Leibniz defined mathematics as the “science of possible worlds.” But what worlds were assumed as the possible ones by thinkers of different eras? It is this question that is considered in this article. According to Einstein, development of science requires “external justification” (observations and experiments that should be explained or predicted) and internal perfection (following the internal logic of this discipline). Mathematics has no “external justification” and experiment cannot refute mathematical theory. In this sense, mathematics is closer to creativity, to art than to natural science. Its connection with culture turns out to be more complex and mediated. To explain this connection, Daniel Bell, within the framework of his theory of social development, puts forward the “axial principle,” according to which the role of science is considered as the main characteristic of society. From this point of view, we single out traditional, industrial and post-industrial societies. Each of these phases has its own ideals, norms, and types of mathematical creativity. In traditional society, following the Pythagorean tradition, mathematics is focused on finding harmony in nature, on identifying unity on the basis of universal relationships determined by the numerical characteristics of the objects under study. As the industrial era approaches, the constructivist, or “project-based,” approach becomes increasingly important. And unity emerges at a higher “meta-level.” The forerunner of this direction is Descartes, who raised the question of finding a single, universal method for solving all mathematical problems. The work traces the change in the formulation of a number of “invariant,” “eternal” mathematical problems as well as the evolution of the concept of “complexity” in the historical retrospective. Close attention is paid to the post-industrial phase of civilizational development and to “computer mathematics,” which has become the basis for formation of virtual reality, which in many respects changes the very direction of progress. As a result of this, the “extraverted orientation” of humanity, the course toward new horizons was replaced by the “introvert” one, which prioritizes tasks associated with comfort, convenience, and consumption. The “change of milestones” that has taken place is traced in a comparison of “big projects” related to mathematics that were put forward in the 1960s, and those that are considered as priorities at present. In fact, we are faced with a “crisis of expectations.” We see a way out of this crisis in a revival of the “Pythagorean tradition,” at a new level. But if in a traditional society the goal of the development of mathematics in alliance with other sciences and arts was to reveal harmony in the natural world, then at the post-industrial phase, the priorities are different. They are associated with computer modeling, understanding, and identifying the foundations of harmony in the human world.