![]() ![]() “ As regards this proposition itself, I hold that it is one of the most general and useful in geometry, inasmuch as it is so general that it applies to all curves, even in the case where they are drawn at will without a specific law and given any figure it exhibits infinitely many others, the dimension of each of which depends on the former and vice versa. And it is the same for other things of that kind, which that most profound gentleman Joachim Jung called true within a tolerance, and which are of the greatest service to the art of discovery even though in my opinion they encompass something of the fictive and imaginary, this can nevertheless be rectified by a reduction to ordinary expressions so readily that no error can intervene.” (GM V 385). See the passage from Observationes quod rationes quoted above, or his 1713 letter to Wolff: “And this is in agreement with the Law of Continuity that I once proposed in Bayle’s Nouvelles Lettres, and applied to the Laws of motion: whence it happens that in continua an exclusive extreme point can be treated as inclusive, so that the ultimate case, even though of a wholly different nature, is subsumed under the general law of the other cases, and at the same time, by a certain paradoxical reasoning and, so to speak, Philosophico-rhetorical Figure, we can understand the point to be included in the line, rest in motion, the special case in the contradistinguished general case, as if the point were an infinitely small or evanescent line, or rest an evanescent motion. We then show how in his mature papers the latter strategy, now articulated as based on the Law of Continuity, is presented to critics of the calculus as being equally constitutive for the foundations of algebra and geometry and also as being provably rigorous according to the accepted standards in keeping with the Archimedean axiom. By a detailed analysis of Leibniz’s arguments in his De quadratura of 1675–1676, we show that Leibniz had already presented there two strategies for presenting infinitesimalist methods, one in which one uses finite quantities that can be made as small as necessary in order for the error to be smaller than can be assigned, and thus zero and another “direct” method in which the infinite and infinitely small are introduced by a fiction analogous to imaginary roots in algebra, and to points at infinity in projective geometry. Thus, one cannot infer the existence of infinitesimals from their successful use. Against this, we show that by 1676 Leibniz had already developed an interpretation from which he never wavered, according to which infinitesimals, like infinite wholes, cannot be regarded as existing because their concepts entail contradictions, even though they may be used as if they exist under certain specified conditions-a conception he later characterized as “syncategorematic”. Some authors claim that when Leibniz called them “fictions” in response to the criticisms of the calculus by Rolle and others at the turn of the century, he had in mind a different meaning of “fiction” than in his earlier work, involving a commitment to their existence as non-Archimedean elements of the continuum. In this paper, we endeavour to give a historically accurate presentation of how Leibniz understood his infinitesimals, and how he justified their use. ![]()
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |