But what about the method used to obtain it? It was severely criticized even in the seventeenth century. Of course the answer is correct since the derivative of x 2 is indeed 2 x. Finally, 2 x + e can be identified with 2 x since e is infinitesimally small compared to 2 x and can therefore be deleted. To the parabola can be identified with the line joining these two points. Is a point on the parabola infinitesimally close to ( x, x 2), hence the tangent line For example, to find the slope of the tangent line (later called the derivative) to the parabola, y = x 2 at the point (x, x 2), seventeenth-century mathematicians would argue as follows: They were indispensable in the calculus of the seventeenth, eighteenth, and early nineteenth centuries. This was also the case for infinitesimals-or differentials, as Gottfried Wilhelm Leibniz (1646-1716) called them. In each case, these numbers were introduced because they turned out to be useful. Thus, the integers were introduced so as to make sense of numbers such as -1, the real numbers to give meaning to numbers like √2, and the complex numbers to accommodate such numbers as √-1. For example, while the positive integers are prehistoric, the other number systems, such as the integers, rational numbers, real numbers, and complex numbers, arose over the centuries as human constructs. This idea of extending a mathematical system in order to obtain a desired property not already present is common and important in mathematics. To accommodate infinitesimals we must extend the real numbers. But such triangles do not exist (in Euclidean geometry)! In the case of infinitesimals, there are no real infinitesimals, since given any positive real number a, a/2 is a smaller positive real. For example, we can define an obtuse-angled triangle as a triangle all of whose angles are greater than 90 degrees. But merely defining a mathematical entity does not guarantee its existence. More precisely, it is a nonzero number smaller in absolute value than any positive real number. BackgroundĪn infinitesimal is an infinitely small number. Since that time, nonstandard analysis has had an important effect on several areas of mathematics as well as on mathematical physics and economics. This changed in 1960, when Abraham Robinson resurrected their use with his creation of nonstandard analysis. Between the mid-1800s and the mid-1900s, however, infinitesimals were excluded from calculus because they could not be rigorously established. cited Klein's racist distinctions within mathematics, and sharpened them into open antisemitism" fabricates a spurious continuity between the two figures mentioned and is thus an odious misrepresentation of Klein's position.The Resurrection of Infinitesimals: Abraham Robinson and Nonstandard Analysis Overviewįor centuries prior to 1800, infinitesimals-infinitely small numbers-were an indispensable tool in the calculus practiced by the great mathematicians of the age. Mehrtens' unsourced claim that Hilbert was interested in production rather than meaning appears to stem from Mehrtens' marxist leanings. Among Klein's credits is helping launch the career of Abraham Fraenkel, and advancing the careers of Sophus Lie, Emmy Noether, and Ernst Zermelo, all four surely of impeccable modernist credentials. Klein and Hilbert were equally interested in the axiomatisation of physics. Hilbert's views on intuition are closer to Klein's views than Mehrtens is willing to allow. Klein's Goettingen lecture and other texts shed light on Klein's modernism. We argue that Klein and Hilbert, both at Goettingen, were not adversaries but rather modernist allies in a bid to broaden the scope of mathematics beyond a narrow focus on arithmetized analysis as practiced by the Berlin school. Some of Mehrtens' conclusions have been picked up by both historians (Jeremy Gray) and mathematicians (Frank Quinn). Historian Herbert Mehrtens sought to portray the history of turn-of-the-century mathematics as a struggle of modern vs countermodern, led respectively by David Hilbert and Felix Klein.
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