What does the saying mean "The exception that proves the rule"?

How can an exception to a rule prove it?

The meaning of this saying is rooted in an important principle of Information Theory which says that "Knowledge only progresses when an experiment fails." Or inversely, "We don't learn anything from our successes." Here is an example:
Teacher: I'm going to give you a series of four numbers based on a pattern. You can give me three test series and I'll tell you if they match the pattern. Then you must tell me what the pattern is. Ready? "12, 14, 16, 18". Okay give me some test series, and I will tell you if they follow the actual pattern or not.

Student: 20, 22, 24, 26

Teacher: Correct.

Student: 32, 34, 36, 38.

Teacher: Correct.

Student 2, 4, 6, 8.

Teacher: Correct. What is the pattern?

Student: Consecutive even numbers.

Teacher: Incorrect. The pattern is this: "Each number must be larger than the previous one." 31, 45, 122, 123" would have also followed the pattern. Or even "1, 2, 3, 4."
This is a well-known example to demonstrate the seductiveness of Confirmation Bias. The student presupposed she knew the pattern and only tested to see if her presupposition was correct. Since all her tests pointed in one direction -- the direction she expected them to point -- she assumed her presumptions were right. Consequently, she learned nothing about the actual pattern. If she had inputted random numbers or if she had inputted numbers she was certain would fail the test, she would have learned at least something. She could have offered the original pattern in reverse order ("18, 16, 14, 12") and learned that the numbers needed to be in ascending order. There was actually very little value to her in offering number patterns that she fully expected to test positive.

And so you see that it was only by discovering patterns that DIDN'T follow the presumed rule (exceptions) that she could PROVE the rule to be correct.

This is a very important wider principle. In life, we only progress by failure. Naturally, it is our desire for our choices to always come up aces. But if they do, we will be the same person in 10 or 20 years that we are today. Why shouldn't we be when the status quo is so successful? We want our children to always succeed, but unless we believe they are fully realized, mature, ideal persons at birth, we should not attempt to ensure that outcome. It can be scary and even risky. But, generally speaking, it is a necessary risk.

In economics it is the same. People deride "Market Failures". But in fact, the value of an undirected, private market over centrally planned economies is not that it never fails. It is the opposite. The value is that it fails over and over, thousands of times simultaneously and often in very big, demonstrative ways. Government planned initiatives almost never fail even if they do not produce positive results. Or, at best they fail in slow motion, over a generation or two, and no one is ever held accountable for it. And that is why they are inefficient. Even if a government planned solution is PERFECT at the time it is designed and implemented, it will not be so in the near future and since it cannot fail, it cannot progress or adapt to change.
“Economic progress, in capitalist society, means turmoil.”
~ Joseph Schumpeter

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