Infinitesimaalsete

Infinitesimaalsete refers to an archaic concept in mathematics dealing with quantities that are infinitesimally small, meaning they are smaller than any positive real number but not zero. This idea was central to the development of calculus by Isaac Newton and Gottfried Wilhelm Leibniz. They used infinitesimals to represent instantaneous rates of change and areas under curves.

The original formulation of calculus with infinitesimals was not rigorously defined according to modern standards. Critics like Bishop Berkeley famously argued against the "ghosts of departed quantities," questioning the logical consistency of operating with numbers that were both infinitely small and yet somehow non-zero. This lack of a solid foundation led to much debate and hindered the acceptance of calculus by some.

Over time, mathematicians developed more rigorous approaches to calculus that avoided the explicit use of infinitesimals. The most prominent of these is the epsilon-delta definition of limits, introduced by mathematicians like Augustin-Louis Cauchy and Karl Weierstrass. This approach provides a precise way to define continuity, derivatives, and integrals without resorting to the vague notion of infinitesimally small quantities.

Despite being largely superseded by modern definitions, the historical concept of infinitesimals played a crucial role in the genesis of calculus. It provided an intuitive, albeit imprecise, framework for understanding the fundamental ideas of change and accumulation that are now cornerstones of mathematics and science. Some modern non-standard analysis approaches have since provided a rigorous framework for infinitesimals, but the historical context remains significant.