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January's topics:
- Gödel's incompleteness theorem in The Guardian
- "Fun with Math" in The New Yorker
- Rock-paper-scissors and evolutionary game theory
- "What physics owes to math"
Gödel's incompleteness theorem in The Guardian
Alex Bellos's Monday puzzle in The Guardian for January 10, 2022 was derived from explanations of Gödel's incompleteness theorem due to the logician Raymond Smullyan. (Smullyan published Forever Undecided: A Puzzle Guide to Gödel in 1987). The setting for the puzzle, as Bellos presents it, is a hypothetical island he calls "If." Natives of If are either "Alethians" or "Pseudians;" they are indistinguishable except that Alethians always tell the truth, while Pseudians always lie. This sounds like the traditional Liar Problem, but there is a wrinkle: on the island is a ledger where every native is listed, along with his or her tribe. Anyone can consult this ledger. You arive on If and a person, Kurt, comes up to you stating: "You will never have concrete evidence that confirms I am an Alethian." The puzzle: is Kurt an Alethian, a Pseudian or neither? Think about it before checking Bellos's solution and before reading on. Now comes the connection with Gödel's incompleteness theorem, which states, as Bellos puts it, "that there are mathematical statements that are true but not formally provable." Suppose you are the first non-native ever to visit If, so you know that everyone you meet is Alethian or Pseudian. Kurt pops up and says, just as before, "You will never have concrete evidence that confirms I am an Alethian." But now just as he speaks the Ledger burns to ashes. Where are we? Kurt cannot be a Pseudian, because with no Ledger that statement has to be true. So you know Kurt is an Alethian. But you can never have concrete evidence of that fact because if you did, his statement would be false, and it can't be false since he is Alethian. Think about it. [Thanks to Jonathan Farley for bringing this item to my attention. -TP]"Fun with Math" in The New Yorker
Dan Rockmore's contribution to "Talk of the Town" in the January 17, 2022 issue of the The New Yorker was an item titled "Fun with Math." He recounts attending "a recent evening of math dinner theater" organized by Cindy Lawrence, director of New York's National Museum of Mathematics. The event featured Peter Winkler, a mathematics professor at Dartmouth and expert on math puzzles. Among the puzzles and phenomena that Winkler served up for discussion during dinner:- "On average, how many cards does it take to get to a jack in a shuffled deck of fifty-two cards?"
- "What's the best way to use two coin tosses to determine which of two coins, one fair and one 'biased,' is fair?"
- Simpson's paradox in statistics, best described by an example: for Berkeley's graduate programs in 1973, overall "men were admitted at a higher rate than women, but, program by program, women were admitted at a higher rate." (See MinutePhysics for a more detailed explanation.)
Rock-paper-scissors and evolutionary game theory
"Non-Hermitian topology in rock-paper-scissors games" by the three Tsukuba physicists Tsuneya Yoshida, Tomonari Mizoguchi and Yasuhiro Hatsugai was published January 12, 2022 in Scientific Reports. This is a physics article, but it applies a nice piece of mathematics, evolutionary game theory, to the familiar rock-paper-scissors game. The game consists of two players; at a signal each shows a clenched fist ("rock"), a flat hand ("paper") or a vertical hand with the first two fingers displayed ("scissors"). The winner (rock smashes scissors, scissors cut paper, paper covers rock) gets one point, the loser loses one. If both players show the same symbol, each gets zero. The article contains this image:

"What physics owes to math"
On January 12, 2022 Le Monde ran a guest article with the title "What physics owes to math" (text in French) written by two physicists, Jean Farago and Wiebke Drenckhan, from the Institut Charles-Sadron in Strasbourg. The authors begin with the famous quote from Galileo about the Book of Nature being written in the language of mathematics, and go on to observe: "The millenia elapsed between the birth of mathematics and its use in physics demonstrate that this contiguity between natural phenomena and the mathematical laws of our human rationality was far from being obvious." Farago and Drenckhan mention that one of the most antonishing examples of the "intimate" relationship between mathematics and physics comes from complex numbers. Starting in the 16th century, mathematicians found that calculating solutions to polynomial equations with whole-number coefficients required the use of an 'imaginary' numberRead other recent Tony's Take columns.