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polymathwannabe comments on What are your contrarian views? - Less Wrong Discussion
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It's hard to give a general answer to this, as different moral realists would answer this question differently. Most would agree that it means that there are facts about one ought to do and not do.
How do you tell if something is a fact?
That depends on what it's a fact about. If it's a fact about the physical world, I use my senses. If it's about mathematics, I use mathematical methods (e.g. proofs). If it's a moral fact, I reason about whether it's something that one should do.
How do you know if your reasoning is correct and someone else's (who disagrees with you) isn't?
By engaging with their arguments, seeing what they're based on, whether they really are what one ought to do, etc.
So, what do you do if you start from the same premises but then diverge? Is there an "objective" way to figure out who is right in absence of some mathematical theory of morality?
If we start with the same premises, we should reach the same conclusions, if I'm interpreting your question correctly. It may help to provide a concrete example of disagreement.