Gödel’s Loophole

At Princeton in the 1940s, Albert Einstein became a close friend of logician Kurt Gödel, whose incompleteness theorems lie at the heart of modern mathematics. Toward the end of his life Einstein said that his “own work no longer meant much, that he came to the Institute merely … to have the privilege of walking home with Gödel.”

In 1947 Einstein and economist Oskar Morgenstern accompanied Gödel to his U.S. citizenship exam because they were concerned about his unpredictable behavior: During his voluminous preparation for the exam, Gödel said, he had uncovered a flaw in the U.S. constitution that could lead to a dictatorship. Einstein and Morgenstern told him that the exam would really be quite simple and urged him not to prepare so extensively.

At the hearing, judge Phillip Forman asked Gödel:

“Now, Mr. Gödel, where do you come from?”

“Where I come from? Austria.”

“What kind of government did you have in Austria?”

“It was a republic, but the constitution was such that it finally was changed into a dictatorship.”

“Oh! That is very bad. This could not happen in this country.”

“Oh, yes,” Gödel said. “I can prove it.”

“So of all the possible questions, just that critical one was asked by the Examinor,” Morgenstern wrote later. “Einstein and I were horrified during this exchange; the Examinor was intelligent enough to quickly quieten Gödel and say, ‘Oh, God, let’s not go into this.'”

The logician got his citizenship and the friends returned to Princeton. What was the flaw that Gödel had found? There’s no record of it in Morgenstern’s account, so we don’t know. Stephen Hawking suggests that it involved the president’s power to fill vacancies during Senate recesses, and Barry University law professor F.E. Guerra-Pujol conjectures that it might involve the constitution’s power to amend itself. Maybe it’s best if we never discover it.

(Thanks, Louis.)

| History · Science & Math · Society

Deep Thinking

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Image: Wikimedia Commons

Natural philosopher John Wilkins’ Mathematical Magick of 1648 contains a startling passage in which he foretells the advantages of a long-range submarine, or “ship, wherein men may safely swim underwater”:

  1. ‘Tis private; a man may thus go to any coast of the world invisibly, without being discovered or prevented in his journey;
  2. ‘Tis safe; from the uncertainty of Tides, and the violence of Tempests, which do never move the sea above five or six paces deep. From Pirates and Robbers which do so infest other voyages; from ice and great frosts, which do so much endanger the passages toward the Poles.
  3. It may be of very great advantage against a Navy of enemies, who by this means may be undermined in the water, and blown up.
  4. It may be of a special use for the relief of any place that is besieged by water, to convey unto them invisible supplies: and so likewise for the surprisal of any place that is accessible by water.
  5. It may be of unspeakable benefit from submarine experiments and discoveries.

Wilkins was aware of Cornelius Drebbel’s primitive sub of 1620, but he looks much farther ahead, seeming to foresee combat submarines and deep-sea exploration vessels.

“I am not able to judge what other advantages there may be suggested, or whether experiment would fully answer to these notional conjectures,” he concluded. “But however, because the invention did unto me seem ingenious and new, being not impertinent to the present enquiry, therefore I thought it might be worth the mentioning.”

(From Joseph J. Thorndike Jr., ed., Mysteries of the Deep, 1980.)

| History · Technology

Face to Face

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Japanese pilot Kaname Harada recalls air combat in World War II:

The initial feeling after shooting down someone was relief, because it was not me who was shot down. My next thought was that I was a better pilot, so I felt superior to the enemy aviator who was shot down. These feelings lasted only for a short time. When we shot at each other, we were at very close range, and during this time I could see my opponent’s face very well. When I saw the enemy’s face, it looked terrible because he was going down. Soon after this I felt very bad, because I could imagine that my opponent had a family of his own, and I killed him. Therefore, to this day I feel very bad about shooting down pilots during the war.

Afterward, when these faces haunted his dreams, Harada became an antiwar activist and even traveled to the United States and Britain to meet some of the pilots he’d fought against. “In general, I have a bad feeling about being involved in the war, and I feel guilty about killing other people in combat,” he said. “I also feel that the war should not have happened in the first place. This is because the governments of countries around the world don’t make an effort to resolve their differences. Instead, they order their armed forces to kill each other. I believe World War II veterans know this best, because we were used as ‘pawns’ by our government to fight a war.”

