Infinitesimal

In math, an infinitesimal is an extremely small quantity that is smaller than any standard real number but not equal to zero, serving as a core concept in calculus for limits and derivatives. In philosophy, it represents the idea of a continuum being made of infinitely small parts, a concept that has been debated for centuries due to its apparent contradiction with concepts of order and rationality, and its association with potential chaos.

Mathematics

Calculus: Infinitesimals were foundational to Leibniz’s development of calculus, representing quantities that are infinitely small for calculations like finding the slope of a curve (derivative) or the area under it (integral).
Real numbers: In the standard theory of real numbers, there is no such thing as an infinitesimal that is “smaller than any positive real number but not zero”.
Modern theory: Modern mathematics, specifically non-standard analysis, has a formal way to include infinitesimals as actual numbers, which are now considered a fully developed and rigorous concept.
Engineering and physics: Infinitesimals are used to model quantities that are so small that their higher powers can be neglected, a principle that remains useful in many applications.

Philosophy

Continuity: Philosophers have debated the nature of infinitesimals and how they relate to continuous lines and spaces. The idea that a line is composed of an infinite number of infinitesimally small, discrete points was highly contentious.
Order vs. chaos: Opponents like the Jesuits in the 17th century saw infinitesimals as a dangerous concept that threatened the orderly, hierarchical view of the world, associating them with the chaos and disorder of change and pluralism.
Authority: The debate was also political, as the new ideas of infinitesimals, and the inductive method they supported, were seen by figures like Thomas Hobbes as a threat to established authority and traditional deductive reasoning.
Foundation of knowledge: Supporters, on the other hand, viewed infinitesimals as the key to scientific progress, a new way to understand the world through a “bottom-up” approach that contrasted with the top-down logic of Euclidean geometry.

AI responses may include mistakes.

[1] https://plato.stanford.edu/entries/continuity/

[2] wikipedia/en/Infinitesimal