The book puts forward the postulate of the functional asymmetry of nature, formed by two universal formation processes - compression and expansion, continuity and discontinuity. The duality of its fractal geometry is substantiated, which consists in the duality of their images - depending on the projection, either continuous sets of full measure or completely discontinuous zero-dimensional sets are obtained. It turns out the main systemic pair of oppositions "material - ideal". Its correlate in science is the “physics - informatics” pair. As a formal analogue of this duality, a model of numerical asymmetry is considered - the union of real R and p-adic numbers Qp into a single self-dual system. It is shown that it logically connects various mathematical results on duality, which are consistent with the binary nature of natural sciences and the phenomenology of general systems theory. Zeno's paradoxes are considered from the point of view of applications of mathematics - as a test for its adequacy to natural sciences. A unified interpretation of all known paradoxes from the point of view of numerical asymmetry is proposed. The possibilities of harmonizing mathematical concepts with the basic concepts of language, biology, consciousness, physics and religious worldview are considered. The book is addressed to applied mathematicians, all researchers who apply mahematics and systems ideas in their work.