In a way that is so obviously true that nobody needs to prove it. An axiom is a starting point you simply accept — in maths, 'two things equal to a third thing are equal to each other' is not proved, it is assumed, and everything else is built on top. So when you say something follows axiomatically, you mean it comes straight out of the definition, not out of evidence: 'A bachelor is axiomatically unmarried.' You could not check this by interviewing bachelors; it is true because of what the word means. This is a formal, thinking-and-arguing word. You meet it in philosophy, maths, law, economics writing — not in conversation. Two ways people use it: (1) 'obviously, by definition' as above; (2) 'without argument, as an assumption' — 'The report axiomatically treats growth as good,' meaning the writer never defends that idea, just starts from it. Notice these two pull in different directions. The first is a compliment to the logic; the second is often a quiet criticism — you are saying someone assumed what they should have proved. Which one is meant usually shows in the tone around it. It is also stretched, quite loosely, to mean nothing more than 'obviously' — 'Axiomatically, you need money to start a business.' Careful writers dislike this, because that one is not true by definition at all; it is just true. But you will see it, and now you know why some readers wince.

不證自明地、公理式地。數學課講的「公理」就是這個 axiom；當某件事是「按照定義就成立」而不是「查證後成立」時，就用這個字。注意它常帶一層「這人根本沒論證，直接當前提用」的暗諷，不完全等於中文口語的「當然」。