Why Is AI Math More Than a Convex Hull?

Alex Kontorovich says AI math is more than a convex hull because known moves, chained together, reach much further than that label allows. Tim Gowers describes the better picture as the semi-group generated by the moves. Putting known moves together is the same notion, but what the space of known moves can accomplish is much bigger than the derogatory convex hull.

People talk about the work dismissively, as if the result already sat in the convex hull. The system does the moves humans already know, does them really well, and runs a really good tree search. On the quasi-RH, the moves taken were all things known to human beings. A very long cascade of the moves is shocking. The sequence looks beautiful enough to prompt the question of how one move followed another, which anthropomorphizes the system.

A balloon makes the convex hull concrete. The inside of the balloon is the convex hull. Mathematicians have thought up many results and ideas, and anything inside is some linear combination of them. The region feels compact, a closed, enclosed area. Inside the area, AI has not really made new progress, because AI has not moved the balloon out and has not given a new idea yet. The label is derogatory because the intuition is that mathematicians could have done the work, since the idea was already quite obvious.