Timeline for Why doesn't mathematics collapse even though humans quite often make mistakes in their proofs?
Current License: CC BY-SA 4.0
48 events
| when toggle format | what | by | license | comment | |
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| Aug 30, 2023 at 18:02 | answer | added | shuhalo | timeline score: 1 | |
| Feb 17, 2022 at 1:52 | answer | added | Timothy Chow | timeline score: 5 | |
| Nov 15, 2021 at 13:07 | comment | added | Pietro Majer | It is true, sadly, that many published papers are flawed with gaps and errors. But important results, in particular those on which other works rely, immediately start being checked line by line by all people interested, who already have a deep insight in the topic. Irrelevant papers may live with even fatal errors, that nobody will possibly notice, but who cares. | |
| Aug 27, 2019 at 18:01 | vote | accept | J. Doe | ||
| Aug 26, 2019 at 15:47 | answer | added | msouth | timeline score: 0 | |
| Aug 20, 2019 at 22:54 | history | edited | Rodrigo de Azevedo | CC BY-SA 4.0 |
If "collapse" is used, I would say that appending "down" is redundant and superfluous.
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| Aug 20, 2019 at 19:12 | comment | added | Robin Saunders | This is a recurring question - I'm surprised (even skeptical) that it hasn't come up on this site before. Here's a good answer on Quora which quotes writings of both Thurston and Tao without requiring you to read all of either: quora.com/… | |
| Aug 20, 2019 at 19:11 | comment | added | Robin Saunders | Perhaps the canonical reference on this subject is Bill Thurston's "On Proof and Progress in Mathematics" (arxiv.org/abs/math/9404236), which is mentioned in Terry Tao's (also excellent) blog post that he linked to in the comments to Mike Shulman's answer, but not directly mentioned in any answers or comments here so far. | |
| Aug 20, 2019 at 18:21 | comment | added | Hao S | Supposedly the reviewers check everything but mistakes still happen and to my knowledge the original papers are still available in online journals and people working in the field just know that there is an error in them Seriously I have no idea what to do if something in a paper doesn't seem quite right maybe it doesn't affect the fundamental thing the paper proves or... | |
| Aug 20, 2019 at 16:01 | comment | added | efkah | Math is based on axioms and laws, just like physics is with Newtons laws. Like something and zero is someting. Every Mathematical structure has some of them, and every proof has to be based on them. So nobody excepts proofs to be forever thruthful, they are just true in regards to the laws they were proofen with. | |
| Aug 20, 2019 at 10:40 | comment | added | xuq01 | I'm surprise that Lakatos' wonderful little book (Proofs and Refutations) has only been mentioned once, in a comment. It's well worth a read. | |
| S Aug 19, 2019 at 16:06 | answer | added | Carsten Führmann | timeline score: 10 | |
| S Aug 19, 2019 at 16:06 | history | protected | CommunityBot | ||
| Aug 19, 2019 at 15:19 | comment | added | Yemon Choi | @Michael on a slight tangent, but hopefully not completely derailing the discussion here: see Kevin Buzzard's remarks in the concluding paragraphs of this blogpost xenaproject.wordpress.com/2019/06/02/… | |
| Aug 19, 2019 at 14:39 | answer | added | Frank Mayer | timeline score: 10 | |
| Aug 19, 2019 at 3:36 | comment | added | Daniel R. Collins | Related? Wigner: "the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it" en.wikipedia.org/wiki/… | |
| Aug 19, 2019 at 1:20 | comment | added | Michael | This question seems to have some assumptions which may not be correct - for instance, that all (or not enough) mathematicians involved in reviewing a proof do not rigorously check it, as well as the notion that the effort in reviewing each proof is equal. I suspect that more famous and/or important/fundamental proofs are going to be checked very well - for instance, Wiles' famous proof was initially found to have some very subtle issues that took some months to iron out. | |
| Aug 18, 2019 at 22:51 | comment | added | irchans | @LSpice Maybe it is different for different branches of mathematics. I have always been under the impression that I needed to verify every line of every proof in the paper I am reviewing, but I did not need to check the proofs in the references. However, my advisor strongly encouraged me to check the references also. I think I've reviewed only 20 papers in my life. It takes me about an hour per page. It is quite possible that I was only given easier papers to review (a few of my former professors are journal editors). I've never been a professor, but occasionally I teach calculus. | |
| Aug 18, 2019 at 22:11 | comment | added | GH from MO | @LSpice: I think that the referee's task is to verify the correctness of a paper, while it is the author's responsibility to publish it. Also, we are human, so we have the right to err. | |
| Aug 18, 2019 at 21:50 | answer | added | Sam Hopkins | timeline score: 14 | |
| Aug 18, 2019 at 21:49 | answer | added | einpoklum | timeline score: 16 | |
| Aug 18, 2019 at 21:30 | answer | added | Francois Ziegler | timeline score: 13 | |
| Aug 18, 2019 at 21:02 | comment | added | LSpice | @irchans, I believe community and editorial consensus is that the referee is not responsible for verifying the correctness of proofs. I try very hard to do so anyway; but I hope I am not the only one who doesn't try transitively to verify the correctness of all the papers cited by the one I'm reading …. | |
