In contrast with some recent theories of infinitesimals as non-Archimedean entities, Leibniz’s mature interpretation was fully in accord with the Archimedean Axiom: infinitesimals are fictions, whose treatment as entities incomparably smaller than finite quantities is justifiable wholly in terms of variable finite quantities that can be taken as small as desired, i.e. syncategorematically. In this paper I explain this syncategorematic interpretation, and how Leibniz used it to justify the calculus. I then compare it with the approach of Smooth Infinitesimal Analysis, (...) as propounded by John Bell. I find some salient differences, especially with regard to higher-order infinitesimals. I illustrate these differences by a consideration of how each approach might be applied to propositions of Newton’s Principia concerning the derivation of force laws for bodies orbiting in a circle and an ellipse. “If the Leibnizian calculus needs a rehabilitation because of too severe treatment by historians in the past half century, as Robinson suggests (1966, 250), I feel that the legitimate grounds for such a rehabilitation are to be found in the Leibnizian theory itself.”—(Bos 1974–1975, 82–83). (shrink)

In contrast with some recent theories of infinitesimals as non-Archimedean entities, Leibniz’s mature interpretation was fully in accord with the Archimedean Axiom: infinitesimals are fictions, whose treatment as entities incomparably smaller than finite quantities is justifiable wholly in terms of variable finite quantities that can be taken as small as desired, i.e. syncategorematically. In this paper I explain this syncategorematic interpretation, and how Leibniz used it to justify the calculus. I then compare it with the approach of Smooth Infinitesimal Analysis (...) (SIA), as propounded by John Bell. Despite many parallels between SIA and Leibniz’s approach —the non-punctiform nature of infinitesimals, their acting as parts of the continuum, the dependence on variables (as opposed to the static quantities of both Standard and Non-standard Analysis), the resolution of curves into infinitesided polygons, and the finessing of a commitment to the existence of infinitesimals— I find some salient differences, especially with regard to higher-order infinitesimals. These differences are illustrated by a consideration of how each approach might be applied to Newton’s Proposition 6 of the Principia, and the derivation from it of the v2/r law for the centripetal force on a body orbiting around a centre of force. It is found that while Leibniz’s syncategorematic approach is adequate to ground a Leibnizian version of the v2/r law for the “solicitation” ddr experienced by the orbiting body, there is no corresponding possibility for a derivation of the law by nilsquare infinitesimals; and while SIA can allow for second order differentials if nilcube infinitesimals are assumed, difficulties remain concerning the compatibility of nilcube infinitesimals with the principles of SIA, and in any case render the type of infinitesimal analysis adopted dependent on its applicability to the problem at hand. (shrink)

Arthur E. Lean’s irreverent and con­troversial essays represent the distillation of many ideas about education—ideas developed during most of a lifetime spent in and about schools. In the second edition of this popular work, to which he has added eight new essays, he presents his latest observations on current ele­ments and programs in education—such as the grading system, academic rank, the teaching process, assessment of edu­cational progress—concluding that many of them are not only unnecessary but actually harmful to the (...) very process which they are supposed to serve. Covering a broad spectrum of current issues in its criticism of American edu­cation at all levels, _And Merely Teach _is a book to be read by veteran educator, beginning teacher, and lay citizen alike. Though critical of the teaching profes­sion, Arthur Lean also offers constructive suggestions in the form of practical solu­tions that are urgently needed to save our ailing educational system. (shrink)

This work examines the nature of what philosophers call de re mental attitudes, paying close attention to the controversies over the nature of these and allied...

Le positivisme, de Comte à Carnap, a marqué une étape dans le développement d’une science spéciale en dehors des sujets traités jusque là par la philosophie. Cette communication montre que la « syntaxe logique » de Carnap comme la « synthèse subjective » de Comte est un hybride, — une tranche de la philosophie en train de devenir science, et essayant, à l’étape intermédiaire de son processus, d’être à la fois science et philosophie.