Southwestern ranch-style house

Pretty cool, and the justification for employing a picture proof for the arithmetic mean/geometric mean inequality (on p 48 of the textbook) is interesting as well:

Maybe you agree that, although each step is believable (and correct), the sequence of all of them seems like magic. The little steps do not reveal the structure of the argument, and the why is still elusive. For example, if the algebra steps had ended with "(a + b)/4 >= sqrt(ab)", it would not have seemed obviously wrong. We would like a proof whose result could not have been otherwise.

A picture is evidently supposed to provide this (admittedly in this case the picture proof is, I don't really want to say more convincing, since the algebraic proof is convincing, but more satisfactory, anyway).

Relatedly I have hit upon a cunningly idiotic proof of the proposition that propositions are zeroth-degree beliefs.  Many people will say that, say, one's belief that p is a first-order belief, and that one's belief that one believes p is second-order. But clearly the only proper thing to say about these beliefs is that the first is an nth-order belief and the second is an n+1th-order belief.  After all, in the first, p could have been "one believes that ρ", and in that case, the post-substitution proposition would clearly not have involved a first-order belief. So obviously when you say something like "he believes that p", you are attributing an belief to him whose order is one plus the order of p. Type safety demands that p also be a belief.