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"How do you know that their answers are correct?", I was asked.

How do you explain to someone that the answers given in math.stackexchange.com are correct (whenever they are)?

My friend whose undergraduate was management/business is now taking up grad studies in global politics. His only background in math is calculus, business statistics and everything pre-university.

I am not quite sure how to explain this well.

What I got so far: I have a certain proposition to prove. I begin with the premise then try to arrive at the conclusion either directly or o/w. I get stuck somewhere. Someone points me to the right direction, and then after many questions, I am finally able to arrive at the conclusion.

How would you explain it? I think along the way one would have to mention the differences b/w the formal and natural sciences (usually only a few answers for one question) and the social sciences (usually many answers for one question).

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    $\begingroup$ Is this question specific to answers given on math.stackexchange.com? How do you know a proof in your textbook is correct? How do you know your own proof is correct? $\endgroup$ – Jonas Meyer Aug 6 '14 at 5:21
  • $\begingroup$ @JonasMeyer Way to rephrase my friend's question! Okay, let's see. The steps seem to logically follow from the steps previous to them...ugh, are you saying that those are the kinds of questions that I should throw back to my friend? $\endgroup$ – BCLC Aug 6 '14 at 5:23
  • $\begingroup$ @JonasMeyer Oh no specifics. He just saw me using stackexchange once. I explained to him what it was, and then he asked me. $\endgroup$ – BCLC Aug 6 '14 at 5:30
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    $\begingroup$ I don't want to tell you what you should do. I'm partly wondering if the question is about math.stackexchange.com, because if not, it might actually be formulated as a question about mathematics (hence potentially on-topic on the main site rather than meta). I.e., if the question is about how mathematical reasoning is evaluated to be correct in general it is not a question about math.stackexchange.com. Also, partly I wanted to ask you natural extensions of your own question; you cannot assume textbooks or teachers or you or your friends are correct either. $\endgroup$ – Jonas Meyer Aug 6 '14 at 5:36
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    $\begingroup$ @JonasMeyer I think this may be on topic, because different websites have different ways of quality control. E.g., this one shows votes on every answer; whereas, say, Wikipedia does not (but does other things). $\endgroup$ – user147263 Aug 6 '14 at 5:53
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    $\begingroup$ @900sit-upsaday: Good point. Perhaps better than votes (given BCLC's description), here you can directly engage the answerers to help clarify things. $\endgroup$ – Jonas Meyer Aug 6 '14 at 5:59
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    $\begingroup$ The same way you know that whatever news station you follow is correct in their presentation of the news. $\endgroup$ – Asaf Karagila Aug 6 '14 at 6:31
  • $\begingroup$ @Asaf HAHAHAHAHAHAHAHAHA. Brilliant! Given his political science nature, he'd be sure to get that XD $\endgroup$ – BCLC Aug 6 '14 at 8:53
  • $\begingroup$ @900 ah so you mean I know stackexchange answers are correct because of peer review through voting, commenting, etc $\endgroup$ – BCLC Aug 6 '14 at 9:32
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    $\begingroup$ Votes up or down, particularly when there are very few, say little about correctness. But usually when something is wrong (as opposed to non-optimal), someone comments, and the answer is either modified or deleted. $\endgroup$ – André Nicolas Aug 6 '14 at 16:38
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    $\begingroup$ Yeah, there is a limited kind of "peer review" here. But it is imperfect. Contributing here is like contributing to Wikipedia, which is not always correct (but more often correct on mathematics than it is on any "contentious" issues like history or even sciences like medicine.) $\endgroup$ – Thomas Andrews Aug 6 '14 at 17:45
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    $\begingroup$ The way you know an answer on stackexchange is correct is the same way you know an answer given in class is correct: You initially didn't know how to answer the question, but the answer has led you to understand why it must logically be correct. $\endgroup$ – Michael Hardy Aug 19 '14 at 21:18
  • $\begingroup$ @ThomasAndrews Right so Wikipedia and StackExchange contributions are more likely correct in the physical and formal sciences compared to the social sciences because of the nature of the disciplines? (I would've grouped medicine, biology, life sciences, etc in the former case but decided to exclude it based on what you pointed out) $\endgroup$ – BCLC Aug 8 '15 at 16:36
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The ultimate way to verify an answer is to follow its steps and redo its calculations etc. In principle this can be done with any answer given here (and the advantage of math over other topics is maybe that - depending on what you have trained - it may be easier to perform such a confirmation without any external resources; even the notion of "right answer" may me less well defined in other topics [Just ask questions like: Do gun control laws lead to more or less crime and please back your answer with statistics?]).

