Is it permissible to translate books that mix correct ideas with atheistic Western ideas, bearing in mind that the purpose of translating them is for commercial gain and not to refute the ideas? Is it permissible for the translator to participate in translating parts of these books that do not contain atheistic ideas? And what is the criterion that determines the permissibility of translating a book?

There is no harm in translating books on psychology, human development, and similar fields that contain beneficial scientific aspects. While such books may contain other elements that pose a jurisprudential problem, translation does not constitute direct or intended assistance in spreading those violations. As for books whose translation is forbidden, they are those authored to call to atheism, disbelief, immorality, or indecency, unless their translation is for a valid purpose, such as refuting and discrediting them.