Re:How to Read Mathematics
by rocqua on Monday May 01, 2017 @08:20AM
(#14236977)
Most mathematical proofs translate quite badly to formal languages. Their formulation often relies strongly on the peculiarities of language.
Consider the "it's easy to see that" from the article. There is no translation of that into a formal language beyond either introducing the claim as an axiom, or actually providing the proof. It also means that every usage of a well known theorem needs to be explicitly mentioned.
The end result would be an incredibly verbose proofs. Moreover, the verbosity of the proof wouldn't relate very well to the complexity of the idea behind the proof. This might bloat rather simple or elegant proofs, and us mathematicians really like our elegant proofs.
Put another way, this would require writing down a lot more than the epiphany that lead to the proof.
In general, it might be nice to somehow make proofs more tree-like rather than linear. That is, show the proof from the highest level, and allow a reader to 'expand' the claims to see their proofs. Down to the 'leaves' which are considered obvious.
Consider the "it's easy to see that" from the article. There is no translation of that into a formal language beyond either introducing the claim as an axiom, or actually providing the proof. It also means that every usage of a well known theorem needs to be explicitly mentioned.
The end result would be an incredibly verbose proofs. Moreover, the verbosity of the proof wouldn't relate very well to the complexity of the idea behind the proof. This might bloat rather simple or elegant proofs, and us mathematicians really like our elegant proofs.
Put another way, this would require writing down a lot more than the epiphany that lead to the proof.
In general, it might be nice to somehow make proofs more tree-like rather than linear. That is, show the proof from the highest level, and allow a reader to 'expand' the claims to see their proofs. Down to the 'leaves' which are considered obvious.