The Radical Mistakes in the Theories on Calculus of Cauchy-Lebesgue System

In this paper it is suggested that the history of calculus consists of three stages including the period of precalculus before I. Newton established his calculus system initially in 1667, the first period of history of calculus from 1667 until A. L. Cauchy published his monograph 'cours d'analyse' in 1821, the second period of history of calculus from 1821 until the novel calculus system was established in 2010 and the third one after 2010.

During the period of 189 years from 1821 to 2010, the theories on calculus developed in this second period has never been justified by the application of mathematics in science and technology, which only demonstrated the usefulness of calculus invented by I. Newton and G. W. Leibniz and developed by the subsequent mathematicians as before. In contrast, the history of 189 years has revealed that the theories on calculus in this second period have blocked the development of mathematics significantly.

In this paper the radical mistakes in the theories on calculus of Cauchy-Lebesgue system were shown and discussed. The preliminary achievements of novel theories on calculus established by the author have been published 1 .

The basic ideas of theories on calculus in the first period (i.e., Newton-Leibniz theories on calculus) are correct although they are not consistent.

, given by Leibniz, but he did not understand Leibniz’s explanation of the differentials dx and dy .

Leibniz said, ‘A differential is like the contact angle of Euclid, which is smaller than any given quantity, but not equal to zero.’ He also said, ‘We consider an infinitesimal quantity as a relative zero, not a simple zero or an absolute zero.’ It should be noted that a worldwide mistake exists that the differential is considered as an arbitrarily small quantity. Indeed, Leibniz indicated that the differential is not zero, or a finite quantity, not to mention infinity, but ‘a relative zero smaller than any given quantity’.

It is a new type of quantity since the concept of modern numbers hasn’t appeared in Leibniz’s time; while the concept of ’the relative zero’ existed in the concepts of all the numbers, including unnormalized numbers of Robinson, from the time of Cantor-Dedekind to 2010. However, Cauchy introduced the idea of the limit to calculus system because he wasn’t able to understand the ideas of Leibniz. Making use of the concept of the limit and the derivative to define the differential violates the original meaning of the differential from Leibniz.

Cauchy defined the differential as a finite quantity, which had changed Leibniz’s idea that the differential was defined as ‘a qualitative zero’, ‘smaller than any given quantity’. It seems that a derivative has been defined perfectly and the Newton-Leibniz equation has been proved by making use of the concept of limit, but there exist radical problems of equating dx and x ∆ when it comes to defining the differential, because the differential (expressed starting with the symbol ’d’) has been defined as the linear main-part of a change (expressed starting with the symbol ’ ∆ ‘) resulting in that dx is not equal to x ∆ . Formulas ( ) ( ) , respectively, without logic [2][3][4] ; the other one is that equations

n n dx x = ∆ are defined without logic, where n is equal to an arbitrarily finitely natural number in both cases [4][5][6][7][8] .

From a viewpoint of argument, Cauchy has defined the differential (e.g., dy and dx ) as the linear main-part of a change (e.g., y

∆ and x ∆ ), therefore, it is not appropriate to define the differential is equal to the change. On the other side, from the differentiable function One assumption is incorrect, thus the other one is also incorrect.

In a word, both of the two assumptions are wrong. From a viewpoint of counterargument, regarding generally differentiable functions

( , ,…, ,…, ) The two assumptions are the most representative in the Cauchy-Lebesgue theories on calculus, which violate the principles of science. However, the defenders of Cauchy-Lebesgue system said, ‘The subject of mathematics is just a formal system, not a physical one. The definition of the differential by Cauchy doesn’t deal with composite functions, therefore, the Cauchy-Lebesgue system is impeccable.’ However, in fact it is not true. Many cases in calculus are related to composite functions including the consistence of the expression of the differential 9 , the rules of derivation of composite functions, implicit functions, parametric equations and polar equations, the two types of Substitution Rule and the method of substitution of variables in differential equations.

Actually there exist mathematicians supporting Cauchy-Lebesgue theories on calculus, who have also discovered the mistakes. Some of them have done some corrections silently although the corrected system is still wrong 2,[9][10][11] . Some of them choose to avoid dealing with differentials 12 .

1. Cauchy has misunderstood the meaning of differentials given by I. N

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