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Are Computers Ready to Create Mathematical Proofs?

Posted by michael on from the GIGO dept.

DoraLives writes "Interesting article in the New York Times regarding the quandary mathematicians are now finding themselves in. In a lovely irony reminiscent of the torture, in days of yore, that students were put through when it came to using, or not using, newfangled calculators in class, the Big Guys are now wrestling with a very similar issue regarding computers: 'Can we trust the darned things?' 'Can we know what we know?' Fascinating stuff."

4 of 441 comments (clear)

  1. Create vs. Verify by Squeamish+Ossifrage · · Score: 5, Informative

    The headline does a slight disservice in describing the article that way: Whether or not computers can create proofs isn't an issue. The problem comes when the resulting proof is too involved to be verified by a human, and so the computer's work has to be trusted.

  2. It was a big help with the 4-color map proof... by The+Beezer · · Score: 5, Informative

    and there are already programs out that help with this. Here's one for example...

  3. Re:Can someone elaborate on... by stephentyrone · · Score: 5, Informative

    It's not hard to *see*. It's hard to *prove*. Very little of mathematics is consumed with proving deep, mystical statements that no one would ever anticipate to be true. Much (maybe most) of mathematics is built around proving (relatively) obvious things. Why bother? because sometimes, relatively obvious things turn out to be false, and there's no way to know that they won't until you've prooved them true. In general, showing that discrete (or semi-discrete) phenomena are optimal is fairly tricky; you can't just appeal to calculus to optimize some function. You often have to somehow break the search space up into a bunch of disjunct cases that span all the possibilities, and be able to prove that they span all possibilities. Then, if you're lucky, you can use some kind of calculus-type argument on the continuous spaces you're left with.

  4. Re:Rumsfeld, anyone? by Tower · · Score: 5, Informative

    Actually, this is three of the four quadrants of knowledge...

    KK | KD
    ___|____
    |
    DK | DD

    So, you can:
    Know that you Know something
    Know that you Don't Know something (the second most common)
    Don't Know that you Don't Know something (most things fall in this category for most people)
    Don't Know that you Know something (the most interesting of the categories)

    Big, huge, red tape operations can easily fall into the latter category (DK)... since it is rather easy for a group to obtain knowledge, yet be unaware of it [political commentary omitted].

    --
    "It's tough to be bilingual when you get hit in the head."