Nous étudions l’apparition des monstres en mathématiques (comme, par exemple, le monstre dit de Weierstrass — une fonction continue qui n’est différentiable en aucun point) au XIXe siècle. Nous traitons aussi la question : comment peut-on distinguer les mathématiques normales (ou naturelles) des mathématiques pathologiques?

The “new math” curriculum, one version of which was developed in the 1950s and 1960s by the School Mathematics Study Group under the auspices of the National Science Foundation, occasioned a great deal of controversy among mathematicians. Well before its rejection by parents and teachers, some mathematicians were vocal critics, decrying the new curriculum because of the way it described the practice and history of the discipline. The nature of mathematics, despite the field’s triumphs in helping to win World War (...) II and its midcentury promotion as the key to a modern technological society, was surprisingly contested in this period. Supporters of the School Mathematics Study Group, like its director, Edward Begle, emphasized the importance of portraying mathematics as a system of abstract structures, while opponents like Morris Kline argued that math was essentially a tool for understanding the natural world. The debate about the curriculum—and the role of mathematicians in its design—was also a debate about the underlying identity of the subject itself. (shrink)

This essay explores the research practice of French geometer Michel Chasles, from his 1837 Aperçu historique up to the preparation of his courses on ‘higher geometry’ between 1846 and 1852. It argues that this scientific pursuit was jointly carried out on a historiographical and a mathematical terrain. Epistemic techniques such as the archival search for and comparison of manuscripts, the deconstructive historiography of past geometrical methods, and the epistemologically motivated periodization of the history of mathematics are shown to have played (...) a crucial role in the shaping of Chasles's own theories. In particular, we present Chasles's approach to the ‘material history’ of algebraic symbolism and argue that it motivated and informed his subsequent invention of a novel notational technology for the writing of geometrical proofs and propositions. In return, this technology allowed Chasles to carry out a programme for the modernization of geometry in keeping with epistemic requirements he had also delineated via a form of historical writing. (shrink)

Lamarck's perspective on change within the organic world, in particular his conception of "la marche de la nature," , crystallized during the last decade of the 18th century and the early years of the 19th. I argue that it should be viewed as resulting in part from interactions with, and transfers from, the social thought—modes of thinking, ways of conceptualizing, models, metaphors and analogies—of the decades before the French revolution and of the revolutionary decade itself. Moreover, Lamarck's involvement with the (...) new institutional frameworks initiated during the revolutionary period had brought him into direct contacts with the then prevalent modes of discourse on things social. (shrink)

Many philosophers regard the persistence of philosophical disputes as symptomatic of overly ambitious, ill-founded intellectual projects. There are indeed strong reasons to believe that persistent disputes in philosophy (and more generally in the discourse at large) are pointless. We call this the pessimistic view of the nature of philosophical disputes. In order to respond to the pessimistic view, we articulate the supporting reasons and provide a precise formulation in terms of the idea that the best explanation of persistent disputes entails (...) that they are pointless. We then show how to answer the pessimistic argument. Taking a well-known mathematical controversy as our paradigm example, we argue that some persistent disputes reflect substantive disagreements at the “meta-analytic” level, i.e., disagreements about the best way, among quite different candidates, to understand the topic at issue, and the best associated cluster of analytic truths one should accept concerning it. Moreover, our concrete example shows that such meta-analytic disagreements can in principle be settled and yield a genuine theoretical (as opposed to merely pragmatic) breakthrough. We conclude optimistically that persistent disputes can be an important means of fostering epistemic progress. (shrink)