I am very willing (maybe too much, that's part of the problem) to accept a "Platonic existence" for *the* integers. I am far from sure however that this does not involve a significant amount of faith.
Indeed. It needs an infinite act of faith. But I have no problem with that ...
There are some objections to it and I am not sure that none of them make sense. Also, as someone said (if anybody has the original reference, in am interested): the desire to believe is a reason to doubt. I think that, even if it is true, arithmetic realism needs to be postulated (or conjectured) since I can't figure how it could be established.
All right. That's why I explicitly put the AR in the definition of computationalism.
About your question "is the universe computable?" the problem depends on what you mean by "universe". The definition you gave recently are based on some first person point of view, and even that answer does not makes things sufficiently less ambiguous to answer. Don't hesitate to try again. You can also read my thesis which bears on that subject (in french). You may be interested in learning that at least the *physical* universe cannot be computable once we postulate the comp hypothesis (that is mainly the thesis that "I" or "You" are computable; + Church thesis + AR). The reason is that with comp, as with Everett (and despite minor errors in Everett on that point), the traditional psycho-parallelism cannot be maintained. See my URL below for more.
Why there is no FAQ? Because we are still discussing the meaning of a lot of terms ....
I agree with you in your critics of Searle. I agree with most critics of Chalmers