But that it is sometimes incorrectly used when applied to a physical problem, and that it can lead to contradictions and false physical results has not been widely acknowledged or appreciated.
A parable can best illustrate this fact:
Long ago, before the internet, before the automobile, and before even the telephone, there was a kingdom with a just and educated king - well trained in the sciences and in mathematics. One day, the day that was chosen for the execution of a convict, the king suddenly felt he wanted to pardon this man. The execution was to take place in a village 20 miles from the king's castle. It would happen in one hour and twenty-five minutes.
The king realized that if he could dispatch a messenger who could travel at 15 miles per hour, the messenger would arrive in one hour and twenty minutes - five minutes before the scheduled event. 15 into 20 is after all 1 hour and a fraction, 5 over 15, or twenty minutes. Unfortunately, all that he had available was a runner who could travel at only 10 miles per hour, but at the half way point, ten miles down the road, there was a village where a rider could carry the pardon the rest of the way at 20 miles per hour. The king calculated that the average speed would be 10 + 20 over 2, or 15 so the rider would get there on time.
When the king later learned that the rider had gotten there five minutes AFTER the scheduled execution, and the criminal was dead, he was angry and upset andlooked around for someone to blame.
Someone pointed out to him that the first half of the trip lasted an hour, and the second half required half an hour a total of one hour and thirty minutes.
So he himself was to blame for having used the wrong calculation.
He could not accept this and ordered the chief mathematician of the realm either to explain this error or lose his head.
The mathematician lost his head.
The moral of this parable is 'look out', the most revered of theoretical physicists, Einstein and Lorentz, and before that Maxwell, have been careless and have come to conclusions that seem warranted mathematically but are 'dead' wrong physically.
The answer is simple but hard to explain: while you can average time, and distance, you cannot simply average a derived quantity like distance divided by time, that is speed or velocity. In fact you can't generally add velocities or speeds, much less add and divide by two! That applies to all equations in which a 'rate' or a change of a 'rate', a second derivative, is used. 'Yes you can' - mathematically - no problem, but look out when you apply it in physics.
If the king had used a 'weighted' average, in his case 1/3 times the first leg of the journey and 2/3 times the second leg he would have gotten the right answer - but who bothers to think that through - no physicist, past or present, is on the record with such a sophisticated solution.
Note: In a case like that of a gun fired on a moving train where you want to know the speed of the bullet with respect to the ground, you CAN add the speed of the train and the speed of a bullet fired in the direction in which the train moves. But if you have two runners running a relay you can't just add their speeds and get anything sensible - much less average the two speeds. The same applies in particle physics where the first particle in an accelerator 'dies', and is replaced by a second particle. To think that the second particle was in turn accelerated somehow, by the first particle, and that it has twice the speed of the first particle is nonsense - and you don't need special relativity to confuse the issue.
Although the above example, a gun fired on a train, seems to make an exception to the rule that you cannot add velocities, a deeper analysis shows that this is not the case. Here we are concerned with the velocity of one entity alone, but wish to relate the velocity of this entity to a different platform or system. The principle that motion is relative is the issue here. It happens that to relate the velocity of the bullet to the ground instead of to the gun we can add the velocity of the train to that of the bullet. If the train were moving in a circle we would need a different algorithm.
He represent the path of a photon or light quantum in one coordinate system, that he calls K and, as well, in another coordinate system he calls K', by writing two equations x = ct for the system K, and x' = ct' for the system K'. We can, for example, think of K as the system in which we deal with units measured in yards, and K' as the system in which we express distance in meters. Einstein does not make clear how the two systems are to be conceived of, or related to each other, but we are at liberty to make this choice. Notice that he also uses t and t' so perhaps we chose seconds as the units in system K and hours in the system K'. What we can then infer from these two formulas, but what is not explicit in the math, is that at the time the light pulse is emitted, t = t' = 0. The origin of time is in both cases chosen to coincide with the emission of the pulse of light.
Remember that in the system K the speed of light is measured in yards per second, whereas in the system K' it is measured in meters per hour. The value of c in the first system does not have the same 'value' as the value of c in the second system - even if the velocity of light is an absolute constant (which it probably is not). But if we represent this absolute constant by c in the system K then we should call it c' in the system K'. With that, however, nothing that follows in Einstein's derivation of the Lorentz Transformation has any meaning at all.
An alternative to interpreting physically what Einstein's mathematical equations could mean, would be to interpret mathematically what Einstein's physics says.
Einstein wants us to believe that c is a universal constant, representing the velocity of light in any system, but in addition, and even more significantly, that the speed of light is independent of the movement of the source. So we could assume that K' is the system in which the source is stationary, but we chose the origin of K' to be at the time the pulse of light is emitted, just as is the case for K. If v is the velocity of the source, or the system K', relative to K than a coordinate value x is represented as
and t is the same for both coordinate systems.
What Einstein wants to say, and what he believes, can best be said in words:
If the speed of the source is v, in the system K, at the time a pulse of light is emitted from the source, then the speed of the emitted light pulse is independent of this speed, v. The pulse after a time t ends up at x, if we think of the origin of the light pulse as x = 0; it ends up at x' if we think of a light pulse as originating at x' = 0, whatever the velocity of the source, v, may be.
But that is not what his mathematics in fact represents.
Whether K' differs from K in the choice of units of measurement, or whether it differs from K in that it is in motion with respect to K cannot be ascertained from the equations Einstein presents. That is what sets physics above mathematics and requires us to express in words what is hidden in mere formulas and equations.
We need to be clear about the consequences of Einstein's belief. After, say, one second, two light pulses, one from the stationary, and one from the moving system (which, we have supposed to start at the same point, x = x' = 0) both end up at the same point, since the speed of light is supposedly independent of v. But one ends up at the point x = c, and the other ends up at the point x'=c. Since it must be the same point, we can conclude that x = x' = c. In other words x' = x + v = x (remember t = 1). That is a contradiction! It cannot happen in a Euclidian system; and that in turn means that the length as measured in the system in which the source is in motion must have shrunk, or perhaps the measurement of time is not the same in the two systems, or both. In other words we need something like a Lorentz Transformation (LT) to distort distance and time, and to rationalize a contradiction, in order to accommodate Einstein's belief.
The alternative is to abandon that belief. In the absence of decisive experimental evidence, and the fact that no error free derivation of the LT exists, that is the path chosen here.