@layman,
The basic flaw in the premises of SR can be found in the claim that the distortion of measuring instruments predicted by the lorentz transformation is "reciprocal."
Let's say two objects are approaching each at the rate of .5c. The observers on each object completely agree about this; it's not a matter of contention. So, what then? According to SR:
1. A will say that he is motionless and that B is the one moving. Based on this, he will conclude that B's clock has slowed down, because, according the to the LT, it always the moving clock which slows down..
2. B will say that he is motionless and that A is the one moving. Based on this, he will conclude that A's clock has slowed down.
Now is this a logical contradiction or a "paradox?" No, not at all. People make opposing assumption every day and that's just par for the course.
So, what's the problem? The problem is that SR says that "both are right." That's what's logically impossible. The two possibilities (either A is the one moving or else B is the one moving) are mutually exclusive. They cannot BOTH be "at rest" or else neither of them would perceive any relative motion between them to begin with.
That means that (at least) one of them is wrong in his assumptions.
As Herbert Dingle noted, long ago, "The theory of special relativity unavoidably requires that A works more slowly than B and B more slowly than A — which it requires no super-intelligence to see is impossible."
So who is supposedly wrong? SR actually answers this question (out of necessity) in its "resolution" of the so-called "twin paradox." It says that the travelling twin is wrong, and was wrong all along when he assumed he was not moving. He is the one who returns younger, and hence he was the one moving, whether he thought so, or not.
The problem here is that SR is forced to renounce its own crucial assumptions, and is therefore self-refuting. Ultimately SR does not even try to claim (because it's impossible) that each twin is younger than the other when they are reunited.
Not a single mathematical symbol is required to see and understand the problem here.