Though "formal science" is not a familiar term to me, I take your word for it and hear what you are saying. However, I always considered mathematics to be it's

Though "formal science" is not a familiar term to me, I take your word for it and hear what you are saying. However, I always considered mathematics to be it's own thing, different than science. Real science is physics and chem and then biology starts to merge more into a body of knowledge. Anything that has "science" in the title isn't really. And in line with that, I consider "computer science" to be a misnomer as it is more accurately categorized as a branch of mathematics. And computer science is, of course, completely distinct from software development, which is a craft. All that said, I would be hesitant to put psychology and psychiatry in the same boat. Neither one is a hard science, but psychiatry is much less likely to be anyone's fall-back major.