## The Sword and the Pen

The sword of the warrior was taken down for the purpose of being polished. It had not been long out of use. The rust was rubbed off, but there were spots that would not go; they were of blood. The sword was placed on the table, near the pen of the warrior’s secretary. The pen took advantage of the first breath of air to move a little further off.

“Thou art right,” said the sword, “I am a bad neighbor.”

“I fear thee not,” replied the pen, “I am more powerful than thou art; but I love not thy society.”

“I exterminate,” said the sword.

“And I perpetuate,” answered the pen; “where are thy victories if I record them not? Even where thou thyself shalt one day be in the lake of oblivion.”

— *The Rural Repository*, Vol. 25, March 31, 1849

## Cancelling Out

Hold on to your seats, boys. This one gets complicated.

It seems that on Aug. 12, 1940, there occurred an automobile collision at Forty-second avenue S. and Forty-second street.

A car driven by Leo R. Johnson, Hopkins, and owned by Frank M. Anderson, 2912 Sixteenth avenue S., was in collision with one driven by Francis D. Hall, 3413 E. Minnehaha parkway.

It happened that both owners were insured with the same company, the Westchester Fire Insurance Co., and both had $50 deductible collision policies.

The company paid $65 to Anderson for damages to his car and $300 to Hall for damages to his car.

Then the company started suit in municipal court to recover the money.

Action was started against Hall to recover the $65 paid for damages to Anderson’s car, charging him with negligence, and against Anderson to recover the $300 paid for damages to Hall’s car, charging Johnson, the driver, with negligence. …

Benjamin Rigler, attorney for Hall, then came into court with an answer denying that his client was negligent and charging that the driver of Anderson’s car was the negligent party.

And A.C. Johnston, attorney for Anderson, came into court with an answer denying that Johnson was negligent in driving and charging that Hall was guilty of contributory negligence.

The insurance company then filed replies to both answers, denying them.

Since it denied the answers, it also denied each answer’s charge that the other party was guilty of negligence, and thus denied its own original complaint.

Today the attorneys moved for dismissal on the ground that the reply to the answers was ‘sham and frivolous.’

Municipal Judge William A. Anderson dismissed both cases.

— William L. Prosser, *The Judicial Humorist*, 1952

## Bertrand’s Paradox

We ask for the probability that a number, integer or fractional, commensurable or incommensurable, randomly chosen between 0 and 100, is greater than 50. The answer seems evident: the number of favourable cases is half the number of possible cases. The probability is 1/2.

Instead of the number, however, we can choose its square. If the number is between 50 and 100, its square will be between 2,500 and 10,000.

The probability that a randomly chosen number between 0 and 10,000 is greater than 2,500 seems evident: the number of favourable cases is three quarters of the number of possible cases. The probability is 3/4.

The two problems are identical. Why are the two answers different?

— Joseph Bertrand, *Calcul des Probabilités* (Trans. Sorin Bangu), 1889

The pigpen is mightier than the sword. — E.E.Cummings

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Many a pen has been stilled by the sword…

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I’m guessing that the problem has to do with the fact that the order of infinity with respect to a line segment is different from the order of infinity of a portion of a plane.

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Well scratch that theory. The two infinities are the same. Just looked it up.

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Really enjoyed the story of the sword and pen. Have to think more about the other two as I’m neither a logical thinker or a mathematician however my brain sometimes likes a challenge!

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Both pen and sword now reside in the museum of technology but the visitor numbers for the sword exhibit is mightier.

I’m no mathematician but isn’t the probability the same in both cases of Bertrand’s Paradox? Not 3/4 but 1/2 both times. Or have I misunderstood?

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Great learning post and loved the story of the sword and the pen

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Thanks for sharing these stories! I look forward to them in my RSS feed every day. I’m surprised by the lawyer story—I would have thought that the two drivers would be accusing each other of being the *only* negligent party, which the insurance company could deny because they were both negligent. Maybe that’s why the judge threw out both cases?

I love Bertrand’s paradox for pointing out that there’s more than one way to choose a number “randomly,” and that if you change the method then various probabilities, averages, etc. might change too. Usually when people say “a random number between A and B” they are referring to the uniform distribution, which (roughly speaking) spreads out the random numbers evenly between A and B. The two problems in Bertrand’s paradox are not equivalent, because choosing a (uniformly distributed) random number between 0 and 100, then squaring it, is not a method of producing a (uniformly distributed) random number between 0 and 10000. There’s a more than 70% chance of getting a number below 5000, for example, since if the original number is anything from 0 to 70 then its square will only range from 0 to 4900.

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