This paper discusses the connection between the actual history of mathematics and Lakatos's philosophy of mathematics, in three parts. The first points to studies by Lakatos and others which support his conception of mathematics and its history. In the second I suggest that the apparent poverty of Lakatosian examples may be due to the way in which the history of mathematics is usually written. The third part argues that Lakatos is right to hold philosophy accountable to history, even if Lakatos's (...) own view of mathematics fails that test. (shrink)

The UK Arts and Humanities Research Council has agreed to fund a series of three meetings with associated publications on mathematical cultures. This note describes the project.

This is a review of What is a Mathematical Concept? edited by Elizabeth de Freitas, Nathalie Sinclair, and Alf Coles. In this collection of sixteen chapters, philosophers, educationalists, historians of mathematics, a cognitive scientist, and a mathematician consider, problematise, historicise, contextualise, and destabilise the terms ‘mathematical’ and ‘concept’. The contributors come from many disciplines, but the editors are all in mathematics education, which gives the whole volume a disciplinary centre of gravity. The editors set out to explore and reclaim the (...) canonical question ‘what is a mathematical concept?’ from the philosophy of mathematics. This review comments on each paper in the collection. (shrink)

The schism between analytic and continental philosophy resists repair because it is not confined to philosophers. It is a local manifestation of a far more profound and pervasive division. In 1959 C.P. Snow lamented the partition of intellectual life in to `two cultures': that of the scientist and that of the literary intellectual. If we follow the practice of most universities and bundle historical and literary studies together in the faculty of humanities on the one hand, and count pure mathematics (...) among the sciences on the other, then it is fair to say that the mutual ignorance and occasional hostility between scientists and humanists decried by Snow is still with us. And it runs through the middle of philosophy. Philosophy aspires to say something about everything, so it is unsurprising that philosophers have reproduced in miniature the division between the arts and the sciences. What is worrying is that we have failed to overcome it. (shrink)

The leading philosophers of science of the first half of the twentieth century had little use for the history of science. There are several possible explanations for this. One is that philosophers of science sometimes (knowingly or not) mimic the methodological habits and values of scientists. Many philosophers of science are motivated by admiration for the perceived rigor and intellectual hygiene of the exact sciences. Historical sense is not normally a cardinal virtue among physicists. Hence, those philosophers who take their (...) methodological cues from their scientific heroes are unlikely to think of philosophy as a historical discipline. Indeed, the idea of a close relationship between history and philosophy may have fallen into disrepute as a consequence of its association with idealist philosophers, of whom Hegel is the most notorious. Then there is Feyerabend's explanation, which is that the philosophy of science was transformed by the spectacular advances in formal logic which took place in the late nineteenth century. The development of modern formal logic made possible a range of hitherto inconceivable projects in the philosophy of science. The most radical of these was the positivist attempt to replace philosophy with the logical analysis of scientific language. Equally dependent on the new logic was Popper's attempt to explain the workings of empirical science using only deductive patterns of inference. According to Feyerabend, the philosophical projects engendered by the advances in formal logic came to dominate so thoroughly that everything else was marginalized (see feyerabend). It was not necessary for philosophers to know anything about the history of science, but they did have to be competent logicians: “As always maturity in a narrow domain means illiteracy elsewhere” (Feyerabend 1981, p. 20). (shrink)

Terence Tao New York, Oxford University Press, 2006xii + 103 pp., ISBN 9780199205615, £37.50 (hardback), ISBN 9780199205608, £12.99 (paperback)This is a book of mathematical problems and their solu...

Evolutionary fitness threats and rewards are associated with subjectively unpleasant and pleasant sensations, respectively. Initially, these correlations appear explainable via adaptation by natural selection. But here I analyse the major metaphysical perspectives on consciousness – physicalism, dualism, and panpsychism – and conclude that none help to understand the adaptive-seeming correlations via adaptation. I also argue that a recently proposed explanation, the phenomenal powers view, has major problems that mean it cannot explain the adaptive-seeming correlations via adaptation either. So the mystery (...) – call it the evolutionary paradox of consciousness – remains. Some have used this mystery to argue for non-naturalistic (e.g. theistic) explanations. But I propose a naturalistic, non-adaptive explanation of the adaptive-seeming correlations: namely, ‘sensational associative learning’ during development. In this perspective, pairing of particular sensations with unconditioned stimuli – fitness rewards or threats – cause the sensations themselves to come to be interpreted as good or bad, respectively. Sensations, like colours, that are not reliably paired with either fitness rewards or threats remain largely unvalenced. Sensational associative learning also provides explanations for adaptive-seeming structural aspects of sensations, such as the observation that sounds of different pitch are experienced as ordinal in correspondence to their wavelengths while the same is not true of colours of different hue. The sensational associative learning perspective appears compatible with physicalism, panpsychism, and dualism (though not epiphenomenalism). (shrink)

Burt's argument relies on a motte-and-bailey fallacy. Burt aims to argue against the value of genetics for social science; instead she argues against certain interpretations of a specific kind of genetics tool, polygenic scores (PGSs). The limitations, previously identified by behavioural geneticists including ourselves, do not negate the value of PGSs, let alone genetics in general, for social science.

