A couple of stray thoughts on the matter:

1) Logic, which is part of rationality, may at first seem completely objective. A + B = C, no matter who you are. Of course, it is generally recognised that each of us has our own perspective shaped by individual psychology and culture, so logic in principle is objective but in practice obviously isn't. But, something that really interests me is that what we call logic may in fact not necessarily be logical at all. Technically there is no real way of proving that logic actually works - that a logical argument necessarily makes sense. The thing is, you can't prove the logical nature of logic itself by means of logic. Maddening. (Actually, Logicomix is a fun read about Bertrand Russell's quest to discover the immutable foundations of logic).

You can't use any mode of knowing to prove itself, since any such mode necessarily rests on particular axioms. For example, you can't scientifically investigate the veracity of science, because science assumes that it already possesses that veracity.

(Of course, I'm not trying to say here that logic and science are useless - far from it. But they're not 'objective', specifically in the sense that they are built on certain axioms.)

1.1) Actually what I really think about is the fact that rationality seems rational in a fundamental way, but there's no rational basis for this.

2) More and more I think about the difference between a rational belief and a rationalised belief. I might believe something that is completely rational without having ever actually reasoned its rational nature myself. It's the gap between a priori and a posteriori. Or something. I mean, I can believe that taking cold showers is good for you without actually understanding why - like if I was raised with that belief, and have simply accepted it all my life.

3) Logic, as I said, is only a part of rationality. By itself it's pretty useless, since you can construct logical arguments that have no actual application to reality or, indeed, that are entirely untrue.

Apologies if these thoughts seem disorganised or... irrational.