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i half disagree. even informal proofs are often very long, but while they may omit some steps omitted, i wouldn't say they're ambiguous. it's just that writin
by auvrw 13y ago
i half disagree. even informal proofs are often very long, but while they may omit some steps omitted, i wouldn't say they're ambiguous. it's just that writing something for a human who can (and in fact will find it easier to) fill in some blanks themselves is different for writing something for a machine, which only understands "how to do" kinds of descriptions.
also, a tangent about Erdos: Erdos was really good at posing problems. if his proofs are concise, it could have something to do with the problems he chose to work on. i'm not saying there's no book, i'm just saying that the most perfect, concise proof of some facts might still turn out to be very long.
- tel 13y agoCertainly agree about Erdos/the Book. It's not an effort to prove that all elegant wonderful proofs are also concise, but instead that there exist elegant, wonderful, and concise proofs which are sometimes found. I always think of human proofs as the words on the page plus a significant guiding practice and convention of mathematics. Some proofs require more of that than others. Internalizing these proofs is highly non-mechanical. But I don't think we disagree here.