Science & Math

Math Notes

math notes

Sure Thing

Relativism either applies to itself or it doesn’t.

If it does, then it’s only relatively true.

If it doesn’t, then there’s an absolute truth.

Wheels Within Wheels

If the moon orbits the earth, always presenting the same face to us, does it rotate on its own axis?

It seems a simple question, but its appearance in the London Times in April 1856 set off a war among the English intelligentsia:

  • “A ship sailing round the world presents to the fishes always the same face as the Moon does to us. Coming home again, it will surely not be said that the ship has performed a [rotation].”
  • “Let him perforate a small ivory ball to represent the Moon, pass a wire through it, and bend this wire into a circle of a foot in diameter, and then push the ball round the circumference. Will there then remain any doubt of her not rotating on her axis?”

The answer, as William James would note in his parable of the squirrel, is that “which party is right depends on what you practically mean” by the term in question. Today we’d say that the moon rotates about its axis in the same time it takes to orbit the earth.

Incidentally, Lewis Carroll submitted two letters, but the Times didn’t print them. Perhaps it’s just as well — he was far ahead of everyone else: “I noticed for the first time the fact that though [the moon] only goes 13 times round the earth in the course of the year, it makes 14 revolutions round its own axis, the extra one being due to its motion round the sun.”

The Streetcar Singularity

‘If a man followed the directions of a street-car company,’ said Jones, ‘he would never enter one of its cars. Once in, paradoxically, he would never leave it. Just read that sign; it says, ‘Passengers are forbidden to enter or leave this car while in motion.’ Now, how in the name of Lindley Murray can a passenger do otherwise than get in motion, while leaving or entering a street car?’

– Marshall Brown, Bulls and Blunders, 1893

Math Notes

math notes


A 12th-century version of the liar paradox:

Socrates swears that he will speak only falsehoods about you.

Then he says, “You are a stone.”

This shows that a man can lie and speak the truth at the same time.


  • SCINTILLESCENT contains 7 pairs of letters.
  • Rub two pennies together and you’ll see a third between them.
  • Charles Darwin and Abraham Lincoln were born on the same day.
  • 1285 = (1 + 28) × 5
  • Squeeze an orange peel into a candle flame and you’ll produce a burst of fire.

A Candid Puzzle

Bertrand Russell admired G.E. Moore’s dedication to the truth.

“I have never but once succeeded in making him tell a lie,” he wrote, “and that was by a subterfuge.

“‘Moore,’ I said, ‘do you always speak the truth?’

“‘No,’ he replied.

“I believe this to be the only lie he ever told.”

Stop the World

Arguments against Galileo:

“Animals, which move, have limbs and muscles; the earth has no limbs or muscles, therefore it does not move.” — Scipio Chiaramonti, University of Pisa, 1633

“Buildings and the earth itself would fly off with such a rapid motion that men would have to be provided with claws like cats to enable them to hold fast to the earth’s surface.” — Libertus Fromundus, Anti-Aristarchus, 1631

“If we concede the motion of the earth, why is it that an arrow shot into the air falls back to the same spot, while the earth and all the things on it have in the meantime moved very rapidly toward the east? Who does not see that great confusion would result from this motion?” — Polacco, Anticopernicus Catholicus, 1644

More recent:

“[Astronomers give the rate of Earth's rotation as 1,000 kilometers per hour.] An aircraft flying at this rate in the same direction as that of the rotation could not cover any ground at all. It would remain suspended in mid-air over the spot from which it took off, since both speeds are equal. There would, in addition, be no need to fly from one place to another situated on the same latitude. The aircraft could just rise and wait for the desired country to arrive in the ordinary course of the rotation, and then land; although it is difficult to see how any plane could manage to touch ground at all on an airfield which is slipping away at the rate of 1,000 kilometers per hour. It might certainly be useful to know what people who fly think of the rotation of the earth.” — Gabrielle Henriet, Heaven and Earth, 1957

See No Spin Zone.

Thanks Anyway

In 1776, Viennese schoolmaster Antonio Felkel factored every number up to 408,000. Few people bought the book, though, so the treasury recalled it and used the paper to make ammunition cartridges.

University of Prague professor J.P. Kulik spent 20 years extending the work to 100,000,000. He published it in six volumes in 1867.

Volume 2 has been lost.

The Collatz Conjecture

Think of any whole number greater than zero.

  • If the number is even, divide it by two.
  • If the number is odd, triple it and add one.

If you apply these rules repeatedly, will you always reach 1? Surprisingly, no one knows.

Paul Erdos said, “Mathematics is not yet ready for such confusing, troubling, and hard problems.”

An Artificial Aurora

Karl Selim Lemström worked a quiet miracle in 1882: He strung conducting wire over the summit of a Lapland mountain and watched it draw down a shaft of light from the night sky — poetic proof that the aurora borealis is an electrical discharge from the upper atmosphere.

See Charged Words.

Satanic Compounds

Here’s a sugar alcohol derived from the North Atlantic seaweed Fucus vesiculosus. It’s called fucitol.

