For almost two centuries, mathematics has played a central role in physics, and mathematical physics has succeeded in becoming the most revered of all the theoretical scientific disciplines. Mathematics, as an abstract logical discipline cannot be challenged. It is a consistent science and a discipline that has no peer in higher educational teachings.
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 and looked 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.