Re "motivation" and "intention" in mathematics, see Imre Lakatos's classic Proofs and Refutations. This is a fabulous work which should be read by everyone with an interest in mathematics. It doesn't really discuss the microstructure of proofs (why was this step taken here), but it does discuss the motivation and intention of proofs, examples, counterexamples, theorems, definitions, etc. To compress the argument to the absolute minimum, he essentially shows that you don't "have" the axioms, rules of inference, theorem, and proof; they are generated by the process of doing math.
The amazing thing about this book is that it manages to be philosophy of mathematics at the same time as history of mathematics (the Euler characteristic in particular). And it is a good read. I can't recommend it too highly.
And you know what, this is not a tangent: it is actually directly relevant to the Ontology forum: for example, how do you deal with exceptional cases (penguins are birds without wings)? Which comes first, definitions (sc. ontologies) or theorems (sc. systems)? Etc.
Imagine you have
the axioms, the rules of inference, the theorem to be proved, and also the
proof. Now, it would be absolutely great to be able to understand the steps
of the proof, the intention of the mathematician, the reason why this and
that step was taken at this and that point.
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