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Insert_Idionym_Here comments on Archimedes's Chronophone - Less Wrong
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It seems that in order to get Archimedes to make a discovery that won't be widely accepted for hundreds of years, you yourself have to make a discovery that won't be widely accepted for hundreds of years; you have to be just as far in the dark as you want Archimedes to be. So talking about plant rights would probably produce something useful on the other end, but only if what you say is honestly new and difficult to think about. If I wanted Archimedes to discover Bayes' theorem, I would need to put someone on the line who is doing mathematics that is hundreds of years ahead of their time, and hope they have a break-through.
I think probability theory would have been very accessible to the Greeks, had they only thought to think about games of chance, which they certainly played. I bet if you'd asked Archimedes 'What odds should you offer on a bet that two dice get seven?', then the whole thing would have come crashing out within a hundred years or so.
So you might want to put him in touch with a modern philosopher trying to take a mathematical approach to something mysterious, say Dennett.