TITLE: what is a valid mathematical proof?
QUESTION [8 upvotes]: from what i have seen in my experience with math we can say that 

a valid proof is one that uses some form of logic (usually predicate logic) and uses logical rules of deduction and axioms or theorems in it's specific field to drive some new sentences that will eventually lead to the proposition we want to prove .

but we know that most of the proofs given in most of the fields (if not all!) are actually in informal language . 
if we accept the definition above then non of these proofs are valid proofs .
how can we extend or change the idea of a valid proof to get to a better definition of a valid mathematical proof ?

REPLY [2 votes]: First of all the distinction is not as important as one would think. When you use an informal formulation it can be considered a short hand description of how you would go about in creating the formal proof - just like a recipe for gingerbread is not a gingerbread, but it allows anyone to produce gingerbread if they're interested in doing so. Note that sharing proof is a matter of transferring knowledge of the proof to a receiver and if the receiver is convinced that he can produce a formal proof, actually produce a formal proof or is convinced that the proof is good enough - then it is good enough.
Second the informal language might look just like an informal language, but there's nothing that prevents a formal system to look just like some resemblance of a natural language. You could "simply" create a bijective map between a subset of the natural language and that of the formal system. Then what looks like a informal proof could be an image of a formal proof or even a formal proof itself.