La publication des travaux de Galois dans le Journal de Liouville (1846) a été annoncée lors d’un débat entre les académiciens Libri et Liouville, dès 1843. Plutôt que de restreindre cette publication à une étude de controverse, nous voudrions montrer qu’elle s’inscrit dans un contexte plus large de redéfinition de l’algèbre, dont ce débat est un révélateur. Pour comprendre la redécouverte de Galois, il faut l’inscrire dans le temps médian des pratiques et des usages mathématiques.

This chapter focuses on alternative logics. It discusses a hierarchy of logical reform. It presents case studies that illustrate particular aspects of the logical revisionism discussed in the chapter. The first case study is of intuitionistic logic. The second case study turns to quantum logic, a system proposed on empirical grounds as a resolution of the antinomies of quantum mechanics. The third case study is concerned with systems of relevance logic, which have been the subject of an especially detailed reform (...) program. Finally, the fourth case study is paraconsistent logic, perhaps the most controversial of serious proposals. (shrink)

(1983). RATIONALISM AND THE STRONG PROGRAMME OF THE SOCIOLOGY OF KNOWLEKE: RECONCILIATION WITHOUT TEARS. Philosophical Papers: Vol. 12, No. 1, pp. 1-31.

Monsters lurk within mathematical as well as literary haunts. I propose to trace some pathways between these two monstrous habitats. I start from Jeffrey Jerome Cohen’s influential account of monster culture and explore how well mathematical monsters fit each of his seven theses. The mathematical monsters I discuss are drawn primarily from three distinct but overlapping domains. Firstly, late nineteenth-century mathematicians made numerous unsettling discoveries that threatened their understanding of their own discipline and challenged their intuitions. The great French mathematician (...) Henri Poincaré characterised these anomalies as ‘monsters’, a name that stuck. Secondly, the twentieth-century philosopher Imre Lakatos composed a seminal work on the nature of mathematical proof, in which monsters play a conspicuous role. Lakatos coined such terms as ‘monster-barring’ and ‘monster-adjusting’ to describe strategies for dealing with entities whose properties seem to falsify a conjecture. Thirdly, and most recently, mathematicians dubbed the largest of the sporadic groups ‘the Monster’, because of its vast size and uncanny properties, and because its existence was suspected long before it could be confirmed. (shrink)

Mathematics used to be portrayed as a deductive science. Stemming from Polya, however, is a philosophical movement which broadens the concept of mathematical reasoning to include inductive or quasi-empirical methods. Interest in inductive methods is a welcome turn from foundationalism toward a philosophy grounded in mathematical practice. Regrettably, though, the conception of mathematical reasoning embraced by quasi-empiricists is still too narrow to include the sort of thought-experiment which Mueller describes as traditional mathematical proof and which Lakatos examines in Proofs and (...) refutations. This paper extends the concept of mathematical reasoning along two further dimensions to accommodate thought-experiment.Keywords: Thought-experiment; Informal proof; Mathematical reasoning. (shrink)

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)

In the sociology of knowledge, the relationship between society and knowledge —or rather what separates them — remains an unsolved problem. A critical analysis of various solutions that we must look for to this problem suggests the plausibility of a passage between social groups, styles of argumentation and objects of knowledge. An empirical model of “decision displacements” is proposed on the basis of a corpus of texts and of observations derived from concrete analysis of a laboratory situation.

This paper introduces what is called the intersubjective interpretation of the probability calculus. Intersubjective probabilities are related to subjective probabilities, and the paper begins with a particular formulation of the familiar Dutch Book argument. This argument is then extended, in Section 3, to social groups, and this enables the concept of intersubjective probability to be introduced in Section 4. It is then argued that the intersubjective interpretation is the appropriate one for the probabilities which appear in confirmation theory whether of (...) a Bayesian or a Popperian variety. The final section of the paper states and tries to answer an objection due to Putnam. (shrink)

ArgumentKuhnian phases of paradigmatic development correspond to characteristic variations of citation measures. These correlations can in turn be predicted from a simple model of human information processing when applied to the common environments of scientists. By combining a scientometric and a human information processing approach to the history of scientific thought, structures of disciplinary development, and in particular paradigmatic cycles, can be more reliably assessed than before. Consequently, the quantitative historian of science is liberated to some extent from the vagaries (...) of qualitative judgment, as exemplified by traditional narrative type approaches to the history of science. (shrink)

