The really simple answer to this question is: because at a high level of abstraction, it isn't useful. Most applications are being specified by people who don't understand in any detail what they want the application to do. They can't even imagine the problem well enough to discuss it let alone specify it. A very large proportion of failed projects I've seen is due to this break down. The people who have the time and…
There is a substantial language barrier; articles like this keep using the word 'proof', and I can't figure out what they mean. Each discipline has it's own standards of proof. In mathematics, the word means that a statement transcends the laws of time and space. In law, it means a group of 'reasonable people' can be convinced. In science it means that there is a theory with supporting data that has not been falsifie…
This was historically and still is largely false. A mathematical proof is a written argument and most proofs are too complex to be "obviously correct", therefore mathematical proofs are usually more like your latter case: It is an argument that convinces people. Mathematics has a long history of doing things "that sound right but haven't been proven yet but it seems to work out ok", e.g. this was done all the time in the earlier days of analysis.