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To begin with, I am aware of these questions, which seems to be related: How do I fix someone's published error?, Examples of common false beliefs in mathematics, When have we lost a body of mathematics because errors were found?, etc...

My background: I am a senior undergraduate student in mathematics. Recently, I got a nice chance in a REU program, and started to read some journal articles. My impression was: any result in modern mathematics critically depends on another result, and that result depends on some other result, and ad infinitum.

On the other hand, some graduate students and professors in my university, who stand in quite intimate relations to me, say that, they do not check every details of proofs when they read mathematical monographs and research articles. They simply do not have enough time to read all the details and fill in the lines. (Clearly, I also do not read all the proofs in detail, if it seems to be so difficult or not much relevant to what I am interested in.)

Finally, I've been heard of some stories on fatal mathematical errors. To be honest, I do not understand what the errors precisely are. What I've been heard about are some "urban legends". (I intentionally didn't write down the details of these urban legends, since if I write down everything I've heard, maybe someone working in the mentioned field may feel insulted...)

For the above reasons, recently I am afraid of the situation where a field in mathematics collapse down because of a single, fatal, but very subtle error in the foundations of that field. In mathematics, everything seems to be so much intertwined, and it seems that no one actually checks every single detail in every mathematical articles.

But the mathematics community seems to be very sound. Maybe at least one of the followings are true:

  • Actually, a typical mathematical result does not depend that much on other results. So whenever if possible, a mathematician can check the details of every results which is of interest to him/her.

  • Strictly speaking, rigor is actually not that important. Even if a mathematical result turns out to be false, there is still something true in the statement. Therefore, only minor changes will be needed, and all the results depending on the turned-out-to-be-false result remains sound.

Here are my questions.

  1. Why the whole mathematics remains so sound, even though humans are imperfect and quite often produce errors? Are my explanations above correct?

  2. If a theorem turns out to be wrong, then mathematicians will try to correct (if possible) all the results depending on that theorem. How hard is this job? Isn't it very tedious and frustrating? I want to hear some personal stories.

  3. As an undergraduate student, I want to know if anybody who is much wiser, older, or experienced, had the same fear as mine. (Again, I want to hear some personal stories.)

  4. As an undergraduate student who will get into a graduate school in the near future, I want to get some advice. Should I stop worrying and believe the authors of the books and articles I read? When should I check all the details, and when should I just accept the theorem as given?

Thanks to everyone for reading my question.

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    $\begingroup$ 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. $\endgroup$ – provocateur Aug 18 at 14:17
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    $\begingroup$ @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. $\endgroup$ – Will Sawin Aug 18 at 15:50
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    $\begingroup$ @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". $\endgroup$ – Terry Tao Aug 18 at 17:05
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    $\begingroup$ Why only mathematics? This applied to all kinds of human activities. $\endgroup$ – Alexandre Eremenko Aug 18 at 18:11
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    $\begingroup$ 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. $\endgroup$ – irchans Aug 18 at 20:01

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In addition to the answers that have already been given, I think another reason that mathematics doesn't collapse is that the fundamental content of mathematics is ideas and understanding, not only proofs. If mathematics were done by computers that mindlessly searched for theorems and proof but sometimes made mistakes in their proofs, then I expect that it would collapse. But usually when a human mathematician proves a theorem, they do it by achieving some new understanding or idea, and usually that idea is "correct" even if the first proof given involving it is not.

One recent and well-publicized story is that told by the late Vladimir Voevodsky in his note The Origins and Motivations of Univalent Foundations. Here's a bit of one story that he tells about his own experience:

my paper "Cohomological Theory of Presheaves with Transfers," ... was written... in 1992-93. [Only] In 1999-2000... did I discover that the proof of a key lemma in my paper contained a mistake and that the lemma, as stated, could not be salvaged. Fortunately, I was able to prove a weaker and more complicated lemma, which turned out to be sufficient for all applications....

This story got me scared. Starting from 1993, multiple groups of mathematicians studied my paper at seminars and used it in their work and none of them noticed the mistake.... A technical argument by a trusted author, which is hard to check and looks similar to arguments known to be correct, is hardly ever checked in detail.

