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Given that somebody claims a proof to a problem and you ask 100 highly qualified mathematicians whether this proof is valid, will they all agree that it's valid or all claim it's invalid?

In other words, is a chain of logic unique and pure? Or is it like the law where when you present very convincing evidence in a court case, there could still be some doubt? Is there some set of rules that make a proof unique and universally understood by a machine?

You could present a proof at the most extreme detail and go into the most fundamental axioms of mathematics, or you could skip key points and just say "clearly, this and that is the case".

My second question is with respect to the format. I always see proofs with text and equations, but you also instead show a graph or a truth table or a drawing. Do illustrations also qualify as proofs (if they are presented in isolation)?

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up vote 2 down vote accepted

A proof should be unambiguously clear to the intended audience. It should leave no room for doubt. To a student who is first studying proofs, it would be a bad idea to confuse an illustration with a formal proof, because that leaves them in doubt as to whether the question has been been rigorously adressed. A professional mathematician who is well-versed in classical arguments, tricks and lingo can fill in the technical gaps left by a picture or proof sketch. Therefore, a drawing can stand as a proof to a professional audience, provided they can interpret it unequivocally.

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The answer to your second question is "yes." See, for example, this wikipedia article on proof without words.

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In the majority of cases, a mathematical proof is a form of communication to other mathematicians. In that sense, anything that effectively conveys the idea of the proof to the recipient in a convincing manner is a legitimate format.

In the study of formal logic, things may be a little different: I have heard the field described as "any mathematics in which proofs are manipulated as mathematical objects in their own right". In such a context, what constitutes a proof may be a good deal stricter – one may come up with definitions along the lines of "a proof in the predicate calculus is a finite sequence of logical formulae, each constructed from its predecessors by one of the following rules..." or "a proof of a proposition in this type system is a value inhabiting a type corresponding to that proposition". The verification of such proofs is then a purely mechanical, automatable process, which is reassuring, but they may be impractically clumsy for some theorems.

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Why the link to Kruskal's tree theorem? – Mario Carneiro Mar 27 '14 at 1:01
Friedman's finite form of it can be proven for each $n$ in Peano Arithmetic, but the shortest possible proof grows impossibly fast, so you wouldn't be able to verify such a proof in a purely mechanical way. – Ben Millwood Mar 27 '14 at 18:32
But the verification is still "efficient" (i.e. polynomial in the input); in this case you are looking at a family of proofs that grow very fast. Presumably the verification is not much harder than the length of the proof (which itself grows super-exponentially as a function of $n$). – Mario Carneiro Mar 27 '14 at 19:04

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