Last time's problem.
     Is mathematics a game? Argument against it. “The Theory of the game is not arbitrary although the game is.” The theory of the game as pure mathematics & physics. Can we say that the fact that you can't mate with … rests on certain physical & certain mathematical facts? Can we say that the possibility of proving so & so in such & such a way rests on a mathematical, logical, fact? Great temptations. This is, of course, restating our old problem.
     Suppose we said: it never happens that A mates B with …. This should be the more modest proposition. But what does it mean? But couldn't we say: it never happens that we say A … B …? Certainly & this is a very important fact but based on what? So now is the theory of the game arbitrary?
     “I believe that Goldbach's theorem will come true”. How is this belief in the end verified? By a proof. By any proof? No. By this particular proof? No. By something we shall recognize as a proof. But isn't the fact that such & such a proof is possible based on a mathematical fact, a mathematical reality? I mean the fact that there is a proof at least somewhere in the region we still recognize as that of proofs? The mathematical fact being that such & such a structure is possible. That it is imaginable. How do we imagine this possibility? What is a structure like which is impossible. Possible = describable.