(Note: this is purely hypothetical at the moment, but I thought of it and it struck me as an interesting dilemma.)
I am an early-career faculty member in linguistics at a big public college in the U.S.; I do quantitative research on language change and dialectology. Most of the reference letters I write, unsurprisingly, are for graduate programs in linguistics (or psychology). However, I occasionally get a request from a student for a letter of recommendation to medical school or nursing. I'm totally happy to provide these.
While tracking down the password for my American Medical College Application Services account just now, it occurred to me that I might someday get a request for e.g. programs in naturopathy or chiropractice. Right now, I wouldn't know how to handle such a thing.
To be clear, I have no medical credentials at all (seriously - I stopped taking biology after my tenth grade pre-AP class). And I can see the argument that it's absolutely none of my business what my students decide to do, and that introducing my personal views into the decision of whether to write a letter is potentially a slippery slope. (For instance, would I refuse to write a letter for a student applying to a graduate program in some country whose military actions I might disapprove of? I hope I wouldn't. Is that much different from this?)
But I'm a big fan of, well, science in general; I'm also keenly interested in the place of science in American society. I regularly read blogs such as Science Based Medicine, which leave me feeling pretty suspicious of most "alternative medicine" and wanting to do what I can to avoid supporting it. If I received a request for a letter to a program in naturopathy or chiropractice or something, I'd probably cringe. I'd rather work out how to deal with this kind of situation in advance in case it ever suddenly presents itself. Any thoughts?