Philosophy of maths questions:
Say I want to ask for an explanation of some mathematical phenomenon (that is, to ask "Why is it true that <X>?"). You might offer a formal proof of <X>, but that doesn't feel like an explanation to me because a proof is essentially a statement that <X> is logically implied by the assumptions. So the proof, as an explanation, is equivalent to "because I chose these assumptions".
Are proofs the only explanations that pure maths has to offer?
Are there other forms of mathematical explanation (e.g. involving reference to assumptions outside the minimal axioms required for a formal proof)?
Is it even sensible to ask these questions in the context of pure maths?
Is it different when we shift to applied maths and have to recognise that the maths is a model of the system of interest (so there's a possibly fallible mapping between the maths and the system of interest and the mathematical axioms presumably correspond to assumed truths in the system of interest)?
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