I don't know any of the details of the mathematics in this story, but the fact that he was able to prove a "weaker and more complicated lemma which turned out to be sufficient for all applications" matches my own experience. For instance, while working on a recent project I discovered no fewer than nine mistaken theorem statements (not just mistakes in proofs of correct theorems) in published or almost-published literature, including several by well-known experts (and two by myself). However, in all nine cases it was simple to strengthen the hypothesis or weaken the conclusion in such a way as to make the theorem true, in a way that sufficed for all the applications I know of.

I would argue that this is because the mistaken statements were based on correct ideas, and the mistakes were simply in making those ideas precise. Or to put it differently, mathematicians get our intuitions from "well-behaved" objects: sometimes that intuition can be wrong for "pathological" objects we didn't know about, but in such cases we simply alter the definitions to exclude the pathological ones from consideration.

On the other hand, people do sometimes get mistaken ideas. For instance, here's another quote from Voevodsky's article:

In October 1998, Carlos Simpson ... claimed to provide an argument that implied that the main result of the "∞-groupoids" paper, which Kapranov and I had published in 1989, cannot be true. However, Kapranov and I had considered a similar critique ourselves and had convinced each other that it did not apply. I was sure that we were right until the fall of 2013 (!!).

I can see two factors that contributed to this outrageous situation: Simpson claimed to have constructed a counterexample, but he was not able to show where the mistake was in our paper. Because of this, it was not clear whether we made a mistake somewhere in our paper or he made a mistake somewhere in his counterexample. Mathematical research currently relies on a complex system of mutual trust based on reputations. By the time Simpson’s paper appeared, both Kapranov and I had strong reputations. Simpson’s paper created doubts in our result, which led to it being unused by other researchers, but no one came forward and challenged us on it.

In this case I do know something about the mathematics involved, and my own opinion is somewhat different from Voevodsky's. In the 2000's I was a graduate student working on higher category theory, and my impression was that in the community of higher category theory it was taken for granted that Simpson's counterexample was correct and the Kapranov-Voevodsky paper was wrong, because the claimed KV result contradicted well-known ideas in the field.

The point here is that a community of people developing ideas together is likely to have arrived at correct intuitions, and these intuitions can flag "suspicious" results and lead to increased scrutiny of them. That is, when looking for mistaken ideas (as opposed to technical slips), it makes sense to give differing amounts of scrutiny to different claims based on whether they accord with the intuitions and expectations of experience.

So what do you do as a student? In addition to the other good advice that's been given, I think one of your primary goals should be to train your own intuition. That way you will be better-able to evaluate whether a given result, or something like it, is probably true, before you decide whether to read and check the proof in detail.

Of course, there is also the position that Voevodsky was led to:

And I now do my mathematics with a proof assistant. I have a lot of wishes in terms of getting this proof assistant to work better, but at least I don’t have to go home and worry about having made a mistake in my work.

I have a lot of respect for that position; I do plenty of formalization in proof assistants myself, and am very supportive of it. But I don't think that mathematics would be in danger of collapse without formalization, and I feel free to also do plenty of mathematics that would be prohibitively time-consuming to formalize in present-day proof assistants.

