In this chapter we discuss the infinitesimal calculus and its discrete analogues.

The origins of calculus

Before the work of Newton and Leibniz established differential and integral calculus, the term calculus referred to any general body of mathematics. Afterwards, it became reserved almost exclusively for what should more properly be called the infinitesimal calculus, or the mathematics of infinitesimals. Infinitesimals are non-zero numbers that are smaller in magnitude than any finite number, the latter being numbers that can be assigned as magnitudes of physically measurable quantities.

Although all mathematical concepts are abstractions, infinitesimals are usually regarded as somehow ‘more abstract’ than finite numbers. The reason for this anomaly is that, whilst we use ordinary integers such as one, two, etc. for counting and fractions such as a half, one third, etc. for dividing up objects such as cakes, we do not encounter situations calling for explicit use of infinitesimals. In those circumstances where we encounter objects that appear very small on ordinary scales, such as atoms in chemistry and angles of optical resolution in astronomy, we use devices such as microscopes and telescopes to magnify them, so that they appear to be of finite size.

Infinitesimals are perhaps best regarded in terms of processes; that is, their value lies in what is done with them rather than with their individual qualities.