A man's (or woman's) word is a peculiar sort of thing. When you "give your word" that something is true, or that something will be done, then you are basically swearing your entire honor that it is, or will be, so.

As long as you don't fail, your word is the most valuable thing you can give. But once you've given your word and failed, your word becomes worthless. Like innocence, once lost, your word can never be recovered.

It's likely that giving his word is what an atheist would do instead of swearing a vow.