Previous studies of corporate environmental and social action identify exactly three similar patterns of activity. They provide divergent structural explanations for these patterns, as networks of institutional constraint, and networks of local inter-dependence, respectively. A theory of sociocultural viability, known in anthropology and policy science as Cultural Theory, explains that social systems consist of four patterns of social interaction, shaped by two distinct structural factors. Our own analysis of 45 items of environmental, social, and governance factors reconcile extant studies’ findings (...) with Cultural Theory’s expectations. Although earlier studies of corporate social action observed only three patterns of activity, our analysis uncovered the fourth, and that diverging structural explanations of earlier studies are partial and complementary, clarified by Cultural Theory. Our call for further study leverages the normative dimension of Cultural Theory, which specifies policy processes that privilege human decency over parochial preference functions. (shrink)

The increased popularity of the label "cultural" within science studies, especially in relation to "cultural studies, " invites consideration of how it is and can be used in historical work. A lot more seems now to be invested in the notion of "cultural history. " This article examines some recent historiography of science as a means of considering what counts as cultural history in that domain and attempts to coordinate it with the sociologically informed studies of the past ten orfifteen (...) years. The label "sociocultural" seems a more useful term by which to capture recent developments. (shrink)

The 19th century saw the development of an eclectic medical marketplace in both the United Kingdom and the United States, with mesmerists, herbalists and hydrotherapists amongst the plethora of medical ‘sectarians’ offering mainstream (or ‘allopathic’) medicine stiff competition. Foremost amongst these competitors were homoeopaths, a group of practitioners who followed Samuel Hahnemann (1982[1810]) in prescribing highly dilute doses of single-drug substances at infrequent intervals according to the ‘law of similars’ (like cures like). The theoretical sophistication of homoeopathy, compared to other (...) medical sectarian systems, alongside its institutional growth after the mid-19th-century cholera epidemics, led to homoeopathy presenting a challenge to allopathy that the latter could not ignore. Whilst the subsequent decline of homoeopathy at the beginning of the 20th century was the result of multiple factors, including developments within medical education, the Progressive movement, and wider socio-economic changes, this article focuses on allopathy’s response to homoeopathy’s conceptual challenge. Using the theoretical framework of Berger and Luckmann (1991[1966]) and taking a Tory historiographical approach (Fuller, 2002) to recover more fully 19th-century homoeopathic knowledge, this article demonstrates how increasingly sophisticated ‘nihilative’ strategies were ultimately successful in neutralising homoeopathy and that homoeopaths were defeated by allopaths (rather than disproven) at the conceptual level. In this process, the therapeutic use of ‘nosodes’ (live disease products) and the language of bacteriology were pivotal. For their part, homoeopaths failed to mount a counter-attack against allopaths with an explanatory framework available to them. (shrink)

First published in 2002. Routledge is an imprint of Taylor & Francis, an informa company.

I argue for the Wittgensteinian thesis that mathematical statements are expressions of norms, rather than descriptions of the world. An expression of a norm is a statement like a promise or a New Year's resolution, which says that someone is committed or entitled to a certain line of action. A expression of a norm is not a mere description of a regularity of human behavior, nor is it merely a descriptive statement which happens to entail a norms. The view can (...) be thought of as a sort of logicism for the logical expressivist---a person who believes that the purpose of logical language is to make explicit commitments and entitlements that are implicit in ordinary practice. The thesis that mathematical statements are expression of norms is a kind of logicism, not because it says that mathematics can be reduced to logic, but because it says that mathematical statements play the same role as logical statements. ;I contrast my position with two sets of views, an empiricist view, which says that mathematical knowledge is acquired and justified through experience, and a cluster of nativist and apriorist views, which say that mathematical knowledge is either hardwired into the human brain, or justified a priori, or both. To develop the empiricist view, I look at the work of Kitcher and Mill, arguing that although their ideas can withstand the criticisms brought against empiricism by Frege and others, they cannot reply to a version of the critique brought by Wittgenstein in the Remarks on the Foundations of Mathematics. To develop the nativist and apriorist views, I look at the work of contemporary developmental psychologists, like Gelman and Gallistel and Karen Wynn, as well as the work of philosophers who advocate the existence of a mathematical intuition, such as Kant, Husserl, and Parsons. After clarifying the definitions of "innate" and "a priori," I argue that the mechanisms proposed by the nativists cannot bring knowledge, and the existence of the mechanisms proposed by the apriorists is not supported by the arguments they give. (shrink)