And its optical isomers are called D-fuc-ol and L-fuc-ol.

The glycoprotein that vampire bats use to prevent their victims’ blood from clotting is called draculin.

And diethyl azodicarboxylate is explosive, shock-sensitive, carcinogenic, and an eye, skin, and respiratory irritant, which helps to justify its acronym: DEAD.

See Juvenile Chemistry.

Science Marches On

An “infallible remedy against epilepsy,” published in Paris in 1686:

Take of common polypody dried and powdered, of moss growing from the skull of a man who died by violent means (criminals preferred), of nail-filings from human hands and feet, two drachms each; piony root half an ounce, and of fresh misletoe half an ounce. Boil them together as the moon wanes; cool, strain, and administer in small doses.

Cited in Charles White, Three Years in Constantinople, 1846.

See Well, Hey!

Math Notes

88 + 88 + 58 + 98 + 38 + 48 + 78 + 78 = 88593477

Long Addition" title="2009-10-23-long-addition

In The Hunting of the Snark, the Butcher confirms for the Beaver that Two and One are Three:

Taking Three as the subject to reason about–
A convenient number to state–
We add Seven, and Ten, and then multiply out
By One Thousand diminished by Eight.

The result we proceed to divide, as you see,
By Nine Hundred and Ninety and Two:
Then subtract Seventeen, and the answer must be
Exactly and perfectly true.

Fittingly for Carroll, the math works:

snark math

Mirror Twins

42263001 is a perfect square, and so is its reversal, 10036224.


If Martians are observing us, how can we show them we’re intelligent?

Carl Friedrich Gauss proposed marking a huge right triangle on the Siberian plain; Austrian astronomer Joseph von Littrow suggested carving a perfect circle in the Sahara and filling it with burning kerosene.

Joseph Pulitzer favored a more direct approach: He wanted to build a huge billboard in New Jersey recommending his newspaper to inquiring Martians.

He pressed the idea until an assistant asked, “What language shall we print it in?”

Perron’s Paradox

Let N be the largest positive integer. Then either N = 1 or N > 1.

If N > 1 then N2 > N, which breaks our definition of N as the largest integer. Therefore N = 1.

“The implications of this paradox are devastating,” writes Laurence Chisholm Young. “In seeking the solution to a problem, we can no longer assume that this solution exists. Yet this assumption has been made from time immemorial, right back in the beginnings of elementary algebra, where problems are solved by starting off with the phrase: ‘Let x be the desired quantity.’”

What’s in a Name

The disciples of Descartes made a perfect anagram upon the Latinised name of their master, ‘Renatus Cartesius,’ one which not only takes up every letter, but which also expresses their opinion of that master’s speciality–’Tu scis res naturae’ (Thou knowest the things of nature).

– William T. Dobson, Poetical Ingenuities and Eccentricities, 1882


A guaranteed way to win at roulette, from Eugene Northrop’s Riddles in Mathematics (1945):

  1. Bet $1 on red.
  2. If you win, go to step 6. If you lose, bet $2 on red.
  3. If you win, go to step 6. If you lose, bet $4 on red.
  4. If you win, go to step 6. If you lose, bet $8 on red.
  5. (And so on.)
  6. When you win, you’ll be $1 ahead. Go back to step 1.

“Theoretically, of course, it is possible for the bank to wipe you out financially. Actually, however, runs of more than 10 or 12 successive blacks or reds are extremely rare, and your stake at the twelfth play would be only $2048. When you do win you will, as before, be $1 ahead of the bank. You can then begin all over again. Simple, isn’t it?”

Benardete’s Paradox

Prometheus angers Zeus, who dispatches an army of demons with these instructions:

  • Demon 1: If Prometheus is not dead in one hour, kill him.
  • Demon 2: If Prometheus is not dead in half an hour, kill him.
  • Demon 3: If Prometheus is not dead in quarter of an hour, kill him.

And so on. When Prometheus is found dead, the council of gods is displeased, but they find it impossible to identify the guilty demon — any suspect can point to an infinity of demons who must have acted before him. Must Zeus go free?

The Morning Star Paradox

As it happens, the morning star and the evening star are both Venus–but the solar system might have evolved so that Mercury, for instance, was the brightest star in the morning sky.

Thus the morning star has a property that the evening star does not have: It’s necessarily identical with the morning star.

And if the morning star and the evening star have different properties, then they’re not the same object after all.

Thinking Back

Can you move an object using only your mind? Of course not. But can you move one in the past?

Since January 1997, the Retropsychokinesis Project at the University of Kent has invited Web visitors to try to influence the replay of a prerecorded bitstream. In other words, they must try to influence an event that has already happened.

The experimenters claim to be agnostic as to whether retroactive causality exists, but “the best existing database suggests that the odds are in the order of 1 in 630 thousand million that the experimental evidence is the result of chance.”

Try it for yourself here — but remember, if you have some skepticism about this, it may only be because someone in the future is influencing you.