Science & Math

Faith No More

Kurt Gödel composed an ontological proof of God’s existence:

Axiom 1. A property is positive if and only if its negation is negative.

Axiom 2. A property is positive if it necessarily contains a positive property.

Theorem 1. A positive property is logically consistent (that is,
possibly it has an existence).

Definition. Something is God-like if and only if it possesses all positive properties.

Axiom 3. Being God-like is a positive property.

Axiom 4. Being a positive property is logical and hence necessary.

Definition. A property P is the essence of x if and only if x has the property P and P is necessarily minimal.

Theorem 2. If x is God-like, then being God-like is the essence of x.

Definition. x necessarily exists if it has an essential property.

Axiom 5. Being necessarily existent is God-like.

Theorem 3. Necessarily there is some x such that x is God-like.

“I am convinced of the afterlife, independent of theology,” he once wrote. “If the world is rationally constructed, there must be an afterlife.”

Hintikka’s Paradox

(1) If a thing can’t be done without something wrong being done, then the thing itself is wrong.

(2) If X is impossible and Y is wrong, then I can’t do both X and Y, and I can’t do X but not Y.

But if Y is wrong and doing X-but-not-Y is impossible, then by (1) it’s wrong to do X.

Hence if it’s impossible to do a thing, then it’s wrong to do it.

Oaks and Acorns

Each of these pairs of numbers contains the 10 digits:

57321 60984
35172 60984
58413 96702
59403 76182

Square any one of them and it will grow into its own 10-digit pandigital number.

In the Dark

http://commons.wikimedia.org/wiki/File:NGC_1672_HST.jpg

“A universe simple enough to be understood is too simple to produce a mind capable of understanding it.” — Cambridge cosmologist John Barrow

Homework

One day while teaching a class at Yale, Shizuo Kakutani wrote a lemma on the blackboard and remarked that the proof was obvious. A student timidly raised his hand and said that it wasn’t obvious to him. Kakutani stared at the lemma for some moments and realized that he couldn’t prove it himself. He apologized and said he would report back at the next class meeting.

After class he went straight to his office and worked for some time further on the proof. Still unsuccessful, he skipped lunch, went to the library, and tracked down the original paper. It stated the lemma clearly but left the proof as an “exercise for the reader.”

The author was Shizuo Kakutani.

Who’s On First?

Stigler’s Law of Eponymy states that “no scientific discovery is named after its original discoverer.” Examples:

  • Arabic numerals were invented in India.
  • Darwin lists 18 predecessors who had advanced the idea of evolution by natural selection.
  • Freeman Dyson credited the idea of the Dyson sphere to Olaf Stapledon.
  • Salmonella was discovered by Theobald Smith but named after Daniel Elmer Salmon.
  • Copernicus propounded Gresham’s Law.
  • Pell’s equation was first solved by William Brouncker.
  • Euler’s number was discovered by Jacob Bernoulli.
  • The Gaussian distribution was introduced by Abraham de Moivre.
  • The Mandelbrot set was discovered by Pierre Fatou and Gaston Julia.

University of Chicago statistics professor Stephen Stigler advanced the idea in 1980.

Delightfully, he attributes it to Robert Merton.

Infallible

[Bertrand] Russell is reputed at a dinner party once to have said, ‘Oh, it is useless talking about inconsistent things, from an inconsistent proposition you can prove anything you like.’ Well, it is very easy to show this by mathematical means. But, as usual, Russell was much cleverer than this. Somebody at the dinner table said, ‘Oh, come on!’ He said, ‘Well, name an inconsistent proposition,’ and the man said, ‘Well, what shall we say, 2 = 1.’ ‘All right,’ said Russell, ‘what do you want me to prove?’ The man said, ‘I want you to prove that you are the pope.’ ‘Why,’ said Russell, ‘the pope and I are two, but two equals one, therefore the pope and I are one.’

— Jacob Bronowski, The Origins of Knowledge and Imagination, 1979

Hidden Order

http://commons.wikimedia.org/wiki/File:Thebault_theorem.png

Erect squares on the sides of any parallelogram and their centers will always form a square.

In any triangle, the midpoints of the sides and the feet of the altitudes always fall on a circle.

Stubborn

Write down any natural number, reverse its digits to form a new number, and add the two:

lychrel number example - 1

In most cases, repeating this procedure eventually yields a palindrome:

lychrel number example - 2
lychrel number example - 3

With 196, perversely, it does not — or, at least, it hasn’t in computer trials, which have repeated the process until it produced numbers 300 million digits long.

Is 196 somehow immune to producing palindromes? No one’s yet offered a conclusive proof — so we don’t know.

Fair Point

One threatening morning as Einstein was about to leave his house in Princeton, Mrs. Einstein advised him to take along a hat.

Einstein, who rarely used a hat, refused.

‘But it might rain!’ cautioned Mrs. Einstein.

‘So?’ replied the mathematician. ‘My hair will dry faster than my hat.’

