The Outer Limits

http://web.mit.edu/philosophy/poster.pdf

In January 2007, inspired by this article by computer scientist Scott Aaronson, philosophers Agustín Rayo of MIT and Adam Elga of Princeton joined in the “large number duel” to come up with the largest finite number ever written on an ordinary-sized chalkboard.

The rules were simple. The two would take turns writing down expressions denoting natural numbers, and whoever could name the largest number would win the duel. No primitive semantic vocabulary was allowed (so that it would be illegal simply to write the phrase “the smallest number bigger than any number named by a human so far”), and the two agreed not to build on one another’s contributions (so neither could simply write “the previous entry plus one”).

Elga went first, writing the number 1. Rayo countered with a string of 1s:

111111111111111111111111111111111111111111111111111111111111

and Elga erased a line through the base of half this string to produce a factorial:

1111111111111111111111111111!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

The two began defining their own functions, and toward the end Rayo wrote this phrase:

The smallest number bigger than any number that can be named by an expression in the language of first-order set theory with less than a googol (10100) symbols.

With some tweaking, this became the winning entry, now enshrined as “Rayo’s number.”

“It was a great game,” Elga said after the match. “Heated at times, but nevertheless, a really great game.”

The use of philosophy was “crucial,” Rayo said. “The limit of math ability was reached at the end. Knowing a bit of philosophy, that was the key.”

Asked whether he thought his entry had set the Guinness world record, “It’s hard to be sure,” Rayo said, “but the number is bigger than any number I have ever seen.”

(Thanks, Erik.)

| Science & Math

Ruly English

In 1957, the U.S. Patent Office wanted to design a computer that could track down earlier references to an idea submitted by an inventor. This is difficult, because patents are described in ordinary English, which uses many ambiguous and imprecise terms. The word glass, for instance, refers to a material, but also to any number of things made of that material, and even to objects that have nothing to do with glass, such as plastic eyeglasses and drinking glasses.

To solve this problem, engineer-lawyer Simon M. Newman planned a synthetic language called Ruly English that gave one and only one meaning to each word. In ordinary English the preposition through has at least 13 meanings; Newman proposed replacing each of them with a new Ruly term with a single meaning. The Ruly word howby, for example, means “mode of proximate cause.” It might replace the unruly terms by(take by force) or with (to kill with kindness) or through (to cure through surgery), but it always has the same basic sense.

Newman had to coin other terms to take account of differing points of view. A watch spring and a bridge girder are both flexible to some degree, but using the word flexible to describe both would leave a computer at a loss as to how they compare. Newman coined the Ruly word resilrig to cover the whole scale, from extreme flexibility to extreme rigidity, adding prefixes such as sli (slightly) and sub (substantially). So in Ruly English a bridge girder would be sliresilrig and a watch spring subresilrig. A computer that knew these terms would not be confused into thinking that a thin bridge girder was more flexible than a rigid watch spring.

“Humans are not expected to read or speak Ruly English,” noted Time in 1958. “To them, unruly English will always be more ruly.”

(Newman describes his plan briefly here. I don’t know how far he got.)

| Language · Technology

“The Kiss”

“I saw you take his kiss!” “‘Tis true.”
“O modesty!” “‘Twas strictly kept:
He thought me asleep — at least, I knew
He thought I thought he thought I slept.”

— Coventry Patmore

| Poems

Podcast Episode 102: The Bunion Derby

https://www.flickr.com/photos/kaibabnationalforest/5734775201
Image: Flickr

In 1928, 199 runners set out on a perilous 3,400-mile footrace across America, from Los Angeles to Chicago and on to New York. The winner would receive $25,000 — if anyone finished at all. In this week’s episode of the Futility Closet podcast we’ll follow the Trans-American Footrace, better known as the Bunion Derby, billed as the greatest footrace the world had ever known.

We’ll also learn some creepy things about spiders and puzzle over why one man needs three cars.

See full show notes …

| Podcast

Loss

https://pixabay.com/en/milky-way-galaxy-night-sky-stars-984050/

From C.S. Lewis’ A Grief Observed, a collection of reflections on the loss of his wife, Joy, in 1960:

It is hard to have patience with people who say ‘There is no death’ or ‘Death doesn’t matter.’ There is death. And whatever is matters. And whatever happens has consequences, and it and they are irrevocable and irreversible. You might as well say that birth doesn’t matter. I look up at the night sky. Is anything more certain that in all those vast times and spaces, if I were allowed to search them, I should nowhere find her face, her voice, her touch? She died. She is dead. Is the word so difficult to learn? …

Talk to me about the truth of religion and I’ll listen gladly. Talk to me about the duty of religion and I’ll listen submissively. But don’t come talking to me about the consolations of religion or I shall suspect that you don’t understand.

