Mathematics and Philosophy (on the Dialectical Development of Mathematics)

The interplay and synthesis of antithetic elements of cognitive activity of mathematicians is considered. The emphasis is on basic antithetic elements such as logic and intuition, generality and individuality, synthesis and construction. Besides, we perform a dialectical analysis of the appearance of new abstract concepts based directly on one's sensory experience or on abstract concepts that have already been used earlier in the solution of mathematical problems. All conclusions from Euclid and Diophantus to problems of foundations in the present state of the art are illustrated by examples.