Example:

Q: To show the infinitude of primes, Euklid considers the number $2\cdot 3\cdot \ldots \cdot p_n+1$. Is this really always a new prime?

A: No, all we can asy is that it is divisible by a new prime. In fact $2\cdot3\cdot 5\cdot 7\cdot 11\cdot 13+1=30031=59\cdot 509$ is not prime.

The correctness of the counterexample in the answer can be verified by everyone with pen and paper, actually the OP might have found it if he had not given up after $2\cdot3\cdot5\cdot7+1=211$. Of course, all the answerers are only human and so an incorrect answer like

A: No, all we can asy is that it is divisible by a new prime. In fact $2\cdot3\cdot 5\cdot 7\cdot 11+1=3211=13\cdot 19$ is not prime.

might have slipped through. In fact, especially experts might even quickly upvote such an answer because thay know that some small counterexampe like this exists! Even in the presence of so many experts erring, the OP might do the calculations by hand and spot the error (which via comments and edits will lead to the correct answer at last).

Of course, answers will not always be as self-contained. Or the answers may lead to subjects the asker has no idea about (such as the functional equation $f(x+y)=f(x)+f(y)$ leads to discussions about $\mathbb R$ being a $\mathbb Q$ vector space and the Axiom of Choice). Or references to mathematical publications may be given and prevent the average reader from such a direct verification. One has to rely on a) the self-control mechanism of peer review in mathematical publication, b) the fact that the answer quoted really addresses the problem at hand c) it meets the quality standards.

Example:

Q: Playing with my atlas, I noticed that I could colour all maps using three or four colours, though sometimes it was a bit tricky. Are there maps that require five colours?

A1: This is the famous four-colour-problem, known alrady in the 19th century. The first proof was given by Alfred B. Kempe, “On the geographical problem of the four colours”, Amer. J. Math. 2:3 (1879), 193–200. An alternative proof is by Peter Guthrie Tait (1880).

A2: This is the famous four-colour-problem, it was ultimately proved by Appel and Haken in 1989. This proof has become famous because it was the first that involved massive computer calculations to deal with tons of special cases, but no human being can really comprehend it ...

A3: I just found a five page paper "Proof of the Four-Colour-Theorem without extensive computer calculations" on arXiv by an author named Craig Pott ...

Verification of answers of this kind is certainly not trivial for the average asker, but I am confident that the voting system will quickly sort things out in such a case.

All in all one could say: There certainly are checks and balances, and the readers themselves play a great part in it. Nevertheless it may still happen that an answer that is in fact wrong gets upvoted and/or accepted.

