Infinitésimale

Infinitésimale refers to a quantity that is smaller than any positive real number but is not zero. The concept of infinitesimals is central to the development of calculus and has been a subject of mathematical debate and refinement throughout history.

In traditional calculus, infinitesimals were used intuitively by mathematicians like Newton and Leibniz. They reasoned about quantities that were "infinitely small" to understand the notion of instantaneous rate of change (derivatives) and the accumulation of quantities (integrals). However, this intuitive use lacked rigorous justification and led to paradoxes.

The development of the epsilon-delta definition of limits by mathematicians like Cauchy and Weierstrass provided a rigorous foundation for calculus without explicitly relying on infinitesimals. This approach defines limits in terms of arbitrarily small positive real numbers (epsilon) and intervals (delta), effectively bypassing the need for a separate class of infinitesimal numbers.

More recently, non-standard analysis, pioneered by Abraham Robinson, has provided a rigorous framework for working with infinitesimals. In this system, infinitesimals are treated as actual numbers within an extended real number system. This approach allows for a more direct translation of intuitive calculus arguments into formal proofs. Despite these formalizations, the term "infinitesimal" is often still used informally in a pedagogical context to explain the underlying concepts of calculus.