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In mathematics, I often meet the expression " 'x' occurs naturally", or " 'x' occurs naturally in 'Y' ".

For example: "You should know why eigenvectors and eigenvalues occur naturally in linear algebra." (Garrity, Thomas A., All the mathematics you missed: but need to know for graduate school).

Does it have to do with physical nature (as in, mapping well some physics' magnitude behaviour)?

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    $\begingroup$ Occurs "naturally" in your context is because of "as a consequence of an unmentioned relevant (and obvious) theorem" In your example about the eigenvalues/vectors: They have to exist because a theorem states that and they are consequently needed to prove other theorems that have logical answers which could not be obtained without them. That's how I look at it... $\endgroup$ – imranfat Oct 30 '15 at 15:58
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    $\begingroup$ I think it is a meaningless statement. I would ignore the word 'naturally' in that sentence. $\endgroup$ – copper.hat Oct 30 '15 at 15:59
  • $\begingroup$ I agree with copper.hat. I think it is more like a rhetorical device (unlike "is naturally isomorphic to"). $\endgroup$ – Stefan Perko Oct 30 '15 at 16:01
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    $\begingroup$ It means that the concept is key to understand several things in a field and/or that it forms part of the answer to several questions in the field. I was tempted to write "natural questions in the field", but it's difficult to specify what kind of mahematical cuestions are "natural". It could be how stuff works, how stuff can be described, how stuff relates to other stuff, etc. $\endgroup$ – dafinguzman Oct 30 '15 at 16:12

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