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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
We also work long hours in a field where, contrary to public perception, the pay is not commensurate with the work we do. Still, the intangible rewards are many in a career I see as being a sacred ...
Then there are people who are really concerned that their good work may not be seen, that they may not be valued, or the compensation they're going to be receiving is not commensurate with their ...
But my personal pain was not commensurate with the systemic social disenfranchisement experienced by actual fat people. After my weight loss, I did not struggle to find clothes to wear in my size ...
The fact that the likelihood function can be defined in a way that includes contributions that are not commensurate (the density and the probability mass) arises from the way in which the likelihood function is defined up to a constant of proportionality, where this "constant" can change with the observation , but not with the parameter .
Often a set of weak diffraction spots appears between the stronger spots belonging to what is referred to as the substructure. In some cases a phase transition occurs, e.g., at higher temperatures, where the superstructure disappears and the material reverts to the simpler substructure. Not all compounds exhibit a superstructure.