In The Structure of Scientific Inference, Mary Hesse has argued for a realist interpretation of science. Her realism, however, is not to be understood in the context of the traditional realist/instrumentalist debate within the philosophy of science. That debate focused on the question of whether or not science did anything other than systematize the empirical data given in our experience. The traditional realist answered “yes” and most frequently described the theoretical statements of science, not only as true, but also as (...) providing descriptions of the world. The instrumentalist on the other hand answered “no” and ascribed no descriptive force to the theoretical statements of science. However, according to Hesse, neither party in the contemporary realism debate assumes the notion of an empirical given. Consequently, the present dispute must be understood as taking place on a level which is logically prior to its predecessor. (shrink)

Tracing Walter Benjamin's convergences with, and divergences from, influential German theorist Carl Schmitt, this edited collection places his thinking in the context of broader 20th century political philosophy of his time, and examines the question of whether Benjamin presents the possibility for a distinctive political theology, mapping the coordinates of this question without collapsing the tensions internal to Benjamin's thought. This volume brings together a host of multifaceted contributions that explore why Benjamin has been a fertile source for thinking about (...) political theology and the possibilities that this new perspective brings to light. (shrink)

This article begins with a criticism of Mclntyre and Gorovitz's account of medical error. Their theory implies that error, at least sometimes, it a necessary consequence of the inductive character of medical inquiry. The counter intuitive consequences of this account suggest that the issues surrounding induction may not be the most fertile area for developing a coherent interpretation of medical error. Given these shortcomings, I develop a new theory which assumes that the best philosophical soil for constructing a theory of (...) medical trror is the problem of universals. I then explain the problem and how the medical universal functions within arguments concerning diagnosis and treatment. A Wittgensteinian solution to the problem is presented which emphasizes the "borderline" character of some, if not many, medical judgments. Next, an argument is offered to establish that questions of medical error are not purely professional questions. This is because the theories used to make such professional judgments permit borderline cases even under the best circumstances. Finally, while there are a number of legitimate responses to this situation, I recommend one which involves legislative action. CiteULike Connotea Del.icio.us What's this? (shrink)

This volume consists of fourteen chapters selected from papers presented at the conference 'Ethics, Medicine and Health Care: An Appraisal of the Thought of H. Tristram Engelhardt, Jr.' along with a response to those chapters by Engelhardt and a Foreword by Laurence B. McCullough. The chapters direct primary attention to various aspects of Engelhardt's philosophy of medicine and bioethics as presented in The Foundations of Bioethics and Bioethics and Secular Humanism: The Search for a Common Morality. Among the topics treated (...) are the economics of health care and the medical profession, the libertarian and communitarian aspects of Engelhardt's thought, the moral status of children, abortion, the moral foundations for a health care system, feminism and clinical epistemology, and the relation between secular and religious moralities. In response to the various challenges posed by the authors, Engelhardt considers the implications of the failure of the modern philosophical project, the role of reason in ethics, and the resolution of conflict among communities that do not share the same moral vision. The book will be of interest to professionals in medicine, philosophy, theology, health policy, and law, and to graduate students in those disciplines. (shrink)

Bas van Fraassen has formulated a semantical or model theoretic analysis of the structure of scientific theories. He contrasts his semantical approach with the syntactic approach of the logical positivists and argues that his theory is preferable on a number of grounds. The aims of this paper are threefold. First, a brief description of van Fraassen's approach is presented. Secondly, his theory is compared with that of the logical positivists and, in so doing, the virtues that van Fraassen believes recommend (...) his theory over the positivist theory are identified. Finally, an objection that is positivist in spirit is raised which suggests that, epistemically, his account is indistinguishable from classical historicism. (shrink)

Ender Wiggin spends much of Ender's Game playing games of one sort or another. These range from simple role‐playing games with his siblings (“buggers and astronauts”), to battleroom contests, to the strange free play Giant's Drink video game in which he must kill a giant and confront his deepest fears. This chapter examines the role that games play in Ender's development as both a military commander and as a human being. It considers a number of interrelated questions: What is a (...) game? Is it really appropriate to call Battle School activities “games,” given how serious they are? What is the relationship between a good game and the development of a good person?. (shrink)

_Lakatos: An Introduction_ provides a thorough overview of both Lakatos's thought and his place in twentieth century philosophy. It is an essential and insightful read for students and anyone interested in the philosophy of science.

