METHODOLOGICAL ANALYSIS OF THE PROBLEMS OF APPLICABILITY OF MATHEMATICS
V. M. Reznikov
Institute of Philosophy and Law of the RAS, st. Nikolaeva, 8, 630090, Novosibirsk, Russia
Keywords: адекватность математической дисциплины, теория вероятностей, математическая и прикладная статистика, философия и методология науки, adequacy of mathematical discipline, theory of probability, mathematical and applied statistics, methodology and philosophy of science
Abstract
The article is devoted to the problem of adequacy of mathematics. This problem is a significant and interdisciplinary one; it covers pure mathematics, applied mathematics, philosophy and methodology of science, pedagogics and etc. There exist purely philosophical approaches to the problem of adequacy of mathematics, which are primarily based on the thesis of indispensability of mathematics in the best and successful natural science theories. However, in our opinion, they have no value for real applications. Firstly, the vast majority of scientific disciplines are not formalized ones. Secondly, less than one percent of all scientific problems are solved on the basis of theories. Thirdly, even in physics the role of theories is restricted only to theoretical problems, while the explanation of concrete physical phenomena requires the use of empirical laws and models of physical theories. The work objective consists in carrying out a methodological analysis of real applications of applied mathematical statistics. The fact is that, in spite of massive application of mathematics in practically all branches of knowledge, modern formal apparatus does not fully conform to the correct application of mathematics. The known conditions of application of mathematics are not stringent, and the correct use of mathematics is rather an art than a science. The problems of correct and effective use of mathematics are topical, because they are important for scientific practice; however, they have not received sufficient attention in the philosophical and methodological literature. The basic idea of the work consists in the demonstration of effectiveness of methodological analysis in the context of real applications of mathematics. In the article, the requirements to application of a mathematical discipline are formulated on the basis of a suggested notion of a basic property of the mathematical discipline. Since the set of the requirements is not complete, a strategy is proposed directed on widening and refinement of the requirements of using mathematics. The strategy consists in defining properties of mathematical objects facilitating or preventing the usage of mathematical apparatus. As a result of this work, the notion of adequacy of mathematical discipline, in particular, of applied statistics is made more precise. The difficulties of application of applied statistics cause the appearance of new promising directions in statistics. All known potentials of applied mathematical statistics is, in some way, connected with the development of general and applied statistical ideas. However, the advancement of the statistical science in general requires improving the methodology of teaching the statistical disciplines and realizing the organizational modification of statistical bodies.
