Very interesting extrapolation, V, and I must say that you've largely encapsulated the reasons why this apparently so simple problem delights and irks me to the same degree.
Okay, I am absolutely no mathematician, but I like to think that I've got the basics fairly well down. But this so simply constructed yet apparently unravellable paradox seems to me to cut across - or at least have implications on - many differing modes of thought: maths, logic, philosophy, even quantum. It's pretty much a thought experiment that involves pure metaphysics.
Indeterminacy does seem to have a major role here. This from Wiki on Heisenberg's Uncertainty Principle:
Wikipedia wrote:In quantum mechanics, the uncertainty principle is any of a variety of mathematical inequalities asserting a fundamental limit to the precision with which certain pairs of physical properties of a particle known as complementary variables, such as position x and momentum p, can be known simultaneously. For instance, in 1927, Werner Heisenberg stated that the more precisely the position of some particle is determined, the less precisely its momentum can be known, and vice versa.
This would seem to have some relevance to - or at least some commonality with - the two envelope problem. The two envelope problem also has
a pair of complementary variables and also features a situation in which nothing can be known about how the two complementary variables relate to each other (i.e. which was the larger and which was the smaller) until after the event (that event being the revealing of what was in both envelopes). Since I'm speaking of quantum, this, as I and Hashi have both said before now, brings Schroedinger's cat very much to mind - the final act of observation (the revealing of what's in the second envelope) collapses the waveform and defines the reality - but only at that precise point. It's only then that indeterminacy/uncertainty finally ends.
Actually, I ought to pose a question resulting from that assertion - presumably you guys also think it makes no difference to things if you open the first envelope and find a $100 bill? Sure, you've now got the first amount clearly defined - it's a hundred bucks - but you still have no clue as to what's in the second envelope and therefore no clue as to whether your hundred bucks is the smaller or the larger amount. To me, indeterminacy still seems to persist completely here.
V, your comment about pre-destination brought to my mind Newton, who as we all know was the champion of the purely mechanical universe where everything that would ever happen was utterly pre-determined from the outset. Of course, Newton injected God into things at the last minute, presumably for fear of burning in Hell (or possibly for being burnt as a heretic). I find it fascinating that such a seemingly simple problem as the two envelope conundrum can cause us to question whether our concept of free will is entirely illusory. As you said:
Vraith wrote:a fair number of people say this basically describes the universe. It APPEARS to us to include "chaos," and "probability" and "chances." But it ACTUALLY is wholly determined. Every single event was absolutely set from the beginning. The chaos and odds are simply due to our permanent and unbridgeable ignorance. One definite answer in any and all possible instances that we can never, even theoretically, know.
At this stage, I need to ask a couple of previously asked questions again:-
A.
Say I give you $100. I then say to you that you can either keep that $100, or you can go for a coin flip, where I'll flip a coin and you call it. If you call it correctly, I'll give you $200 instead of your original $100. If you call it incorrectly, I'll give you $50 instead of your original $100.
There is no doubt as to the best thing to do here, right? Or is there?
C.
Say I give you $100. I then say to you that you can either keep that $100, or you can reveal an already randomly flipped coin under a cup. If that coin shows heads, I'll give you $200 instead of your original $100. If that coin shows tails, I'll give you $50 instead of your original $100.
Is this - as Hashi maintains - exactly the same as A?
And if my point about indeterminacy above holds water (and needless to say, I'm unsure if it does), what's the significant difference between either A or C above and the original two sealed envelope puzzle? I mean, in all cases, pairs of complementary variables apply, with nothing being fully determined i) until the coin has been called and flipped [in A], or ii) until the coin has been revealed [in C], or iii) until the second envelope has been opened [in the original problem]. Nothing gets determined for certain until that final point in all examples. Is this significant?
??? I thought the general opinion was that there was no advantage or disadvantage to swapping? But hang on... I'm going to combine your thoughts with V's in a moment.
):-
