Vector Calculus and Maxwell's Equations: Logic Errors in Mathematics and Electrodynamics

The critical analysis of the foundations of vector calculus and classical electrodynamics is proposed. Methodological basis of the analysis is the unity of formal logic and rational dialectics. The main results are the following statements: (1) a vector is a property of the motion and of the interaction of material objects, i.e. the concept of a vector is the concept of a physical property. Therefore, the concept of a vector is a general and abstract concept; (2) a vector is depicted in the form of an arrow (i.e., “straight-line segment with arrowhead”) in a real (material) coordinate system. A vector drawn (depicted) in a coordinate system does not have the measure “meter”. Therefore, a vector is a pseudo-geometric figure in a coordinate system. A vector is an imaginary (fictitious) geometric figure; (3) geometrical constructions containing vectors (as pseudo-geometric figures) and vector operations in a coordinate system are fictitious actions; (4) the scalar and vector products of vectors represent absurd because vectors (as abstract concepts, as fictional geometric figures that have different measures) cannot intersect at the material point of the coordinate system; (5) the concepts of gradient, divergence, and rotor as the basic concepts of vector analysis are a consequence of the main mathematical error in the foundations of differential and integral calculus. This error is that the definition of the derivative function contains the inadmissible operation: the division by zero; (6) Maxwell’s equations – the main content of classical electrodynamics – are based on vector calculus. This is the first blunder in the foundations of electrodynamics. The second blunder is the methodological errors because Maxwell’s equations contradict to the following points: (a) the dialectical definition of the concept of measure; (b) the formal-logical law of identity and the law of lack of contradiction. The logical contradiction is that the left and right sides of the equations do not have identical measures (i.e., the sides do not have identical qualitative determinacy).

Thus, vector calculus and classical electrodynamics represent false theories. 

Keywords: general mathematics, pure mathematics, philosophy of mathematics, methodology of mathematics, general applied mathematics, dimensional analysis, Euclidean and projective geometries, vector, vector calculus, vector fields, history of mathematics, higher education; general physics, theoretical physics, mathematical physics, classical electromagnetism, classical electrodynamics, Maxwell equations, applied classical electromagnetism, engineering, history of science; formal logic, dialectics, epistemology, philosophy of science.

MSC: 00A05, 00A30, 00A35, 00A69, 00A73, 00A79, 03A05, 03A10, 03B30, 03B42, 35Q61, 37C10, 51A05, 53A45, 97A30, 97B40, 97G70, 97M50, 51P05.

PACS: 41.20.-q, 41.20.Jb, 41.90.+e, 01.40.-d, 01.55.+b, 01.65.+g, 01.70.+w, 01.90.+g, 02.40.Dr, 03.50.-z, 03.50.De, 03.65.Ca.