Differential calculus: a gross error in mathematics

Author(s): Temur Z Kalanov

A detailed proof of the incorrectness of the foundations of the differential calculus is proposed. The correct methodological basis for the proof is the unity of formal logic and rational dialectics. The proof leads to the following irrefutable statement: differential calculus represents a gross error in mathematics and physics. The proof of this statement is based on the following irrefutable results:

1. The standard theory of infinitesimals and the theory oflimits underlying the differential calculus are grosserrors. The main error is that infinitesimal (infinitelydecreasing) quantities do not take on numerical valuesin the process of tending to zero. The number “zero” isnot a permissible value of infinitesimal quantity. Theconcepts of “infinitesimal quantity”, “movement”,“process of tendency”, and “limit of tendency” aremeaningless concepts in mathematics: they are notmathematical concepts because the mathematicalformalism does not contain movement (process);

2. The concepts of “increment of argument” and“increment of function” are the starting point of thedifferential calculus. The gross error is that theincrement of argument is not defined. An indefinite(undefined, uncertain, ambiguous, undetermined)increment of an argument is a meaningless quantity(concept);

3. The definition of the derivative of a function is a grosserror. The derivative is the limit of the ratio of thefunction increment to the argument increment under
the following conditions:
a)The argument increment is not equal to zero;
b)The increment of the argument tends to zero andreaches the value “zero”. In this case, thefollowing logical contradiction arises: theincrement of the argument is both not equal tozero and equal to zero;

4.The differentials of the argument and the function - asinfinitesimal quantities - do not take on numerical values.This means that the differentials of quantities have neither quantitative nor qualitative determinacy. In this case, thedifferentials of quantities are meaningless symbols. Thegeometric and physical interpretations of the derivative area gross error;

5.The definition of the total differential of a function of two(many) variables is a gross error because the definitioncontains a formal-logical contradiction, i.e. the definition asthe sum of partial differentials does not satisfy the formal-logical law of the lack (absence) of contradiction;

6.The theory of proportions completely refutes the theory ofdifferential calculus.
Thus, differential calculus does not satisfy the criterion of truth and is not correct scientific (mathematical) theory.

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