The problem of the bases of mathematics directly mentions questions of existence and the ontological status of mathematical object and is defined by the content of two basic contradictory methodological conceptions - realism and constructivism. It is shown that at the heart of their opposition dialectic relations of categorial pairs «individual?general», «subject?object», «essence?phenomenon» lie. Ways of overcoming of the arisen contradictions on the basis of ideas of modern schools of constructive realism, structuralism and naturalistic constructivism are considered.