I made fun of philosophers last week regarding the Monte Hall problem and a friend pointed out that it actually first appeared in the mathematical recreations literature. Which is true, but philosophers tend to be the ones who run with these problems, spinning out endless papers for the Journal of the Otherwise Unemployable and its sibling publications Dazed, Confused, and Metaphysics.
And in this case the variant of problem I want to consider is due to Raymond Smullyan, who is a philosopher, not a mathematician, so I feel justified in making fun of philosophers in this case. Or any case, really.
The problem is the "Two Envelopes Paradox" and again illustrates the difficulties of thinking about conditional probability. It also illustrates a quirk of human psychology that I do not understand, which is the Persistence of Paradoxes.
That is, once humans have decided that a paradox exists, nothing can be done to clarify it for them. You can explain to the end of the Earth, and even if they briefly understand what the solution is, they will eventually put that knowledge and understanding aside and go back to being fascinated by the object of their obsession.
In the case of Smullyan's two-envelope problem, the idea is that there are two envelopes, one with an amount A and the other with an amount A/2. So if A is $10 then A/2 is $5. I like working with concrete values in these things because I'm not a very good abstract thinker and have a lot of suspicion of abstraction, as I think it is primarily a tool for lying (mostly to ourselves) and only secondarily useful for thinking clearly. The less abstract thought we can get away with the better.
The paradox is that two players choose the envelopes at random and are asked if they want to switch. Should they?
The argument goes like this: supposed one player has chosen the envelope with $10. Then he stands to lose $5 on the swap. On the other hand, suppose the envelope has $5 in it. Then he stands to gain $5 on the swap. Since $5 = $5, the amount gained or lost is equal, and the two cases have equal probability, so there is no benefit to swapping, which is exactly what our naive intuition tells us must be the case. Wait, wasn't there supposed to be a paradox in here somewhere?
The construction of the "paradox" turns on the conflation of the condition where the player chooses the envelope with $10 in it, and the condition where the player chooses the envelope with $5 in it. There is a little bit of algebraic legerdemain involved, which has no purpose other than hiding this fact.
The curious thing is that fifteen or twenty years ago I edited the Wikipedia page on the paradox to make this point manifest. At the time the way the paradox was stated, after mentioning that A might be $10, it had something along the lines of "the player either chooses the envelope with A or A/2 in it..." which I edited to read "the player either chooses the envelope with A (in just the case where the envelope contains $10) or A/2 (in just the case where the envelope contains $5) in it..."
By separating the concrete conditions in this way the paradox evaporated, and there was a flurry of activity on the Wikepedia "Talk" page for the article (which I checked on a few weeks later but did not take part in, having committed my act of epistemic vandalism and moved on) discussing What Is To Be Done? After all, an article on a paradox that now looked, well… kind of stupid... wasn't going to make anyone happy.
Eventually--I checked back a few years later--there was a "Self Solving" section added to the article that described the difficulty, but subsequent edits have revised that out of existence. Although at least the "Simple Solution" section in the current version retains the gist of my point, which is that the value of "A" as it is used in the argument has to subtly change from clause to clause, because different clauses refer to different conditions:
The famous mystification is evoked by the mixing up of two different circumstances and situations, giving wrong results. The so-called "paradox" presents two already appointed and already locked envelopes, where one envelope is already locked with twice the amount of the other already locked envelope. Whereas step 6 boldly claims "Thus the other envelope contains 2A with probability 1/2 and A/2 with probability 1/2.", in the given situation, that claim can never be applicable to any A nor to any average A.
This seems to me to be clear and obvious, but from the extensive and ongoing discussion of the "problem" it is apparent that this is... not universally the case.
For a long time I tended to think that people persist in these discussions because they enjoy the feeling of mystification, like some kind of epistemic drunkenness, but I'm no longer at all sure that's the case. It starts to look to me like they really don't get it, despite the fairly impressive mathematical machinery brought to the table: the "fast" intuitive system of our brain is just not capable of internalizing the rules for reasoning about conditional probabilities, and the discipline of formally splitting conditional problems up and thinking about them in terms of multiple unrelated situations is excruciatingly hard to learn.
People have a strong desire to have a feeling of understanding, which is quite different from having an understanding. Things we don't understand are dangerous, at least potentially, but we don't actually need to really understand most of them to deal with them. But we need to find a box to put them in, a place to hang them, so we know where to find them, where they fit, if we need to come back to them later. A feeling of understanding is what we get when we find such a place.
Paradoxes resist this process: they are a kind of minor epistemic trauma.
Ordinary trauma leads to post-traumatic stress disorder (PTSD) precisely because it doesn't fit anywhere. We can't process it or integrate it into our identity, at least insofar as our identity is that of a healthy, functional, capable being, which is the kind of identity most of us strive to maintain. As such, traumatic memories tend to roll around loose in our brain, bashing into stuff and causing pain at strange and inopportune times.
Paradoxes function similarly: they can't be made to fit into any sensible logic that is within the capacity of most people to grasp, so they continue to float free, and often appear at the edge of our attention when we should be thinking about something else. Maintaining the category of "paradox" gives us someplace to put them.
Me, I don't really do paradoxes, because I've spent too many decades standing face to face with the big one: reality itself, as revealed by something called Bell's Paradox, which--all going well--I'll have something to say about next week.
Bell's Paradox is not really a paradox any more, although the resolution is as incomprehensible as the paradox itself, because it reveals something about the world we live in that cannot--not "does not", but "cannot"--make any sense, because the aspects of reality involved violate the conditions that a thing must fulfill to be made sense of.
This will segue "World of Wonders" in a direction I've had in mind for a long time: that of explicating the nature of knowable reality, and what lies beyond it. Stay tuned!
I really like the direction of this and your last post. I have only encountered these so-called paradoxes in discussions with people trying to look clever. They get the explanation wrong every time. It makes me think a lot about the so-called 'skeptic' community. It's got nothing to do with the advancement of reasoning and is compulsively looking for clever things to say about everyday ideas. Paradox is TikTok for intellectuals.
Looking forward to reading more about what is behind knowable reality!