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    $\begingroup$ Agreed; very good answer. I particularly agree that that's probably how most mistakes get caught: not by nose-to-the-grindstone checks of logical correctness, but by having sensitive mathematical noses that detect when something smells just a little bit fishy. Human intuition is so much faster than human logic. Moreover, you have to develop the inner quietude to listen to those whisperings (especially when it comes to catching your own mistakes!). $\endgroup$ – Todd Trimble Aug 18 at 16:16
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    $\begingroup$ +1. I would also add that sometimes published, true mathematical results do contradict the community intuition and experiences; however, in these cases, mathematicians do check the published result very carefully, in order to check correctness and to update their intuition. $\endgroup$ – mathworker21 Aug 18 at 19:05
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    $\begingroup$ I have some discussion of these points at the end of my blog post terrytao.wordpress.com/career-advice/… . Mathematicians with a "post-rigorous" mindset can make formal errors, but these errors tend not to propagate all the way towards collapse, because of the soundness of the intuition guiding them. $\endgroup$ – Terry Tao Aug 18 at 21:05
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    $\begingroup$ The classic and beautiful book Proofs and Refutations by Imre Lakatos discusses the nature of mathematical proofs (and their iterative refinement), using Euler's polyhedron formula $V-E+F=2$ as an example.You can be wrong, but still sort-of right. $\endgroup$ – Fab Aug 18 at 21:22
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    $\begingroup$ @AndrejBauer Because mathematicians have to communicate with one another, and pure, intuitive ideas rarely transfer from one mind to another unchanged. We need a protection against such mutation, and formalism and rigidity is the way the mathematical community has chosen to go. At least that's my two cents. $\endgroup$ – Arthur Aug 19 at 11:59
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  1. Redundancy is one big source of self-healing. A result with three different proofs is rather unlikely to be wrong. Also, people try to apply fresh results; wrong results often lead to contradictions when applied, alerting mathematicians to their wrongness. Same for proofs: Mistakes in proofs are often spotted when someone tries to adapt the proof to other questions.

  2. This is tricky. These days, using Google Scholar's "cited by" feature and various other backlink aggregators, you can get a list of papers/book that reference a given paper. Thus, if you find an error in the literature, you can track down where the "corruption" has spread. But getting corrections published is very difficult. Ted Hill and Nikolai Mnev are known for having struggled through the whole process of correcting someone else's false claims, but lots of people end up staying silent or (these days) just posting what they know somewhere on a forum like MathOverflow when someone stumbles upon the same problem. Then there are situations where no specific error can be pinpointed, but important material is simply imprecise and unreadable; fields often linger in such a limbo until someone does the thankless job of building the foundations underneath them. Katrin Wehrheim is one example of this.

  3. This question of mine got 41 votes, so yes, this is a fairly well-acknowledged problem.

  4. Ask your advisor and others. You definitely want to understand all proofs in undergraduate and lower-level graduate classes; they aren't particularly likely to be wrong, but you'll use the ideas anyway. As for advanced theory you rely upon, it depends.

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    $\begingroup$ For 4, I either agree and want to clarify or disagree slightly. You should understand all proofs presented in those classes, but you should trust the professor / the author about most of the theorems which are used in the classes but not proved, as sometimes happens when the proof of a result is too long and difficult or too far afield. It's definitely good to sometimes go beyond what is taught but in general the professor or textbook author probably decided what proofs to include based exactly on what will be useful to you in the future. $\endgroup$ – Will Sawin Aug 18 at 12:58
  • $\begingroup$ Redundancy (1.) is good when the several proofs are by different authors. Several proofs in one paper, on the other hand, are not a good sign (as Arnol’d relished in pointing out). $\endgroup$ – Francois Ziegler Aug 18 at 23:33
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This is a broad question, but you may find it helpful to read The Existential Risk of Math Errors. It suggests a certain robustness of the mathematical edifice, which I actually think extends to the natural sciences as a whole. (Newtonian mechanics is "wrong" in a fundamental sense, but neither the development of relativistic mechanics nor the discovery of quantum mechanics has caused the collapse of classical mechanics.)

This quote in particular from Gian-Carlo Rota bears on your points 1 and 2:

When the Germans were planning to publish Hilbert’s collected papers and to present him with a set on the occasion of one of his later birthdays, they realized that they could not publish the papers in their original versions because they were full of errors, some of them quite serious. Thereupon they hired a young unemployed mathematician, Olga Taussky-Todd, to go over Hilbert’s papers and correct all mistakes. Olga labored for three years; it turned out that all mistakes could be corrected without any major changes in the statement of the theorems. There was one exception, a paper Hilbert wrote in his old age, which could not be fixed; it was a purported proof of the continuum hypothesis, you will find it in a volume of the Mathematische Annalen of the early thirties. At last, on Hilbert’s birthday, a freshly printed set of Hilbert’s collected papers was presented to the Geheimrat. Hilbert leafed through them carefully and did not notice anything.

