From Gábor J. Székely’s Paradoxes in Probability Theory and Mathematical Statistics, via Mark Chang’s Paradoxology of Scientific Inference:
A, B, C, D, and E make up a five-member jury. They’ll decide the guilt of a prisoner by a simple majority vote. The probability that A gives the wrong verdict is 5%; for B, C, and D it’s 10%; for E it’s 20%. When the five jurors vote independently, the probability that they’ll bring in the wrong verdict is about 1%. But if E (whose judgment is poorest) abandons his autonomy and echoes the vote of A (whose judgment is best), the chance of an error rises to 1.5%.
Even more surprisingly, if B, C, D, and E all follow A, then the chance of a bad verdict rises to 5%, five times worse than if they vote independently, even though A is nominally the best leader. Chang writes, “This paradox implies it is better to have your own opinion even if it is not as good as the leader’s opinion, in general.”
When in very good spirits he would jest in a delightful manner. This took the form of deliberately absurd or extravagant remarks uttered in a tone, and with a mien, of affected seriousness. On one walk he ‘gave’ to me each tree that we passed, with the reservation that I was not to cut it down or do anything to it, or prevent the previous owners from doing anything to it: with those reservations it was henceforth mine. Once when we were walking across Jesus Green at night, he pointed at Cassiopeia and said that it was a ‘W’ and that it meant Wittgenstein. I said that I thought it was an ‘M’ written upside down and that it meant Malcolm. He gravely assured me that I was wrong.
— Norman Malcolm, Ludwig Wittgenstein: A Memoir, 1958
Botanist George B. Hinton named the plant species Salvia leninae Epling after a saddle mule, Lenina, who had helped him to gather more than 150,000 specimens in the mountains of western Mexico.
He wrote, “What is more deserving of commemoration than the dignity of long and faithful service to science, even though it be somewhat unwitting — or even unwilling?”
See Rigged Latin.
In summer 1940, Germany demanded access to Swedish telephone cables to send encoded messages from occupied Norway back to the homeland. Sweden acceded but tapped the lines and discovered that a new cryptographic system was being used. The Geheimschreiber, with more than 800 quadrillion settings, was conveying top-secret information but seemed immune to a successful codebreaking attack.
The Swedish intelligence service assigned mathematician Arne Beurling to the task, giving him only a pile of coded messages and no knowledge of the mechanism that had been used to encode them. But after two weeks alone with a pencil and paper he announced that the G-schreiber contained 10 wheels, with a different number of positions on each wheel, and described how a complementary machine could be built to decode the messages.
Thanks to his work, Swedish officials learned in advance of the impending invasion of the Soviet Union. Unfortunately, Stalin’s staff disregarded their warnings.
“To this day no one knows exactly how Beurling reasoned during the two weeks he spent on the G-Schreiber,” writes Peter Jones in his foreword to The Codebreakers, Bengt Beckman’s account of the exploit. “In 1976 he was interviewed about his work by a group from the Swedish military, and became extremely irritated when pressed for an explanation. He finally responded, ‘A magician does not reveal his tricks.’ It seems the only clue Beurling ever offered was the remark, cryptic itself, that threes and fives were important.”
In 1962 mycologist R.W.G. Dennis reported a new species of fungus he had observed growing in Lancashire and East Africa. He called it Golfballia ambusta:
The unopened fruit body evidently closely resembles certain small, hard but elastic, spheres employed by the Caledonians in certain tribal rites, practised at all seasons of the year in enclosures of partially mown grass set apart for the purpose. The diameter of the volva is approximately 3 cm., its surface smooth or regularly furrowed, becoming much wrinkled after dehiscence, its texture extremely hard and tough. A gelatinous stratum, so characteristic of other phalloids, is wanting. The appearance and texture of the immature gleba is still unknown but at maturity it is extruded as a column, thickly set with short strap-like processes of an elastic consistency, each scarcely 1 cm. long and 1.5 mm. wide, abruptly truncated at the free end. As with other phalloids, there is a strong and distinctive odour, in this instance not unpleasant and identified independently by several observers as reminiscent of old or heated india-rubber. This is probably a reliable and important diagnostic character. Taste not recorded but probably mild; the fruit bodies are unlikely to be toxic but may well prove inedible from their texture. Spores have not been recovered and the means of reproduction therefore remains unknown.
It seems to be very prolific in America as well.
