The Real World

I had a growing feeling in the later years of my work at the subject that a good mathematical theorem dealing with economic hypotheses was very unlikely to be good economics: and I went more and more on the rules — (1) Use mathematics as a shorthand language, rather than an engine of inquiry. (2) Keep to them until you have done. (3) Translate into English. (4) Then illustrate by examples that are important in real life. (5) Burn the mathematics. (6) If you can’t succeed in 4, burn 3. This last I did often.

— Alfred Marshall, in a letter to A.L. Bowley, Jan. 27, 1906

| Quotations · Science & Math

Podcast Episode 136: The Boston Molasses Disaster

https://commons.wikimedia.org/wiki/File:BostonMolassesDisaster.jpg

In 1919 a bizarre catastrophe struck Boston’s North End: A giant storage tank failed, releasing 2 million gallons of molasses into a crowded business district at the height of a January workday. In this week’s episode of the Futility Closet podcast we’ll tell the story of the Boston Molasses Disaster, which claimed 21 lives and inscribed a sticky page into the city’s history books.

We’ll also admire some Scandinavian statistics and puzzle over a provocative Facebook photo.

See full show notes …

| Oddities · Podcast

New Light

https://commons.wikimedia.org/wiki/File:Trial_by_Jury_Usher.jpg

Our legal system assumes that a defendant is innocent until proven guilty beyond a reasonable doubt. But what constitutes a reasonable doubt? Law professors Ariel Porat and Alon Harel suggest that an “aggregate probabilities principle” might help to determine whether an accused party is innocent or guilty.

Suppose we’ve decided that the evidence must indicate a probability of 95 percent guilt before we’re willing to declare a defendant guilty. Mr. Smith is accused of two separate crimes, with a 90 percent probability of guilt in each case. Under the 95 percent rule he’d be acquitted of both crimes. But Porat and Harel point out that there’s a 10 percent chance that Smith is innocent of each crime, and aggregating the probabilities gives a 0.10 × 0.10 = 0.01 chance that Smith is innocent of both — that is, there’s a 99 percent chance that he’s guilty of at least one of the offenses.

On the other hand, consider Miller, who is also accused of two different crimes. Suppose that the evidence gives a 95 percent probability that he committed each crime. Normally he’d be convicted of both offenses, but aggregating the probabilities gives a 0.95 × 0.95 = 0.9025 chance that he’s guilty of both offenses, and hence he’d be acquitted of one.

In A Mathematical Medley (2010), mathematician George Szpiro points out that this practice can produce some paradoxical outcomes. Peter and Paul are each accused of a crime, each with a 90 percent chance of being guilty. Normally both would be acquitted. But suppose that each was accused of a similar crime in the past, Peter with a 90 percent chance of guilt and Paul with a 95 percent chance. Accordingly Peter was acquitted and Paul went to prison. But historically Peter has now been accused of two crimes, with a 90 percent chance of guilt in each case; according to the reasoning above he ought to be convicted of one of the two crimes and hence ought to go to jail today. Paul has also been accused of two crimes, with a 0.95 × 0.90 = 0.855 chance that he’s guilty of both. He’s already served one prison term, so the judge ought to acquit him today.

Szpiro writes, “Thus we have the following scenario: in spite of the evidence being identical, the previously convicted Peter is acquitted, while Paul, with a clean record, is incarcerated.”

(Ariel Porat and Alon Harel, “Aggregating Probabilities Across Offences in Criminal Law,” Public Law Working Paper #204, University of Chicago, 2008; George Szpiro, A Mathematical Medley, 2010.)

| Crime · Society

Crying Wolf

https://commons.wikimedia.org/wiki/File:Stanis%C5%82aw_Witkiewicz_-_Noc_ukrai%C5%84ska_1895.jpg

A puzzle from reader Paul Sophocleous:

Van Helsing, who is of course famous for his part in the destruction of Dracula, has had many other encounters with supernatural creatures. In the early hours of one morning, he was woken by a loud knock at the door. “Come quickly!” cried the chief of police. “There’s been a ghastly attack at the manor house on the hill!”

Van Helsing dressed hurriedly and followed the chief. A grisly sight met him when he arrived. The front door of the house was open, and the beam of light that came from within shone on the body of a young man lying on the path. His throat had been torn out viciously, as though he had been attacked by some kind of hideous wild beast. Van Helsing looked around, but the grounds were dark, since the moon had set some time before, and he could see nothing else.

