Mathematicians sometime talk about algebra and geometry being dual to each other. One way to formalise this is by talking about opposite categories. If the objects of a category act like algebras, then in the opposite category they act like spaces.
But the category of finite dimensional vector spaces is its own opposite! This suggests that linear algebra is in some sense the place where algebra and geometry meet. Perhaps that explains why it's so tractable and efficacious.
@OscarCunningham - I agree that this is one of the great things about finite-dimensional vector spaces. Another nice feature of vector spaces, at least in classical logic, is their "total freedom".
@johncarlosbaez There's a funny nLab article somewhere where they explain that without excluded middle the category of pointed sets isn't equivalent to the category of sets and partial functions. They don't even think that every pointed set has a basis!
@OscarCunningham@johncarlosbaez : You need Excluded Middle to show that every vector space over F_1 has a basis, and you need the Axiom of Choice to prove that every vector space over any other field has a basis.
@johncarlosbaez@TobyBartels@OscarCunningham Yep! But can one prove things about that object in enough detail to know that you need Excluded Middle to show that every vector space over it has a basis?
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