In this paper we discuss a number of issues related to the development of modern science, first of all mathematics and physics (typological classification of scientists, the phenomenon of «sleeping beauties», the role of analogy in exact sciences, bureaucratic pressure on the science, etc.). The description of different types of researchers was considered by the participants of research for a quite long time. The most well-known division of the “classics” and “romantics” proposed by the German physicist and chemist Ostwald. Dyson proposed his own division into "birds" and "frogs." We will offer our own version of the classification with the important addition that such types are rarely found in absolutely pure form and usually each scientist is a “mixture” of such types. The concept of “sleeping beauties” recently introduced by science historians refers to that long-known phenomenon in the history of science when the same result, often in the same form, is opened independently by several researchers separated from each other by both time and space. On a whole number of examples we will show the fundamental, in our opinion, role played by the revealing of analogies in the process of mathematical work. It occurs first at the level of intuition, and only then is line up as a necessary logical reasoning. The unprecedented and utterly illiterate intervention of the bureaucracy in the work of scientists, which penetrates modern science with incredible speed, gives grounds for rather pessimistic conclusions about its future. These considerations reflect a fairly long personal experience of the author, working in mathematics and interested in its philosophical foundations. The texts under consideration by F. Dyson and I. R. Shafarevich, in various ways addressing the issues mentioned above, serve as a starting point for presentation of the author’s own thoughts.