What I'm reading about now

So there are three choices you can make when it comes to something like inference to the best explanation.

1) You can give a good justification for the principle: I'm still looking to find one. The problem with justifying inference to the best explanation is that it's pretty much trying to justify inductive inference (meaning, trying to justify my belief that since the sun rises everyday it's also going to rise tomorrow). Hume provided arguments that make it difficult to justify inductive inference, though we can do our best to describe our practice. Then you can check to see if your justification justifies anything else by the way, such as inference to things that are necessary for ethical reasoning.

2) You can give an unconvincing justification: One could complain that this is what Enoch does. I tend to agree.

3) You can refuse to give a justification: after all, justification has to end somewhere. You (probably) can't have everything justified. The chain has to run out somewhere (probably). But then the challenge is, how do you avoid mayhem? If inference to the best explanation is unjustifiable, then is there anything wrong with us taking other wild and crazy principles as primitives?

Sorting this out is one of the current challenges I'm trying to learn more about.

Another related challenge is trying to figure out whether one implicitly assumes some normative stuff when you accept inference to the best explanation. After all, aren't we presupposing certain values when we talk about the 'best' explanation? Aren't we presupposing our ability to deliberate and come to conclusions? Doesn't this mean that there are "oughts" hidden in our scientific talk, and if those "oughts" are OK and in our ontology, then does it help out for believing in moral "oughts"?

And how the hell can I bring math back into this (after all, my primary goal starting off wasn't to defend moral realism or to explore scentific anti-realism, but to understand math and ethics better through close analysis of realism/anti-realism debate. Is that goal still possible?)