I read the following: http://www.feynmanlectures.caltech.edu/I_15.html#Ch15-S4
"Now let us see what happens to the moving clock. Before the man took it aboard, he agreed that it was a nice, standard clock, and when he goes along in the space ship he will not see anything peculiar. If he did, he would know he was moving—if anything at all changed because of the motion, he could tell he was moving. But the principle of relativity says this is impossible in a uniformly moving system, so nothing has changed."
In my point of view Feynman uses the principle of relativity as a kind of an axiom for his derivation of time dilation.
Question: How do we know that the principle of relativity is true and can use it like an axiom for new phenomena? If we had treated Newton's laws of motion as axioms we would never have found out how nature actually works. In other words, what is the law behind the fact that you cannot know how fast you are moving? How can you use this assumption as axiom for e.g. the behavior of clocks as Feynman does further down in the link?
P.S.: Feel free to edit since I'm not a native English speaker. Nevertheless I want to improve my English and it would be great if you pointed out my errors.