I was going to post a snarky comment to the effect that if you discard outright religious views and cognition motivated by them, there doesn't seem to be much left to non-naturalism. But the Stanford Encyclopedia of Philosophy says it better:
There may be as much philosophical controversy about how to distinguish naturalism from non-naturalism as there is about which view is correct.
So non-naturalism looks like mostly a combination of religion and arguments over the meaning of the word "natural". As in, if we found evidence that spirits of the dead affected physical events, that would promote them to the status of natural physical phenomena. So of course everything that exists is "natural" - according to some definitions of the word.
What is there to non-naturalism that is worth the time of seriously investigating it?
How would you investigate it?
Despite being (IMO) a philosophy blog, many Less Wrongers tend to disparage mainstream philosophy and emphasize the divergence between our beliefs and theirs. But, how different are we really? My intention with this post is to quantify this difference.
The questions I will post as comments to this article are from the 2009 PhilPapers Survey. If you answer "other" on any of the questions, then please reply to that comment in order to elaborate your answer. Later, I'll post another article comparing the answers I obtain from Less Wrongers with those given by the professional philosophers. This should give us some indication about the differences in belief between Less Wrong and mainstream philosophy.
Glossary
analytic-synthetic distinction, A-theory and B-theory, atheism, compatibilism, consequentialism, contextualism, correspondence theory of truth, deontology, egalitarianism, empiricism, Humeanism, libertarianism, mental content externalism, moral realism, moral motivation internalism and externalism, naturalism, nominalism, Newcomb's problem, physicalism, Platonism, rationalism, relativism, scientific realism, trolley problem, theism, virtue ethics
Note
Thanks pragmatist, for attaching short (mostly accurate) descriptions of the philosophical positions under the poll comments.
Post Script
The polls stopped rendering correctly after the migration to LW 2.0, but the raw data can be found in this repo.