The article describes A.N.Kolmogorov's views on the development of mathematical giftedness in children and young people analyzing reminiscences of his pupils, his own comments on the development of mathematical giftedness in different sources, his correspondence with a psychologist, expert on mathematical ability V.A.Krutetsky. In this contest of sorts between the psychologist and the mathematician it is Kolmogorov's ideas that are more in agreement with modern views on the nature of giftedness.