On Thu, Dec 15, 2011 at 12:19 PM, Joe Niederberger <firstname.lastname@example.org> wrote: >> The ability to generate proofs is quite different from >> the ability to work backward from a conjecture still in > need of proof. > > You obviously didn't get the import of Godel's Lost Letter. Or, you have actually proved P != NP yourself. Congrats, enjoy your new found fame and riches - when's the big announcement? >
That wasn't on my required reading list. I have fame and riches to spare so that's not of interest either.
> But, while you are still slumming here - could you spell out for us slumdogs exactly what is that difference Kirby? > If you mean that mathematicians don't work by simply listing out proof in such a sequence, I'm afraid there's no prize for that. But its certainly not clear that they don't work by *some* algorithm, hidden from us though it may be by that damned black box. >
The difference is you can't just plug in a well known conjecture, such as this thing you hope to prove, and get back a proved theorem necessarily, even in Euclidean geometry, let alone number theory. We don't have any software that does that effectively.
We have ways of churning out endless reams of things called "proofs" in various languages such as WFF 'N PROOF, if which I am an acknowledged past master -- but who cares? Not me.
> Regardless, there are many logical consequences to the simple fact that the proofs of a formal system are enumerable, your observations notwithstanding. > > Joe N
I guess I'm more in the Sir Roger Penrose camp. He sees computers as blindly dumb compared to humans because the latter have a track record of doing non-computable things. You can imagine a "black box" with an algorithm inside if you like, but it's just a fantasy, science fiction.
I don't like science fiction that does too much to mislead the public. I get the joke with The Turk and in retrospect totally admire the perpetrators of that hoax. On the other hand, a lot of these TV cop shows like NUMB3RS inspire undeserved awe of disciplines that are nowhere near as evolved as the screenwriters make it seem. But do mathematicians complain? No, of course not. Like doctors and lawyers, they like to be lionized, and when the TV show is over, the math teacher turns to his family and says: "yep, that's what I do for a living, in case you wondered."
It's fun to have a superman complex. But a problem for many students today is they grow up watching 'West Wing' and other such fiction, and then think the day job is supposed to correspond, as if Hollywood fantasies were the real world. That leads to major confusions. I'm not interested in encouraging them too much (a little, but not too much).