INFINITESIMAL, number that is closer to zero than to any finite number but is not yet zero. This is sometimes expressed as 1/∞.
The infinitesimal has fascinated mathematicians and philosophers ever since the ancient Greek philosopher Zeno of Elea (c. 490–430 BCE) formulated his famous paradox: Imagine Achilles is in a race with a tortoise. The tortoise is given a head start, so Achilles must first catch up before he can overtake the tortoise. However, by the time Achilles has caught up, the tortoise will have moved a little further, and Achilles will need to catch up again. Even as the tortoise’s lead gets smaller and smaller––approaching infinitely small––Achilles must always first catch up before he can overtake the tortoise. Since this process can repeat forever, the paradox concludes, Achilles will never overtake the tortoise.
The issue here is that space, as well as time and numerical values in the abstract, are conceived of as unbroken continua and therefore infinitely divisible. For example, if you divide one second in half forever, you will not reach a unit of time that cannot be divided. You will not locate the moment in which one second definitively changes into the next; you will only get infinitely closer to it.
Aristotle argued against this theory of continua and asserted that there are indeed very small but discrete units of space and time that cannot be further divided. His theory, known as “atomism,” was picked up by medieval Christians who wanted to confine the quality of infinity to God. Renegade mathematicians like Galileo and Bonaventura Cavalieri experimented with infinitesimals until the entire concept was banned in Italy by the Catholic Church in 1632 for promoting paradox and disorder.
In the late seventeenth century infinitesimals became essential to the development of calculus, invented simultaneously but independently by Isaac Newton and Gottfried Wilhelm Leibniz. In modern calculus the use of infinitesimals has largely been replaced with the use of limits, a concept developed in the nineteenth century by Karl Weierstrass that makes calculating derivatives easier and more precise. In the 1960s a more reliably accurate way to use infinitesimals and other “hyperreal numbers” was developed by Abraham Robinson.