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I watched a talk given by Kevin Buzzard, he said he had lost faith in human maths since he found a mistake in a paper written by an expert. I later found that people have found fatal mistakes in well-accepted papers (such as Vladimir Voevodsky's paper "Cohomological Theory of Presheaves with Transfers" and the widely-believed-to-be-true Busemann-Petty Theorem in higher dimensions, although I don't understand a single word in these papers). Given that the modern process of verifying a paper is simply exposing it to as many human eyes as possible, and there are a bunch of papers which no one even bothered to verify but still got cited (said by Kevin Buzzard), I'm tempted to ask the following questions:

  1. As recent maths papers use more and more citations, would the referees and journal editors really trace down the citations to their earliest origins, where they could find a satisfying proof (eg. can be understood by any maths graduate, or even better can be verified by a computer proof assistant) to every result in this "recursive" citation process?

  2. As maths papers become longer, how more prone to mistakes do you think they are compared to papers in the previous centuries? eg. The classification of finite simple groups took tens of thousands of pages, even John Conway said mistakes are bound to happen.

  3. In general, how would you justify the reliability of modern maths in an objective or even "quantifiable" way? eg. Have there been someone using statistical or mathematical tools to estimate the probability that a modern maths paper contains fatal mistakes?

I just find it disappointing that mathematics, which is supposed to emphasise rigour more than any other subject other than Logic itself, is based solely on human verification.

HIH
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    I do not find this disappointing. The interplay between rigor and being convincing to fellow humans is exactly what makes math interesting. You can observe this every day here on MSE. – Kurt G. Sep 29 '22 at 11:22
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    Compared to other disciplines (sociology for example), mathematics remain a very privilegized domain where something blattantly disputable cannot be published... – Jean Marie Sep 29 '22 at 12:44
  • @SassatelliGiulio Could you explain in detail the developments in the past 80 years that ensure the validity of maths papers? – HIH Sep 29 '22 at 18:41
  • The answer to your question 1. is simply no because that's impossible. Math papers are prone to errors and always will be. What I think matters more is if important theorems are well understood by a large enough community. This is mostly not a matter of writing a single error free math paper. It is rather a process of maintenance over time not unlike maintaining the infrastructure that we live in. – Kurt G. Sep 30 '22 at 08:34
  • @KurtG. Thank you for your reply! I hope I didn't misunderstand what you said, so rather than ensuring every paper is error free (which I agree is impossible), a more practical way is to do our best to ensure most of the important theorems are correct. In that sense, would you say it is necessary to use "non-human" tools, eg. computer proof assistants, to verify the proofs of these theorems? I know their reliability still depends on humans but I think it doesn't hurt to introduce one more tool to verify proofs. – HIH Sep 30 '22 at 15:46
  • Which technical progress in human history was ever necessary ? That's a very philosophical question. I rather think that, with the progress in AI, that automatic proof verification systems will sooner or later become more wide spread simply because will become technically possible. At least as far as math is concerned we can be quite sure that this technical progress is more beneficial than not. One year ago there was an article in Quantamagazine you might find interesting. – Kurt G. Sep 30 '22 at 17:35
  • This MO post is quite interesting. – Kurt G. Oct 05 '22 at 14:02

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