Philosophers have argued about them for centuries. Scientists and engineers employ them routinely, while thousands of Calculus students every year are taught that they do not exist. Some mathematicians acknowledge them as a useful fiction, others reject them as a useless curiosity, while a small but enlightened third group accord them the same ontological status as transcendental and complex numbers, and with their aid have proved a variety of new, true, and beautiful mathematical theorems.
Infinitesimals are numbers which are smaller in absolute value than every positive real number. Other than the number 0, it is difficult to imagine that any such object can exist. However, thanks to the work of 20th century mathematical logicians (notably Abraham Robinson, who created the coherent and powerful methodology of Nonstandard Analysis) we now know that it is consistent to assume that the infinitesimals do indeed exist.
This site is devoted to infinitesimals past and present, with a special emphasis on nonstandard analysis, the legacy of Abraham Robinson.