Relative Motion 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.
 A: Experiments that have been conducted so far have shown no measurable difference in the observed speed of light. That means that the hypothesis that the speed of light is constant in any inertial frame seems like a good starting point for formulating a theory. 
What is more interesting is that the theory of relativity was able to predict certain phenomena that had not been observed - for example time dilation. Clocks that traveled on commercial airliners in the famous 1970 Hafele-Keating experiment - one going around the earth in the E-W direction and the other in the W-E direction - were shown to have drifted from the clocks that were left behind, by the amount predicted. The clock traveling westward had gained 273+-7 ns, while the one traveling east lost 59+-10 ns. 
As you know, GPS relies on measurements of signals from multiple satellites, and accurate knowledge of the "time" on the satellite is essential for this calculation. The fact that GPS works so very well is a daily reminder that both special and general relativity are really quite good models for some aspects of how the real world works.
A: We don't know whether the principle of relativity is true. It is just an assumption which so far is giving results that agree with experiment. But we are ready to throw the principle of relativity away without regret, if we measure something contradicting it.
If we declared the Newton's laws as axioms, we would still discover non-Newtonian physics. Through measurement. The measuring devices precise enough to see the problem with Newton's physics have been around for many decades already. Having Newton's physics as axioms would not prevent us from measuring that Newton was wrong.
A: As mentioned by the popular physicist Brian Greene, all objects are constantly moving within that 4 dimensional structure known as Space-Time. The speed of that constant motion is the speed of light, be it motion across space, across time, or across Space-Time in any other possible direction.
If you analyze the outcome of such a phenomena, you end up with Special Relativity and all of its equations.
One also discovers that all of the equations are valid between a body at rest in space and a body in motion across space, and one discovers that all of the equations are also valid between two bodies that are both in motion across space but at different velocities.
Thus in turn, since the equations are valid in both cases, you can not truly tell if one of the two bodies actually is truly at rest in space. 
Thus, absolute rest can not be detected. Thus you are left with the limits of relativity instead.
A: The relevant foundational principles and definitions of relativity are not subject to experimental tests; but they must be adopted and used in the first place in order for experimental tests to be conducted. Namely experimental tests of expectations or hypotheses concerning models; such as cosmological models and astronomical models dealing with the distribution of matter, radiation etc. on the largest scales, all the way to models about properties of particles at the smallest/"elementary" scales. And inbetween for instance hypotheses/expectations/models about whether given clocks might "keep ticking regularly" and (if so) "at equal rates", especially while being separated from each other.
One particularly basic and important definition is how participants under consideration ought to determine whether they were (pairwise) "at rest to each other", or not. It is a matter of definition (of what's meant and universally understood as "having been at rest to each other") that,


*

*if $A$ and $B$ found having been and remained "at rest to each other" (throughout some trial), and

*if $A$ and $J$ did not find having been "at rest to each other" (ever, in that trial),


then 


*

*it is guaranteed that $B$ and $J$ did not find having been "at rest to each other" (in that trial) either, and

*there may or may not be some other participant(s), e.g. $K$, such that $J$ and $K$ found having been and remained "at rest to each other" (throughout the trial under consideration). And if so, then $A$ and $K$ did not find having been "at rest to each other" (ever, in that trial), and $B$ and $K$ did not find having been "at rest to each other" (ever, in that trial), and $A$ and $B$ are said to have "moved uniformly" with respect to $J$ and $K$, and vice versa, $J$ and $K$ are said to have "moved uniformly" with respect to $A$ and $B$, throughout that trial. 
