“A Rough Justice”

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British inventor Sir Robert Watson-Watt pioneered the development of radar, a contribution that helped the Royal Air Force win the Battle of Britain. Ironically, after the war he was pulled over for speeding by a Canadian policeman wielding a radar gun. His wife tried to point out the absurdity of the situation, but the officer wasn’t interested, and the couple drove away with a $12.50 fine. Watson-Watt wrote this poem:

Pity Sir Robert Watson-Watt,
strange target of this radar plot

And thus, with others I can mention,
the victim of his own invention.

His magical all-seeing eye
enabled cloud-bound planes to fly

but now by some ironic twist
it spots the speeding motorist

and bites, no doubt with legal wit,
the hand that once created it.

Oh Frankenstein who lost control
of monsters man created whole,

with fondest sympathy regard
one more hoist with his petard.

As for you courageous boffins
who may be nailing up your coffins,

particularly those whose mission
deals in the realm of nuclear fission,

pause and contemplate fate’s counter plot
and learn with us what’s Watson-Watt.

(Thanks, Chris.)

Augury

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Ohio State University philosopher Stewart Shapiro relates a puzzling experience that a friend once encountered in a physics lab. “The class was looking at an oscilloscope and a funny shape kept forming at the end of the screen. Although it had nothing to do with the lesson that day, my friend asked for an explanation. The lab instructor wrote something on the board (probably a differential equation) and said that the funny shape occurs because a function solving the equation has a zero at a particular value. My friend told me that he became even more puzzled that the occurrence of a zero in a function should count as an explanation of a physical event, but he did not feel up to pursuing the issue further at the time.

“This example indicates that much of the theoretical and practical work in a science consists of constructing or discovering mathematical models of physical phenomena. Many scientific and engineering problems are tasks of finding a differential equation, a formula, or a function associated with a class of phenomena. A scientific ‘explanation’ of a physical event often amounts to no more than a mathematical description of it, but what on earth can that mean? What is a mathematical description of a physical event?”

What right do we have to presume that the natural world will hew to mathematical laws? And why does the universe oblige us so graciously by doing so? Repeatedly, mathematicians have developed abstract structures and concepts that have later found unexpected applications in science. How can this happen?

“It is positively spooky how the physicist finds the mathematician has been there before him or her.” — Steven Weinberg

“I find it quite amazing that it is possible to predict what will happen by mathematics, which is simply following rules which really have nothing to do with the original thing.” — Richard Feynman

“One cannot escape the feeling that these mathematical formulae have an independent existence and intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than was originally put into them.” — Heinrich Hertz

“The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve.” — Eugene Wigner

(From Stewart Shapiro, Thinking About Mathematics, 2000; also his paper “Mathematics and Reality” in Philosophy of Science 50:4 [December 1983].)

Small Business

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Image: Wikimedia Commons

To help interest young students in chemistry, James Tour of Rice University devised “NanoPutians,” organic molecules that take the form of stick figures. The body is a series of carbon atoms that join two benzene rings; the arms and legs are acetylene units, each terminating in an alkyl group; and the head is a 1,3-dioxolane ring.

This gets even better — by using microwave irradiation, Tour found a way to vary the heads, creating a range of NanoProfessionals:

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Image: Wikimedia Commons

The synthesis is detailed on the Wikipedia page.

Equal Opportunity

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Can two dice be weighted so that the probability of each of the numbers 2, 3, …, 12 is the same?

Click for Answer

Cause for Alarm

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Sherlock Holmes is walking through the valley of Reichenbach Fall. On a clifftop overhead, Moriarty has perched a boulder. When he pushes it, it will have a 90 percent chance of killing Holmes. Just as he is about to send it over the edge, Watson arrives at the clifftop. Watson can’t see Holmes, so he’s not able to push the boulder safely clear, but he reasons that it’s better to push the boulder in a random direction than to let Moriarty aim it carefully. So he pushes the boulder off the cliff in such a way that Holmes’ chance of dying is reduced to 10 percent.

Unfortunately the boulder crushes Holmes anyway. Watson’s push decreased the chance of Holmes’ death, but it also caused it.

What are we to make of this? Generally speaking, it seems true to say that Pre-emptive pushing prevents death by crushing. That is, Watson’s push was of the sort that made it less likely that Holmes would die — if the scenario were re-enacted many times, with the boulder pushed sometimes by Watson, sometimes by Moriarty, Watson-type pushes would result in fewer deaths. But it also seems true to say that Watson’s pushing the rock caused Holmes to die. But cause and prevent are antonyms. How can both of these statements be true?

(Christopher Read Hitchcock, “The Mishap at Reichenbach Fall: Singular vs. General Causation,” Philosophical Studies, June 1995.)

Half and Half

A bisecting arc is one that bisects the area of a given region. “What is the shortest bisecting arc of a circle?” Murray Klamkin asked D.J. Newman. Newman supposed that it was a diameter. “What is the shortest bisecting arc of a square?” Newman answered that it was an altitude through the center. Finally Klamkin asked, “And what is the shortest bisecting arc of an equilateral triangle?”

“By this time, Newman had suspected that I was setting him up (and I was) and almost was going to say the angle bisector,” Klamkin writes. “But he hesitated and said let me consider a chord parallel to the base and since this turns out to be shorter than an angle bisector, he gave this as his answer.”

Was he right?

Click for Answer

Proizvolov’s Identity

List the first 2N positive integers (here let N = 4):

1, 2, 3, 4, 5, 6, 7, 8

Divide them arbitrarily into two groups of N numbers:

1, 4, 6, 7

2, 3, 5, 8

Arrange one group in ascending order, the other in descending order:

1, 4, 6, 7

8, 5, 3, 2

Now the sum of the absolute differences of these pairs will always equal N2:

| 1 – 8 | + | 4 – 5 | + | 6 – 3 | + | 7 – 2 | = 16 = N2

(Presented by Vyacheslav Proizvolov as a problem in the 1985 All-Union Soviet Student Olympiads.)