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    $\begingroup$ Rota is unreliable here: Hilbert’s published theorems sometimes were incorrect as stated. See McLarty on the note in the Annalen about his famous result on invariants. webusers.imj-prg.fr/~michael.harris/theology.pdf $\endgroup$ – Matt F. Aug 18 at 13:38
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    $\begingroup$ Rota is a great story-teller, but generally speaking people should fact-check him on his pronouncements. $\endgroup$ – Todd Trimble Aug 18 at 16:04
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    $\begingroup$ @MattF: The word "major" in "major changes" is doing a lot of work, I suspect. An author's close collaborators usually have a good idea of what the author was meaning to say, and thus can fix all sorts of imprecisions that an outsider would be stumped by. $\endgroup$ – darij grinberg Aug 18 at 19:41
  • $\begingroup$ I take issue with "Newtonian mechanics is "wrong"". Rather, it's imprecise. $\endgroup$ – Pablo H Aug 20 at 13:22
  • $\begingroup$ "wrong" in the sense that it assumes position and velocity of a particle can be determined simultaneously, or in the sense that it assumes that velocities are additive. $\endgroup$ – Carlo Beenakker Aug 20 at 13:50
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If a result is not used much, then its veracity does not matter much for the rest of mathematics.

Otherwise, there may be several proofs of the result, which makes it much more probable (in a general sense) that the result is true. Importantly, there is usually some explanation or understanding of why the result is true, that is, the ideas behind the result.

Also, people try and do construct counterexamples to disprove a result, if they do not see why the result should be true.

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It might feel like the chain of citations goes ad infinitum, but of course this can't literally be the case. Indeed as long as mathematics is (at least primarily) an endeavor by and for humans, the entire chain of reasoning must be graspable by about the time a mathematician reaches PhD level. Much of it will not yet be accessible to an advanced college student such as the OP, and modern mathematics has gone in so many directions that no one mathematician can grasp more than a small sliver of the frontier $-$ which is why modern research mathematics requires specialization, and a mathematical community large enough to support such a large frontier. Inevitably errors occur, and a few of those propagate for some time before being caught. But the enterprise as a whole is self-correcting (as already explained in several ways in other answers).

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Well, because of a multitude of reasons, including:

  1. Mathematicians rarely make mistakes in simple proofs. And a lot of the basics of math are simple theorems that follow easily from the formal definitions and axioms.
  2. Mistakes get corrected before claims gain acceptance:

    2.1 Most people double-check and re-check their own proofs before claiming to have proved something significant (although some don't - I've heard a rumor that Saharon Shelah had a bunch of papers with errors and brushed this off by saying they could easily be rectified, so it doesn't matter.)

    2.2 Mathematicians are a community and check each other's work - so there's much less chance of mistakes escaping scrutiny.

  3. Doubt: People - mathematicians and users of math - don't just accept new claims at face value. Even if they don't have the time to check proofs themselves, they'll treat new claims/results as somewhat suspect until sufficient (or apparently sufficient) verification happens.

  4. Using an erroneous theorem usually leads to obvious problems, so whoever adopts an erroneous theorem as valid typically falls flat on their face, and this easily leads to doubt being cast on their assumptions - namely, the faulty theorem.

