2 sounds wrong to me - like you're trying to explain why having a consistent internal belief structure is important to someone who already believes that.

The things which would occur to me are:

• If both of you are having reactions like this then you're dealing with status, in-group and out-group stuff, taking offense, etc. If you can make it not be about that and be about the philosophical issues - if you can both get curious - then that's great. But I don't know how to make that happen.
• Does your friend actually have any contradictory beliefs? Do they believe that they do?
• You could escalate - point out every time your friend applies a math thing to social justice. "2000 people? That's counting. You're applying a math thing there." "You think this is better than that? That's called a partial ordering and it's a math thing". I'm not sure I'd recommend this approach though.