infinitésimales

Infinitesimals are quantities that are smaller than any positive real number but not equal to zero. The concept of infinitesimals has a long and complex history in mathematics. Initially, they were used intuitively by mathematicians like Isaac Newton and Gottfried Wilhelm Leibniz in the development of calculus. They treated infinitesimals as infinitely small quantities that could be used in calculations involving limits and derivatives.

However, the lack of a rigorous definition for infinitesimals led to philosophical and mathematical challenges. Critics, such as Bishop Berkeley, argued that the use of infinitesimals was logically inconsistent, referring to them as "ghosts of departed quantities." This criticism spurred mathematicians to find a more solid foundation for calculus.

The development of the epsilon-delta definition of limits by Augustin-Louis Cauchy and Karl Weierstrass in the 19th century provided a rigorous way to handle concepts like continuity, derivatives, and integrals without explicitly using infinitesimals. This approach, based on real numbers, became the standard in mainstream calculus.

In the 20th century, Abraham Robinson developed a formal system called non-standard analysis. This system provides a rigorous framework for working with infinitesimals and infinite numbers. Non-standard analysis allows for the construction of an "enlarged" number system that includes both standard real numbers and these infinitesimals. While non-standard analysis is a powerful tool, it has not replaced the traditional epsilon-delta approach in most introductory calculus courses.