Science & Math

Single Cases

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If we roll a fair die an infinite number of times, the outcome 4 occurs in 1/6 of the cases. In this light we can say that the probability of rolling a 4 with this die is 1/6. But suppose that, instead of repeating the experiment forever, we roll the die only once. Now it still seems natural to say that there’s a 1/6 chance of rolling a 4, but in fact either we’ll roll a 4 … or we won’t. Can it make sense to assign a probability to a single outcome? Charles Sanders Peirce writes:

If a man had to choose between drawing a card from a pack containing twenty-five red cards and a black one, or from a pack containing twenty-five black cards and a red one, and if the drawing of a red card were destined to transport him to eternal felicity, and that of a black one to consign him to everlasting woe, it would be folly to deny that he ought to prefer the pack containing the larger proportion of red cards, although, from the nature of the risk, it could not be repeated. It is not easy to reconcile this with our analysis of the conception of chance. But suppose he should choose the red pack, and should draw the wrong card, what consolation would he have? He might say that he had acted in accordance with reason, but that would only show that his reason was absolutely worthless. And if he should choose the right card, how could he regard it as anything but a happy accident? He could not say that if he had drawn from the other pack, he might have drawn the wrong one, because an hypothetical proposition such as, ‘if A, then B,’ means nothing with reference to a single case.

Peirce’s solution to this problem is curiously humanistic. Our inferences must extend to include the interests of all races in all epochs. A soldier storms a fort knowing that he may die but that his zeal, if carried through the regiment, will win the day. The man trying to draw a red card “cannot be logical so long as he is concerned only with his own fate” but “should care equally for what was to happen in all possible cases … and would draw from the pack with the most red cards.”

“He who would not sacrifice his own soul to save the whole world, is, as it seems to me, illogical in all his inferences, collectively.”

Pursuit of Truth

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Can animals reason without using language? Sextus Empiricus writes:

[Chrysippus] declares that the dog makes use of the fifth complex indemonstrable syllogism when, on arriving at a spot where three ways meet …, after smelling at the two roads by which the quarry did not pass, he rushes off at once by the third without stopping to smell. For, says the old writer, the dog implicitly reasons thus: ‘The animal went either by this road, or by that, or by the other: but it did not go by this or that, therefore he went the other way.’

So, perhaps. There’s a limit, though.

The Lakes of Wada

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Find a square island and establish a blue lake on it, bringing blue water within a certain distance of every point on the island’s remaining dry land. Then create a red lake, bringing red water even closer to every point on the remaining land, and a green lake bringing green water still closer.

If you continue this indefinitely, irrigating the island more and more aggressively from each lake in turn, you’ll reach the perplexing state where the three lakes have the same boundary. Japanese mathematician Kunizo Yoneyama offered this example in 1917.

Miquel’s Theorem

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Draw any triangle, pick a point on each side, and connect these in pairs to the vertices using circles as shown.

The circles will always intersect in a single point.

Further, the angles marked in green will all be equal.

Jury Duty

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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.”

Possession

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

Special Thanks

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.

Quick Thinking

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.”

(Thanks, John.)

Fore!

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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.)

Building Codes

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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.)

Fish Story

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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?

Specialists

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

Misc

  • 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

First Principles

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

Crowd Control

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In July 1968, ethologist John B. Calhoun built a “mouse utopia,” a metal enclosure 9 feet square with unlimited food, water, and nesting material. He introduced four pairs of mice, and within a year they had multiplied to 620. But after that the society began to fall apart — males became aggressive, females began neglecting their young, and the weaker mice were crowded to the center of the pen, where resources were scarce. After 600 days the females stopped reproducing and the males withdrew from them entirely, and by January 1973 the whole colony was dead. Even when the population had returned to its former levels, the mice’s behavior had remained permanently changed.

There were no predators in the mouse universe; the only adversity was confinement itself. Calhoun felt that his experiment held lessons as to the potential dangers of human overpopulation, and he urged his colleagues to study the effects of high population density on human behavior. “Our success in being human has so far derived from our honoring deviance more than tradition,” he said. “Now we must search diligently for those creative deviants from which, alone, will come the conceptualization of an evolutionary designing process. This can assure us an open-ended future toward whose realization we can participate.”

(Thanks, Pål.)

Periodic Labels

I think this first appeared in the puzzle newsletter The Ag Mine — 12 chemical elements can be spelled using element symbols:

ArSeNiC
AsTaTiNe
BiSmUTh
CArBON
CoPPEr
IrON
KrYPtON
NeON
PHOsPHORuS
SiLiCoN
TiN
XeNoN

See Transmutation.

Club Science

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From Sir Edward Victor Appleton’s speech at the 1947 Nobel Banquet:

Ladies and gentlemen, you should not … overrate scientific methods, as you will learn from the story of a man who started an investigation to find out why people get drunk. I believe this tale might interest you here in Sweden. This man offered some of his friends one evening a drink consisting of a certain amount of whisky and a certain amount of soda water and in due course observed the results. The next evening he gave the same friends another drink, of brandy and soda water in the same proportion as the previous night. And so it went on for two more days, but with rum and soda water, and gin and soda water. The results were always the same.

He then applied scientific methods, used his sense of logic and drew the only possible conclusion — that the cause of the intoxication must have been the common substance: namely the soda water!

That’s from Ronald Clark, Sir Edward Appleton, 1971. Clark adds, “Appleton was pleased but a little surprised at the huge success of the story. Only later did he learn that the Crown Prince drank only soda water — ‘one of those unexpected bonuses which even the undeserving get from Providence from time to time,’ as he put it.”

