Definition of Derivative Function: Logical Error in Mathematics

The critical analysis of the foundations of the differential calculus is proposed. Methodological basis of the analysis is the unity of formal logic and of rational dialectics. It is shown that differential calculus is fictitious mathematical theory because the concept of the limiting process is the starting point for definition of the derivative function. The passage to the limit “zero” in the definition of the derivative function signifies that the variable quantity takes the only essential value “zero”. This fact leads to the following errors. (1) The definition of the derivative function is based on the violation of the necessary and sufficient condition for the validity of the relationship between the increment of the function argument and the increment of the function because the increment of the function is divided by the zero increment of the argument in the case of the limiting process. (2) The definition of the derivative function is based on the contradiction which is that the increment of the argument is both zero and not zero in the same relationship. This contradiction represents a violation of the formal-logical law of identity and of the formallogical law of the lack of contradiction. (3) The definition of the differential of function is based on two contradictory (mutually exclusive) features: the differential of the argument is not zero while the increment of the argument in the definition of the derivative function is zero.

The purpose of this work is to propose the critical analysis of the foundations of differential calculus within the framework of methodological basisthe unity of formal logic and of rational dialectics. The critical analysis is based on the dialectical principle of functional connection and of movement.

The Principle of Functional Connection and of Movement
Movement is Change in General In other words, movement is a change in state. The principle of movement (change) is a theoretical generalization of practice and represents a concretization of the laws of dialectics and formal logic. The principle of functional connection and of movement in mathematics is formulated as follows. a) If the continuous function y of one argument x is given, then the function   c) The change in the numerical values of the function y is characterized by the increment y  of the function.
The quantity y  takes certain numerical values. The definition of the function increment is the following: the function increment is the difference of two numerical values of the function. Therefore, the dimension of the quantity y  is identical to the dimension of the quantity y . e) The coefficient k of the relative increment (i.e., the ratio of the quantity of the increment of the function to the quantity of the increment of the argument) is defined by the following relationship: These algebraic relationships express arithmetic relationships between numbers. k) Proportion is the only correct relationship between changes in the values of the argument and of the function:

Definition of the Derivative Function
As is known [15,16] represents the continuous function y of one argument x , then the derivative function is defined as follows: are the increments of the argument and of the function, respectively. As is known [15,16], are the differentials of the argument and of the function, respectively. The differential y d is a function of two variable quantities x and x d which are independent of each other.
The essence of the concept of the derivative function becomes the apparent (obvious, evident, certain) fact in the following example.
Example x , then one obtains the following equality: Thus, the above example discovers (ascertains, reveals, detects) a formal-logical error in differential calculus.

Logical Errors in the Definition of the Derivative Function
To understand the essence (nature) of the error in the definition of the derivative function, one must know how the computer performs the calculations. As is known, a programmer and a computer perform the concretization of mathematical (quantitative) relationships expressed in terms of letters and symbols of operations. The computer cannot perform, for example, the operation of addition But in order to analyze and to understand why this condition is not satisfied in the definition of the derivative function, one must know the formal logic. The formal-logical analysis is not accessible to a computer because a computer cannot operate with concepts. Formal logic (as the science of the laws of correct thinking) operates with concepts and is accessible only to man.
The formal-logical errors in the definition of the derivative function are as follows.
1) In accordance with the law of identity, the object x  of thought must be identical with itself in the process of reasoning: 2) In accordance with the law of lack of contradiction, it is not permitted that the same object of thought contains two contradictory features at the same time, in the same sense or in the same relation. But . This signifies that movement (or the cause of movement) does not exist (i.e., 0   x ), but such a feature (property) of movement as movement speed (i.e., derivative) exists. This is physical absurdity. Newton probably did not understand that the properties (speed, acceleration) of motion do not exist if motion does not exist. Thus, the absurdity in the form of differential and integral calculus entered in mathematics. The absurdity took an elegant form (shape) thanks to the canon of differential calculus which was created by logician G. Leibniz. (For the first time, Leibniz's canon was published in the journal Acta Eruditorum, Leipzig, 1684). But Leibniz could not find, understand, and detect Newton's logical errors. 2. Today, mathematicians and physicists all over the world use differential and integral calculus. They believe in the correctness, firmness, and inviolability of this theory. Therefore, scientists do not work for mastery of the correct methodological basis of science: the unity of formal logic and of rational dialectics. The unity of formal logic and of rational dialectics is also a criterion of truth. But errors in science (for example, physics) often arise because of the existence of methodological errors in mathematics and the "mindless, thoughtless application of mathematics" (L. Boltzmann). Is there "problem of existence of science for science" today? As the history of science shows, scientists are in no hurry to cast doubts on old theories within the framework of the correct criterion of truth because they are afraid to loss prestige and well-being.

Conclusion
Thus, the critical analysis of the foundations of differential calculus, carried out within the framework of the correct methodological basis, leads to the following statements: 1) If the continuous function of one argument is given, then this function is a mathematical (quantitative) representation of the dialectical principle of the functional connection. The dialectical principle of the quantitative change in the functional connection is that a change (increment) in the argument leads to a change (increment) in the function. 2) The necessary and sufficient condition for the validity of the relation between the increment of the function argument and the increment of the function is that the increment of the argument must be non-zero in all cases. But this condition is violated in determination of the derivative function: in the case of the passage to limit "zero", the increment of the function is divided by the zero increment of the argument.
3) The definition of the derivative function contains the contradiction which is that the increment of the argument is both zero and non-zero in the same relationship. This contradiction represents a violation of the formal-logical law of identity and the formal-logical law of the lack of contradiction. 4) In accordance with the formally-logical law of the lack of contradiction, one and the same object of thought should not contain two contradictory features at the same time, in the same sense or in the same relation. But the definition of the differential of function contains two contradictory (mutually exclusive) features that cannot belong to the same relationship: the differential of the argument is not zero, but the increment of the argument is zero in the definition of the derivative function. Thus, differential calculus is a fallacious mathematical theory because it contains formal-logical errors.