He published it originally under the pseudonym N.W. Clerk, a pun on the Old English for “I know not what scholar.”

| Death · Literature · Religion

“The Pythagorean Curiosity”

waterhouse pythagorean curiosity

Here’s the item I mentioned in Episode 99 of the podcast — New York City engineer John Waterhouse published it in July 1899. It’s not a proof of the Pythagorean theorem, as I’d thought, but rather a related curiosity. It made a splash at the time — the Proceedings of the American Society of Civil Engineers said it “interested instructors of geometry all over the country, bringing many letters of commendation to him from prominent teachers.” Listener Colin Beveridge has been immensely helpful in devising the diagram above and making sense of Waterhouse’s proof as it appears on page 252 of Elisha Scott Loomis’ 1940 book The Pythagorean Proposition. Click the diagram to enlarge it a bit further.

  1. Red squares BN = AI + CE — Pythagoras’s theorem
  2. Blue triangles AEH, CDN, BMI are all equal in area to ABC, reasoning via X and Y and base sides.
  3. Green angles GHI and IBM are equal and green triangle GHI is congruent to IBM (side angle side), so IG = IK = IM. IH′K is congruent to IHK as angle HIK = angle HIG and the adjacent sides correspond. This means G and K are the same distance from the line HH′, so GK is parallel to HI. Similarly, DE is parallel to PF and MN is parallel to LO.
  4. GK = 4HI, because TU=HI, TG = AH (HTG congruent to EAH) and UK = UG (symmetry). Similarly, PF = 4DE. Dark blue triangles IVK and LWM are equal, so WM = VK. Similarly, OX = QD (dark green triangles PQD and NXO are congruent). Also, WX=MJ and XN=NJ, so M and N are the midpoints of WJ and XJ. That makes WX=2MN, so LO = 4MN.
  5. Each of the trapezia we just looked at (HIKG, OLMN and PFED) have five times the area of ABC.
  6. The areas of orange squares MK and NP are together five times EG. This is because:
    • the square on MI is (the square on MY) + (the square on IY) = (AC2) + (2AB)2 = 4AB2 + AC2.
    • the square on ND is (the square on NZ) + (the square on DZ) = (AB2) + (2AC)2 = 4AC2 + AB2
    • the sum of these is 5(AB2 + AC2) = 5BC2, and BC = HE.
  7. A′S = A′T, so A′SAT is a square and the bisector of angle B′A′C′ passes through A. However, the bisectors of angle A′B′C′ and A′C′B′ do not pass through B and C (resp.) [Colin says Waterhouse’s reasoning for this is not immediately clear.]
  8. Square LO = square GK + square FP, as LO = 4AC, GK = 4AB and FP = 4BC.
  9. [We’re not quite sure what Waterhouse means by “etc. etc.” — perhaps that one could continue to build squares and triangles outward forever.]
| Science & Math

Boontling

Boonville, Calif., has a dwindling language all its own. “Boontling” grew up as a jargon among residents of Anderson Valley around the turn of the 20th century. It includes more than a thousand words and phrases but is dying out among the small population. A brief glossary:

applehead – a young girl
belhoon – a dollar
Bill Nunn – syrup
boshin’ – deer hunting
bucky walter – a pay telephone
can-kicky – angry
dicking – cheating at cards
forbes – a half dollar
glow worm – a lantern
greeley – a newspaper or reporter
harpin’ tidrick – a lengthy discussion
high pockets – a wealthy person
killing snake – working very hard
pearlin – light rain
skee – whiskey
tobe – tobacco
walter – a telephone
zeese – coffee

“A few of us try to keep our skills sharp on the teleef [telephone],” resident Bobby Glover told the San Francisco Chronicle in 2001. “We’re adding new words now that the old-timers are gone.”

Thanks to the efforts of a number of researchers, the jargon has been pretty well documented now — the Chronicle even managed to translate “Old Mother Hubbard”:

The old dame piked for the chigrel nook for gorms for her bahl belljeemer
The gorms had shied, the nook was strung, and the bahl belljeemer had neemer.

| Language

Twos and Tens

The Wikipedia page for 1024 gives a handy technique for estimating large powers of 2 in decimal notation. For exponents up to about 100,

210a+b ≈ 2b103a.

For example, 235 = 34359738368 ≈ 32 × 109 = 32000000000.

This works because 210 ≈ 103. 3a gives a good estimate of the number of digits for exponents up to 300.

(Thanks, Stephen.)

| Science & Math