| Aug 18, 2019 at 20:43 | answer | added | larspars | timeline score: -1 | |
| Aug 18, 2019 at 20:41 | history | became hot network question | |||
| Aug 18, 2019 at 20:01 | comment | added | irchans | The mathematicians that I know well are rather careful. I got my PhD about 25 years ago. When I review an article, I carefully read every line of the proofs. If there is a reference to a proof in another article that I don't know, I get that article. The biggest problem is when there is a reference to an article written in another language. If it's French or German, then there's a chance I can translate it. I guess what I'm saying is that some of us check things carefully and occasionally we do find errors in published works and then we publish corrections. | |
| Aug 18, 2019 at 18:46 | comment | added | Alan | fun-with-arithmetics.blogspot.com/2013/09/… makes you wonder are all humans religious?! :-D | |
| Aug 18, 2019 at 18:11 | comment | added | Alexandre Eremenko | Why only mathematics? This applied to all kinds of human activities. | |
| Aug 18, 2019 at 17:36 | comment | added | Will Sawin | @TerryTao Good point. I think this depends somewhat on the choice of the field - analytic number theorists consider implications between two widely believed conjectures to be very valuable, depending on the conjectures and techniques used, but practitioners of the Langlands correspondence are generally less interested in these statements. | |
| Aug 18, 2019 at 17:34 | answer | added | Noam D. Elkies | timeline score: 21 | |
| Aug 18, 2019 at 17:05 | comment | added | Terry Tao | @WillSawin On the other hand, a "theorem" in the edifice of mathematical literature that is not fully proven, but is nevertheless true and mutually reinforcing with the rest of the edifice, is not particularly likely to cause a "collapse". Fixing such a "theorem" with an actual proof remains an important task, but not an existential one - the worst case is that the "theorem" gets demoted to a "widely believed conjecture". | |
| Aug 18, 2019 at 16:50 | review | Close votes | |||
| Aug 19, 2019 at 15:23 | |||||
| Aug 18, 2019 at 16:01 | answer | added | Mike Shulman | timeline score: 226 | |
| Aug 18, 2019 at 15:50 | comment | added | Will Sawin | @provocateur A problem with this argument is if we want our theorems to be proven/provable, and not just true, then the mutually reinforcing character is not so helpful - the theorem could reinforce the others because it is true, not because the proof is right. | |
| Aug 18, 2019 at 14:17 | comment | added | provocateur | It is not as if the dependence of one theorem on another goes one way; most mathematical bodies of knowledge consist of mutually re-inforcing theorems that collectively paint a picture about some class of mathematical object. Given the mutually reinforcing character of these theorems, a 'false' theorem is generally not likely to 'fit' with the remainder of mathematical knowledge properly, and will be spotted relatively quickly. Of course, that's just a general consideration, and there can certainly be exceptions. But it gives one way of thinking about the OP's question. | |
| Aug 18, 2019 at 14:14 | comment | added | user44143 | @J.Doe, the headline question here asks for explanation, which is also the main theme of the answers so far; if you want advice, that seems worth a separate post. | |
| Aug 18, 2019 at 13:56 | comment | added | J. Doe | Then how does an average mathematician proceed with her/his own work? If you have enough time and energy, can you please offer me an answer or a comment, which will be useful not only for me, but for other young students in mathematics? | |
| Aug 18, 2019 at 13:43 | comment | added | user44143 | If that’s what you had in mind, then mathematics has always been collapsed! | |
| Aug 18, 2019 at 13:39 | comment | added | J. Doe | By "collapse", I had in my mind a hypothetical situation where an average mathematician being not sure whether a statement in a literature is absolutely correct or not. Admittedly, this is a vague description, and I feel sorry for this. @Matt F. | |
| Aug 18, 2019 at 13:31 | comment | added | user44143 | If the alternative to collapse is that all published proofs are fully correct, then the collapse is already here. So what alternative to collapse do you want to explain: the persistence of funding for math, the reliability of math in engineering, the rarity of major errors in textbooks, ...? | |
| Aug 18, 2019 at 13:18 | comment | added | Steve Huntsman | mathoverflow.net/questions/282742/endless-controversy | |
| Aug 18, 2019 at 12:52 | answer | added | Iosif Pinelis | timeline score: 23 | |
| Aug 18, 2019 at 12:52 | history | made wiki | Post Made Community Wiki by Todd Trimble | ||
| Aug 18, 2019 at 12:50 | answer | added | darij grinberg | timeline score: 57 | |
| Aug 18, 2019 at 12:40 | answer | added | Carlo Beenakker | timeline score: 33 | |
| Aug 18, 2019 at 12:39 | comment | added | Will Sawin | It's certainly possible that there are undiscovered errors which mess up whole series of results. The only cases we know for sure this hasn't happened are cases which are formally verified. | |
| Aug 18, 2019 at 12:35 | review | First posts | |||
| Aug 18, 2019 at 13:16 | |||||
| Aug 18, 2019 at 12:32 | history | asked | J. Doe | CC BY-SA 4.0 |