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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 implementat

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.

jll29 2y ago
> definition-theorem-reason-proof Along the lines of your own argument: even better would be reason0-definition-reason1-theorem-reason2-proof
gowld 2y ago
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 2y ago
  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 2y ago
  By Gödel's first incompleteness theorem there are true statements that cannot be proven (without adding new axioms).

jll29 2y ago

> definition-theorem-reason-proof Along the lines of your own argument: even better would be reason0-definition-reason1-theorem-reason2-proof

gowld 2y ago

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 2y ago
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 2y ago
By Gödel's first incompleteness theorem there are true statements that cannot be proven (without adding new axioms).