Best-Laid Plans

Launched in November 1981, the Soviet Union’s Venera 14 probe carried a spring-loaded arm to test the soil of Venus.

The craft journeyed for four lonely months to reach its destination, descended safely through the hostile atmosphere, and landed securely on the surface.

The spring-loaded arm plunged downward — into a camera lens cap, which had just fallen there.

(Thanks, Merv.)

Fair Enough

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Apocryphal but entertaining: During one of Norbert Wiener’s talks on cybernetics, a student raised an esoteric point.

Wiener said, “Why, that’s as improbable as a bunch of monkeys having typed out the Encyclopaedia Britannica.”

The student said brightly, “But that’s happened once, anyway.”

Progress

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T.H. Huxley defined “four stages of public opinion” of a new scientific theory:

  1. Just after publication — The novelty is absurd and subversive of religion and morality. The propounder both fool and knave.
  2. 20 years later — The novelty is absolute truth and will yield a full and satisfactory explanation of things in general. The propounder man of sublime genius and perfect virtue.
  3. 40 years later — The novelty won’t explain things in general after all and therefore is a wretched failure. The propounder a very ordinary person advertised by a clique.
  4. A century later — The novelty is a mixture of truth and error. Explains as much as could reasonably be expected. The propounder worthy of all honour in spite of his share of human frailities, as one who has added to the permanent possessions of science.

J.B.S. Haldane had a more concise list:

  1. This is worthless nonsense.
  2. This is an interesting, but perverse, point of view.
  3. This is true, but quite unimportant.
  4. I always said so.

Law and Order

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Niels Bohr liked westerns but found them exasperating. After one feature he told his friends, “I did not like that picture, it was too improbable. That the scoundrel runs off with the beautiful girl is logical, it always happens. That the bridge collapses under their carriage is unlikely but I am willing to accept it. That the heroine remains suspended in midair over a precipice is even more unlikely, but again I accept it. I am even willing to accept that at that very moment Tom Mix is coming by on his horse. But that at that very moment there should be a fellow with a motion picture camera to film the whole business — that is more than I am willing to believe.”

He did approve of movie gunfights, where the villain always draws first and yet the hero always wins. Bohr reasoned that the man who draws first in a gunfight is using conscious volition, where his opponent is relying on reflex, a much faster response. Hence the second man should win.

“We disagreed with this theory,” wrote George Gamow, “and the next day I went to a toy store and bought two guns in Western holders. We shot it out with Bohr, he playing the hero, and he ‘killed’ all his students.”

New Units

Since Helen’s face launched a thousand ships, Isaac Asimov proposed that one millihelen was the amount of beauty needed to launch a single ship. And one negative helen is the amount of ugliness that will send a thousand ships in the other direction.

When the taciturn Paul Dirac was a fellow at Cambridge, the dons defined the dirac as the smallest measurable amount of conversation — one word per hour.

Robert Millikan was said to be somewhat conceited; a rival suggested that perhaps the kan was a unit of modesty.

And a bruno is 1158 cubic centimeters, the size of the dent in asphalt resulting from the six-story free fall of an upright piano. It’s named after MIT student Charlie Bruno, who proposed the experiment in 1972. The drop has become an MIT tradition; last year students dropped a piano onto another piano:

Lecture Notes

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“While I am describing to you how Nature works, you won’t understand why Nature works that way. But you see, nobody understands that.” — Richard Feynman

“I am no poet, but if you think for yourselves, as I proceed, the facts will form a poem in your minds.” — Michael Faraday

“Now, this case is not very interesting,” said Bell Labs mathematician Peter Winkler during a lecture at Rutgers. “But the reason why it’s not interesting is really interesting, so let me tell you about it.”

Ernest Rutherford addressed the Royal Institution in 1904:

I came into the room, which was half dark, and presently spotted Lord Kelvin in the audience and realised that I was in for trouble at the last part of the speech dealing with the age of the Earth, where my views conflicted with his. To my relief Kelvin fell fast asleep, but as I came to the important point, I saw the old bird sit up, open an eye, and cock a baleful glance at me. Then a sudden inspiration came and I said Lord Kelvin had limited the age of the Earth, provided no new source was discovered. That prophetic utterance referred to what we are now considering tonight, radium! Behold! the old boy beamed upon me.

When Antonie van Leeuwenhoek declined to teach his new methods in microbiology, Leibniz worried that they might be lost. Leeuwenhoek replied, “The professors and students of the University of Leyden were long ago dazzled by my discoveries. They hired three lens grinders to come to teach the students, but what came of it? Nothing, so far as I can judge, for almost all of the courses they teach there are for the purpose of getting money through knowledge or for gaining the respect of the world by showing people how learned you are, and these things have nothing to do with discovering the things that are buried from our eyes.”

Great Minds

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In 1784, French architect Étienne-Louis Boullée proposed building an enormous cenotaph for Isaac Newton, a cypress-fringed globe 500 feet high. A sarcophagus would rest on a raised catafalque at the bottom of the sphere; by day light would enter through holes pierced in the globe, simulating starlight, and at night a lamp hung in the center would represent the sun.

“I want to situate Newton in the sky,” Boullée wrote. “Sublime mind! Vast and profound genius! Divine being! Newton! Accept the homage of my weak talents. … O Newton! … I conceive the idea of surrounding thee with thy discovery, and thus, somehow, surrounding thee with thyself.”

As far as I can tell, this is unrelated to Thomas Steele’s proposal to enshrine Newton’s house under a stone globe, which came 41 years later. Apparently Newton just inspired globes.