## Mathematicians’ Graves

Archimedes wanted no other epitaph than a sphere inscribed within a cylinder — he had determined the sphere’s relative volume and considered this his greatest achievement.

Henry Perigal’s tomb in Essex displays his graphic proof of the Pythagorean theorem (left).

Gauss wanted to be buried under a heptadecagon, which he’d shown can be constructed with compass and straightedge. (The stonemason demurred, fearing he’d produce only a circle.)

And Jakob Bernoulli opted for a logarithmic spiral and the words *Eadem mutata resurgo*—the motto means “I shall arise the same though changed.”

## Benardete’s String Paradox

Let us take a piece of string. In the first half minute we shall form an equilateral triangle with the string; in the next quarter minute we shall employ the string to form a square; in the next eighth minute we shall form a regular pentagon; etc. ad infinitum. At the end of the minute what figure or shape will our piece of string be found to have assumed? Surely it can only be a circle. And yet how intelligible is that process? Each and every one of the polygons in our infinite series contains only a finite number of sides. There is thus a serious conceptual gap separating the circle, as in the limiting case, from each and every polygon in the infinite series.

— Jose Amado Benardete, *Infinity: An Essay in Metaphysics*, 1964

## Math Notes

Turn each of these palindromes “inside out” and their sum remains the same:

13031 + 42024 + 53035 + 57075 + 68086 + 97079 = 31013 + 24042 + 35053 + 75057 + 86068 + 79097

Remarkably, this holds true even if you square or cube them:

13031^{2} + 42024^{2} + 53035^{2} + 57075^{2} + 68086^{2} + 97079^{2} = 31013^{2} + 24042^{2} + 35053^{2} + 75057^{2} + 86068^{2} + 79097^{2}

13031^{3} + 42024^{3} + 53035^{3} + 57075^{3} + 68086^{3} + 97079^{3} = 31013^{3} + 24042^{3} + 35053^{3} + 75057^{3} + 86068^{3} + 79097^{3}

From Albert Beiler, *Recreations in the Theory of Numbers*, 1964.

## Misc

- Douglas Adams claimed that the funniest three-digit number is 359.
- Romeo has more lines than Juliet, Iago than Othello, and Portia than Shylock.
- Friday the 13th occurs at least once a year.
- “By nature, men love newfangledness.” — Chaucer
- John was the only apostle to die a natural death.

## The Paradox of Dives and Lazarus

The Gospel of Luke contains a parable about a rich man and a beggar. Both men die, and the rich man is consigned to hell while the beggar is received into the bosom of Abraham. The rich man pleads for mercy, but Abraham tells him that in his lifetime he received good things and the beggar evil things: “now he is comforted and thou art tormented.” The rich man then begs that his brothers be warned of what lies in store for them, but Abraham rejects this plea as well, saying, “If they hear not Moses and the prophets, neither will they be persuaded though one rose from the dead.”

Now, writes E.V. Milner:

Suppose … that this last request of Dives had been granted; suppose, in fact, that some means were found to convince the living, whether rich men or beggars, that ‘justice would be done’ in a future life, then, it seems to me, an interesting paradox would emerge. For if I

knewthat the unhappiness which I suffer in this world would be recompensed by eternal bliss in the next world, then I should be happy inthisworld. But being happy in this world I should fail to qualify, so to speak, for happiness in thenextworld. Therefore, if there were such a recompense awaiting me, its existence would seem to entail that I should at least be not wholly convinced of its existence.

“Put epigrammatically, it would appear that the proposition ‘Justice will be done’ can only be true for one who believes it to be false. For one who believes it to be true justice is being done already.”

## Countdown

Write the numbers 82 to 1 in descending order and string them together:

8281807978777675747372717069686766656463626160595857565554535251504948474645444-3424140393837363534333231302928272625242322212019181716151413121110987654321

The resulting 155-digit number is prime.

## Benardete’s Book Paradox

Here is a book lying on a table. Open it. Look at the first page. Measure its thickness. It is very thick indeed for a single sheet of paper — one half inch thick. Now turn to the second page of the book. How thick is this second sheet of paper? One fourth inch thick. And the third page of the book, how thick is this third sheet of paper? One eighth inch thick, etc. ad infinitum. We are to posit not only that each page of the book is followed by an immediate successor the thickness of which is one half that of the immediately preceding page but also (and this is not unimportant) that each page is separated from page 1 by a finite number of pages. These two conditions are logically compatible: there is no certifiable contradiction in their joint assertion. But they mutually entail that there is no last page in the book. Close the book. Turn it over so that the front cover of the book is now lying face down upon the table. Now, slowly lift the back cover of the book with the aim of exposing to view the stack of pages lying beneath it. There is nothing to see. For there is no last page in the book to meet our gaze.

— Patrick Hughes and George Brecht, *Vicious Circles and Infinity*, 1978

## The Answer Wheel

Multiply 212765957446808510638297872340425531914893617 by any number from 2 to 46 and you’ll find the product on the ring above.

Einstein wrote, “Pure mathematics is, in its way, the poetry of logical ideas.”

## The Unrepentant Liar

“According to [Bertrand] Russell’s treatment the sentence within the rectangle of Fig. 1 is meaningless, and may be called a pseudo-statement, because it is a version of the liar-paradox. But Russell’s treatment is unsatisfactory because it resolves the original paradox at the price of a new one. For, if the sentence of Fig. 1 is meaningless we must admit, since we observe that there are no other sentences within the rectangle, that it is false that there is a genuine or meaningful statement within the rectangle of Fig. 1. And, if there is no statement within the rectangle of Fig. 1 then *it is false that there is a true statement within the rectangle of Fig. 1*. The italicized part of the preceding sentence will be recognized as identical with (even if a different token of) the sentence within the rectangle of Fig. 1. And since the italicized sentence is true, and therefore a meaningful statement, the sentence within the rectangle is not a pseudo-statement either. Thus, if the sentence in question is meaningless, then it is meaningful and vice versa.”

— A.P. Ushenko, “A Note on the Liar Paradox,” *Mind*, October 1955

## 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

“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

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:

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

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

## Words and Numbers

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

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.