Abuse of notation is only a surface phenomenon. The real magic behind mathematics is in a safe and productive abuse of abstraction.
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for example when translating between algebra and geometry
Can you give me an example of an abuse of abstraction? Do you mean something like the invention of fields by abstracting over the reals?
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maybe proof of existence of eigenvalue for complex vector spaces “abusing” the fundamental theorem of algebra?
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It's not an abuse, is it? Just compute the characteristic polynomial, and there's always a root, ergo always an eigenvector.
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