I have checked both the question “Examples of common false beliefs in mathematics”, which was asked over on M.O. back in May 2010, and all answers proposed there. Really I would like to know the mathematical reasons why this question is so highly rated (+593 as of Dec 2016) although it does not refer to any open problems in mathematics.

My question here: Why, in the view of the mathematical community, is this question of false beliefs important, particularly for mathematics?

Thank you for any help.


closed as off-topic by bof, suomynonA, Rohan, user91500, Alex M. Dec 31 '16 at 11:19

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question is not about mathematics, within the scope defined in the help center." – bof, suomynonA, Rohan, user91500, Alex M.
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    $\begingroup$ I think a lot of people hanging out here have had to teach math at some point in their life. Hence the importance of the question, some mistakes occurring repeatedly $\endgroup$ – Vincent Dec 5 '16 at 17:21
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    $\begingroup$ I would add that whatever fallacy leads people to bad intuition is worth thinking about. After all, a seductive error probably comes up over and over in various contexts and you'll want to spot it. $\endgroup$ – lulu Dec 5 '16 at 17:23
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    $\begingroup$ however it's not touched any open problem in mathematics There are several big-list entries about open problems on MO, see here for example. $\endgroup$ – dxiv Dec 5 '16 at 17:35
  • $\begingroup$ I meant in the list montioned in the question" mathoverflow.net/questions/23478/…" , don't touch any open problem and don't touch any of them $\endgroup$ – zeraoulia rafik Dec 5 '16 at 17:44
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    $\begingroup$ Right, I don't see why a question about common false beliefs would touch open problems. The link I posted goes to a list specifically about open questions, where those are discussed. $\endgroup$ – dxiv Dec 5 '16 at 17:55

I think that your question needs to be answered in two parts:

  • Why is the question highly rated?
  • Why does it not refer to open questions in mathematics?

I shall answer those in reverse order.

Why does it not refer to open questions?

I think that you ask this because of a misunderstanding: the question is specifically about beginners’ mistakes (for some advanced value of “beginners”), which they abandon “when their mistake is pointed out”. It is, therefore, not about conjectures which were popularly believed and then disproved; nor is it about conjectures which may yet be disproved.

N.B. The odd answer mentions a misconception that something is an open question!

Why is still a good question (despite not referring to open questions) is the subject of the next part.

Why is the question highly rated?

The questioner says they are interested in “beliefs many intelligent people have while learning mathematics, … and … why they have these beliefs”. They also call it “more like a psychological question than a mathematical one” and ask for cases from “reasonably advanced mathematics” where “the reasons they are found plausible are quite varied”.

As mentioned in comments, knowing of these misconceptions, how they arise and how they may be shown wrong can be helpful:

  • To avoid comparable pitfalls – in some cases the very same one – in one’s own thinking.
  • To understand how one’s pupils go wrong and to set them right again.

I think, moreover, that people sometimes find it entertaining and even satisfying to look back on their own and others’ mistakes because:

  • It reminds them how far they have come.
  • Spotting an error is a sort of problem-solving.
  • Identifying the point at which an argument fails may improve one’s understanding.
  • The flash of understanding is comparable to the delighted surprise when one understands a joke; more so because of the erroneous twist than when one finds or follows a valid proof.

One reason is because whereas bad teachers ignore the reader's psychology and just do definition/theorem/proof ad nauseum, good teachers on the other hand adopt a more holistic approach in which a reasonable level of motivation is given, and common misconceptions are discussed and dealt with.


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