Arthur and Robert are identical twins. One always lies, and the other always tells the truth, but you don’t know which is the liar. One day you meet one of them and want to find out whether it’s Arthur or Robert. But you can ask only one yes/no question, and the question can’t contain more than three words. What question will do? Alternatively, suppose you want to find out whether it’s Arthur or Robert who’s truthful. What three-word yes/no question will reveal the answer?
Late
Argentina had a surprise on July 10, 1945: The German submarine U-530 turned up at Mar del Plata and surrendered. Commander Otto Wermuth said that he’d received orders on May 8 to cease hostilities and proceed to the nearest United Nations port for surrender. He’d thought this was an enemy trick and decided to intern his submarine and crew in a neutral country. He chose Argentina thinking that it had not declared war and turned up there two months after the German surrender.
That created a fertile field for speculation that the sub had been transporting Nazi gold or leaders to South America — Wermuth was blamed for sinking the Brazilian cruiser Bahia (later disproven), and one reporter even claimed that he’d seen a mysterious sub putting ashore an officer and a civilian who might have been Adolf Hitler and Eva Braun.
“In answer to questions, WERMUTH said that he did not know of any other submarines which were headed for Argentina, that he had been in touch with no other submarines,” read the intelligence report. “He added the somewhat enigmatic remark, however, that if any more were coming they would arrive within a week of his arrival. The reason for this statement was not given.”
Löb’s Paradox
A paradox by the German mathematician Martin Löb:
Let A be any sentence. Let B be the sentence: ‘If this sentence is true, then A.’ Then a contradiction arises.
Here’s the contradiction. B makes the assertion “If B is true, then A.” Now consider this argument. Assume B is true. Then, by B, since B is true, A is true. This argument shows that, if B is true, then A. But that’s exactly what B had asserted! So B is true. And therefore, by B, since B is true, A is true. And thus every sentence is true, which is impossible.
(Lan Wen, “Semantic Paradoxes as Equations,” Mathematical Intelligencer 23:1 [December 2001], 43-48.)
Black and White
Unquote
“My own opinion is that belief is the death of intelligence. As soon as one believes a doctrine of any sort, or assumes certitude, one stops thinking about that aspect of existence. The more certitude one assumes, the less there is left to think about, and a person sure of everything would never have any need to think about anything and might be considered clinically dead under current medical standards, where the absence of brain activity is taken to mean that life has ended.” — Robert Anton Wilson
Outreach
The Statue of Liberty’s disembodied arm stood in Madison Square from 1876 to 1882. It had been agreed that Frédéric Bartholdi would create the statue while the United States paid for the pedestal. Americans were a bit behindhand in offering donations, so Bartholdi sent along the arm and torch to help inspire contributions.
It took six years of benefit concerts, auctions, souvenir photos, and other mementos, but the full statue was finally dedicated on Liberty Island on October 28, 1886.
Proof Without Words
In the January 2001 issue of the College Mathematics Journal, University of Calgary mathematician Jonathan Schaer offers this simple proof that arctan 1 + arctan 2 + arctan 3 = π. The sum of the angles of this large triangle is 180°. And the diagram shows that its lower left angle is arctan 3, its lower right angle is arctan 2, and its top angle is part of an isosceles right triangle. So arctan 1 + arctan 2 + arctan 3 = 180°, or π radians. “No words, even no symbols!”
(“Miscellanea,” College Mathematics Journal 32:1, 68-71.)
Rotating Office
Actress Ilka Chase married Louis Calhern in 1926, but he divorced her the following year to marry Julia Hoyt.
Sorting through her possessions afterward, she discovered a set of engraved calling cards that she’d had printed with the name Mrs. Louis Calhern.
“They were the best cards — thin, flexible parchment, highly embossed — and it seemed a pity to waste them, and so I mailed the box to my successor,” she wrote later.
“But aware of Lou’s mercurial marital habits, I wrote on the top one, ‘Dear Julia, I hope these reach you in time.’
“I received no acknowledgment.”
(From Chase’s 1945 autobiography Past Imperfect.)
In a Word
volant
adj. engaged in flight
espieglerie
n. impish or playful behaviour; mischief
apopemptic
adj. pertaining to leave-taking or departing
réclame
n. publicity or notoriety
In 1953, 61-year-old British ace Christopher Draper flew an Auster monoplane under 15 of the 18 bridges on the Thames, negotiating 50-foot arches at 90 mph.
“I did it for the publicity,” he told the press. “For 14 months I have been out of a job, and I’m broke. I wanted to prove that I am still fit, useful and worth employing. … It was my last-ever flight — I meant it as a spectacular swan song.” He was fined 10 guineas.
Night Life
Suppose that your dream-life underwent a remarkable change. Suppose that on going to bed at home and falling asleep, you found yourself to all appearances waking up in a hut raised on poles at the edge of a lake. A dusky woman, whom you realize to be your wife, tells you to go out and catch some fish. The dream continues with the apparent length of an ordinary human day, replete with an appropriate and causally coherent variety of tropical incident. At last you climb up the rope ladder to your hut and fall asleep. At once you find yourself awaking at home, to the world of normal responsibilities and expectations. The next night life by the side of the tropical lake continues in a coherent and natural way from the point at which it left off. Your wife says, ‘You were very restless last night. What were you dreaming about?’ and you find yourself giving her a condensed version of your English day. And so it goes on. Injuries given in England leave scars in England, insults given at the lakeside complicate lakeside personal relations. One day in England, after a heavy lunch, you fall asleep in your armchair and dream of yourself, or find yourself, waking up in the middle of the night beside the lake. Things get too much for you at the lakeside, your wife has departed with all the cooking-pots, and you suspect that she is urging the villagers to sacrifice you to the moon. So you fall on your fish-spear and from that moment on your English slumbers are disturbed no more than in the old pre-lakeside days.
— Anthony Quinton, Thoughts and Thinkers, 1982
“Is such a two-space reality conceivable?” asks Peg Tittle in What If…: Collected Thought Experiments in Philosophy (2016). “That is, is it conceivable that we could live in two different but real spaces?”
See Figure and Ground.