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Learning new math concepts often comes down to identifying the "primitive" object and then figuring out how the new concept extends this primitive (credit to @geomblog for impressing this idea). Having multiple interpretations makes it much easier to see extensions
Sure - a good example would be learning about Hilbert spaces. The primitive object is a vector space, which we extend to be complete and have norms (making it a Banach space), and also have a dot product (giving us the full definition of a Hilbert space)
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