Romantic mathematical art. II (Q2019380)

This column by Jim Henle is the second part of ``Romantic Mathematical Art''. The first part focused on the infinite; the second part on impossibility and mystery. The author starts with the liar paradox, citing examples of reasoning where the impossible is not avoided or denied, but somehow becomes art. The story of Euathlus, a law student of Protagoras, is particularly fun. He swore that he would pay for the lessons after winning his first case. However, he didn't dedicate himself to law after his studies. His teacher sued him, but that's when the paradox arose: if the student were to win the case, he would not have to pay anything, of course. If he were to lose the case, would there be any way in which he would have broken his oath, since he had sworn to pay after winning his first case? At the end of the day, the student didn't pay, not even a penny! To ``solve'' some paradoxes, truth values can be denied, and the truth values of questions can be distinguished from the truth values of answers. The self-referential paradox called ``Miniac'' is particularly engaging. Then, the author mentions the Axiom of Choice and some paradoxes related to knowledge, that can be ``explained'' and ``solved'' through the notion of infinity. According to Henle, in fact, ``the art of the impossible touches the art of infinity.'' Other fields such as Statistics and Politics (voting systems) are not spared by the irony of paradoxes. The explanation of a magic trick with cards ends the column, adding a touch of magic to a brilliant discussion for the sake of mathematical pleasure.