The world is four-dimensional - Hermann Minkowski's irrefutable proof
Note 1: The main reason for using the expression "irrefutable proof" is that experiments would be impossible if it were wrong - see below.
Note 2: The strong formulation of this piece of foundational knowledge ("irrefutable proof") might look to some colleagues not sufficiently "politically correct." Although the reason for choosing this formulation seems quite transparent - to invite and even provoke colleagues to challenge the proposed piece of foundational knowledge - it is worth saying two more things:
Thanks to Hermann Minkowski we already possess a major piece of foundational knowledge - the reality of spacetime or the World as Minkowski called it.
Here is Minkowski's most general proof that the world is four-dimensional. Lorentz formally suggested that observers in relative motion have different times. Einstein postulated that the times of different observers are equally good, that is, each observer has his own time, and that for Einstein meant that time is relative.
Minkowski demonstrated that if observers in relative motion have different times, they inescapably have different spaces as well (since space is a set of simultaneous events and different times imply different simultaneity, i.e., different spaces):
"Hereafter we would then have in the world no more the space, but an infinite number of spaces analogously as there is an infinite number of planes in three-dimensional space. Three-dimensional geometry becomes a chapter in four-dimensional physics. You see why I said at the beginning that space and time will recede completely to become mere shadows and only a world in itself will exist."
As many spaces are possible in a four-dimensional world, Minkowski's irrefutable proof that the world is four-dimensional becomes self-evident:
If the real world were three-dimensional, there would exist a single space, i.e. a single class of simultaneous events (a single time), which would mean that simultaneity and time would be absolute in contradiction with both the theory of relativity and, most importantly, the experiments which confirmed its predictions (e.g., it is an experimental fact, used every second by the GPS, that observers in relative motion have different times, which is impossible in a three-dimensional world).
In other words, the relativistic experiments and the theory of relativity itself are impossible in a three-dimensional world.
To see specifically how the experiments which confirmed the kinematic relativistic effects would be impossible if the world were not four-dimensional (i.e., if spacetime were just a mathematical space), consider Minkowski's own explanation of length contraction (which is the accepted explanation); start at the last paragraph on p. 79.