The difference is the difference between Intrinsic and extrinsic properties.
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Intrinsic properties of the language can be distinguished from extrinsic properties of a particular grammar.
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Variable stars have periodic or random changes in luminosity because of intrinsic or extrinsic properties.
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His theorema egregium gives a method for computing the curvature of a extrinsic properties of the ambient space.
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On the other hand, extrinsic properties relying on an embedding of a surface in Euclidean space have also been extensively studied.
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:Your question hits upon one of the fundamental issues in the development of mathematics : the distinction between intrinsic and extrinsic properties of objects.
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An example of an extrinsic property is " fatherhood " : something is a father only if that something is a male and has a child.
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Manifold theory has come to focus exclusively on these intrinsic properties ( or invariants ), while largely ignoring the extrinsic properties of the ambient space.
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An " intrinsic property " is just a property that something has independently of anything else; an " extrinsic property " is had only in relation to something.
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For example, density is a physical intrinsic property of any physical object, whereas weight is an extrinsic property that varies depending on the strength of the gravitational field in which the respective object is placed.
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