Penyngton

Maybe it would be topical to use a "herd immunity" argument. We all need to have privacy to protect those who are vulnerable as it's impossible to only protect the "at risk" people.

Thanks for your reply. I'll definitely send you an e-mail in the near future!

As a mathematics PhD student, I wish there were some way to practise my writing independently of my supervisor, like the Code Review Stack Exchange but for proofs (and definitions)! I suppose the community would be small, and perhaps people are expected to be confident at this before starting a PhD, but still I wish it existed.

I'm not sure if this quite matches your request (I suppose much hinges on the word "important"), but here's a story of a fairly famous and long-standing error: https://www.jamesrmeyer.com/ffgit/godel-error-2.html Personally, I'd say the fact that someone bothered to discover it was false fifty years after it was published shows its importance.

The short version is that Gödel claimed in a paper that he had an algorithm to determine satisfiability for a certain logic, but the algorithm actually worked for a slightly different logic instead. People basically took Gödel's word for it for fifty years until a logician called Goldfarb found the mistake.

EDIT: As a disclaimer, I don't have any association with or endorsement of the website I've linked to. I found it just now because I needed a source for the story that I've known for many years (I think I first heard it in person, but it's definitely contained in the book `The Classical Decision Problem' by Börger, Grädel and Gurevich). I skimmed the post and it seemed to cover the relevant details, but that's the extent of my knowledge of the site and its contents.

I like doing courses in French on <https://www.fun-mooc.fr/fr/>. They have courses to learn French, but also on many other subjects.

Yes, certainly things like corruption, jealousy and ego games do exist in pure maths, but I just wanted to contest the other comment's assertion that pure mathematics is completely closed to outsiders, let alone ones who could resolve the Riemann Hypothesis.

Furthermore, I could believe that the above issues and, as you also mention, reproducibility, are worse in pure mathematics or theoretical CS than in other fields because so little is at stake. Which is not to say I think most maths papers are false, but just that they're not all _entirely_ true either. Their audience is so small and the readers/reviewers "know what they mean" anyway, most of the time. And others are just false and no-one noticed because they couldn't be bothered to check the details, because the result was inconsequential to their own work and the author was a nice enough chap...

That's not my experience at all. To be fair, I've never solved the Riemann Hypothesis, but in my experience it is possible to find professional academic mathematicians who were willing to meet me (as an outsider) to discuss their work and my ideas in a serious way.

Obviously, you have to bear in mind that high-profile mathematicians do get contacted by cranks and they have to weigh up what's worth spending their time on, but I have no doubt that if I did chance upon a genuine proof of the Riemann Hypothesis, I'd be able to find a decently respected mathematician to look at it and help me to publish it in a form acceptable to the academic community.

On the other hand, I agree with your comment about formal proof systems, and I'd love to use them in my (pure mathematical) research, but I've found the usability isn't there for me yet.

Here <https://cacm.acm.org/magazines/2017/8/219606-the-science-of-...> is an article about a computer-generated proof of a combinatorial problem in number theory. Although you could characterise the proof as "just long", I think that the article may offer a different perspective on why proofs of this nature are interesting and useful.