The problem with Brute Facts Going back briefly to the beginning:
"His analysis is that there has to be some things for which there is no explanation that explain the things that do have explanations. Something must be brute."
But is that sound? I'm not sure. Suppose there exist no things that lack an explanation, at least in principle. Then there would be an infinite regress of both things to be explained and of explanations. Suppose we accept that possibility. In that scenario, whenever we find an explanation we also find a new thing to be explained. Well, that's exactly the way it's been for us so far. It might be that way just as long as we choose to go on and are able to find new explanations. It's an infinite regress. But it is not a vicious regress as far as I can see.
I think we are tempted by the notion of brute facts because it opens up the possibility that in the future anything that can be explained will have been explained; and that anything that has not been explained is beyond explanation. It's a comforting thought, perhaps. But there's no reason to suppose that we will ever reach that happy state. And even if we did reach it, we would never know that we had reached it, because we could never be quite certain that the things we presume to be 'brute facts' are not, after all, explicable by something else.
So perhaps the whole 'brute fact' idea is an illusion.
I think it's a different notion from the idea of underlying assumptions that held to be true beyond doubt. They are in a different case, I think. The argument there is that unless we hold some things to be true and beyond doubt we cannot even begin to make sense of any questions or uncertainties. I think there's a good case to be made for that view. But that's not about 'brute facts'.