(From Ron Werneth, Beyond Pearl Harbor, 2008.)

| History

Missive

In May 1936 a publisher invited Albert Einstein to contribute a message to be sealed in a metal box in the cornerstone of a new library wing in his country home, to be opened a thousand years hence. He sent this:

Dear Posterity,

If you have not become more just, more peaceful, and generally more rational than we are (or were) — why then, the Devil take you.

(From Helen Dukas and Banesh Hoffmann, eds., Albert Einstein, the Human Side, 1979.)

| History · Science & Math

Self-Regard

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Aristotle referred to humans as the “state-building animal.” Other names our species has proposed for itself, in various writings:

Homo absconditus: “man the inscrutable”
Homo adorans: “worshipping man”
Homo aestheticus: “aesthetic man”
Homo amans: “loving man”
Homo animalis: “man with a soul”
Homo avarus: “man the greedy”
Homo creator: “creator man”
Homo demens: “mad man” (the only creature with irrational delusions)
Homo discens: “learning man”
Homo domesticus: “domestic man” (because he builds his environment)
Homo duplex: “double man” (because of his animal and social tendencies)
Homo economicus: “economic man”
Homo educandus: “to be educated” (we require education to reach maturity)
Homo ethicus: “ethical man”
Homo excentricus: “not self-centered” (we’re capable of objectivity and self-reflection)
Homo faber: “toolmaker man”
Homo ferox: “ferocious man” (T.H. White)
Homo grammaticus: “grammatical man”
Homo humanus: “human man” (as opposed to Homo biologicus)
Homo hypocritus: “hypocritical man” (Robin Hanson, who also called us “man the sly rule bender”)
Homo imitans: “imitating man” (capable of learning by imitation)
Homo inermis: “helpless man” (devoid of animal instincts)
Homo investigans: “investigating man” (curious and capable of learning by deduction)
Homo laborans: “working man” (capable of dividing labor and specializing)
Homo logicus: “the man who wants to understand”
Homo loquens: “talking man”
Homo loquax: “chattering man” (Henri Bergson)
Homo ludens: “playing man” (Schiller)
Homo mendax: “lying man” (able to tell lies)
Homo metaphysicus: “metaphysical man” (Schopenhauer)
Pan narrans: “storytelling ape” (Terry Pratchett)
Homo necans: “killing man”
Homo patiens: “suffering man” (Viktor Frankl)
Homo pictor: “depicting man”
Homo poetica: “man the poet”
Homo religiosus: “religious man”
Homo ridens: “laughing man”
Homo reciprocans: “reciprocal man” (a cooperative actor)
Homo sanguinis: “bloody man”
Homo sciens: “knowing man”
Homo sentimentalis: “sentimental man” (empathizing and idealizing emotions)
Homo socius: “social man”
Homo sociologicus: “sociological man” (prone to sociology)
Homo technologicus: “technological man”
Homo viator: “man the pilgrim” (on his way toward finding God)

Wikipedia lists 72 of these. Douglas Adams wrote, “Human beings, who are almost unique in having the ability to learn from the experience of others, are also remarkable for their apparent disinclination to do so.”

| History · Society

Sea Rules

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Articles observed by the crew of Welsh pirate Bartholomew Roberts (1682-1722):

  1. Every man has a vote in affairs of moment; has equal title to the fresh provisions, or strong liquors, at any time seized, and may use them at pleasure, unless a scarcity makes it necessary, for the good of all, to vote a retrenchment.
  2. Every man to be called fairly in turn, by list, on board of prizes because, (over and above their proper share) they were on these occasions allowed a shift of clothes: but if they defrauded the company to the value of a dollar in plate, jewels, or money, marooning was their punishment.
  3. No person to game at cards or dice for money.
  4. The lights and candles to be put out at eight o’clock at night: if any of the crew, after that hour still remained inclined for drinking, they were to do it on the open deck.
  5. To keep their piece, pistols, and cutlass clean and fit for service.
  6. No boy or woman to be allowed amongst them. If any man were to be found seducing any of the latter sex, and carried her to sea, disguised, he was to suffer death.
  7. To desert the ship or their quarters in battle, was punished with death or marooning.
  8. No striking one another on board, but every man’s quarrels to be ended on shore, at sword and pistol.
  9. No man to talk of breaking up their way of living, till each had shared £1,000. If in order to do this, any man should lose a limb, or become a cripple in their service, he was to have 800 dollars, out of the public stock, and for lesser hurts, proportionately.
  10. The captain and quartermaster to receive two shares of a prize: the master, boatswain, and gunner, one share and a half, and other officers one and a quarter.
  11. The musicians to have rest on the Sabbath Day, but the other six days and nights, none without special favour.

