the most influential articles in ...

In this paper we propose to argue for two claims. The first is that a sizeable group of epistemological projects – a group which includes much of what has been done in epistemology in the analytic tradition – would be seriously undermined if one or more of a cluster of empirical hypotheses about epistemic intuitions turns out to be true. The basis for this claim will be set out in Section 2. The second claim is that, while the jury is (...) still out, there is now a substantial body of evidence suggesting that some of those empirical hypotheses are true. Much of this evidence derives from an ongoing series of experimental studies of epistemic intuitions that we have been conducting. A preliminary report on these studies will be presented in Section 3. In light of these studies, we think it is incumbent on those who pursue the epistemological projects in question to either explain why the truth of the hypotheses does not undermine their projects, or to say why, in light of the evidence we will present, they nonetheless assume that the hypotheses are false. In Section 4, which is devoted to Objections and Replies, we’ll consider some of the ways in which defenders of the projects we are criticizing might reply to our challenge. Our goal, in all of this, is not to offer a conclusive argument demonstrating that the epistemological projects we will be criticizing are untenable. Rather, our aim is to shift the burden of argument. (shrink)

Recent years have seen an explosion of empirical data concerning arithmetical cognition. In this paper that data is taken to be philosophically important and an outline for an empirically feasible epistemological theory of arithmetic is presented. The epistemological theory is based on the empirically well-supported hypothesis that our arithmetical ability is built on a protoarithmetical ability to categorize observations in terms of quantities that we have already as infants and share with many nonhuman animals. It is argued here that arithmetical (...) knowledge developed in such a way cannot be totally conceptual in the sense relevant to the philosophy of arithmetic, but neither can arithmetic understood to be empirical. Rather, we need to develop a contextual a priori notion of arithmetical knowledge that preserves the special mathematical characteristics without ignoring the roots of arithmetical cognition. Such a contextual a priori theory is shown not to require any ontologically problematic assumptions, in addition to fitting well within a standard framework of general epistemology. (shrink)

Nous examinons les conditions de vérité pour des attributions de savoir dans le cas des connaissances mathématiques. La disposition d’une démonstration formalisable semble être un critère naturel :(*) X sait que p est vrai si et seulement si X en principe dispose d’une démonstration formalisable pour p.La formalisabilité pourtant ne joue pas un grand rôle dans la pratique mathématique effective. Nous présentons des résultats d’une recherche empirique qui indiquent que les mathématiciens n’employent pas certaines spécifications de (*) quand ils attribuent (...) du savoir. De plus, nous montrons que le concept de savoir mathématique qui est à la base de l’emploi effectif du mot « savoir » de la pratique mathématique est tout à fait compatible avec certaines intuitions philosophiques mais apparaît comme différent des concepts philosophiques formant la base de (*). (shrink)

Nous examinons les conditions de vérité pour des attributions de savoir dans le cas des connaissances mathématiques. La disposition d’une démonstration formalisable semble être un critère naturel :(*) X sait que p est vrai si et seulement si X en principe dispose d’une démonstration formalisable pour p.La formalisabilité pourtant ne joue pas un grand rôle dans la pratique mathématique effective. Nous présentons des résultats d’une recherche empirique qui indiquent que les mathématiciens n’employent pas certaines spécifications de (*) quand ils attribuent (...) du savoir. De plus, nous montrons que le concept de savoir mathématique qui est à la base de l’emploi effectif du mot « savoir » de la pratique mathématique est tout à fait compatible avec certaines intuitions philosophiques mais apparaît comme différent des concepts philosophiques formant la base de (*). (shrink)

We argue that mathematical knowledge is context dependent. Our main argument is that on pain of distorting mathematical practice, one must analyse the notion of having available a proof, which supplies justification in mathematics, in a context dependent way.

Bibliography of works in experimental philosophy.

The form of nominalism known as 'mathematical fictionalism' is examined and found wanting, mainly on grounds that go back to an early antinominalist work of Rudolf Carnap that has unfortunately not been paid sufficient attention by more recent writers.

