Infinitesimale

Infinitesimale refers to a quantity that is smaller than any positive real number but not equal to zero. The concept of infinitesimale is central to the development of calculus. Historically, mathematicians like Archimedes used methods that hinted at infinitesimale ideas, but it was Gottfried Wilhelm Leibniz who explicitly introduced the notion of infinitesimale quantities in his work on calculus.

Leibniz viewed infinitesimale as quantities that could be manipulated algebraically, such as being added, subtracted, multiplied, and divided, yet remained so small that they could be considered "less than any assignable quantity" but "greater than nothing." This intuitive understanding allowed for the development of differential and integral calculus, enabling the calculation of slopes of curves and areas under them.

However, the rigorous foundation of infinitesimale proved challenging. Critics, most notably Bishop George Berkeley, argued that the reasoning involving infinitesimale was logically inconsistent, referring to them as "the ghosts of departed quantities." This criticism highlighted the need for a more solid mathematical framework.

The development of limits by mathematicians like Augustin-Louis Cauchy and Karl Weierstrass in the 19th century provided this rigorous foundation. The epsilon-delta definition of a limit allows calculus to be understood and formalized without recourse to actual infinitesimale quantities. Instead, the behavior of functions as their inputs approach a certain value is analyzed. Despite this, the intuitive idea of infinitesimale continues to be a useful pedagogical tool in introducing calculus concepts. Modern non-standard analysis, developed by Abraham Robinson in the 20th century, has provided a formal system where infinitesimale quantities can be rigorously defined and manipulated.