  5. New mistakes don't invalidate past work. If a field of mathematics has some valid, no-mistakes basis, then our mistakes building on that basis don't invalidate it. At most, we may get confused and doubt some of it - but that will only make us double-check its validity or look for a counter-example - which will fail.
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    $\begingroup$ #4 is the most important point on this page. Math is all just theory until someone decides that it is useful and actually applies it to some real problem in the real world. If the math is broken then it simply won't work as expected. It's like a bug in a computer program. Until you use the part of the program where the bug is, you might well not know that it is there. Once the bug is known, then it can be analyzed and corrected. Same with math. $\endgroup$ – J... Aug 19 at 12:06
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    $\begingroup$ @J... on #4: I’m not sure we have any example of mathematical mistake revealed by “real world” application (see e.g. 1, 2, 3, 4,...) $\endgroup$ – Francois Ziegler Aug 19 at 12:49
  • $\begingroup$ @FrancoisZiegler Not typically all the way down to the engineering or physics level, of course, but plenty of incorrect proofs are only discovered by counterexample (ie: by the theorem breaking when applied to a specific instance of the general case it is meant to describe - when the theorem is applied to some specific problem). I mean to distinguish this disproof-by-accident from someone simply reading the proof and finding an error. I think there are plenty of examples of theorems only being discovered wrong when they've been taken from the toolbox to apply to a new problem. $\endgroup$ – J... Aug 19 at 13:30
  • $\begingroup$ @J... Yes; the point (e.g. Einstein’s, at link 4) is: better not to invoke “the real world”. $\endgroup$ – Francois Ziegler Aug 19 at 13:41
  • $\begingroup$ @FrancoisZiegler Fair enough - maybe my own biases. I just figured mathematicians considered anything "real world" once you let x have some value. ;) $\endgroup$ – J... Aug 19 at 13:45
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While the responses to this question have so far have generally (and correctly) focused on the robust nature of the edifice of mathematics as a whole, it might be worth pointing out by way of contrast that sometimes there are indeed doubts about foundational issues within a specific subfield (because, e.g., an "important" paper is known to have flaws), and that can be extremely deleterious to the field in question. Something along these lines is discussed in this Quanta magazine article: https://www.quantamagazine.org/the-fight-to-fix-symplectic-geometry-20170209/. (And I would say that what is described there is not even the most extreme example of what can happen.)

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    $\begingroup$ (This case is also mentioned by @darijgrinberg.) $\endgroup$ – Francois Ziegler Aug 18 at 22:31
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    $\begingroup$ Surely this example is an argument for the robustness of mathematics: people were long aware there were technical issues and either avoided using the problematic techniques by working in restricted settings (e.g. exact manifolds, which left, and still leaves, plenty of interesting problems) or by developing new approaches like polyfolds/implicit atlases/d-manifolds when they really wanted the full strength/generality. The subject was able to absorb and digest these ideas, became stronger for it and did not collapse. $\endgroup$ – Jonny Evans Aug 19 at 20:25
  • $\begingroup$ @JonnyEvans: in the "Keynesian long run" it is probably true that significant errors will be corrected; but in the short term fields where the status of the main questions are in doubt can fail to attract new students, lay dormant for years, etc. $\endgroup$ – Sam Hopkins Aug 20 at 1:22
  • $\begingroup$ But in this instance (symplectic geometry) the field continued to attract students and stayed extremely active. The technical issues affected a small number of papers which were treated with caution. $\endgroup$ – Jonny Evans Aug 20 at 6:30
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Further to M. Shulman’s advice to “develop your intuition”, it’s probably worth adding that this is often done by understanding many examples, special cases whose moving parts are already transparent to you. You get something simpler and more robust. And psychologically at least, “trust” in a result often relies on familiarity with a library of such cases, more than on line-checking a general proof. (I would also guess that many innocuous “errors” only reflect some overshoot in generality when streamlining things for publication, and that is why not everything collapses.) Or, in the polemical words of Arnol’d (2004):

There are two principal ways to formulate mathematical assertions (problems, conjectures, theorems,...): Russian and French. The Russian way is to choose the most simple and specific case (so that nobody could simplify the formulation preserving the main point). The French way is to generalize the statement as far as nobody could generalize it further.

(That’s not to say that “top-down” never wins — e.g. reputedly, only the discovery of Bott periodicity settled a “spirited controversy” on specific homotopy groups (1959, p. 355 and Math review).)