(R.W.G. Dennis. A remarkable new genus of phalloid in Lancashire and East Africa, Journ. Kew Guild. 8, 67 (1962): 181-182.)
Writing on “The Sagacity of the Bees” in fourth century, Pappus of Alexandria argued that bees had contrived the hexagonal shape of their honeycomb cells “with a certain geometrical forethought.” Irregularly shaped cells “would be displeasing to the bees,” he wrote, and only triangles, squares, or hexagons could fill the space regularly. “The bees in their wisdom chose for their work that which has the most angles, perceiving that it would hold more honey than either of the two others.”
In 1964, in a charming address titled “What the Bees Know and What They Do Not Know,” Hungarian mathematician László Tóth told the American Mathematical Society that he had found a slight improvement on the classic honeycomb design: Instead of closing the bottom of each cell with three rhombi, as bees do, it’s more efficient to use two hexagons and two rhombi.
But, he added immediately, “We must admit that all this has no practical consequence. By building such cells the bees would save per cell less than 0.35% of the area of an opening (and a much smaller percentage of the surface-area of a cell). On the other hand, the walls of the bee-cells have a non-negligible width which is, in addition, far from being uniform and even the openings of the bee-cells are far from being exactly regular. Under such conditions the above ‘saving’ is quite illusory. Besides, the building style of the bees is definitely simpler than that described above. So we would fail in shaking someone’s conviction that the bees have a deep geometrical intuition.”
(László Fejes Tóth, “What the Bees Know and What They Do Not Know,” Bulletin of the American Mathematical Society, 1964, 468-81.)
UPDATE: Wait — maybe they’re not as smart as we thought. (Thanks, Vic.)
David Hume argued that reports of miracles can never be credited, because the weight of human experience must always favor a more natural explanation. “Nothing is esteemed a miracle, if it ever happen in the common course of nature. It is no miracle that a man, seemingly in good health, should die on a sudden: because such a kind of death, though more unusual than any other, has yet been frequently observed to happen. But it is a miracle, that a dead man should come to life; because that has never been observed in any age or country. There must, therefore, be a uniform experience against every miraculous event, otherwise the event would not merit that appellation.”
The sun is said to have danced in the sky in 1917. Well, which is more likely, that such an extraordinary event actually occurred, or that it was really a mass hallucination, an optical illusion, or any of a hundred more familiar explanations? A miracle, a suspension of natural law, is always the least likely possibility, so as rational creatures we must always reject it.
But Alfred Russel Wallace objected, “Such a simple fact as the existence of flying fish could never be proved, if Hume’s argument is a good one; for the first man who saw and described one, would have the universal experience against him that fish do not fly, or make any approach to flying, and his evidence being rejected, the same argument would apply to the second, and to every subsequent witness.”
Hume’s argument, he said, was “radically fallacious,” because if it were sound “no perfectly new fact could ever be proved, since the first and each succeeding witness would be assumed to have universal experience against him.” Who’s right?
I don’t know who came up with this — it’s been bouncing around science journals for 50 years:
hydromicrobiogeochemist: one who studies small underwater flora and their relationship to underlying rock strata by using chemical methods
microhydrobiogeochemist: one who studies flora in very small bodies of water and their relationship to underlying rock strata by using chemical methods
microbiohydrogeochemist: one who studies small flora and their relationship to underlying rock strata by using chemical methods and SCUBA equipment
biohydromicrogeochemist: a very small geochemist who studies the effect of plant life in hydrology
hydrobiomicrogeochemist: a very small geochemist who studies wet plants
biomicrohydrogeochemist: a very small, wet geochemist who likes lettuce
- WEALTH is an anagram of THE LAW.
- U.S. Navy submarines observe an 18-hour day.
- Joaquín Rodrigo wrote his compositions in Braille.
- 45632 = –45 + 63×2
- “Thy modesty’s a candle to thy merit.” — Henry Fielding
This prudence of not attempting to give reasons before one is sure of facts, I learnt from one of your sex, who, as Selden tells us, being in company with some gentlemen that were viewing, and considering something which they called a Chinese shoe, and disputing earnestly about the manner of wearing it, and how it could possibly be put on; put in her word, and said modestly, Gentlemen, are you sure it is a shoe? — Should not that be settled first?
— Benjamin Franklin, letter to Mary Stevenson, Sept. 13, 1760