History and Philosophy of Modern Mathematics was first published in 1988. Minnesota Archive Editions uses digital technology to make long-unavailable books once again accessible, and are published unaltered from the original University of Minnesota Press editions. The fourteen essays in this volume build on the pioneering effort of Garrett Birkhoff, professor of mathematics at Harvard University, who in 1974 organized a conference of mathematicians and historians of modern mathematics to examine how the two disciplines approach the history of mathematics. In (...) History and Philosophy of Modern Mathematics, William Aspray and Philip Kitcher bring together distinguished scholars from mathematics, history, and philosophy to assess the current state of the field. Their essays, which grow out of a 1985 conference at the University of Minnesota, develop the basic premise that mathematical thought needs to be studied from an interdisciplinary perspective. The opening essays study issues arising within logic and the foundations of mathematics, a traditional area of interest to historians and philosophers. The second section examines issues in the history of mathematics within the framework of established historical periods and questions. Next come case studies that illustrate the power of an interdisciplinary approach to the study of mathematics. The collection closes with a look at mathematics from a sociohistorical perspective, including the way institutions affect what constitutes mathematical knowledge. (shrink)

There is an urgent need in philosophy of mathematics for new approaches which pay closer attention to mathematical practice. This book will blaze the trail: it offers philosophical analyses of important characteristics of contemporary mathematics and of many aspects of mathematical activity which escape purely formal logical treatment.

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)

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.

Experimental Philosophy: Volume 2 contains fourteen articles -- thirteen previously published and one new -- that reflect the fast-moving changes in the field over the last five years. The field of experimental philosophy is one of the most innovative and exciting parts of the current philosophical landscape; it has also engendered controversy. Proponents argue that philosophers should employ empirical research, including the methods of experimental psychology, to buttress their philosophical claims. Rather than armchair theorizing, experimental philosophers should go into the (...) field to research how people actually think and reason. In a sense this is a return to a view of philosophy as the progenitor of psychology: inherently concerned with the human condition, with no limits to its scope or methods. In the course of the last decade, many experimental philosophers have overturned assumptions about how people think in the real world. This volume provides an essential guide to the most influential recent work on this vital and exciting area of philosophical research. (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 book Bruno Latour brings together these different approaches to provide a lively and challenging analysis of science, demonstrating how social context..

In this ambitious study, David Corfield attacks the widely held view that it is the nature of mathematical knowledge which has shaped the way in which mathematics is treated philosophically and claims that contingent factors have brought us to the present thematically limited discipline. Illustrating his discussion with a wealth of examples, he sets out a variety of approaches to new thinking about the philosophy of mathematics, ranging from an exploration of whether computers producing mathematical proofs or conjectures are doing (...) real mathematics, to the use of analogy, the prospects for a Bayesian confirmation theory, the notion of a mathematical research programme and the ways in which new concepts are justified. His inspiring book challenges both philosophers and mathematicians to develop the broadest and richest philosophical resources for work in their disciplines and points clearly to the ways in which this can be done. (shrink)

The first edition of this book profoundly challenged and divided students of philosophy, sociology, and the history of science when it was published in 1976. In this second edition, Bloor responds in a substantial new Afterword to the heated debates engendered by his book.

In this ambitious study, David Corfield attacks the widely held view that it is the nature of mathematical knowledge which has shaped the way in which mathematics is treated philosophically, and claims that contingent factors have brought us to the present thematically limited discipline. Illustrating his discussion with a wealth of examples, he sets out a variety of new ways to think philosophically about mathematics, ranging from an exploration of whether computers producing mathematical proofs or conjectures are doing real mathematics, (...) to the use of analogy, the prospects for a Bayesian confirmation theory, the notion of a mathematical research programme, and the ways in which new concepts are justified. His highly original book challenges both philosophers and mathematicians to develop the broadest and richest philosophical resources for work in their disciplines, and points clearly to the ways in which this can be done. (shrink)