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Euclid did not use any concept of real number, but he used a notion of congruence of line segments, and of one such segment being longer or shorter than another. That a / b is rational is a necessary and sufficient condition for the existence of some real number c, and integers m and n, such that a = mc and b = nc.
In 1962, Thomas Kuhn and Paul Feyerabend both independently introduced the idea of incommensurability to the philosophy of science. In both cases, the concept came from mathematics; in its original sense, it is defined as the absence of a common unit of measurement that would allow a direct and exact measurement of two variables, such as the prediction of the diagonal of a square from the ...
Commensurability (astronomy), whether two orbital periods are mathematically commensurate. Commensurability (crystal structure), whether periodic material properties repeat over a distance that is mathematically commensurate with the length of the unit cell. Commensurability (economics), whether economic value can always be measured by money
In ethics, two values (or norms, reasons, or goods) are incommensurable (or incommensurate, or incomparable) when they do not share a common standard of measurement or cannot be compared to each other in a certain way. There is a cluster of related ideas, and many philosophers use the terms differently. On one common usage:
"Throughout the year, the agency was called on to do more without commensurate funding, working within the confines of strained resources and competing priorities while steadfastly supporting the ...
Two groups G 1 and G 2 are said to be (abstractly) commensurable if there are subgroups H 1 ⊂ G 1 and H 2 ⊂ G 2 of finite index such that H 1 is isomorphic to H 2. [1] For example: A group is finite if and only if it is commensurable with the trivial group.
The commensurate total on America's southwestern border is over 1.5 million. On drugs, the US Drug Enforcement Administration has often downplayed Canada's role.
“If you look at a state like Nebraska, which was mentioned (in the study), their rates are commensurate with the risks. Nebraska experiences frequent severe weather, including tornadoes, strong ...