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I may be misunderstanding what you are really thinking here, but I see all of these operations as algorithms; procedures used to simplify and solve specific types of problems but have contributed to the development of mathematics in general. ...
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i agree, though I am trying to say something more specific than that. they all involve approximating one thing with something else that has a more tractable structure.
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I wonder whether the language of structuralism would help you here?
https://dl.dropboxusercontent.com/u/...%201997%29.pdf
Might give you some things to chew on.