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

Pour y voir clair, il suffit souvent de changer la direction de son regard

www.xkcd.com/386/

www.xkcd.com/386/

^{27}drunkscriblerian26th Feb 2011 07:18:22 PM from Castle Geekhaven , Relationship Status: In season

Street Writing Man

@U: despite not being a math major, I get the joke. d >.< b

*If I were to write some of the strange things that come under my eyes they would not be believed.*

~Cora M. Strayer~

^{28}storyyeller26th Feb 2011 07:19:09 PM from Appleloosa , Relationship Status: RelationshipOutOfBoundsException: 1

More like giant cherries

Yeah, the professor should have said "You're almost certainly mistaken, because I have two proofs which I've carefully checked. As an expert on the subject and in particular the material I just covered, the Bayesian probability that I am in fact wrong is extremely small."
But that takes too long to say.

edited 26th Feb '11 7:20:52 PM by storyyeller

NOT THE BEES

What are statistics, if not many personal experiences?

edited 26th Feb '11 7:20:21 PM by Pykrete

^{30}drunkscriblerian26th Feb 2011 07:20:48 PM from Castle Geekhaven , Relationship Status: In season

Street Writing Man

@Pykrete:
What I'd like to tell a lot of people, but I know they wouldn't listen.

*If I were to write some of the strange things that come under my eyes they would not be believed.*

~Cora M. Strayer~

NOT THE BEES

It should be noted that one should be careful not to equate them haphazardly — one's personal experience may not be indicative of the general pool of them, and may be the exception to the rule.
Still, those statistics

*are*coming from*somewhere*.I've heard people say they've had "counter-examples" in a math class and be wrong. Unfortunately I dunno any specific examples, but it's very easy to think you've disproved something that you actually haven't.
Besides that, anecdotal evidence is entirely useless as evidence. Anecdotes aren't totally useless as

*arguments*, but in the way you see them most of the time, as*evidence*, they are complete and total crap. EDIT: A more thorough explanation, now that I think about it: There are two types of claims, as far as anecdotal evidence is concerned: Claims that something happens*often*, and claims that something happens*at all*. Anecdotal evidence is totally useless to prove "often" because your anecdote could be entirely true and yet your claim could still be wrong easily; you may well be an anomaly. It might seem like anecdotal evidence would be useful to prove "at all", but in fact it isn't because things you need to prove happen*at all*are things like "ghosts exist" which if they really did happen at all would have much more solid proof than the account of some guy on the internet. Besides that, with anecdotes there's always the solid chance that the anecdote isn't actually true; not that the person is lying but that they misremember something or they overestimate how often they've heard something or any number of other problems that can happen when you store your information in your brain and nowhere else.edited 26th Feb '11 7:31:06 PM by BlackHumor

I'm convinced that our modern day analogues to ancient scholars are comedians. -0dd1

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

Pour y voir clair, il suffit souvent de changer la direction de son regard

www.xkcd.com/386/

www.xkcd.com/386/

Besides that, anecdotal evidence is entirely useless as evidence. Anecdotes aren't totally useless as arguments, but in the way you see them most of the time, as evidence, they are complete and total crap.

The problem that I run into, however, is that there exist certain individuals on the internet who confuse arguments with evidence. And vice versa. There's plenty of people out there that refuse to listen to the anecdotes pertaining to the negative (and smaller) percent, and wind up completely dismissing it entirely.
For example: 83% of Americans are above the poverty line. I tell them [anecdote about living in poverty] and because it's statistically unlikely to happen, they determine that I'm lying out my ass.
"I don't know how I do it. I'm like the Mr. Bean of sex." -Drunkscriblerian

NOT THE BEES

Uchuu: Oh certainly. But it can give you quite a useful perspective with which to look for secondary factors or other explanations within your statistics.
For example.
I conduct a study that shows 50% of traffic accidents involve SUV's and conclude that the car model is extremely unsafe. Someone hands me an anecdote that the SUV driver they met on the road this morning was watching Madagascar on the road and merging without looking. I'm inspired to do a followup study on driver recklessness and find an unusual amount of it among SUV drivers. I revise my conclusions accordingly, as showoff reckless drivers tending to purchase SUV's in the first place could correlate with its accident rate better than the make of the vehicle.
An anecdote isn't proof, but it is an anonymous tip that you might want to look for something else.

edited 26th Feb '11 7:38:52 PM by Pykrete

^{36}storyyeller26th Feb 2011 07:37:36 PM from Appleloosa , Relationship Status: RelationshipOutOfBoundsException: 1

More like giant cherries

^^^ I personally love proving teachers wrong, but I've been wrong many times before as well. And that includes thinking I've found a counter example when I really haven't.
While either could easily be the case, I'd say that based purely on the wording in the problem, I think the student is more likely to be wrong.

edited 26th Feb '11 7:37:53 PM by storyyeller

I think you are missing the point. The truth value of the original statement is undecidable, but giving out a second proof is NEVER the correct way to disprove a counter example. The reason? Incompleteness theorem and consistency. Read it. Understand it. (If you don't want to peruse it, the short version is that any sufficiently complex mathematical system could have two contradictionary statements that are BOTH "true", i.e. that both follow from the basic axioms)

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. :)

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

Pour y voir clair, il suffit souvent de changer la direction de son regard

www.xkcd.com/386/

www.xkcd.com/386/

Uh, no; that would mean the sophisticated mathematical system is essentially crap. Because two contradictory statements actually being true is obviously not possible meta-mathematically, one of them must actually (meta-mathematically) be false, which means that it's possible to prove a false statement true in the system, which means the system is worthless.
What Godel proved is that you cannot have a system that is consistent (no two contradictory statements can be proven true) and complete (all statements that are true can be proven).
Most mathematicians prefer, and most mathematical systems are, incomplete rather than inconsistent.

I'm convinced that our modern day analogues to ancient scholars are comedians. -0dd1

I really would like to argue about this in German -.-
There are two parts about the incompleteness theorem - one addresses the completeness, the other one says if you have any system that includes the natural numbers with addition and multiplikation, then it cannot be proven that this system is consistent, i.e. that it doesn't lead to contradictions (second incompleteness theorem). That means we don't know that it doesn't lead to contradictions, it could. So different valid logical deduction could be contradictionary.

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.

[edit]

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.

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.

[edit]

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

Pour y voir clair, il suffit souvent de changer la direction de son regard

www.xkcd.com/386/

www.xkcd.com/386/

^{40}storyyeller26th Feb 2011 08:55:00 PM from Appleloosa , Relationship Status: RelationshipOutOfBoundsException: 1

More like giant cherries

The thing though is that consistency can be ignored for all practical purposes, because if the natural numbers are not consistent, then we've basically reached the apocalypse anyway.
It's like in physics where you can never be absolutely sure that there aren't invisible gremlins messing with your particle detectors, but you ignore the possibility anyway.
Also, Constructivism sucks.

edited 26th Feb '11 8:55:30 PM by storyyeller

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