Thus, Pourciau ( 2001) argues that Newton possessed a clear kinetic conception of limit similar to Cauchy’s, and cites Newton’s lucid statement to the effect that “Those ultimate ratios … are not actually ratios of ultimate quantities, but limits … which they can approach so closely that their difference is less than any given quantity…” See Newton ( 1946, p. The latter belief is contrary to a consensus of modern scholars. Three additional significant aspects of Berkeley’s criticism could be mentioned: (3) a belief in naive indivisibles (this a century after Cavalieri), i.e., a rejection of infinite divisibility that was already commonly accepted by mathematicians as early as Wallis and others (4) empiricism, i.e., a belief that a theoretical entity is only meaningful insofar as it has an empirical counterpart, or referent this belief ties in with Berkeley’s theory of perception which he identifies with a theory of knowledge (5) Berkeley’s belief that Newton’s attempt to escape a reliance on infinitesimals is futile (see the epigraph to this Sect. After all, if an infinitesimal is merely meant as shorthand for talking about relations among sets of real values, what is the point of the lingering doubts expressed by Leibniz as to the ontological legitimacy of infinitesimals? Certainly the absence of a concrete individual counterpart of the bald king is a closed and shut question, rather than being “open to question”. Leibniz’s observation that the metaphysical (i.e., ontological) status of infinitesimals is “open to question” should apparently have put to rest any suspicions as to their alleged syncategorematic nature. His status transitus is something between real values of the variable, on the one hand, and its limiting real value, on the other. But Leibniz clearly does not have real values in mind when he exploits the term status transitus. In Arthur and Levey’s interpretation, the infinitesimal “serves to reveal logical relations” by tacitly encoding a quantifier applied to ordinary real values. A syncategorematic expression serves to reveal logical relations among those parts of the sentence which are referential. Thus, the phrase ‘the present king of France is bald’ is a syncategorematic expression, in that it doesn’t refer to any concrete individual. Our argument strengthens the conception of modern infinitesimals as a development of Leibniz’s strategy of relating inassignable to assignable quantities by means of his transcendental law of homogeneity.Ī syncategorematic expression has no referential function. We show, moreover, that Leibniz’s system for differential calculus was free of logical fallacies. We argue that Leibniz’s defense of infinitesimals is more firmly grounded than Berkeley’s criticism thereof. Leibniz’s infinitesimals are fictions, not logical fictions, as Ishiguro proposed, but rather pure fictions, like imaginaries, which are not eliminable by some syncategorematic paraphrase. We argue that Robinson, among others, overestimates the force of Berkeley’s criticisms, by underestimating the mathematical and philosophical resources available to Leibniz. Inspite of his Leibnizian sympathies, Robinson regards Berkeley’s criticisms of the infinitesimal calculus as aptly demonstrating the inconsistency of reasoning with historical infinitesimal magnitudes. A notable exception is Robinson himself, whose identification with the Leibnizian tradition inspired Lakatos, Laugwitz, and others to consider the history of the infinitesimal in a more favorable light. Robinson’s hyperreals, while providing a consistent theory of infinitesimals, require the resources of modern logic thus many commentators are comfortable denying a historical continuity. It is continuous over a closed interval if it is continuous at every point in its interior and is continuous at its endpoints.Many historians of the calculus deny significant continuity between infinitesimal calculus of the seventeenth century and twentieth century developments such as Robinson’s theory. A function is continuous over an open interval if it is continuous at every point in the interval.Discontinuities may be classified as removable, jump, or infinite.For a function to be continuous at a point, it must be defined at that point, its limit must exist at the point, and the value of the function at that point must equal the value of the limit at that point.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |