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This is motivated by a recent post on MSE which asks for fake proofs that there are only finitely many primes.

The tag wiki of says:

"seemingly flawless arguments are often presented to prove obvious fallacies (such as $0=1$). This is the appropriate tag to use when asking "Where is the proof wrong?" about proofs of such obvious fallacies."

From this I assume that the tag should be used when someone were given a proof to a false statement and were trying to figure out the flaw.

I voted to close that post, but retracted the vote since I am not so sure. If every answers to that post introduce a fake proof and point out why the answer is wrong, then it seems the post would be a useful one (as a duplicate target).

What is your opinion on that?

Edit: There are not many discussions on meta about this tag. The two I found here, here are concerned with the distinction between this tag and .

Depending on the discussion here, we might need to edit the tag wiki to reflect that.

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    $\begingroup$ Yes, they are off topic. If the question asked is "I was asked to explain the error in the following post... ", that's fine. But seeking erroneous proofs that are falacious seems entirely off topic. $\endgroup$
    – amWhy
    Commented Jul 13, 2021 at 21:33
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    $\begingroup$ The tag wiki for "fake-proofs" states that the tag is meant to find flaws in fake proofs of well known fallacies. It is not supposed to generate more such proofs. $\endgroup$
    – Paramanand Singh Mod
    Commented Jul 14, 2021 at 1:57
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    $\begingroup$ For what it's worth, one of the most popular questions on MathOverflow (905 net upvotes!) is mathoverflow.net/questions/23478/… (examples of common false beliefs in mathematics). This has not caused the site to become a circus. $\endgroup$
    – Gerry Myerson
    Commented Jul 14, 2021 at 7:28
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    $\begingroup$ @GerryMyerson: I really enjoyed that MO thread (at least those examples which were within my grasp). But that is under "big list". If the intent of question (on main) under discussion to get some nice examples we can add "big list" tag. $\endgroup$
    – Paramanand Singh Mod
    Commented Jul 14, 2021 at 8:52
  • $\begingroup$ @ParamanandSingh The current tag wiki do not discuss this use of the tag. This meta post is meant to discuss whether this post is on-topic. If yes, we may edit the tag wiki to reflect that. (I have just edited this meta post) $\endgroup$
    – Arctic Char
    Commented Jul 14, 2021 at 8:54
  • $\begingroup$ @ArcticChar: yes editing tag wiki may be an option (after a proper discussion). $\endgroup$
    – Paramanand Singh Mod
    Commented Jul 14, 2021 at 10:31
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    $\begingroup$ Despite the accepted answer I don’t think there can be a clearly selected answer so I would suggest the big list tag and perhaps community wiki (like the MO post) so that the points motivation doesn’t enter the picture. A good fake proof IMO is an interesting presentation of a surprising fact (ie excellent pedagogy), or a wrong extrapolation past the usual assumptions that we forgot or don’t bother to check (and it is very instructive to see why the assumptions are needed). So I would like the question to stay. $\endgroup$
    – Calvin Khor
    Commented Jul 14, 2021 at 13:03

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Honestly, this kind of question does not seem to make sense to me. Every fake proof does something invalid along the way, from which point you could deduce anything and everything, not just a single false conclusion. So if you actually allow this kind of question (despite it already being opinion-based and hence intrinsically off-topic), you would also have to allow questions for "fake proofs that every group is abelian" and "fake proofs that every field has characteristic zero" and "fakes proof that every set is non-empty" and "fake proofs that there is a real number whose square is negative" and ...

For example, just look at the currently accepted answer, which simply 'reduces' the 'problem' to the fake proof that everything is equal, which plainly has nothing to do with primes. So how then is it a good answer to "fake proofs that there are finitely many primes", as opposed to simply being an answer to "fake proofs of any kind whatsoever"?

