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Commensurable
Two concepts or things are commensurable if they are measurable or comparable by a common standard. Commensurability most commonly refers to commensurability (mathematics). It may also refer to: * 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 * Commensurability (ethics), the commensurability of values in ethics * Commensurability (group theory), when two groups have a subgroup of finite index in common * Commensurability (philosophy of science) * Unit commensurability, a concept in dimensional analysis that concerns conversion of units of measurement * Apples and oranges, common idiom related to incommensurability {{disambiguation it:Incommensurabilità simple:Incommensurabi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commensurability (mathematics)
In mathematics, two non-zero real numbers ''a'' and ''b'' are said to be ''commensurable'' if their ratio ' is a rational number; otherwise ''a'' and ''b'' are called ''incommensurable''. (Recall that a rational number is one that is equivalent to the ratio of two integers.) There is a more general notion of commensurability in group theory. For example, the numbers 3 and 2 are commensurable because their ratio, , is a rational number. The numbers \sqrt and 2\sqrt are also commensurable because their ratio, \frac=\frac, is a rational number. However, the numbers \sqrt and 2 are incommensurable because their ratio, \frac, is an irrational number. More generally, it is immediate from the definition that if ''a'' and ''b'' are any two non-zero rational numbers, then ''a'' and ''b'' are commensurable; it is also immediate that if ''a'' is any irrational number and ''b'' is any non-zero rational number, then ''a'' and ''b'' are incommensurable. On the other hand, if both ''a'' and ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commensurability (group Theory)
In mathematics, specifically in group theory, two groups are commensurable if they differ only by a finite amount, in a precise sense. The commensurator of a subgroup is another subgroup, related to the normalizer. Commensurability in group theory 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. For example: *A group is finite if and only if it is commensurable with the trivial group. *Any two finitely generated free groups on at least 2 generators are commensurable with each other. The group ''SL''(2,Z) is also commensurable with these free groups. *Any two surface groups of genus at least 2 are commensurable with each other. A different but related notion is used for subgroups of a given group. Namely, two subgroups Γ1 and Γ2 of a group ''G'' are said to be commensurable if the intersection Γ1 ∩ Γ2 is of finite index in both Π... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commensurability (philosophy Of Science)
Commensurability is a concept in the philosophy of science whereby scientific theories are said to be "commensurable" if scientists can discuss the theories using a shared nomenclature that allows direct comparison of them to determine which one is more valid or useful. On the other hand, theories are incommensurable if they are embedded in starkly contrasting conceptual frameworks whose languages do not overlap sufficiently to permit scientists to directly compare the theories or to cite empirical evidence favoring one theory over the other. Discussed by Ludwik Fleck in the 1930s, and popularized by Thomas Kuhn in the 1960s, the problem of incommensurability results in scientists talking past each other, as it were, while comparison of theories is muddled by confusions about terms, contexts and consequences. Introduction of the term In 1962, Thomas Kuhn and Paul Feyerabend both independently introduced the idea of incommensurability to the philosophy of science. In both case ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dimensional Analysis
In engineering and science, dimensional analysis is the analysis of the relationships between different physical quantities by identifying their base quantities (such as length, mass, time, and electric current) and units of measure (such as miles vs. kilometres, or pounds vs. kilograms) and tracking these dimensions as calculations or comparisons are performed. The conversion of units from one dimensional unit to another is often easier within the metric or the SI than in others, due to the regular 10-base in all units. ''Commensurable'' physical quantities are of the same kind and have the same dimension, and can be directly compared to each other, even if they are expressed in differing units of measure, e.g. yards and metres, pounds (mass) and kilograms, seconds and years. ''Incommensurable'' physical quantities are of different kinds and have different dimensions, and can not be directly compared to each other, no matter what units they are expressed in, e.g. metres and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Incommensurability
