Re: Envelope Paradox
[ QUOTE ]
These have all been proven rigorously, but people still argue because the only way these are intuitive is if you have studied this before. Math is hard.
[/ QUOTE ]
I have great respect for phzon and PairTheBoard, but this one statement is not only dead wrong, but very dangerous.
Skill in math can be good, but it often induces a special kind of blindness. Problems are simplified so they fit in neat mathematical frameworks, then some mathematicians can only see the math, not the original problem. This leads to people insisting they have the only rational solution to a problem, when what they should say is, they have a rational solution to a simplified problem which may or may not give insight into the real problem.
In this case, it's very easy to come up with mathematical formulations that justify always switching, and formulations that argue you shouldn't always switch. It's easy to come up with real-world examples where each type of formulation is a good model. That's why we call this a paradox. Explaining one side or the other, either in rigorous math or exhaustive words, misses the point. The original problem already did that better.
People who try to "resolve" or "explain" paradoxes don't understand them. No one believes both sides are true simultaneously. The point is to understand both arguments, not to decide which one is right but to see the limits of each. They can't both be true all the time in full generality. To learn from this paradox, you have to understand the force of both sides, and think about which one to apply in different situations.
The paradox was invented by Belgian mathematician Maurice Kraitchik. He was not an idiot. He was good at hard math. Yet he still saw the force of both arguments. He introduced it because both arguments are used all the time both by statisticians and in informal reasoning. Both are valuable tools in some situations. But it took this example to get people to admit that neither argument is universal, that you have to be careful using either one.
Sure, you can redefine the problem to bring it into the realm where one or the other argument is stronger. That's easy. The hard part is to define the precise border between the realms.
|