![]() ![]() For them analogies with current concepts may be helpful the key here is to remember that they are analogies. Who is the historian writing for, after all? Not Leibniz or his contemporaries, but for “us”, one’s egocentrically defined audience of fellow specialists and (one hopes) fellow philosophers. Like cicerones of every sort, the historian needs to be flexible. To describe the actual rank and function of Agamemnon would require explanation one could translate basileus as “priest/judge/warrior having supreme authority in all things” (following Aristotle Politics iii.15) but that would be cumbersome and still not exact. But clearly when Homer refers to him as King he doesn’t have in mind the titular head of a constitutional monarchy. Infinitesimal Differences makes compulsory reading for anyone interested in how Leibniz defined and manipulated infinitesimals as geometrical, algebraic, and physical magnitudes, and how Leibniz's ideas on infinitesimals were received-and often misunderstood-by his contemporaries, including. ![]() 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.Agamemnon is King (for so we translate basileus) by virtue of his similarity with the Kings of England (the Anglo-Saxon kings in fact called themselves “basileus”). As a philosopher, Leibniz thought deeply about the nature of the infinite and the infinitesimal. Leibniz infinitesimals free#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. In the engineer’s practical treatment of the differential calculus, an infinitesimal is a number so small that its square and all higher powers can be neglected. 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. 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. ![]() 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. ![]() Many historians of the calculus deny significant continuity between infinitesimal calculus of the seventeenth century and twentieth century developments such as Robinson's theory. In this paper I offer a defence of Leibnizs interpretation of infinitesimals as fictions, arguing that with it Leibniz provides a sound foundation for his. ![]()
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