On the other hand, it’s possible that objective morality exists but is not empirically obtainable knowledge in nature. If that was the case, the only other way I can imagine for that knowledge to work is by some kind of enlightenment or grace—an inherent inner knowledge that we all possess, or can possess if we achieve the right state of mind, not from observation of the outside world, but by introspection.
Mathematical knowledge is not empirical. By your reasoning, does mathematical knowledge therefore “work by some kind of enlightenment or grace”?
This sense of “real” is contested in contemporary philosophy. As a sketch, here are a few things that are often considered quite plausible:
There are abstract objects and abstract facts
Plausible examples: numbers (or sets), shapes, propositions
Abstract objects are real
Moral facts are abstract facts
If (1-3) are plausible, then it’s also plausible that (4) moral facts are real. I’m not exactly offering this as an argument, but merely a set of (what contemporary philosophers consider to be, at least) plausible theses. A philosopher who accepts (4) will usually be referred to as a moral non-naturalist.
This is an interesting question for moral non-naturalists; but they would respond that this question is no more mysterious for morality than it is for mathematics (or any other abstract domain). And they’d certainly reject that moral facts (and mathematical facts) are “just” reflections of our beliefs and preferences.
Of course, many philosophers will reject 1-4 above, as well as this thought of yours. Such philosophers think that moral facts simply reduce to natural facts. E.g., facts about wrongness reduce to facts about units of pain, in the same way that facts about water reduce to facts about H2O. If they’re right about that, then all we need to do is figure out how to measure pain-units—or whatever natural property morality corresponds to.