The well-known expression “the exception that proves the rule”, mentioned last week, cannot be taken literally: exceptions question or weaken the rules, and in the rigorous domain of mathematics they invalidate them. Like many proverbs and sayings, it is a “poetic” expression, which implies that the fact that we perceive something as exceptional shows that the opposite is the most common, that is, the rule.
As for other versions of Russell’s paradox, my favorite is what we might call the library version:
Let us call self-referential (A) the books in whose pages the book itself is mentioned and non-self-referential (NA) the books in which there is no mention of the book itself. Most are NA, but the A are not scarce either, and there are illustrious examples: in the second part of Don Quixote, without going any further, it alludes to the appearance in book form of the episode of the windmills and other feats of the first part, and Dante cites his own work in canto XXIV of Purgatorio. And now let’s try to make the complete catalog of NA books. Is the catalog itself an A or NA book? If it’s NA, it should be in the NA catalog of books, and therefore mentions itself, then it’s A. But if it’s A, it shouldn’t be listed in the NA catalog…
Aliens and sexist kings
And for paradoxes, those of probability, which we have often dealt with in previous installments. Coincidentally (or maybe not), Ignacio Alonso uploaded a NMM riddle to the comments section last week that has raised some questions -and a long debate- as to how to approach it:
A flotilla of alien ships takes all the people in a house following a strict rule of separation of the sexes, so that only men or only women can go on the same spaceship. So the crew of the first ship choose one person at random and then another; if the second is of the same sex as the first, they choose a third, but if it is of the other sex, they discard it and leave with only one prey, and so on until they abduct all those present, who are 5 men and 8 women . What is the probability that the last abductee will be a woman?
It seems reasonable to think that any one of the 13 people in the house has the same probability as any other of being the last, and since 8 of those people are women, the requested probability is 8/13. But why so many statements for such a simple solution? Is there any fallacy in the above reasoning?
However, it is also worth asking what is the minimum and maximum number of ships needed to complete the abduction, and what is the expected number.
This problem is reminiscent of that macho king who wanted to reduce the proportion of women in his kingdom and, to do so, issued an order according to which all couples had to stop procreating as soon as they had a girl. In this way, the king reasoned, there would be families with several sons and at most one daughter, but none with several daughters, so the proportion of males would increase considerably. What was the result of his demographic measurement?
Carlo Frabetti is a writer and mathematician, member of the New York Academy of Sciences. He has published more than 50 popular science works for adults, children and young people, including ‘Damn Physics’, ‘Damn Mathematics’ or ‘The Great Game’. He was a screenwriter for ‘The Crystal Ball’.