{∅, {∅}, {∅, {∅}}} : Rage Against the Meat Grinder

From : Schopenhauer and the Geometry of Evil

Returning to Schopenhauer’s  refutation of  Leibniz’s optimism, his  qualitative verbal reasoning can easily be recast in terms of high-dimensional geometry.  Let the goodness g of a possible world   X  be approximated to lowest order as

g(X) = 1 - q(X),

where  q  is a positive definite quadratic form in the d-dimensional real variable X.

Possible worlds correspond to X values where g is positive, lying under a paraboloidal cap centered on the optimum,   g(0)=1,  with negative values of  g  representing impossible worlds.  Leaving out the impossible worlds, simple integration, of the sort Leibniz invented, shows that the average of g over possible worlds is  1-d/(d+2).   So if there is one variable, the average world is 2/3 as good as the best possible, while if there are 198 variables the average world is only 1% as good.  Thus, in the limit of many dimensions, the average world approaches  g=0,  the worst possible.