02 Vanishing Ghosts: The Berkeley Paradox
After Newton and Leibniz invented the general method of calculus, many mathematicians applied this method in many fields with good results, which further strengthened the confidence of the mathematical community in the calculus method.
But at this time, a philosopher came forward and accused Newton and Leibniz of fundamental flaws in their calculus theories. He was the Bishop of Ireland, Berkeley. At this time, the "second mathematical crisis" broke out.
Berkeley is a famous philosopher whose philosophical view is that "to be is to be perceived", that matter is nothing, and that the so-called material entity is nothing more than an abstract concept that does not exist. Berkeley later became a clergyman who tried to reconcile the sharp contradictions between religion and science, establishing a new theoretical basis for theology, and embracing both theology and science. In 1734, while serving as Bishop of Croin, Berkeley made a strong critique of Newton's and Leibniz's calculus methods in The Analyst.
Berkeley first pointed out that a series of concepts in calculus, such as the number of streams, instantaneous, vanishing amounts, initial and final ratios, infinitesimal increments, instantaneous velocity, etc., are quite vague. For example, for instantaneous velocity, Berkeley argues that since velocity is inseparable from space and time intervals, it is impossible to imagine an instantaneous velocity with zero time. And for infinitesimals, it is necessary to suppose that there are some infinitesimal quantities, and that there are infinitesimal quantities that are smaller than them, and that the smallest finite quantities can never be obtained after infinite multiplication. He considers these claims to be rather unreasonable and absurd.
Second, Berkeley pointed out some problems in the calculus method. For example, in Newton's flow arithmetic, some small increments of change are sometimes zero and sometimes non-zero in the derivation of the formula, which is quite flexible, which is completely unrigorous. Berkeley sneered, "What are these vanishing increments? They are neither finite, nor infinitesimal, nor zero, and can we not call them ghosts of vanishing quantities?" Leibniz's method had a similar problem, and Berkeley argued that Leibniz's approach of "ignoring higher-order infinitesimal quantities to eliminate errors" was based on the principle of error, and through "false offsets" he arrived at the conclusion he wanted. Berkeley listed 67 questions in one go, which can be described as a knife to the point.
In Berkeley's view, there is no essential difference between mathematicians and theologians, and they both hold the same faith. He wrote: "Do the mathematicians who are cautious about religious doctrine have the same rigor in their own science?
The essence of Berkeley's questioning is whether infinitesimal quantities are actually zero in calculus, because in Newton's and Leibniz's methods, infinitesimal quantities are sometimes zero and sometimes non-zero. Because of the great usefulness of calculus, mathematicians generally supported it, so Berkeley's doubts were called the "Berkeley paradox".
Similar to the resolution of the first mathematical crisis, calculus has become the driving force behind the continuous progress of mathematics.
The above is an excerpt from Wu Hanxin's "Calculation"! Transferred from the blog point of view, [Meet Mathematics] has been forwarded with permission.
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