Simply put, mathematics is entirely built on consensus, because there's no other way for it to be practised (apart of course from purely formal/mechanical proofs)
Student: Then really what is a proof?
Ideal Mathematician: Well, it's an argument that convinces someone who knows the subject.
Student: Someone who knows the subject? Then the definition of proof is subjective; it depends on particular persons. Before I can decide if something is a proof, I have to decide who the experts are. What does that have to do with proving things?
I.M.: No, no. There's nothing subjective about it! Everybody knows what a proof is. Just read some books, take courses from a competent mathematician, and you'll catch on.
Student: Are you sure?
I.M.: Well, it is possible that you won't, if you don't have any aptitude for it. That can happen, too.
Student: Then you decide what a proof is, and if I don't learn to decide in the same way, you decide I don't have any aptitude.
I.M.: If not me, then who?
Davis, P. J., & Hersh, R. (1998). The ideal mathematician. New directions in the philosophy of mathematics, 177-184.
[...] many great and important theorems don’t actually have proofs. They have sketches of proofs, outlines of arguments, hints and intuitions that were obvious to the author (at least, at the time of writing) and that, hopefully, are understood and believed by some part of the mathematical community. [...]
In any case, there is a web of semi-proved theorems throughout mathematics. Our knowledge of the truth of a theorem depends on the correctness of its proof and on the correctness of all of the theorems used in its proof. It is a shaky foundation. [...]
We (the mathematical community) believe that the proofs are correct because a political consensus has developed in support of their correctness. [...]
How do we recognize mathematical truth? If a theorem has a short complete proof, we can check it. But if the proof is deep, difficult, and already fills 100 journal pages, if no one has the time and energy to fill in the details, if a “complete” proof would be 100,000 pages long, then we rely on the judgments of the bosses in the field. In mathematics, a theorem is true, or it’s not a theorem. But even in mathematics, truth can be political.
Melvyn B. Nathanson, Desperately seeking mathematical truth, Notices Amer. Math. Soc. 55:7 (2008), 773.
This question brings to the fore something that is fundamental and pervasive: that what we are doing is finding ways for people to understand and think about mathematics.
The rapid advance of computers has helped dramatize this point, because computers and people are very different. For instance, when Appel and Haken completed a proof of the 4-color map theorem using a massive automatic computation, it evoked much controversy. I interpret the controversy as having little to do with doubt people had as to the veracity of the theorem or the correctness of the proof. Rather, it reflected a continuing desire for human understanding of a proof, in addition to knowledge that the theorem is true. [...]
Mathematical knowledge and understanding were embedded in the minds and in the social fabric of the community of people thinking about a particular topic. This knowledge was supported by written documents, but the written documents were not really primary. [...]
When people are doing mathematics, the flow of ideas and the social standard of validity is much more reliable than formal documents. People are usually not very good in checking formal correctness of proofs, but they are quite good at detecting potential weaknesses or flaws in proofs. [...]
Mathematicians can and do fill in gaps, correct errors, and supply more detail and more careful scholarship when they are called on or motivated to do so. Our system is quite good at producing reliable theorems that can be solidly backed up. It’s just that the reliability does not primarily come from mathematicians formally checking formal arguments; it comes from mathematicians thinking carefully and critically about mathematical ideas.
Thurston, W. P. (1995). On proof and progress in mathematics. For the learning of mathematics, 15(1), 29-37.
As I’ve blogged about before, proof is a social construct: it does not constitute a proof if I’ve convinced only myself that something is true. It only constitutes a proof if I can readily convince my audience, i.e. other mathematicians, that something is true. Moreover, if I claim to have proved something, it is my responsibility to convince others I’ve done so; it’s not their responsibility to try to understand it (although it would be very nice of them to try). [...]
I recently described a proof to be a convincing argument of why you think something is true. I’ll stick to that definition in spite of a few commenters who want there to be axioms or postulates, because I really don’t think that’s what happens in real life (which is a good thing! It would be an incredibly boring life!). Since I’m a utilitarian, I only care about and only want to discuss what actually happens.
The above definition immediately begs the question, convincing to whom? Can a proof to someone be a non-proof to someone else? Absolutely, proofs are entirely context-driven. If I’m trying to prove something to you and you remain unconvinced, then it is no proof, even if I’ve used the same argument before successfully.
https://mathbabe.org/2012/11/14/the-abc-conjecture-has-not-been-proved/ + https://mathbabe.org/2012/08/06/what-is-a-proof/