Is there a good name for a mathematical theorem being true "only because of an astronomically large computation"? I kind of want to say "approximately independent", since the limit as the computation goes to infinity would be actually independent.
Oh, *that's* what you meant by independence! But I'm too semantics-oriented to back you on that. If the axioms constrain a truth semantically, it's not independent. The size of the proof is just an interesting syntactical side fact, even if the proof length is growing quickly.
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Obviously I agree about the definition of independent. But saying I can't define topological neighborhoods of independent seems weird, sort of like ruling out the phrase "near zero".
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Perhaps you can define them by what they aren't. As concision is a property of beautiful proofs (Hardy), those in The Book (Erdos), these are not-beautiful, or not in The Book.
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I disagree with that aesthetic. The fact that some things are only knowable by computation is one of the most beautiful things about the universe.
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I kinda get it...but still prefer the Hardy aesthetic (eg, proof of infinite primes). Yet I'm happy for you, as a world of people who shared my preferences would not be as interesting, beautiful or productive as a world with diverse ones.
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Worse, a world of people who shared your preferences that only simple arguments are beautiful would be pretty limited if we get access to a lot more computational resources.
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To drop the false contrarianism, there's no disagreement here. The Book contains the most elegant proofs of all theorems. Some of those elegant proofs are still super long, as shown by one of the fairly short Book proofs. Part of liking the Book is liking what is says is true.
End of conversation
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