He stepped inside and found that several officers of the local constabulary were comforting a woman who appeared to be the maid. “It was horrible!” she cried. “I came down here after hearing some racket outside, and I found the young master at the door. ‘There’s something out there,’ he told me, ‘some beast, and I mean to drive it off.’ And he had in his hand the poker from the fireplace as a weapon. But when he opened the door, it was on him in a flash, a great beast, all hairy and shaggy, bigger than a man it was!”

Van Helsing stepped forward. “What was it?” he demanded.

The maid let out a little scream and gasped, “It was a werewolf!” And with that she fainted dead away.

“Could it be, Van Helsing?” said the chief, sounding worried.

Van Helsing shook his head. “Not a chance.”

Why not?

Click for Answer
| Puzzles

Taste

https://commons.wikimedia.org/wiki/File:PGWodehouse.jpg

Cheering news from India. My press clipping agency has sent me a letter from the correspondence column of an Indian paper about a cow that came into the bungalow of a Mr. Verrier Elwyn, who lives at Patengarth, Mandla District, and ate his copy of Carry On, Jeeves, ‘selecting it from a shelf which contained, among other works, books by Shakespeare, Thomas Hardy and Henry Fielding.’ A pretty striking tribute I look on that as.

— P.G. Wodehouse to William Townend, Sept. 3, 1929

| Literature

Traffic Planning

traffic planning - cars

Towns A and B are connected by two roads. Suppose that two cars connected by a rope of length 2r can travel from A to B without breaking the rope. How can we prove that two circular wagons of radius r, moving along these roads in opposite directions, will necessarily collide?

traffic planning - wagons

This can be solved neatly by creating a configuration space. Map each road onto a unit segment, and set these up as two sides of a square. The northern car’s progress is reflected by a point moving up the left side of the square, and the southern car’s by a point moving from left to right along the bottom. Now the motion of the two cars from A to B is represented by a continuous curve connecting (0,0) and (1,1).

The wagons start from opposite towns, so their motion is represented by a curve from (0,1) to (1,0), and it’s immediately clear that the two curves must intersect. The intersection point corresponds to the collision of the wagons.

This example, by N. Konstantinov, is reportedly common in Russian mathematical folklore; I found it in Serge Tabachnikov’s 2005 book Geometry and Billiards (of all places).

| Science & Math

Rough

https://commons.wikimedia.org/wiki/File:Blank_map_of_Canada.svg
Image: Wikimedia Commons

SIR, — While surveying in northern Labrador I had occasion to visit the island of Nukusustok, a few miles to seaward of the village of Nain. On the slope of a hill, and about 300ft. inland, I found a golf ball in good condition. How did the ball come to be there, and so far inland? It is possible that the ball was driven by a golfer from an Atlantic liner during practice, drifted northward past Greenland, and was finally carried ashore by the Labrador current which runs from north to south along the Labrador coast.

I have sent the ball to Dunlops, the makers, who suggest that it was probably carried so far inland by a sea bird. Perhaps some of your readers could help in explaining the mystery.

Yours faithfully,

Thos. O. Hampson

— The Field, June 29, 1935

| Oddities

A Compensatory Harmonica

https://www.flickr.com/photos/soppyfrog/3931666786
Image: Flickr

A problem from the American Mathematical Monthly, March 1930:

Two men jointly own x cows. They sell these for x dollars per head and use the proceeds to buy some sheep at $12 per head. Their income from the cows isn’t divisible by 12, so they buy a lamb with the remainder. Later they divide the flock so that each man has the same number of animals. This leaves the man with the lamb somewhat short-changed, so the other man gives him a harmonica. What’s the harmonica worth?

Click for Answer
| Puzzles

The Spring (Arrangements) Bill

https://pixabay.com/en/spring-awakening-spring-1197602/

In 1936 English humorist A.P. Herbert found himself sitting in Parliament as an independent member for Oxford University. He drafted the following bill in verse to honor the new season — it’s a shame that it wasn’t enacted:

Whereas in every lawn and bed the plucky crocus lifts his head, and to and fro sweet song-birds go, the names of which we do not know:

Whereas the woods no more are dumb, the Boat Race and the Budget come, the Briton swells his manly chest, his mate, as eager, scrubs the nest, and Spring, with light but lavish hand, is spreading madness o’er the land:

It is expedient — but in rhyme — to legislate for such a time: Be it enacted, therefore, by our King with Lords and Commons in a fairy ring, assembled joyously at Westminister (or any other place that they prefer):

Provision for a Season Called Spring

1. (i) It shall be lawful everywhere for citizens to walk on air, to hang their hats upon the trees and wander hatless if they please: and notwithstanding any cracked provision in a previous Act, to give a constable a kiss is not felonious after this.

