Continuing with our thoughts from yesterday.... What are we saying when we say we know that 45 + 303 = 348? That depends on what we understand numbers, and arithmetic, to be. I think the sensible course is to adopt an operational/pragmatic view of mathematical knowledge according to which it is a shorthand for facts concerning the way the world works. In G.E. Moore's spirit, I might hold up a hand, "here is one hand." Then I might hold up another, :"Here, also, is a hand." Finally, with both of them in my field of sight, I can say, "there are two hands here." So we can take 1 + 1 = 2 as settled. But 45 + 303 might be tedious to prove in that way, with some object other than hands (marbles?). Further, there would be the threat of false refutations. I might lose track of the first marble before I collected the 348th. Then I'd count them all and declare that the real answer to the question must be 347! A naive empiricism about such matters becomes