Differentiation Of Quantities Research Articles

Differential Calculus: a gross error in mathematics ∗

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 of limits underlying the differential calculus are gross errors. The main error is that infinitesimal (infinitely decreasing) quantities do not take on numerical values in the process of tending to zero. The number “zero” is not a permissible value of infinitesimal quantity. The concepts of “infinitesimal quantity”, “movement”, “process of tendency”, and “limit of tendency” are meaningless concepts in mathematics: they are not mathematical concepts because the mathematical formalism does not contain movement (process); (2) the concepts of “increment of argument” and “increment of function” are the starting point of the differential calculus. The gross error is that the increment 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 gross error. The derivative is the limit of the ratio of the function 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 and reaches the value “zero”. In this case, the following logical contradiction arises: the increment of the argument is both not equal to zero and equal to zero; (4) the differentials of the argument and the function - as infinitesimal quantities - do not take on numerical values. This means that the differentials of quantities have neither quantitative nor qualitative determinacy. In this case, the differentials of quantities are meaningless symbols. The geometric and physical interpretations of the derivative are a gross error; (5) the definition of the total differential of a function of two (many) variables is a gross error because the definition contains a formal-logical contradiction, i.e. the definition as the 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 of differential calculus. Thus, differential calculus does not satisfy the criterion of truth and is not correct scien tific (mathematical) theory.

Opportunities for recognising, assessing, and providing online support among elementary school students with dyscalculia

Mathematics is present throughout our lives. Already in infancy, children become acquainted with quantities and the differentiation of quantities (Starkey & Cooper, 1995; Huntley et al., 2000; Csépe et al., 2007, 2008). According to von Aster and Shalev (2007), the quantitative core system (cardinality) develops in infancy. An infant is capable of subitising, estimating, and comparing the quantities of a small number of items. After learning from their environment in early childhood, children perceive quantities and then understand the strategy needed for counting and are able to count verbally. They use different classifications to determine the quantities. Kindergarten prepares the knowledge needed for the application of the concept of numbers and lays the foundations for the concept of operations. As a continuation of knowledge acquisition of knowledge, in primary school the establishment of a unified and broad foundation takes place with a focus on the development and consolidation of the concept of numbers and operations. Number words representing quantities are associated with digits, therefore school children express them with symbols without the presence of quantities (von Aster & Shalev, 2007; Jármi, 2013, p. 50). This developmental process allows for the learning of written operation performance in the lower grades, while the internal number line (analogous quantity representation) develops in the child, which strengthens spatial thinking. In the initial stage of the lower school grades, active knowledge acquisition dominates, which is the first level of abstraction of mathematical activities. In the grades that follow, the constantly and spirally repeating, expanding nature of knowledge strengthens the fourth level of abstraction, i.e., the independent application of symbols.