infinitesimaalseid

Infinitesimaalseid, often translated as infinitesimals, are quantities that are infinitely small but not zero. Historically, they were a foundational concept in the development of calculus, particularly by Gottfried Wilhelm Leibniz. Infinitesimals were used to represent the "infinitely small changes" in variables, such as dy and dx, which were then used in the process of differentiation and integration.

The intuitive use of infinitesimals allowed mathematicians to perform calculations that seemed to work, even though the rigorous justification was lacking. This led to a period of criticism, most famously by Bishop Berkeley, who argued that the reasoning involving infinitesimals was logically inconsistent, likening it to the ghosts of departed quantities.

In modern mathematics, the concept of infinitesimals has been rigorously formalized. Non-standard analysis, developed by Abraham Robinson in the 1960s, provides a framework where infinitesimals can be treated as actual numbers within an extended real number system. In this system, an infinitesimal is a non-zero number whose absolute value is less than any positive real number. This formalization resolved the logical paradoxes associated with the historical use of infinitesimals and solidified their place in mathematical analysis. While standard calculus based on limits is more commonly taught today, the underlying principles of infinitesimals are still relevant in advanced mathematical fields.