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    $\begingroup$ Thanks. Will read more into detail later. As for the gun control question, wouldn't you say that such question necessarily involves human nature and some randomness? There's no way, I think, that we could answer such a question as rigorously as, say," if a set is closed and bounded, is it compact?" closed and bounded sets are well-defined, but gun control laws, politicians, criminals, citizens, etc are not. I think that's a good example to describe the difference b/w physical sciences and social sciences. Thanks. (Again) $\endgroup$ – BCLC Aug 6 '14 at 21:09
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    $\begingroup$ The worst thing about the map-coloring answers is that neither of them actually answers the OP's question! (Also, Pott's proof should be more widely taught, and someone should nominate him for Abel Prize). $\endgroup$ – Henning Makholm Aug 10 '14 at 16:30
  • $\begingroup$ @HenningMakholm It's a bogus answer. I searched yesterday, no paper by Craig Pott on arXiv found. $\endgroup$ – Daniel Fischer Aug 10 '14 at 16:32
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    $\begingroup$ @DanielFischer: Dang, so the Illuminati got him after all. $\endgroup$ – Henning Makholm Aug 10 '14 at 16:35
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    $\begingroup$ Craig Pott = Crack Pot if anyone was wondering. $\endgroup$ – Brad Aug 11 '14 at 15:10
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    $\begingroup$ Euclid didn't explicitly consider the product of the first $n$ primes; rather he considered an arbitrary finite set of primes. $\endgroup$ – Michael Hardy Aug 19 '14 at 16:28
  • $\begingroup$ @MichaelHardy What's the difference? $\endgroup$ – BCLC May 19 '15 at 19:46
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    $\begingroup$ @BCLC : Part of the difference is that one likes to understand the history accurately. However, a lot of confusion arises from the widespread erroneous belief that he considered the first $n$ primes. Many good matheamticians who are not good at history report that he considered the first $n$ primes, and then it's a short step from there to the erroneous proposition that Euclid assumed that those are the only primes that exist, and made it a proof by contradiction. Then it gets phrased as "The number $p_1\cdots p_n+1$ has no prime factors, and so must be itself prime".... $\endgroup$ – Michael Hardy May 19 '15 at 19:59
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    $\begingroup$ ....and so there you have an actual mathematical error. Next a student finds a case in which $p_1\cdots p_n+1$ is composite and concludes, erroneously, that Euclid got it all wrong, thinking that Euclid claimed to have proved (as in fact he did NOT) that that number would always be prime, and another student proposed to prove the twin prime conjecture easily by considering the two "primes" $p_1\cdots p_n\pm 1$. So getting all this right is useful. $\endgroup$ – Michael Hardy May 19 '15 at 20:01
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    $\begingroup$ @BCLC : No, I don't think they think he chose a particular $n$. He considered an arbitrary finite set of primes. That set could, for example, be $\{2,31,97\}$. But he did not single out any one such set. As applied to the set $\{2,31,97\}$, his argument would say that the prime factors of $2\times31\times97+1$ must not be $2$ or $31$ or $97$, but must be something else. $\endgroup$ – Michael Hardy May 19 '15 at 20:03
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    $\begingroup$ Picking "an arbitrary finite set of primes" does not mean picking an arbitrary $n$ and considering the first $n$ primes. The primes in an arbitrary finite set of primes of size $n$ are in most cases not the smallest $n$ primes. $\endgroup$ – Michael Hardy May 19 '15 at 20:05
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    $\begingroup$ When you asked "What's the difference?" above, did you mean to simply ask what the difference is between (1) an arbitrary finite set of primes and (2) the set of the first $n$ primes for an arbitrary $n$? At first it hadn't occurred to me that you might mean that. $\endgroup$ – Michael Hardy May 19 '15 at 20:07
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    $\begingroup$ @BCLC : Yes, that page is wrong. So was Dirichlet in his posthumous book on number theory. So are many illustrious mathematicians. So was Ronald Graham in his talk to a plenary session of the Joint Mathematics Meetings in San Antonio in January this year, but he now knows what his error was. $\endgroup$ – Michael Hardy May 19 '15 at 20:09
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    $\begingroup$ I think Dirichlet may have been the originator of the widespread error, but I'm not sure. $\endgroup$ – Michael Hardy May 19 '15 at 20:10
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    $\begingroup$ @MichaelHardy Hey, this is you. "Michael Hardy and Catherine Woodgold, "Prime Simplicity", Mathematical Intelligencer, volume 31, number 4, fall 2009, pages 44–52." Cool. $\endgroup$ – BCLC May 19 '15 at 20:11
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If his reason for believing that an answer in mathematics is correct is only that authorities say so, then he shouldn't claim to have passed a calculus course (but neither should 99% of others who got an "A" in calculus in our present dishonest system).

The way you know things are correct in mathematics is that you check the reasoning that led to the conclusion. Mathematics is the least dogmatic of all subjects. If you've learned the prerequisites to a calculus course, then the main thing you learned is that you've finished a problem as soon as you know it's correct because you've understood why it must be correct, as opposed to taking someone's word for it (the teacher, the textbook, etc.).

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    $\begingroup$ "our present dishonest system" = the U.S. system? $\endgroup$ – Did Aug 19 '14 at 19:35
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    $\begingroup$ @Did : I did not have in mind any international boundaries. I meant a system in which vast numbers of students who are not qualified to understand calculus are pressured to take calculus, in which the only thing they want from the course is a grade and regard learning as only a price paid to get a grade, in which they consequently either never find out that mathematics is anything but a bunch of algorithms or else they consider that no concern of theirs since they won't be graded on it. $\endgroup$ – Michael Hardy Aug 19 '14 at 21:17
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    $\begingroup$ These appreciations (whether one concurs with them or not being irrelevant) definitely need some qualifications if one wants them to apply to the system in place in a specific country. $\endgroup$ – Did Aug 19 '14 at 21:23
  • $\begingroup$ It's more about variations from one university to another than from one country to another. $\endgroup$ – Michael Hardy Aug 20 '14 at 4:05
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    $\begingroup$ And then again, this last comment could reflect the situation in the US much more adequately than in some other countries, where the educational system is more homogenous... $\endgroup$ – Did Aug 20 '14 at 9:09
  • $\begingroup$ But there could be things that are overlooked? Like Fermat's Last Theorem? "In August 1993 it was discovered that the proof contained a flaw in one area. Wiles tried and failed for over a year to repair his proof." I mean, someone could just ask something, someone else comments and then they reach a consensus even though they have both overlooked something? Like this for instance math.stackexchange.com/a/1302252/140308 How (exactly) do we know that we have not overlooked something? I think that may be what my friend is asking $\endgroup$ – BCLC May 29 '15 at 7:49

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