A mathematical proof is perfect. You have assumptions and definitions which are all written out clearly. And then you have conclusions. The final conclusion proves what was to be proven. A simple example using the definition of an even number and the distribution law (an assumption) is this.
Consider two even integers x and y. Since they are even, they can be written as x=2a and y=2b respectively for integers a and b. Then the sum x + y = 2a + 2b = 2(a + b). From this it is clear x+y has 2 as a factor and therefore is even, so the sum of any two even integers is even. Quod erat demonstrandum.
After my first degree I decided to switch to physics. And physics was some kind of shock. Those physicists had a completely different idea about what a proof is! They would state a hypothesis and then make an experiment. And if the experiment was consistent with the hypothesis they would say that they had proven it. In my opinion that was stupid. Nothing was proven - if anything the hypothesis wasn't refuted.
And I have been right. It's just that those physicists, of course, knew that they hadn't proven anything. It was just an imperfect term they used. This is the reason why Mathematics isn't part of the natural sciences, but part of the humanities. Mathematics makes (arguably arbitrary) assumptions and then researches all possible logical conclusions which follow. While this is often extremely useful for the natural sciences, at the end of the day, Mathematics is not grounded in the natural world, but only in the human brain.
If you want to find something out about the natural world you usually don't have the luxury of knowing all circumstances. You can't just make assumptions, because, well, these assumptions can be wrong. Is the woman you consider your mother really your mother? You can't prove this! Yes, you can make a DNA test and the scientists may tell you that the probability of her being your mother is 99.99% percent, but that's not a proof. It is a typical result of the natural sciences.
It is the kind of logic humans use intuitively to some degree. It can also be defined in strict mathematical terms - using some assumptions which can't be proven.
Back to MMOs. Let's assume you are the developer of a successful game with exactly one million players. It is January and you change one feature. The player numbers drop to ten thousand within one week and stay there for the rest of the year. You are bankrupt. Was the change of the feature the reason for your failure?
You can't prove this! It could have been anything! Maybe aliens invaded the planet at just this time and stayed undetected. One of the things they did before they left was to make exactly 990.000 players of your game stop playing. Can you prove that this didn't happen?
When discussing things like this, one often encounters people arguing that correlation isn't proof. And, of course, they are right. Just because there is some chronological correlation between you changing the feature and your players stopping to play, doesn't mean that this is necessarily the reason. It could be anything!
But if humanity had only ever accepted mathematical proofs and not used Bayesian Analysis - in an intuitive way - we would still live in caves. Sure, maybe your computer doesn't work the way it does due to the scientific models applied. Maybe the models are completely wrong. Maybe it is just a coincidence and tomorrow the computer will stop working. Maybe you are just dreaming. Can you prove that any of this is wrong?
N. Taleb's famous example of the goose says that the goose looks at her past and concludes that she will continue to live a well fed life until a natural death. The next day she is butchered and eaten. Sometimes you miss critical information. Sometimes the people with doubts are correct.
But fact is that our common experience has shown that while correlation isn't proof, it is still a useful hint. Yes, sometimes it is wrong. Sometimes there is a common cause you didn't consider. Even though the number of people drowning increases at the same time that they like to buy ice cream, ice cream doesn't lead to people drowning. It's the Summer.
Maybe the aliens didn't only manipulate your players, but manipulated you, too! They made you change this feature which didn't cause anybody to stop playing. And then they also made your players stop playing! And now you think that the change of the feature is the reason, while, in fact, there is a hidden common cause that you didn't consider! Stupid you!
I think it's pretty obvious what I am saying here. First, I don't have all the information of the universe - and if I did, I arguably needed to have a brain as large as the universe. None of us have all the possibly relevant information. Certainly the developer does not! The trick is to make good decisions facing imperfect information.
Second, I don't want to prove anything on this blog unless I clearly state it. I am specifically looking at you, Azuriel :). All I do is making educated guesses - just like everybody else.
Third, my insights as a blogger can be interesting for developers not only in spite of me having less information, but because I have less information. Insights from the outside are useful for human beings, because we tend to become absorbed in our environment and ignore obvious things. We aren't computers. There's something called psychology. More information doesn't necessarily improve our decisions. Metrics aren't necessarily good: they can make you ignore more relevant but less quantifiable information. And those recent page hits from Anaheim tell me that Blizzard employees actually enjoy reading blogs written by bloggers who have less information than they do. I like that.