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I just started learning college mathematics and one of the things I don't like is giving proofs by counterexamples. My question is how is disproving by giving counterexample is seen by advanced 'mathematicians', is it a bit of an 'easy way out' when we are looking to prove and disprove something. Shouldn't we look for a general structure because of which the statement is not true?

For eg. while proving that every subring of a Notherian ring may not be Notherian, I never saw a general proof, all I saw a bunch of counterexamples. Shouldn't we look for what reason a general subring of Notherian ring fails to be Notherian?

Andrew Miller
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It's not regarded as an easy way out in any pejorative sense. It serves at least two important functions: Efficiently tells you the truth (falsity) of a claim; and it often informs intuitions. There is a very good textbook called Counterexamples in Real Analysis, which is excellent in providing you a sense of what certain terms mean by showing you what they don't imply. You can otherwise be very confused about how much information is contained in saying that one object has a given property.

Of course counter-examples are in a sense very incomplete. You will not understand Real Analysis by just reading the book of counterexamples. But it certainly gives you a healthy dose of circumspection that you would not get as efficiently by other means.

Now certainly you would like to know "the reason" why some thing has counter-examples, if that is worth-while and a good use of time. But some "behavior" is "not pathological" and so there really isn't, as far as we can tell, a reason for its behavior. Is the distance between primes always 2-3=1? No, because 5-3 = 2. Why? Meh. If you can figure out the exact pathology of the distance between primes, collect your infinity billion dollars.

Addem
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    There are actually a bunch of books "Counterexamples in ____", the most famous being "Counterexamples in Topology" by Steen and Seebach Jr. – Batman Mar 10 '14 at 18:30
  • Well I think the prime example misses the point as you can easily describe all counterexamples. – user2345215 Mar 10 '14 at 18:41
  • Is an easy description the same as giving a reason? I don't think so, unless you're suggesting that one prove the distance between primes p<q is 1 just if p=2, q=3. But if you insist on having something not so easily described, the example can be easily tweaked: Is the distance between odd primes always greater than 2? No, because 5-3=2. But what is the pathology for the (not easily described, except as "pairs of primes of distance 2") counter-examples? – Addem Mar 10 '14 at 18:49
  • Ok, that is very fair. For disproving your "Prime Distance Example" I will also give the same counterexamples. But whether the subring of a Noetherian ring is whether Noetherian or not and in many other problems I would like to go for the bigger picture and would like to give a general proof. I don't know - giving a counterexample in these cases is never going to satisfy me...but I admit that might not be good use of time – Andrew Miller Mar 10 '14 at 18:52
  • Yeah, I think at this point, there are two factors: what is your class getting at, and what are your mathematical tastes. Like you, I tend to strongly prefer completeness to other values like breath or rigor, although naturally I like both breath and rigor (and other things like beauty). So I'd probably be with you in wanting a more complete analysis of the phenomena. But your professor may have other goals for the class, and it's hard to say that his goals are wrong, rather than just different from yours. – Addem Mar 10 '14 at 20:49
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Really this all depends on how you see the world.

Firstly, finding a counterexample can be difficult - it can be an exercise in mathematical imagination. And it can show the way forward - the history of attempts to define continuity, or to prove the continuum hypothesis, for example, shows that counterexamples can open the way to fruitful mathematical ideas.

Secondly, trying to construct a counterexample can lead to a better understanding of the problem even to the extent of generating ideas for a proof.

The statement "a subring of a Noetherian ring need not be Noetherian" only needs one example for it to be true. If it were a fruitful idea, one could define and study the class of "Noetherian Rings all of whose subrings are also Noetherian". It turns out that submodules seem to be more significant than subrings in the Noetherian context - in that modules encapsulate the issues which the Noetherian property most naturally clarifies and handles.

Mark Bennet
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Proof by counterexample is great! It is easy in the sense that you only need to find a single counterexample and you're done, but what's wrong with that?

I don't think there is any virtue in avoiding disproving something with a counterexample. It is simply one of the tools in the toolbox. To try to avoid using it would be to make things harder on yourself.

Carser
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    Ok, you disproved the statement, aren't you bit curious about why the statement is not true? – Andrew Miller Mar 10 '14 at 18:30
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    @AndrewMiller Sometimes there is more to the story. Sometimes there just isn't. If it's a good counterexample, it might already give you an idea of whether it's worth investigating further. – Joshua Pepper Mar 10 '14 at 18:31
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    I agree with @JoshuaPepper. Although I understand what you're saying about curiosity, sometimes things are just not true, and understanding HOW not true they are is not necessarily useful. – Carser Mar 10 '14 at 18:34