[xanderlewis]

For proofs, I find it a good idea to memorise (or at least implicitly retain) the reason a result is true. So, yes, an outline, but minus any of the implementation details of the proof. I kind of think every book in the definition-theorem-proof style should really be definition-theorem-reason-proof.

The reason part being essentially a one or two line natural language summary of ‘why the proof works’ — something that is almost always possible and is enlightening and conducive to efficient memorisation, but that for some reason is very rarely written down explicitly.

[gowld]

I think a better word is "motivation" -- why we chose this option at this juncture instead of many other options. Yes, it's a "reason", but "reason" already means something else.

The "Reason" as result is true is that it follows from the previously established axioms via logical reasoning.

[xanderlewis]

Motivation is important too, but it’s not what I meant. A very simple example would be

Theorem: Every subspace Y of a second-countable topological space X is second-countable.

Reason: Intersecting each set in a basis for X with Y yields a basis for Y.

Proof: [formal symbolic stuff involving open sets and unions, and mentioning cardinality, etc.]

(I’m not claiming ‘reason’ is the best word for this — it probably isn’t. But it’s not the same thing as motivation.)

> The "Reason" as result is true is that it follows from the previously established axioms via logical reasoning.

One could argue this is not the reason a result is true; it’s the reason we know it’s true. The fact that true statements follow from established truths by logical reasoning is more a property of the formal system (which hopefully is sound and consistent) than it is to do with the notion of truth itself.

[Koshkin]

By Gödel's first incompleteness theorem there are true statements that cannot be proven (without adding new axioms).

[jll29]

> definition-theorem-reason-proof

Along the lines of your own argument: even better would be

reason0-definition-reason1-theorem-reason2-proof