Terence Tao: belief

Transcript context

…Oh, but you’re saying that twin prime, you can’t use the same tools. It doesn’t appear random almost. Well, we don’t know. We believe the primes behave like a random set. And so the reason why we care about the twin prime conjecture is a test case for whether we can genuinely confidently say with 0% chance of error that the primes behave like a random set. Random versions of the primes we know contain twins at least with 100% probably, or probably tending to 100% as you go out further and further. So the primes, we believe that they’re random. The reason why arithmetic progressions are indestructible is that regardless of whether it looks random or looks structured like periodic, in both cases the arithmetic progressions appear, but for different reasons. And this is basically all the ways in which the thing was… There are many proofs of this sort of arithmetic progression-type theorems. And they’re all proven by some sort of dichotomy where your set is either structured or random and in both cases you can say something and then you put the two together. But in twin primes, if the primes are random, then you are happy, you win. If the primes are structured, they could be structured in a specific way that eliminates the twins. And we can’t rule out that one conspiracy. And yet you were able to make, as I understand, progress on the K-tuple version…

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