I should have mentioned before that Tim Gowers’s blog is running installments of a “conversation” on complexity lower bounds. It’s structured as a dialogue between three characters, a cheerful mathematical optimist who likes to suggest approaches to problems, a more sceptical mathematician who knows a bit of theoretical computer science (and is tagged with a “cool” smiley), and an occasionally puzzled onlooker who chips in asking for more details and gives a few comments from the sidelines. We’re just on instalment IV, and there are oodles of comments on the previous instalments.
This is fascinating stuff for philosophers of maths, in both form and content — though I don’t begin to pretend to be following all the ins and outs. In form, because it’s always intriguing to see mathematical work-in-progress, exploring ideas, guesses, dead-ends (live mathematics as an activity, if you like, as opposed to the polished product presented according to the norms for “proper” publication). And in content, because you begin to get a sense of why something that initially seems as though it ought to be easy to settle (P = NP?) is really hard.