Mathematical analysis was originally created to solve concrete problems, such as calculating areas, finding the maximum of a function, or determining the tangent to a curve. This theory has a long history, developing continuously since Antiquity; at its heart lies the concept of infinity. Following Aristotle, infinity may be viewed as either actual or potential. Belief in actual infinity means accepting that infinity truly exists and may be used without restriction, provided due care is taken. Accepting infinity only as potential, by contrast, amounts more or less to regarding it as an ideal construct: useful, certainly, but not something that actually exists.
Aristotle, a Greek philosopher of the 4th century BCE.
In his Physics, he was the first to distinguish between actual and potential infinity.
Until Cauchy's time, this dividing line more or less separated mathematicians into two camps. The most reluctant rejected the very idea of infinitesimals—that is, "infinitely small" quantities. Yet differential and integral calculus had a pressing need for such objects, so those most wary of infinity reasoned in terms of limits, confining themselves to potential infinity. Their opponents accepted that infinitesimals existed and therefore used them as genuine tools of actual infinity.
