Notion and role of 'proofs' in philosophy

[under development.]

Within philosophical jargon, a 'proof' is (the demonstration of) the derivation of one hypothesis (the hypothesis (to be) proven) from other hypotheses (the premises).

The relevance (or lack of relevance) of proving

Much ado was and is made in philosophy about the ability to prove certain things (including, for instance, the existence of god, the existence of the world, the existence of ourselves [our selves?], etc.).

What seems to be neglected, however, when such exceptional ado is being made is that as compared with the opposition between a very plausible and a wildly implausible view, the opposition between a proven and an unproven view is rather marginal. Our inability to prove certain things (such as the existence of the world, or the existence of ourselves [our selves?]), is only mderately relevant if what we aim at is significant knowledge. Even if, for example, we were unable to prove our own existence, the hypothesis that we do not exist is less plausible to such an extent that any serious consideration of the actual question is comparatively devious.

Two caveats about the 'proving' of facts

(1) An important step in epistemology was the recognition (going back to Hume and others, taken up prominently by Popper) that you cannot rest a proof on (inexhaustive) inductive evidence. For a reasoning based on induction to be a (valid)* proof, the induction must be exhaustive. If you want to prove that all swans are white, the premises employed must take all swans into consideration. Considering that the empirical investigation of general matters (of the colour of swans, to take a philosophers' toy example) seldom, if ever, rests on exhaustive evidence (on the examination of the colours of all particular swans), verification in the sense of proved-ness is usually, if not always, unavailable. This is basically what motivates the transition from the verification of general hypotheses (Which is usually unavailable) to the falsification of general hypotheses.

@Poppers two problems! 

(2) Within philosophical jargon, a 'proof' is a matter of relations between intensions, rather than relations between facts!