Infinitesimálisok

Infinitesimálisok are quantities that are smaller than any positive real number but are not equal to zero. In traditional calculus, infinitesimals were used as the fundamental building blocks of differentiation and integration. For example, a derivative was thought of as the ratio of an infinitesimal change in y to an infinitesimal change in x, denoted as dy/dx. Similarly, an integral was conceived as a sum of infinitely many infinitesimally small quantities.

While infinitesimals provided an intuitive way to understand calculus, their existence and manipulation led to logical difficulties and paradoxes. Critics, most notably Bishop Berkeley, argued that infinitesimals were neither zero nor a positive quantity, making their use mathematically unsound. This led to the development of the rigorous epsilon-delta definition of limits by mathematicians like Augustin-Louis Cauchy and Karl Weierstrass in the 19th century. This new foundation allowed calculus to be developed without relying on the concept of infinitesimals, establishing a more solid logical framework.

However, the idea of infinitesimals has resurfaced in modern mathematics through the development of non-standard analysis by Abraham Robinson in the 1960s. Non-standard analysis provides a formal system where infinitesimals can be rigorously defined and used. This approach offers alternative proofs for many calculus theorems and can sometimes simplify complex arguments. Despite their historical controversy, infinitesimals continue to be a topic of interest in mathematical logic and the foundations of analysis.