**Update**: I edited the question as I saw it was closed. Let's see if with some improvements it can be considered worth reopening... (I already accepted an answer, but I'd like to see something more specific, if possible).




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**Premise**: I think that a reasonable definition of what a "correct proof" is goes like: "one which is able to convince most mathematicians in the field that a [formal proof](https://en.wikipedia.org/wiki/Formal_proof) can be written down, if one really wants". I don't remember where I read this first, but it seems sensible to me.


Now, suppose that, aimed at proving the true statement $A$, you write something like:



> 
> Since $B$ is true, then it clearly follows $C$, so that, as $D$ is always impossible in our assumptions, $A$ follows.
> 
> 
> 


Suppose also that every single statement in this proof is correct, in the sense that $B$ and $C$ *are* true and $D$ *is* false in the proposed assumptions. Clearly, in a reasonable mathematical paper, the reader would be able to follow the reasoning by the author, so that the connection between $B,C,D$ and $A$ would be more or less clear. But in general, especially if the author is in a hurry (or if he's V.I. Arnold...), that connection can be highly non-obvious. It's also safe to assume that you cannot replace meaningfully $B,C,D$ by randomly chosen true/true/false statements in the previous proof. For instance, you can't write something like:



> 
> Since $57$ is not prime, then it clearly follows that every irrational rotation is ergodic, so that, as no locally compact Hausdorff space can be meagre, Gauss's Theorema Egregium follows
> 
> 
> 


...and be taken seriously.


However, unless a formal proof is written down, there is always some degree of mathematical understanding which is left to the reader and it can be negligible or relevant, depending on the style of the writer. Take the following sentence:



> 
> Since $|M|>10^{34}$, then $|M|> 10^{50}$, so that...
> 
> 
> 


Without context, this is simply nonsense. Even if you know that $M$ is a sporadic group, if you're in the 1950s the average reader would most probably frown upon this. However, [today](https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups) this makes sense. So it seems to me that the sociological aspects of the question about what a correct/wrong proof is cannot be easily walked around.


**My question**: has any well-known mathematician addressed the problem of how to meaningfully talk about "wrong proofs"? In particular, in cases in which the thesis *is* true (and provable) under the given assumptions, and all the statements in the proof are true as well?



What's a wrong proof?
You might be interested in the paper *[What do mathematicians mean by proof? A comparative judgement study of students’ and mathematicians’ views](https://doi.org/10.1016/j.jmathb.2020.100824)* by Davies, Jones and Alcock. (More generally, I consider the work of Lara Alcock to be interesting and insightful about a range of issues in university-level mathematical pedagogy.)


