The Imaginary and the Real in Mathematics and Its Applications

The article analyzes the evolution of several mathematical concepts, focusing on the combination of aspects of the imaginary and the real in the development of these concepts and related applications. These are trigonometric functions, graph theory, hypercomplex numbers, Dirac delta function, and multidimensional spaces. Based on the examples considered, we built two models of the evolution of mathematical concepts. The first model appeared earlier. The starting point for its development is practical needs. The second model starts from abstract mathematical concepts. This model appeared only on the basis of the established mathematical culture of the scientific community. Both models have a “wave-spiral” character. These include loops and dead-end branches. Using the constructed models, the authors draw certain conclusions about the problems of modern mathematics in the aspect of its interaction with applied sciences, reveal the gap between the level of development of modern mathematical theory and the level of demand for mathematical ideas in the applied sciences, including economics and quality management theory. We consider computational intelligence, which includes mathematical and computer modeling, as an example of an organic combination of the imaginary and the real. We built a model of the evolution of a productive idea “From the idea to the product.” Computational intelligence, which uses machine learning and artificial intelligence, plays the main role in this model. We analyze in detail mathematical models and computational experiments in bioengineering. The basis of non-destructive computational experiments are hypothetical models that belong to the realm of the imaginary. In this case, a computational experiment is the only possible means that allows, for example, to simulate the process of tissue regeneration after injury, or to create an optimal form of a prosthesis, which already belongs to the realm of the real. Mathematical and computer modeling in this area led to the emergence of a new science - mechanobiology. Researchers predict a great future for this science against the backdrop of increasing convergence of various fields and methods of scientific knowledge.