(ii) All citizens who choose to ride on taxi-tops and not inside: and those who do not use their votes because they’re busy painting boats: and any miscreant who hums, instead of doing dismal sums: whoever does a silly thing need only answer “‘Tis the Spring”: and this shall be a good defence in any court with any sense:

Provided that, in late July, this Act, of course, does not apply.

Financial Provisions

2. If any person feels he must get out of London now or bust, because the Spring is in his bones, but he must work for Mr. Jones, it shall be lawful for the same to give the Treasury his name, and say “Upon sufficient grounds I want about a hundred pounds”: and there shall not be any fuss concerning sums expended thus.

Repeal of Redundant Statutes

3. Subsection (i) of Section Four of any Act that seems a bore, and all the Acts concerning beer, and every Act that is not clear (always excepting Schedule A), shall be repealed and thrown away.

House of Commons — Reform of Procedure — Music etc.

4. (i) There shall be banks of maidenhair arranged about the Speaker’s chair: and roses white and roses red shall hang above the Speaker’s head: like some tremendous window-box, the Galleries be gay with phlox: and goldfish, lovely but aloof, shall swim above the glassy roof.

(ii) From now until the First of June all speeches shall be sung (in tune). The Speaker shall determine what hon. Members are in tune or not.

(iii) When in Committee of Supply the House may hum (but not too high). The Clerk-Assistant-at-the-Table shall choose the key (if he is able).

(iv) A band shall nearly always play (not on the first Allotted Day) behind the Speaker’s Chair at three and on the Terrace after tea.

Saving for Committees

5. On any day in May or June Committees shall adjourn quite soon: Provided, if the cuckoo call, Committees shall not sit at all.

Sittings of the Upper House

6. The House of Lords shall never sit on sunny days till after Whit: and they shall rise, if they have met, when it is foggy, fine or wet.

Termination of Official Report

7. (i) Except as hereinafter hinted, Hansard shall not again be printed, and save as in this Act is learned, all previous Hansards shall be burned.

(ii) It is a pity, history teaches, to make reports of people’s speeches, and afterwards to be unkind, simply because they change their mind. It is a most disgusting thing to make such comments in the Spring: so, as from when this Act is passed, that day’s Report shall be the last.

(iii) And as regards exceptions, see Subheading (a) of Schedule B.

Powers and Duties of Departments

8. (i) The secretary of State for Home Affairs shall now proceed to Rome, to Moscow, Washington, Cathay, or anywhere that’s far away, and not return to English skies until the Speaker certifies that Spring has ceased to be a fact under the Moss (Collection) Act.

(ii) Meanwhile o’er all his grim domain a lovely golden girl shall reign: and this delicious creature shall give golden parties in the Mall (paying the bills, if she is dunned, from the Consolidated Fund). The Civil Service, hand in hand, shall dance in masses down the Strand: and all the Cabinet shall wear wild dandelions in their hair.

(iii) It shall be deemed that every one has come into the world for fun. This shall be printed on the wall of every office in Whitehall.

Penalties for Certain Expressions

9. (i) No kind of crisis shall excuse a man exploring avenues: no lesser doom does he deserve when he is straining every nerve: and special punishment is earned by those who leave no stone unturned.

(ii) The penalty for each offence shall be elastic but immense.

(iii) A pension shall reward the man who modestly does all he can.

Interpretation

10. (i) The greatest care has been employed to make this measure null and void: not one expression in this Act means anything it means in fact.

(ii) Examples we decline to give: the lawyers, after all, must live.

Application

11. This Act applies and shall be good where anybody thinks it should:

Provided that, if strong objection should be expressed to any Section, that Section shall not have effect except for those who don’t object.

SCHEDULE B (a)

Any speech, motion, question, amendment or interruption by

A.P.H.

| Humor · Poems