Platonism in mathematics

The article discusses the origin and justification of the term "mathematical Platonism" in the research program of the modern philosophy of mathematics, originated with Frege. It is shown that intuitive understanding of mathematical knowledge as a description of mathematical reality, independent of human consciousness, adopted by community of working mathematicians is consistent with a number of aspects of Plato's philosophy, in particular his concept of the existence of extrasensory world of ideal objects. I discuss various objections to mathematical Platonism, in particular nominalistic view of mathematics and the problem of epistemic access to abstract objects of mathematics. As a response to these objections I consider the radical kind of Platonism, the so-called “Full-blooded Platonism” that expands the existence of mathematical objects to potentially viable entities that largely change the original Frege’s research program.