BackgroundThe evolution of the female orgasm in humans and its role in romantic relationships is poorly understood. Whereas the male orgasm is inherently linked to reproduction, the female orgasm is not linked to obvious reproductive or survival benefits. It also occurs less consistently during penetrative sex than does the male orgasm. Mate-choice hypotheses posit that the wide variation in female orgasm frequency reflects a discriminatory mechanism designed to select high-quality mates.ObjectiveWe aimed to determine whether women report that their orgasm frequency (...) varies between partners, whether this variation reflects mates' personal characteristics, and whether this variation reflects own and partner sexual behaviour during intercourse.DesignWe collected survey data from 103 women who rated the extent to which their orgasm frequency varied between partners, the characteristics of previous sexual partners who induced high-orgasm frequency and those who induced low-orgasm frequency,... (shrink)

It is argued in this study that (i) progress in the philosophy of mathematical practice requires a general positive account of informal proof; (ii) the best candidate is to think of informal proofs as arguments that depend on their matter as well as their logical form; (iii) articulating the dependency of informal inferences on their content requires a redefinition of logic as the general study of inferential actions; (iv) it is a decisive advantage of this conception of logic that it (...) accommodates the many mathematical proofs that include actions on objects other than propositions; (v) this conception of logic permits the articulation of project-sized tasks for the philosophy of mathematical practice, thereby supplying a partial characterisation of normal research in the field. (shrink)

This paper assumes the success of arguments against the view that informal mathematical proofs secure rational conviction in virtue of their relations with corresponding formal derivations. This assumption entails a need for an alternative account of the logic of informal mathematical proofs. Following examination of case studies by Manders, De Toffoli and Giardino, Leitgeb, Feferman and others, this paper proposes a framework for analysing those informal proofs that appeal to the perception or modification of diagrams or to the inspection or (...) imaginative manipulation of mental models of mathematical phenomena. Proofs relying on diagrams can be rigorous if it is easy to draw a diagram that shares or otherwise indicates the structure of the mathematical object, the information thus displayed is not metrical and it is possible to put the inferences into systematic mathematical relation with other mathematical inferential practices. Proofs that appeal to mental models can be rigorous if the mental models can be externalised as diagrammatic practice that satisfies these three conditions. (shrink)

This collection presents significant contributions from an international network project on mathematical cultures, including essays from leading scholars in the history and philosophy of mathematics and mathematics education.​ Mathematics has universal standards of validity. Nevertheless, there are local styles in mathematical research and teaching, and great variation in the place of mathematics in the larger cultures that mathematical practitioners belong to. The reflections on mathematical cultures collected in this book are of interest to mathematicians, philosophers, historians, sociologists, cognitive scientists and (...) mathematics educators. (shrink)

This paper assumes the success of arguments against the view that informal mathematical proofs secure rational conviction in virtue of their relations with corresponding formal derivations. This assumption entails a need for an alternative account of the logic of informal mathematical proofs. Following examination of case studies by Manders, De Toffoli and Giardino, Leitgeb, Feferman and others, this paper proposes a framework for analysing those informal proofs that appeal to the perception or modification of diagrams or to the inspection or (...) imaginative manipulation of mental models of mathematical phenomena. Proofs relying on diagrams can be rigorous if it is easy to draw a diagram that shares or otherwise indicates the structure of the mathematical object, the information thus displayed is not metrical and it is possible to put the inferences into systematic mathematical relation with other mathematical inferential practices. Proofs that appeal to mental models can be rigorous if the mental models can be externalised as diagrammatic practice that satisfies these three conditions. (shrink)

After the publication of The structure of scientific revolutions, Kuhn attempted to fend off accusations of extremism by explaining that his allegedly “relativist” theory is little more than the mundane analytical apparatus common to most historians. The appearance of radicalism is due to the novelty of applying this machinery to the history of science. This defence fails, but it provides an important clue. The claim of this paper is that Kuhn inadvertently allowed features of his procedure and experience as an (...) historian to pass over into his general account of science. Kuhn’s familiar claims, that science is directed in part by extra-scientific influences; that the history of science is divided by revolutionary breaks into periods that cannot be easily compared; that there is no ahistorical standard of rationality by which past episodes may be judged; and that science cannot be shown to be heading towards the Truth—these now appear as methodological commitments rather than historico–philosophical theses.Author Keywords: Kuhn; Koyré; Butterfield; Historicism; Revolution. (shrink)

The metaphor of scaffolding has become current in discussions of the cognitive help we get from artefacts, environmental affordances and each other. Consideration of mathematical tools and representations indicates that in these cases at least (and plausibly for others), scaffolding is the wrong picture, because scaffolding in good order is immobile, temporary and crude. Mathematical representations can be manipulated, are not temporary structures to aid development, and are refined. Reflection on examples from elementary algebra indicates that Menary is on the (...) right track with his ‘enculturation’ view of mathematical cognition. Moreover, these examples allow us to elaborate his remarks on the uniqueness of mathematical representations and their role in the emergence of new thoughts. (shrink)

The view that a mathematical proof is a sketch of or recipe for a formal derivation requires the proof to function as an argument that there is a suitable derivation. This is a mathematical conclusion, and to avoid a regress we require some other account of how the proof can establish it.