Personal Experience vs. Statistical Evidence:
The point is that a counter example indicates that the proof is wrong. Saying "I have a second one" would only mean that that one is wrong as well. In order to refute the student he would have to argue that his example is in fact not a counter example.
The professor is wrong that he thinks a second proof would actually address the problem.
edited 26th Feb '11 7:12:59 PM by Uchuujinsan
edited 26th Feb '11 7:20:52 PM by storyyeller
edited 26th Feb '11 7:20:21 PM by Pykrete
edited 26th Feb '11 7:31:06 PM by BlackHumor
That's not really how math works. Seriously. (especially because both the proof and the counter example could be valid under the used set of assumptions, a reason why constructivism exists.) And if you argue by experience, in my experience the student actually is right because the professor forgot to add a condition to his initial statement. Favored candidates are forgetting to limit the theorem to finite vectorspaces. @Pykrete
The difference lies in the amount of "pollution" you avoid with formalizing the process. Personal experience is often not analyzed with such a scrutiny.
edited 26th Feb '11 7:29:39 PM by Uchuujinsan
edited 26th Feb '11 7:38:52 PM by Pykrete
edited 26th Feb '11 7:37:53 PM by storyyeller
I honestly wouldn't treat even most of the strongest statistical research as more than a tip hinting at the right direction. The reason for this is the correlation/causation problem. A statistic can only show correlation. Not disaggreeing with you, just extending your statement. :)
edited 26th Feb '11 7:59:56 PM by Uchuujinsan
That's the main point of mathematical constructivism (I linked it earlier), the reason why it exists - it doesn't use proofs by contradiction. Normal math just assumes(!) that the used set of axioms are consistent. If they are consistent, that means that the complement of a contradiction must be true. (if we know the proof to be true, because we checked, and the existence of a counter example is a contradiction, then the complement, i.e. no counter example exists, must be true). This does not work if the axioms are not consistent and we normally don't know if they are.
Don't confuse the fact the first incompleteness theorem requires(!) either a contradiction or incompleteness with the different statement that in an incomplete system the existence of a contradiction is only undecidable and no longer required.
edited 26th Feb '11 8:31:24 PM by Uchuujinsan
edited 26th Feb '11 8:55:30 PM by storyyeller
You need to Get Known to get one of those.