This terminology has puzzled me for a while and I haven't seen it actually discussed anywhere. Why does the language indicate relative positions of some space or operator and the objects they deal with?

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    $\begingroup$ I wonder if it is related to the terminology of a ring $F[x]$ of polynomials ‘over’ the field $F$, and who was first responsible for these locutions. $\endgroup$
    – MJD
    Jan 27, 2014 at 14:25
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    $\begingroup$ Related would be: do they use something like "over" for this in other languages? $\endgroup$
    – GEdgar
    Jan 27, 2014 at 15:48
  • $\begingroup$ @GEdgar That's a good point - do we know if "over" is purely an Anglophone construct or does the notion exist in other natural languages as well? $\endgroup$
    – user98131
    Jan 27, 2014 at 15:56
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    $\begingroup$ @GEdgar (and Alexander Yasgar), it's also used in German, "Sei $V$ ein Vektorraum über $K$". $\endgroup$ Jan 29, 2014 at 21:27
  • $\begingroup$ I really don't see why this should raise eyebrows at all. What could it be called instead that you couldn't ask this question about? A space... "around" a field? With? About? It had to be called something. $\endgroup$
    – Jack M
    Jan 30, 2014 at 11:42


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