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    $\begingroup$ In this quote, Arnold does not really pay attention on the accuracy of his formulations (as actually everywhere in his non-mathematical declarations), but it is precisely this style that clearly describes the difference between Russian and French mathematics: it would be more accurate to say that Russian mathematics (and not only mathematics, in general Russian culture) seeks to turn everything into a theatrical performance, while French culture with its old meritocratic tradition eschews such immediacy. :) $\endgroup$ – Sergei Akbarov Aug 19 at 11:22
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Thought provoking question..

Perhaps, the reason why a mistakenly accepted result does not lead to collapse of the edifice of mathematics is because mathematics is supported, not by a chain of reasoning, but by a dense network.

In other words, it has massive redundancy, like a building that continues to stand despite one bad brick.

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To widen the angle a bit, I would like to point out that there is a certain analogy between mathematics and software. Programs are formal constructs that are composed and processed according to formal rules, like mathematical proofs. In fact, for particularly "clean" types of software, for example proof checkers based on dependent type theory, programs are proofs, according to the propositions-as-types paradigm. And just like ordinary software is organized in say, classes and modules, mathematics is organized in propositions and even whole libraries of propositions ("topology", "group theory") that are "exported", like modules.

Now, the world has a lot of buggy software. Sometimes this can lead to catastrophe. But catastrophe is remarkably rare. Because, the more heavily the world relies on a piece of software--that is, the greater the "user base"--the more likely will critical bugs be found and fixed. Alternatively, a critical bug might only do harm when the consumer of a module uses that module in an unusual way. (Called "edge case" in software engineering.)

It would not be surprising if a similar effect stabilizes mathematics--the software that runs on our minds.

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  • $\begingroup$ Also: the greater the user base, the more likely will critical bugs be exploited by hackers. It's not all stability in this analogy. $\endgroup$ – Matt F. Aug 19 at 17:31
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    $\begingroup$ I think that this is a very good analogy, and I would take it a step further. It's the modularity of software, rather than the redundancy (as mentioned by others) that gives it a lot of robustness. The internals of a module are shielded from the rest of the world by its interface. Often, a bug in a module affects only certain parameter values and so the damage the bug does to the whole edifice is limited. One can often fix a bug just by patching a module, without having to rewrite the entire software from scratch. $\endgroup$ – Timothy Chow Aug 19 at 19:58
  • $\begingroup$ @MattF. I suspect, somewhat undermining myself, that hacking is an aspect where my analogy breaks down. From the top of my head, I can't think of major cases where mathematics was hacked. But I might be overlooking something. $\endgroup$ – Carsten Führmann Aug 20 at 10:23
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If this is something you want to make a difference in, one avenue of attack would be to look into Lamport's "Structured Proof" idea. You are young, and you have come of age at a time that hypertext is readily available and widely used as a medium of academic communication.

Your question prompted me to ask this one, which resulted in me finding this reference after having lost it for 20 years. (So, thanks :). If this is something that you particularly want to dedicate time to, Lamport has laid out a path for you. You could

  1. Do all your proofs in the structured format that Lamport has laid out. He references using HTML for this many times, but I don't know if he has done a proof-of-concept. If not, that could be a contribution you make.

  2. When you come up to the point that you are able to do so, you can attempt to do your own study of existing literature, and find out whether his initial one-third number is actually valid. I don't know of anyone that has tried to validate that. (I don't think you would have much luck at this until you have a field that you know well enough to be able to understand existing proofs well.)

See also this redux by Lamport where he gives the same talk 20 years later or so, and also you can search around in his list of publications for references to "101" to see his various mentions of the structured proof concept as he discusses his other work.

If you start now, and you come up with a good way to do a "progressive disclosure"-type interface to a structured proof, it's possible that you could get good enough at doing proofs this way that it becomes second nature, and may actually be faster (it seems like it will almost definitely be more accurate). I'm saying "faster" is a possibility because it might allow you to clear the noise in your head about a sub-part of the proof, knowing that you can just come back to that bit of it. It could make it easier to think about the whole thing. All the speculative statements in this answer are just that--speculation, I haven't tried any of this myself.

I feel like this is a humanly possible effort, and if this question about the soundness of mathematics is something you are highly interested in (I phrase it that way because it will undoubtedly be a lot of work, and an uphill battle against culture and norms), this is potentially a way you could do something about it.

