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?
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$\begingroup$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.
$\endgroup$ 5 $\begingroup$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.
$\endgroup$ $\begingroup$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.
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