In the 1800’s the problems were solved by banishing infinitesimals and defining derivatives by limits. But Leibniz and his followers were unable to construct a self-consistent extension of the real numbers, and the arguments persisted for more than a century. It follows that the reciprocal of an infinitesimal (that is, 1 divided by it) must be larger than any real number, and therefore infinite. These were quantities that are smaller than any real number, yet larger than zero. Leibniz investigated the idea of extending the usual number system (what we call the real numbers) to a larger set containing infinitesimals. Are infinitesimals finite, and so non-zero, or vanishingly small, and so zero? He mocked their use, describing them as “the ghosts of departed quantities” (see references to Berkeley below). He compared the mathematical use of infinitesimals to arguments in theology. Perhaps the most vitriolic criticism came from George Berkeley, who was at one time Bishop of Cloyne. Their work gave rise to heated arguments about infinitesimal quantities. But how small is small?Ĭalculus was devised independently by Isaac Newton and Gottfried Wilhelm Leibniz. It is given by the difference in position (a length) divided by the small interval between measurements (a time). The speed is then the rate of change of this function and is called the derivative of the function. If the position of the car is given at each moment, we say it is a function of time. The branch of mathematics that deals with changes is calculus or, more grandly, analysis. The instantaneous speed requires that the measurement interval is vanishingly small.īishop George Berkeley, and his polemic, The Analyst. But, strictly speaking, this is an approximation: what we have is the average speed for the second over which it is measured. Suppose the car moves twenty metres in that second then we say the speed is twenty metres per second. To get the speed at any instant, we measure the change of position over a small interval of time, say one second. If a car is accelerating, its speed is changing from moment to moment. However, this circumvented rather than solved the problem of infinitesimals. There was no rigorous theory available until the nineteenth century when Cauchy, Bolzano, Weierstrass and others made the concept of a limit precise. Euler manipulated them with his usual flair, but lesser mortals stumbled into contradiction and paradox. The debate about infinitesimals rumbled on for centuries after they were used by Leibniz. Are there infinitesimal numbers lurking amidst the real numbers? And if we admit infinitely small quantities, how do we deal with their multiplicative inverses, which must be infinite? Serious challenges arise when we try to answer these questions. But many people have a sneaking suspicion that there is “something” between the number with all those 9’s after the 2 and the number 3, that is not quite “reached” in the limit. Abraham Robinson (1918-1974) and his book, first published in 1966.Ī few weeks ago, I wrote about Hyperreals and Nonstandard Analysis , promising to revisit the topic.
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