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    $\begingroup$ +1. I had commented that one should not encourage users to manufacture fake proofs. Your answer details why one shouldn't. $\endgroup$
    – Paramanand Singh Mod
    Commented Jul 17, 2021 at 15:46
  • $\begingroup$ I am superficially aware that you can deduce anything from a contradiction. I don't know how to explain exactly how a "fake proof" is different. I don't think that the accepted answer is a "good fake proof". I would prefer if it had a faulty implication "higher up", rather than at the level of induction which would allow it to poke at some "higher mathematics". I guess other people (barring that OP) also feel the same since the other answer has many more votes, 5+accept vs 18 votes. (though, I suppose that one is more like "heuristics" than a "fake proof") $\endgroup$
    – Calvin Khor
    Commented Jul 19, 2021 at 7:04
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    $\begingroup$ @CalvinKhor: Votes on Math SE mean very little more than a popularity contest among mathematically untrained people. You will see a very different voting if only experts were to vote. Based on my logic background, I do know how exactly 'good fake proofs' are different, and the currently accepted answer in this example is definitely not one... Your intuition is not far off, since basically you are saying the same as me; namely the error in the fake proof has nothing to do with primes. $\endgroup$
    – user21820
    Commented Jul 19, 2021 at 8:12
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    $\begingroup$ @CalvinKhor: Even though, from a logic perspective, a false statement implies anything, in my experience that's not how people think about proofs in practice. There are numerous examples of fake proofs that warn the reader not to divide by zero, to check the hypotheses of the theorems they are applying, to not blindly apply rules outside their "domain of validity", and to not place too much faith in heuristic arguments. To me, there's a clear difference between these proofs and fake proofs such as $1=2\implies5=7$, which obviously have no instructive value. $\endgroup$
    – Joe
    Commented Jul 19, 2021 at 16:01
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    $\begingroup$ And Jair Taylor's answer is a very good example of a fake proof that does have instructive value. Of course, there's a degree of subjectivity to this, but to me the post about primes does not cross the threshold for being too opinion-based for this site. $\endgroup$
    – Joe
    Commented Jul 19, 2021 at 16:03
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    $\begingroup$ @Joe: The question has to be much more specific than it currently is, before it can be not too opinion-based. This isn't a site where only good answers are given, so a broad subjective question will likely garner bad answers (as we see it did here). $\endgroup$
    – user21820
    Commented Jul 19, 2021 at 16:20
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    $\begingroup$ @user21820: I understand your view. But the question did also receive good answers, and to me this outweighed the bad ones. In this context, a bad answer is just a fake proof without much instructive value, not something which is mathematically unsound. So I don't see how these bad answers would make the question as a whole detrimental to this site. $\endgroup$
    – Joe
    Commented Jul 21, 2021 at 12:45
  • $\begingroup$ @Joe: Bad answers mislead people. Since many have big trouble distinguishing good from bad mathematics, bad answers would be detrimental if they are highly upvoted, which has happened here. In any case, it is clear that what constitutes a good answer is a bit too subjective, which supports my point that the question is opinion-based. See, I don't even agree that the asker's first example is a 'good fake proof'. $\endgroup$
    – user21820
    Commented Jul 23, 2021 at 9:32
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The help page says, among other on-topic areas there are:

  • Understanding mathematical concepts and theorems.
  • Mathematical problems such as one might come across in a course or textbook.

Any modern "intro to proof" textbook has a nontrivial number of questions asking students to analyze incorrect proofs, precisely for the purpose of understanding better the proof techniques involved. Research papers will often include likely-but-wrong corollaries of theorems (and show why they are wrong!), so that the reader is warned against them and better understand subtle ideas.

Of course there is a danger in "random" wrong proofs. But it is useful pedagogically, as well as for understanding in general, so with proper guideposts to avoid abuse, it seems on-topic.

Alternately, if the consensus is that they are not on-topic, perhaps systematically asking them to post on MESE instead (with explicit education focus, however) could be acceptable.

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    $\begingroup$ I am not sure if these are welcome on MESE though. $\endgroup$
    – Arctic Char
    Commented Jul 22, 2021 at 13:53
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    $\begingroup$ That is a good question, but I presume that if they are properly worded with teaching intent, they would be. Just asking for a list of fake proofs without pedagogical context, probably not, that's true. $\endgroup$
    – kcrisman
    Commented Jul 23, 2021 at 1:41

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