Indivisibilities

Indivisibilities refer to the property of certain objects, goods, or mathematical constructs that cannot be divided into smaller parts without altering their essential character or utility. The concept appears in several fields, notably economics, mathematics, and philosophy. In general usage, an indivisible item is one that must be consumed or used as a whole; dividing it would either destroy its value or render it unusable.

In economics and operations research, indivisibilities give rise to non-convexities in preferences and production sets. Many goods are not easily or ethically divisible: a house, a concert ticket, a single license, or a machine. As a result, firms and consumers face discrete choices and often rely on binary or integer programming. Indivisibilities complicate pricing, allocation, and welfare analysis because marginal reasoning fails for whole units. They also affect market design and procurement, leading to mechanisms that allocate whole units rather than fractional quantities; in cost sharing, indivisible costs require special methods to split expenses fairly.

In mathematics and the history of calculus, "indivisibles" denotes infinitesimal-like elements used by Cavalieri and others to compute areas and volumes by summing slices. Though controversial at the time, the method influenced the development of integral calculus and the concept of limits, culminating in rigorous integral theory.

Other uses include philosophical discussions on the indivisible nature of matter or of basic units in set theory and logic, though these are more specialized.