Throughout the history of human thought, the nature of infinite and infinitesimal quantities has been investigated by the best minds, and their actual use in calculus led to the modern development of mathematics. However, in the first years of the last century, the infinitesimals were banned from the world of mathematics, as their use was considered contradictory. It was only in the 1960s, when Robinson introduced nonstandard analysis, that the use of infinitesimal and infinite numbers in calculus was finally placed on rigorous foundations.

In this talk, I will present some historical observations on infinitesimal numbers, and then introduce the fundamental ideas of nonstandard analysis. In the final part of the seminar, I will also show some applications of nonstandard analysis in the discrete field of combinatorics, presenting new results recently proven in arithmetic Ramsey theory, including the following: In every finite coloring (partition) of the natural numbers, one can always find a monochromatic pattern of the form a, b, a+b, b/a.