I follow my unprovable moral axioms because I believe, on moral reflection, that they are morally compelled. If this sounds terribly circular, that's because it is, but I don't see[why] that should be problematic.
To me, it both is and isn't problematic. It's not because, for day-to-day life it works. At the same time, I'm discomforted by having so much based on such an inherently paradoxical belief. (On a certain level, I'm also disappointed that the limits of my own reason require me to accept belief without any sort of empiricism.)
All this is by way of a longwinded introduction to a series of posts at Volokh asking for responses on this issue, both from a religious and non-religious perspectives. (As well as a 'fight the hypothetical' complaint thread. My profs used to say that if you want to fight the hypothetical, pay more tuition. Law profs tend to be dorks. Make of all this what you will.)
There's quite a few very articulate responses from very smart people who seem to have put a lot of thought into the matter. There is also mine...