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Sometimes these collapses happen. Ole Peters and Murray Gell-Mann have a 2016 paper about a mistake in Expected Utility Theory, which they track down to a mistake in a Bernoulli paper from 1738.

I'm an outsider looking in, but it sure looks like much of that entire field is in serious trouble. There's a post on this here: https://ergodicityeconomics.com/2018/02/16/the-trouble-with-bernoulli-1738/

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  • $\begingroup$ While I know almost nothing about economics, my impression is that this critique is not accepted by mainstream economists. $\endgroup$ – Andy Putman Aug 18 at 22:21
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    $\begingroup$ There is no mistake in EUT, and it is not in trouble from the Peters and Gell-Mann paper. Mathematically, the theory is pretty easy and well-understood, so a mathematical mistake is about as likely as a mistake in the proof of the Sylow theorems. Conceptually, while the original von Neumann-Morganstern approach assumes that you just know the probabilities ex-ante, there's a separate formulation by Savage that leads to expected utility where probabilities are purely subjective beliefs, and a separate formulation by Anscombe-Aumann with a mixture of subjective and objective probabilities. $\endgroup$ – arsmath Aug 19 at 10:11
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    $\begingroup$ EUT isn't even state of the art in economics, because it doesn't describe how people actually behave. But this is not a mathematical or conceptual error, but rather an empirical failure. Ironically, the Peters-Gell Mann paper does even worse along this dimension -- their preferred variant implies people would be willing to take much bigger risks than they really are. $\endgroup$ – arsmath Aug 19 at 10:30
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    $\begingroup$ They also have a misapprehension that bounded utility functions are ruled out in economics. The von Neumann-Morgenstern axioms do give you bounded utility functions (for technical reasons), but unbounded utility (such as the log utility that is their preferred variant) is routine. $\endgroup$ – arsmath Aug 19 at 10:34
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    $\begingroup$ Bernoulli's error documented in the blog is unrelated to the main claim of Peters and Gell-Mann, namely that one-time transactions are the wrong model for real-life economic decisions, right? $\endgroup$ – darij grinberg Aug 19 at 14:29
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Why the whole mathematics remains so sound, even though humans are imperfect and quite often produce errors? Are my explanations above correct?

Who says that it is wholly sound? If most of your premises are mostly correct, perhaps we can expect to be mostly right in our conclusions most of the time. This does not imply total correctness.

In order to answer your questions about the prevalence or extensiveness of corruption through a lack of rigor, we would need to compile a "branching and reuse factor" for propositions in a logic tree. This would be a very interesting project but time has not yet permitted at least for me. One could compute for a given axiom or premise the number of downstream claims for a given proof that are affected, so at least this metric defines a simple possible avenue for investigation.

All claims are atoms with respect to their truth value. A given claim is either true or false. Any claim derived from a false parent claim (including an axiom) is suspect.

Random sampling is a reasonable strategy for an under-resourced organization to "spot-check" a corpus it deems too large for thoroughness. However, we should admit that in practice, many errors go unnoticed due to this practice, and systemic blind spots also often develop, so that the sampling isn't truly random. This is more problematic in non-applied fields because the errors are more difficult to detect through sanity checks.

If a theorem turns out to be wrong, then mathematicians will try to correct (if possible) all the results depending on that theorem. How hard is this job? Isn't it very tedious and frustrating? I want to hear some personal stories.

Updating all downstream dependencies can be either monumental or trivial in terms of the logic, depending on the nature of the discovered error, and whether a solution adds complexity or simplifies. In terms of the workload, either way it is very large. I work in computer programming, and one incorrect assumption can mean that you've designed the entire thing wrong and a re-design is needed to correct the fatal flaw. A single logical contradiction ruins an entire project; as in the infamous Mars lander metric conversion error.

As an undergraduate student, I want to know if anybody who is much wiser, older, or experienced, had the same fear as mine. (Again, I want to hear some personal stories.)

