Modern sciences, using logic as a tool of cognition, often come across theoretical contradictions or contradictions between the consequences of theory and the verbalized results of experience, experiment. This is due to logical errors in the construction of judgments, imperfection of currently existing scientific methods or insufficient accuracy of the instruments used in experiments, as well as inadequacy of the accepted idealization, that is, incorrect axiomatization of theories.
The presence of a paradox stimulates new research, a deeper understanding of the theory, its "obvious" postulates, and often leads to its complete revision.
Paradoxes reveal the deep currents of the cognitive process. Proclaiming about the imminent trouble in science, they at the same time decisively push it forward and precisely by bringing new, even more paradoxical ideas.
Examples of paradoxes in science are Russell's Paradox, Banach-Tarski's Paradox, Garai's Paradox, Smale's Paradox, Hausdorff's Paradox, EPR Paradox, Cosmological Paradoxes.
Paradoxes in science, on the one hand, denote a crisis of old scientific knowledge (which explains the negative attitude of many scientists to this phenomenon), on the other hand, they contribute to the development of a new one, this proves their usefulness and even a certain necessity. Here you can see the choice between peace and development. If science, as it should, prefers development, then scientists should take paradoxes for granted and try to use the knowledge they give, despite its unfamiliarity and contradiction to the established rules. It is this understanding of the paradox that is assumed to be optimal for the development of science at the present time.