Mark Adkins got to the point:

my basic point: the
mathematical basis for "time dilation" disappears altogether when the
principle of the relativity of simultaneity is properly (universally)
applied, that is to say, applied to include (rather than exclude as a
special case) the clock synchronization procedure.

Say what? Two different frames, say A and B, already have their
own convention to determine simultaneity - that's part of the
definition of an inertial frame in special relativity. Time
dilation is simply the determination by B that A's clock rate,
although lasting exactly one tick per second by A's measurements,
is longer when measured by the clocks in the B frame.

The mathematical basis for time dilation is fairly simple
geometry. In fact, the implications of the underlying geometry
are far more interesting, physically, than all the gee-whiz
stories about twin parodoxes or semantic nonsense about how
clocks are slowed by their velocity. Relativity is fundamentally
about symmetry, and symmetry is described by geometry.


---Tim Shuba---