The critical analysis of the foundations of the standard theory of harmonic oscillations is proposed. The unity of formal logic and rational dialectics is methodological basis of the analysis. The analysis leads to the conclusion that this theory represents gross error. The substantiation (validation) of this statement is the following main results. I. In the case of the material point suspended on the elastic spring, the linear differential equation of harmonic oscillations is the equation (condition) of balance of Newton’s force (Newton’s second law) and "Hooke’s force" ("Hooke’s law" as pseudolaw). This equation contains the following gross methodological errors: (a) the differential equation of motion of the material point does not satisfy the dialectical principle of the unity of the qualitative and quantitative determinacy of physical quantities (i.e., Newton’s force and Hooke’s force). In other words, the left and right sides of the differential equation (i.e., the equation of balance of the forces) have no identical qualitative determinacy: the left side of the the equation of balance of the forces represents Newton’s force, and the right side of the the equation of balance of the forces represents the "Hooke’s force" (as pseudolaw); (b) the sum of Newton’s force and the "Hooke force" (as pseudolaw) in the the equation of balance of the forces is equal to zero. This means that the sum of the numerical values of Newton’s force and "Hooke’s force" (as pseudolaw) is equal to zero. Consequently, the numerical values of Newton’s force and "Hooke’s force" (as pseudolaw) are equal to zero in the region of neutral real numbers. This means that the equation of balance of the forces is incorrect; (c) "Hooke’s force" (as pseudolaw) in the equation of balance of the forces represents the product of the spring constant (coefficient of stiffness of the spring) and the coordinate of the material point. In this case, "Hooke’s force" (as pseudolaw) does not represent Hooke’s law. "Hooke’s force" (as pseudolaw) contradicts to Hooke’s law because the coordinate of rhe material point does not determine the spring constant (coefficient of stiffness of the spring). "Hooke’s force" (as pseudolaw) has the dimension of Newton’s force. But, as the practice of measurement of Hooke’s force with the help of a dynamometer shows, the dynamometer readings are real neutral numbers with the dimension "kilogram-force"...(Truncated by xiXra Admin)