![]() ![]() To be an infinitesimal, as far as Leibnitz is concerned, is to be constructed in a certain way by a certain series of changes in another function, but it says nothing of the quantity itself. Infinitesimal to him, therefore, as far as his published work is concerned, refers to any quantity always decreasing, in the same way as Hinchin in his 1950's calculus textbook presented the matter. Obviously, not all infinitesimals are identical. And because the angle of the geodesic where it intersects each side of the triangle is NOT in general the same, the sides of the triangle constructed are NOT, except in a special case of the construction, equal. Both denominator and numerator become smaller as the geodesic approaches, but never become null. He defines each infinitesimal as a fraction constructed by such motion. Whatever can be constructed by such a motion is an infinitesimal, he says. In this case: the denominator cannot be zero, if the system is to be consistent, so the motion of the geodesic is restricted in that it cannot touch the vertex. But he defines a fraction where the numbers are lengths of the sides of the triangle so constructed. He moves the geodesic continuously toward a vertex of the triangle, so creating another triangle whose sides get shorter and converge to zero (if the geodesic touches the vertex). (But he treats them inconsistently across his entire body of work.)īasically, he constructs a right triangle in cartesian coordinates, and intersects it with a geodesic in two places. Perhaps this may be taken as his "official" opinion, seeing that he published it. Leibnitz in a published short paper, on what infinitesimals are, constructs them in a similar (but not always quite the same) way. ![]() So it's arbitrarily small but never null. Wallis in his book on integration, which is prior both Newton and Leibnitz, uses the concept of a right hand limit of a quantity that goes to zero. (Not surprising, since most of the papers are his notes, written "for the desk", not intended by him for publication.) ![]() My impression is that Leibnitz usually offers several approaches that are alternatives, and are not consistent. HOPOS (Journal of the International Society for the History of Philosophy of Science) just published our rebuttal of syncategorematist theories that seek to sweep Leibnizian infinitesimals under a Weierstrassian rug. Would it be accurate to affirm that this is the dominant interpretation as far as Leibnizian infinitesimals are concerned? I was wondering about the current status of Ishiguro's interpretation of Leibnizian infinitesimals, among Leibniz scholars. Ishiguro's interpretation usually goes under the name "syncategorematic". This involves intepreting infinitesimals as non-designating terms, which correspond to propositions quantified over ordinary Archimedean magnitudes. Here she presents an interpretation of Leibniz's infinitesimals in the spirit of a conceptual framework developed by Russell, involving the so-called "logical fictions". ![]() Of particular interest, as far as the history of mathematics is concerned, is her Chapter 5. Japanese scholar Hide Ishiguro published a book in 1990 entitled "Leibniz's philosophy of logic and language" (second edition). ![]()
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