Two concepts or things are commensurable if they are measurable or comparable by a common standard. Commensurability most commonly refers to commensurability (mathematics). It may also refer to: * 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) Commensurability in economics arises whenever there is a common measure through which the value of two entities can be compared. Commensurability has two versions: * Strong commensurability arises when it is possible to give cardinal values to ent ..., whether economic value can always be measured by money * Commensurability (ethics), the commensurability of values in ethics * Commensurability (group theory), when two groups have a subgroup of finite index in common * Commensurability (philosop ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commensurability (ethics)
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: * Two values (for example, freedom and security) are ''incommensurable'' when they cannot be 'traded off' against each other: for example, if there is no set amount of freedom that would compensate for a certain loss of security, or vice versa. * Two options or choices are ''incommensurate'' or ''incomparable'' if and only if: it is not true that one is better, that the other is better, or that they are exactly equally good. This page is concerned almost entirely with the second phenomenon. For clarity, the term 'incomparable' is used. Trichotomous Comparisons and Small Improvement Arguments In terminology due to Ruth Chang, the three ''trichotomous comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commensurability (astronomy)
Commensurability is the property of two orbiting objects, such as planets, satellites, or asteroids, whose orbital periods are in a rational proportion. Examples include the 2:3 commensurability between the orbital periods of Neptune and Pluto, the 3:4 commensurability between the orbital periods of the Saturnian satellites Titan and Hyperion, the orbital periods associated with the Kirkwood gaps in the asteroid belt relative to that of Jupiter, and the 2:1 commensurability between Gliese 876 b and Gliese 876 c. Commensurabilities are normally the result of an orbital resonance, rather than being due to coincidence. See also * Harmonic *Ratio In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ... References {{DEFAULTSORT:Commensurability (Astronomy) Asteroids Celestial me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Apples And Oranges
A comparison of apples and oranges occurs when two items or groups of items are compared that cannot be practically compared, typically because of inherent, fundamental and/or qualitative differences between the items. The idiom, ''comparing apples and oranges'', refers to the apparent differences between items which are popularly thought to be incomparable or incommensurable, such as apples and oranges. The idiom may also be used to indicate that a false analogy has been made between two items, such as where an ''apple'' is faulted for not being a good ''orange''. Variants The idiom is not unique to English. In Quebec French, it may take the form (to compare apples with oranges), while in European French the idiom says (to compare apples and pears) or (to compare cabbages and carrots). In Latin American Spanish, it is usually (comparing potatoes and sweet potatoes) or commonly for all varieties of Spanish (comparing pears with apples). In some other languages the ter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superstructure (condensed Matter)
In solid state physics, a superstructure is some additional structure that is superimposed on a higher symmetry crystalline structure. A typical and important example is ferromagnetic ordering. In a wider sense, the term "superstructure" is applied to polymers and proteins to describe ordering on a length scale larger than that of monomeric segments. In crystals In a crystal, a superstructure manifests itself through additional reflections in diffraction patterns, e.g., in low energy electron diffraction ( LEED) or X-ray diffraction experiments. 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. The superspots in diffraction patterns represent a ''modulation'' of the substructure that causes the inhe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commensurability (economics)
Commensurability in economics arises whenever there is a common measure through which the value of two entities can be compared. Commensurability has two versions: * Strong commensurability arises when it is possible to give cardinal Cardinal or The Cardinal may refer to: Animals * Cardinal (bird) or Cardinalidae, a family of North and South American birds **''Cardinalis'', genus of cardinal in the family Cardinalidae **''Cardinalis cardinalis'', or northern cardinal, the ... values to entities as a consequence of utilising a given property to measure entities. Thus we can say "This is two and a half times more valuable than that." This implies value monism. * Weak commensurability arises when it is only possible to apply ordinal values to entities as a consequence of utilising a given property to rank entities, i.e., it is sufficient to say "This is more valuable than that." This is consistent with value-pluralism. While weak commensurability is a form of strong comparabil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
