incommensurability

Incommensurability denotes the lack of a common measure for two quantities, so that their ratio cannot be expressed as a rational number. In mathematics, this phrase most often describes two magnitudes whose ratio is irrational, implying there is no common unit that measures both with a finite, integral ratio.

The classic example is the diagonal of a square and its side: the length of the diagonal is incommensurable with the side because their ratio is sqrt(2), which is irrational. This discovery in ancient Greek geometry showed that not all magnitudes share a common measure and contributed to the development of irrational numbers. More generally, two lengths are incommensurable if their ratio is irrational; commensurate if the ratio is rational.

In philosophy of science, incommensurability refers to rival theories or paradigms whose underlying concepts, standards, and methods cannot be fully translated into one another. Changes in scientific theories, according to some philosophers (notably Thomas Kuhn), may involve incommensurable vocabularies, measurements, or problem definitions, complicating cross-theory comparison and communication. Critics argue that incommensurability is often overstated and that translation and calibration across paradigms are possible to varying degrees.