That’s from Charles Johnson’s A General History of the Pyrates (1724), via naval historian David Cordingly’s Under the Black Flag (1995). In the early years of the 18th century, Cordingly writes, a pirate captain “had absolute power in battle and when ‘fighting, chasing, or being chased,’ but in all other matters he was governed by the majority wishes of the crew.”

| Crime · History

Bullseye

During World War I, British physicist G.I. Taylor was asked to design a dart to be dropped onto enemy troops from the air. He and a colleague dropped a bundle of darts as a trial and then “went over the field and pushed a square of paper over every dart we could find sticking out of the ground.”

When we had gone over the field in this way and were looking at the distribution, a cavalry officer came up and asked us what we were doing. When we explained that the darts had been dropped from an airplane, he looked at them and, seeing a dart piercing every sheet remarked: ‘If I had not seen it with my own eyes I would never have believed it possible to make such good shooting from the air.’

(The darts were never used — “we were told they were regarded as inhuman weapons and could not be used by gentlemen.”)

(From T.W. Körner, The Pleasures of Counting, 1996.)

| History · Science & Math · Technology

The Martians

In the first half of the 20th century, a considerable number of famous scientists emigrated from Hungary to the United States, including physicists Eugene Wigner, Edward Teller, and Dennis Gabor and mathematicians Theodore von Kármán, John von Neumann, Paul Halmos, George Pólya, and Paul Erdős. Most were Jewish refugees from Nazi Germany, but they had surprising further similarities — many had been born near Budapest, had shown an early interest in chemistry, and had studied physics at German universities before emigrating to America.

One of their number, Leo Szilard, joked that he knew the reason: They were all descended from a Martian scout force that had landed on Earth in that period. The Martians had left eventually, but not before impregnating some Earth women.

The “Martians” adopted Szilard’s name because in many ways they felt themselves to be outsiders in America: All were brilliant, spoke English with a strong accent, and came from a small little-known country.

When Enrico Fermi posed his famous paradox — if intelligent aliens are as common as we believe, why haven’t we encountered one? — Szilard answered, “They are among us — but they call themselves Hungarians.”

(Thanks, Rini.)

| History · Science & Math

Weather Station Kurt

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In July 1977 geomorphologist Peter Johnson stumbled across an old weather station in northern Labrador. It turned out to be an automatic station that had been set up secretly by a German submarine crew in 1943 so that Germany might have notice of impending weather systems. The Allies had never discovered it and it had stood unregarded for 30 years after the war’s end.

“Weather Station Kurt” was probably designed to operate automatically for about six months, transmitting readings on temperature, wind direction, strength, and humidity every three hours until its batteries failed in the cold.

All the witnesses to its installation died when the submarine, U-537, was sunk in the Java Sea. Their work marks the only known armed German military operation on land in North America during World War II.

The station is now on display at the Canadian War Museum.

| History · Technology

Signing Off

donne letter

In England and France, from the 1500s to the 1700s, the placement of a letter writer’s signature on the page conveyed meaning as to his relationship with the recipient. In his 1568 letter-writing manual Enimie of Idlenesse, William Fulwood writes that the subscription and signature of the letter “must be doone according to the estate of the writer, and the qualitie of the person to whom wee write: For to our superiors wee must write at the right side in the neither end of the paper, saying: By your most humble and obedient sonne, or seruant, &c. Or, yours to commaund, &c. And to our equals we must write towards the middest of the paper, saying: By your faithfull friend for euer, &c. Or, yours assured, &c. To our inferiours wee may write on high at the left hand, saying: By yours, &c.”

In 1601 John Donne married Anne More without the blessing of her father, who was lieutenant of the Tower of London. Donne was incarcerated, and in his letters begging for clemency he crammed his signature into the bottom right-hand corner of the page to signal his self-abasement.

(From Sam Willis and James Daybell, Histories of the Unexpected, 2018.)

| History · Society