Here’s a brain teaser I wish I’d invented in time to include it in The Big Questions:
John and Mary live in an isolated village where they have no access to reference materials, no contact with the outside world, and nobody to talk to except each other. One day an anthropologist arrives in this village, sits down for coffee with John and Mary, and quizzes them about their knowledge of the world. John says he’s sure that men have walked on the moon; Mary says she’s sure they haven’t. Never having discussed this issue before, each of them is astonished and flabbergasted by the others’ apparent ignorance. Rather than risk losing all respect for each other, John and Mary agree never to speak of the subject again. But the anthropologist mentions that she’ll be stopping by once a day from now on, and will be glad to know if either of them ever has a change of mind on this topic. If so, the anthropologist will inform the other. Otherwise, the anthropologist will never bring it up either.
The next day (a Monday) nobody’s mind has changed, and therefore the subject is not discussed. The same thing happens on Tuesday, Wednesday and Thursday. Can this go on forever?
The surpising answer, under quite general assumptions about the way people learn, is that eventually, if John and Mary care about the truth (as opposed to, say, winning a debating point) then somebody’s mind must change.
But how can this be? If nobody’s mind changed on Tuesday, Wednesday or Thursday, and if no new information or argument ever gets brought into the picture, how can Friday be different?
Here’s how: On Monday, all John knows is that when it comes to moon landings, Mary disbelieves. By Tuesday, he knows that her disbelief is strong enough to stand up in the face of his belief—which is something he didn’t know on Monday. By Wednesday, he knows that her disbelief is strong enough to stand up in the face of his continued belief in the face of her continued disbelief—which is something he didn’t know on Tuesday. And so forth.
Each time a day goes by with no minds changed, John and Mary learn something new about the strength of each others’ beliefs—and therefore find it harder to maintain their own beliefs, because, after all, there’s always a chance the other guy is right. And the surer the other guy is, the better that chance—at least on the assumption that the other guy is at least trying to arrive at the truth.
The fact that John or Mary must eventually break down in the face of this barrage takes some work to prove; the first round of that work was done by the Nobel prize winning economist Robert Aumann, who took the first step toward showing that it’s essentially impossible for honest truthseekers to “agree to disagree”. Other economists, especially the always innovative Robin Hanson, then took up the baton and pushed these results much further. You can read all about it in Chapter 8 of The Big Questions.
Of course, in the real world, John and Mary never change their minds; they just go off to sulk, become increasingly embittered and eventually stop talking to each other altogether. The disturbing implication of Aumann’s theorem is that therefore John and Mary cannot both be honest truthseekers. Nor can almost anyone else who’s ever been party to an ongoing disagreement. Which, most disturbingly of all, would seem to include me.
It occurs to me that this brain teaser has more than a little in common with a brain teaser that recently got a lot of play over on the blog of the Fields-medal winning mathematician Terence Tao:
There is an island upon which a tribe resides. The tribe consists of 1000 people, with various eye colours. Yet, their religion forbids them to know their own eye color, or even to discuss the topic; thus, each resident can (and does) see the eye colors of all other residents, but has no way of discovering his or her own (there are no reflective surfaces). If a tribesperson does discover his or her own eye color, then their religion compels them to commit ritual suicide at noon the following day in the village square for all to witness. All the tribespeople are highly logical and devout, and they all know that each other is also highly logical and devout (and they all know that they all know that each other is highly logical and devout, and so forth).
Of the 1000 islanders, it turns out that 100 of them have blue eyes and 900 of them have brown eyes, although the islanders are not initially aware of these statistics (each of them can of course only see 999 of the 1000 tribespeople).
One day, a blue-eyed foreigner visits to the island and wins the complete trust of the tribe.
One evening, he addresses the entire tribe to thank them for their hospitality.
However, not knowing the customs, the foreigner makes the mistake of mentioning eye color in his address, remarking “how unusual it is to see another blue-eyed person like myself in this region of the world”.
What effect, if anything, does this faux pas have on the tribe?
Once again the key to the brain teaser is that silence conveys information, silence in the face of silence conveys even more information, and silence in the face of silence in the face of silence conveys even more. One difference is that Tao’s brain teaser is self-contained; you ought to be able to figure it out without reference to journal articles. The agree-to-disagree problem lies deeper, but is also, I suspect, more profound.