On physical limitations of mathematical constructions used in mathematical models

摘要

Physical limitations of mathematical constructions are discussed, which should be taken into account in developing or modifying mathematical models. We begin with consideration of the method of describing physical objects using numbers and restrictions followed from this method. Next, we formulate some general recommendations concerning procedures for modifying mathematical models. Since models of physical phenomena are considered, it is natural to provide a physical interpretation for each stage of the model development. Unfortunately, some of transformations used are treated as purely technical tricks, and therefore the question of the physical meaning is not raised in such cases. The lack of physical meaning of some mathematical procedures does not make them unambiguously unacceptable. However, this marks out the place that requires a reasonable interpretation because the final result should possess the physical meaning. Finally, we discuss the issues related to the dimensionality of the space of places of a model. The above-mentioned physical limitations often are left without necessary attention. Sometimes this leads to various undesirable consequences, which may include excessive complication of the problem, an implicit substitution of the declared problem with another one or, finally, the absence of solution of the formulated problem.