infinitesimals

Infinitesimals are quantities that are greater than zero but smaller than any positive real number. They played a foundational role in the development of calculus by Isaac Newton and Gottfried Wilhelm Leibniz, who used infinitesimals to express instantaneous rates of change and areas under curves. The notion was informal and controversial because it seemed to violate the Archimedean property of the real numbers, leading to difficulties in establishing rigorous arguments.

In the 19th century, the foundations of calculus were recast using the epsilon-delta approach, which avoided actual infinitesimals by defining limits. Nevertheless, the intuitive appeal of infinitesimals persisted, and several modern frameworks reintroduce them in a rigorous way. The most developed is nonstandard analysis, which extends the real numbers to the hyperreal system and treats certain nonzero infinitesimals as actual numbers.

In nonstandard analysis, infinitesimals and infinite numbers satisfy the transfer principle, which ensures that every first-order statement true for real numbers holds for hyperreals as well. The standard part map associates each finite hyperreal with the unique real number infinitely close to it. This provides a precise basis for the differential calculus that closely mirrors the historical use of infinitesimals, including Leibniz's dx and dy.

Other approaches exist, such as synthetic differential geometry, which uses nilpotent infinitesimals in a topos-theoretic setting. Infinitesimals appear in asymptotic methods and perturbation theory in physics and applied mathematics. Today, infinitesimals are used mainly as a heuristic or as part of a rigorous framework like nonstandard analysis, rather than as actual numerical entities in standard real analysis.