I would like to agree with this, but still feel that the difference between necessary and contingent truth needs to be accounted for somehow. I can, like Eliezer, vaguely imagine a world in which 2+2=3 in Peano arithmetic, but I don’t think I could precisely model it in the way I could precisely model a world with different laws of physics. I might be able to, by a huge amount of Dark-Lords-of-the-Matrix-type meddling, precisely model a world in which some things whose closest equivalents in our world would be implementations of Peano arithmetic output 2+2=3, but (a) that just seems too much like cheating to mean much (b) the world’s fundamental laws would still have to be consistent with 2+2=4 – if it had physics like ours, 2 joules and 2 more joules would still make 4 joules; if it were a cellular automaton, 2 living cells and 2 more living cells bordering a cell would be 4 living cells.
I would like to agree with this, but still feel that the difference between necessary and contingent truth needs to be accounted for somehow. I can, like Eliezer, vaguely imagine a world in which 2+2=3 in Peano arithmetic, but I don’t think I could precisely model it in the way I could precisely model a world with different laws of physics. I might be able to, by a huge amount of Dark-Lords-of-the-Matrix-type meddling, precisely model a world in which some things whose closest equivalents in our world would be implementations of Peano arithmetic output 2+2=3, but (a) that just seems too much like cheating to mean much (b) the world’s fundamental laws would still have to be consistent with 2+2=4 – if it had physics like ours, 2 joules and 2 more joules would still make 4 joules; if it were a cellular automaton, 2 living cells and 2 more living cells bordering a cell would be 4 living cells.
(See also: Math is Subjunctively Objective.)