I had the same fear and completed a master's degree that didn't shake the foundations too terribly, but I don't know about your other qualifications. There is much more of work to do to make our mathematics and reasoning sound. An example would be the Barber Paradox. As far as I know, it has no official or widely accepted explanation or solution, but it proves that at least one of the axioms of reasoning used to produce it must be incorrect, because it yields a contradiction given only a simple and "reasonable" supposition. What I have found is that "flat" math is simply insufficient to model reality, which comprehends dynamical systems, hence the contradiction. Any proof that is derived from the erroneous premise that causes the contradiction should be reworked with a corrected premise.

As far as I am aware, the state-of-the-art in mathematics teaching has not corrected this fundamental and glaring error. There exist numerous theories and "fringe" ideas. The fact that an idea is on the fringe with respect to popularity does not disprove its validity or its philosophical soundness.

As an undergraduate student who will get into a graduate school in the near future, I want to get some advice. Should I stop worrying and believe the authors of the books and articles I read? When should I check all the details, and when should I just accept the theorem as given?

No, don't stop worrying, don't just follow the feet in front of you, and especially when your work depends on some prior work (it always does), you should check the work you are building on. You should check the foundations of the science you are using. The greatest breakthroughs can only be found after we fix our fatally flawed assumptions. Otherwise we're just building a castle in the air. You might still find some handy connections that are consequent from what is sound in the logical base, but the best discoveries will be missed so long as your first premises are impure and unsound.

One of your biggest hurdles will be to get buy-in from advisers. Most of them probably won't trust you to tinker with the foundations of science and will probably ask you just to go the blind trust route. For purposes of gaining industry support, it would need to be demonstrated that your innovation has a significant practical impact, otherwise I don't think most people will care.

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  • $\begingroup$ Not really "blind" trust, considering that there are some reasons to trust... $\endgroup$ – paul garrett Aug 19 at 14:24
  • $\begingroup$ It is blind in the sense that you don't really know what you are getting, although some indications might look good, lack of thoroughness rightly leads to uncertainty, and that should be unsettling. If peer reviewers were to map out all of the areas they checked for soundness and leave big orange flags on the areas they hadn't checked and on all unverified assumptions, we'd likely have a much better idea of what we are getting (It still wouldn't completely relieve the possibility of human error). Putting a stamp on the whole thing as acceptable is to turn a blind eye to those possible flaws. $\endgroup$ – pygosceles Aug 19 at 14:29
  • $\begingroup$ I feel that the downvotes are justified due to the misinformation about incompleteness (the "Barber Paradox" is not something that needs to be answered, it's an illustration of Gödel's theorem--the fact that I can pose a question that can't be answered doesn't invalidate any of the questions that can be answered. We're not throwing out English simply because the paradox can be presented in English.), but it's unfortunate because the idea of a dependency tree of theorems is a very interesting one. $\endgroup$ – msouth Aug 26 at 14:35
  • $\begingroup$ The Barber Paradox is simply a paradox. It is not necessarily the case that every paradox is an illustration of Gödel's incompleteness/inconsistency theorem. A paradox can arise from any contradiction in one's premises. Only a subset of paradoxes can be attributed (loosely) to Gödel's theorem, and Gödel's theorem only applies to a subset of rational systems. Therefore questioning the rational basis of a set of axioms that includes a contradiction is valid; otherwise the OP's question would be trivial to answer. See math.stackexchange.com/a/1590932/695864 $\endgroup$ – pygosceles Aug 26 at 15:29
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Large areas of mathematics do collapse, now and then. And this, since mathematics exists.

When such things happen, no one really knows why. The last experts in these fields are usually quite evasive, with explanations of type "perhaps not very interesting" or "does not lead anywhere" and so on. Then activity stops, and since these dying areas are often technically very complicated, no one gets back to them, afterwards. Maybe some of these areas simply collapsed because they are wrong? Who knows.

In any case, Calculus 1 and Calculus 2 seem correct, by some kind of magic. So with the definition "serious math = Calculus 1 and 2", math does not collapse, indeed :)

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