– Howard Whitley Eves, In Mathematical Circles: Quadrants III and IV, 1969

Math Notes

12 = 3 × 4; 56 = 7 × 8

Words and Numbers

http://commons.wikimedia.org/wiki/File:Number-line.gif

In English, the name of each integer shares a letter with each of its neighbors. ONE shares an O with TWO, TWO shares a T with THREE … and so on to infinity.

Huth’s Moving Star

http://www.sxc.hu/photo/172339

In late 1801, Johann Bode, director of the Berlin Observatory, received a curious series of letters from astronomer Hofrath Huth in Frankfort-on-the-Oder. On Dec. 2 Huth had seen something new in the sky, “a star with faint reddish light, round, and admitting of being magnified.” But it wasn’t a star: On subsequent nights he watched it drift slowly to the southwest, growing gradually fainter, and by Jan. 6 it had disappeared. Huth concluded that he was watching an object recede from Earth.

Unfortunately, Bode was busy with other things, and the weather was too cloudy for him to confirm Huth’s observations. Also, the positional data that Huth had provided were somewhat poor.

Huth wasn’t a nut: Among other things, he co-discovered Comet Encke in 1805. And Nature noted later that he had alerted Bode to the object in time for the director to witness it himself if the skies had been clear. But as it happened, Huth was the only one to witness the curious object, whatever it was. And, whatever it was, it has not returned.

Still Waters

Gauss’ scientific diary was a great boon to mathematical historians, but his notes could be frustratingly cryptic. On July 10, 1796, he made this entry:

ΕΥΡΗΚΑ! num = Δ + Δ + Δ

He had discovered that every positive integer is the sum of at most three triangular numbers.

Among the 146 entries, two remain completely opaque. On Oct. 11, 1796, Gauss had written:

Vicimus GEGAN.

And on April 8, 1799:

gauss diary entry

No one knows what either of these means — if they had mathematical significance, it was lost with Gauss.

So it goes. Dirichlet was famously uncommunicative, not even informing his family that his wife had given birth. His father-in-law later complained that he “should at least have been able to write 2 + 1 = 3.”

Fitting

In the Dewey decimal system, books on number theory are labeled 512.81.

512 = 29 and 81 = 92.

Misc

  • Jimmy Carter was the first U.S. president born in a hospital.
  • Hamlet has 1506 lines, fully 39 percent of the play.
  • 736 = 7 + 36
  • NOOK combines two antonyms.
  • “Everything that deceives may be said to enchant.” — Plato

Symmetric Milestones

What’s unusual about these numbers?

palindromic primes in arithmetic progression 1

Each series is spaced evenly on the number line:

palindromic primes in arithmetic progression 2

Each number is a palindrome.

And each is prime.

Education Reconsidered

Reflect, Socrates; you may have to deny your words.

I have reflected, I said; and I shall never deny my words.

Well, said he, and so you say that you wish Cleinias to become wise?

Undoubtedly.

And he is not wise as yet?

At least his modesty will not allow him to say that he is.

You wish him, he said, to become wise, and not to be ignorant?

That we do.

You wish him to be what he is not, and no longer to be what he is?

I was thrown into consternation at this.

Taking advantage of my consternation he added: You wish him no longer to be what he is, which can only mean that you wish him to perish. Pretty lovers and friends they must be who want their favourite not to be, or to perish!

— Plato, Euthydemus

Math Notes

9 + 9 = 18; 9 × 9 = 81
24 + 3 = 27; 24 × 3 = 72
47 + 2 = 49; 47 × 2 = 94
497 + 2 = 499; 497 × 2 = 994

Two Milestones

The date 11/19/1999 contained only odd digits. Less than three months later, 2/2/2000 contained only even.

That’s a rare coincidence. It had been 1111 years since the last all-even date … and it’ll be 1111 more before the next all-odd one.

Skyward

In 1907, Massachusetts physician Duncan MacDougall conceived a singular experiment. When he observed that a patient at his Haverhill hospital was nearing death, he installed him in a specially constructed bed in his office and measured his weight both before and after death. With six such weighings he determined that humans lose between 0.5 and 1.5 ounces at death.

“Is the soul substance?” he wrote. “It would seem to me to be so. … Here we have experimental demonstration that a substance capable of being weighed does leave the human body at death.”

Similar experiments with 15 dogs showed no change in mass, proving, he decided, that dogs have no souls. MacDougall’s findings were written up briefly in the New York Times and occasioned a flurry of correspondence in American Medicine, but after that they were largely forgotten. But who knows? Perhaps he was right.

Math Notes

math notes

Pop Physics

A can of Diet Coke floats in water, while regular Coke sinks:

Why? The Diet Coke contains 190 mg of aspartame, but the regular Coke contains 39 grams of sugar. So the regular Coke is denser.

E Pluribus Unum

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111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
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111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111
11111111111111111111111111111111111111111 is prime.