The ArgumentL. E. J. Brouwer and David Hubert, two titans of twentieth-century mathematics, clashed dramatically in the 1920s. Though they were both Kantian constructivists, their notoriousGrundlagenstreitcentered on sharp differences about the foundations of mathematics: Brouwer was prepared to revise the content and methods of mathematics (his “Intuitionism” did just that radically), while Hilbert's Program was designed to preserve and constructively secure all of classical mathematics.Hilbert's interests and polemics at the time led to at least three misconstruals of intuitionism, misconstruals which (...) last to our own time: Current literature often portrays popular views of intuitionism as the product of Brouwer's idiosyncratic subjectivism; modern logicians view intuitionism as simply applying a non-standard formal logic to mathematics; and contemporary philosophers see that logic as based upon a pure assertabilist theory of meaning. These pictures stem from the way Hilbert structured the controversy.Even though Brouwer's own work and behavior occasionally reinforce these pictures, they are nevertheless inaccurate accounts of his approach to mathematics. However, the framework provided by the Brouwer-Hilbert debate itself does not supply an adequate correction of these inaccuracies. For, even if we eliminate these mistakes within that framework, Brouwer's position would still appear fragmented and internally inconsistent. I propose a Kantian framework — not from Kant's philosophy of mathematics but from his general metaphysics — which does show the coherence and consistency of Brouwer's views. I also suggest that expanding the context of the controversy in this way will illuminate Hilbert's views as well and will even shed light upon Kant's philosophy. (shrink)

These introductory remarks are unorthodox in many respects. The deviance from usual practice is justified by the extreme importance I attach to the subject matter of this special issue. I want to convey to the reader a sense of why I think controversies, particularly in science, are so crucial, and to propose a different way of thinking about them. This mandates, in the limited space available, a compact presentation, omitting supporting arguments and necessary elaboration — for which the reader is (...) referred to the bibliography. (shrink)

RésuméCet article est consacré à l’histoire de la théorie locale des courbes “à double courbure”. Initiée par Clairaut en 1731, cette théorie se développe en parallèle à la théorie des surfaces et trouve son achèvement avec les formules de Serret et Frenet et leur interprétation par Darboux, en 1887. Au delà de l’analyse des contributions de nombreux mathématiciens, parmi lesquels Monge bien sûr mais aussi Fourier, Lagrange et Cauchy, notre étude donne un regard particulier sur l’évolution conjointe de l’Analyse et (...) de la Géométrie, dans une longue période riche de nombreuses remises en cause théoriques. (shrink)

"The development of the calculus during the 17th century was successful in mathematical practice, but raised questions about the nature of infinitesimals: were they real or rather fictitious? This collection of essays, by scholars from Canada, the US, Germany, United Kingdom and Switzerland, gives a comprehensive study of the controversies over the nature and status of the infinitesimal. Aside from Leibniz, the scholars considered are Hobbes, Wallis, Newton, Bernoulli, Hermann, and Nieuwentijt. The collection also contains newly discovered marginalia of Leibniz (...) to the writings of Hobbes."--BOOK JACKET. (shrink)

This collection of essays examines the ways in which disputes and controversies about the application of scientific knowledge are resolved. Four concrete examples of public controversy are considered in detail: the efficacy of Laetrile, the classification of homosexuality as a disease, the setting of safety standards in the workplace, and the utility of nuclear energy as a source of power. The essays in this volume show that debates about these cases are not confined to matters of empirical fact. Rather, as (...) is seen with most scientific and technical controversies, they focus on and are structured by complex ethical, economic, and political interests. Drs. Engelhardt and Caplan have brought together a distinguished group of scholars from the sciences and humanities, who sketch a theory of scientific controversy and attempt to provide recommendations about the ways in which both scientists and the public ought to seek more informed resolutions of highly contentious issues in science and technology. Scientific Controversies is offered as a contribution to the better understanding of the roles of both science and nonscientific interests in disputes and controversies pertaining to science and technology. (shrink)

ABSTRACT 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)

Controversies are pervasive in the history of science. History is thus here also at odds with science's images. According to both traditional and contemporary views of science, there are no scientific controversies sui generis. In traditional images of science controversies are external to science proper; in some contemporary views nothing about controversies in science specifically distinguishes them from controversies in other domains. According to one traditional image, science progresses from common ground to conclusions according to secure procedures such that there (...) is no place for disagreement nor, therefore, for controversy. According to another traditional image of science, there are no such secure procedures. On the contrary: one does and even should jump to conclusions. Whereas here subsequent criticism is called for, it seems that controversy is not. This image is built on the assumption that refutations are clear cut, so that producing a valid refutation does not require a lengthy exchange of arguments. (shrink)