Achilles Recalled

A fragment from Lewis Carroll, Nov. 22, 1874:

A. And thus your favourite paradox, my dear D., is finally disproved of, and Achilles and the Tortoise will walk off hand in hand. No argument of any sort can be maintained, which would prove him not to overtake it.

D. No mathematical argument, you mean; for, if you permit me a classical one, I will contend that the Tortoise was nothing but the “Testudo” of the ancients, a machine of common use in Sieges — that it was at that moment moving against the walls of Troy — and that the true reason why Achilles did not overtake it was simply that he was sulking in his tent and never went near it.

S. I beg to limit this discussion to mathematical argument.

D. Be it so. And the mathematical argument you dispose of, as I understand you, by the assertion that we find ourselves at last among indivisible distances and indivisible periods of time, and thus you propose to plunge us, however reluctant we may be to take the leap, into the dark abyss of the Inconceivable?

S. That is my solution of the paradox.

D. Granting, for argument’s sake, that the paradox is thus finally disposed of, let me ask you a question or two. These indivisible distances — are they equal, or unequal?

S. Am I bound to choose one or other of these categories?

D. I fear I can offer you no third.

S. Well then, as I do not clearly see what you are aiming at, I will, for the present, say “unequal,” reserving to myself however the right of substituting “equal” should I see reason to do so.

D. The privilege is an unusual one, but I will not object to your exercising it. Let them then be: unequal. Now take two of these unequal distances: lay them side by side, so as to coincide at one end: will they coincide at the other end also?

S. Surely not.

D. There will therefore be a difference between them: and this difference, being homogeneous with the things differing, will itself be a distance?

S. I cannot deny it.

D. Divisible, shall we say? Or indivisible?

S. (laughing) Indivisible, of course. You would not wish me to imagine a divisible distance less than an indivisible one?

D. You shall please yourself in that matter. Let me now add together these two lesser indivisible distances. Will their sum total be divisible or indivisible, think you?

S. (after a pause) It occurs to me that I would rather take the other horn of your dilemma, and say that these indivisible distances are all equal.

D. With all my heart. They shall now be all equal. And we will suppose that Achilles has just passed over one of the indivisible distances. What time would you say that he occupied in doing so?

S. An indivisible time, clearly.

D. But the Tortoise had previously passed over the same indivisible distance: how long do you suppose he took to do it?

S. As he travelled at only half the pace of Achilles, it is evident that he required two of our indivisible periods of time.

D. No doubt. But now tell me — at the end of the first of these indivisible periods of time, where had the Tortoise got to?

S. I will trouble you to pass the wine. I think I should like another half-glass of sherry.

Two Chess Problems

In a chess game, White plays 1. f3 2. Kf2 3. Kg3 4. Kh4. Black’s fourth move checkmates White. What is the game?

1. f3 e5 2. Kf2 Qf6 3. Kg3 Qxf3+! 4. Kh4 Be7#

suicide chess problem

White’s play is so suicidal that the task sounds easy, but “this problem is almost impossibly difficult because Qxf3+ is such a horrible move by normal chess standards,” writes former U.S. champion Stuart Rachels. “It is hard for a competent player to consciously consider it.”

Here’s another problem by Sam Loyd:

loyd mate in 2

White to mate in two moves. In 1907 J.H. Blackburne chose this as one of his all-time favorite problems. “It was first published in this country about fifty years ago, and greatly puzzled the solvers of that day, the idea then being entirely new.”

It’s a perfectly fair two-mover — there’s no trickery.

Click for Answer

Hidden Depths

Image: Flickr

It is familiarly said that beer … is an acquired taste; one gradually trains oneself — or just comes — to enjoy that flavor. What flavor? The flavor of the first sip? No one could like that flavor, an experienced beer drinker might retort: ‘Beer tastes different to the experienced beer drinker. If beer went on tasting to me the way the first sip tasted, I would never have gone on drinking beer! Or to put the same point the other way around, if my first sip of beer had tasted to me the way my most recent sip just tasted, I would never have had to acquire the taste in the first place! I would have loved the first sip as much as the one I just enjoyed.’ If we let this speech pass, we must admit that beer is not an acquired taste. No one comes to enjoy the way the first sip tasted. Instead, prolonged beer drinking leads people to experience a taste they enjoy, but precisely their enjoying the taste guarantees that it is not the taste they first experienced.

– Daniel Dennett, “Quining Qualia,” from Consciousness in Contemporary Science, 1988

In a Word

v. to leave abruptly

The Paradox of Self-Deception

If ever a person A deceives a person B into believing that something, p, is true, A knows or truly believes that p is false while causing B to believe that p is true. So when A deceives A (i.e., himself) into believing that p is true, he knows or truly believes that p is false while causing himself to believe that p is true. Thus, A must simultaneously believe that p is false and believe that p is true. But how is this possible?

– Alfred R. Mele, “Two Paradoxes of Self-Deception,” in Self-Deception and Paradoxes of Rationality, 1998

A New Man

In 1944, a San Francisco judge refused to let Tharnmidsbe L. Praghustspondgifcem change his name.

He’d asked to change it to Miswaldpornghuestficset Balstemdrigneshofwintpluasjof Wrandvaistplondqeskycrufemgeish.

The man, whose given name was Edward L. Hayes, had requested the first change in order “to do better in my business and economic affairs.” Evidently he felt he hadn’t gone far enough.

Three Cheers

In 1900 Edward Elgar invited three ladies, teachers of English, French, and German, to a rehearsal of The Dream of Gerontius at a Birmingham school. They sent him this letter of thanks:

My cher Herr!

We sommes so full de Dankbarkeit and débordante Entzücken and sentons so weak et demütig that la Kraft of une Sprache seems insuffisante auszüdrücken our sentiments. Deshalb we unissons unsere powers et versuchen to express en Englisch, French, and Allemand das for que wir feel n’importe quelle Sprache to be insuffisante. Wie can nous beschreiben our accablante Freude and surprise! Wir do pas wissen which nous schützen most: notre Vergnügen to-morrow, ou die fact, que von all gens Sie thought à uns.

We sommes alle three fières und happy, et danken you de ganz our cœur.

Elgar passed it on to the Musical Times, which published it, calling its form of expression “somewhat Tower of Babelish.”

Express Delivery

Why spend money on cat food when there’s a more immediate solution? Leo Voelker’s 1979 invention simultaneously curbs the local sparrow population and keeps the local cats occupied.

The birds enter the housing at the top but can escape only through the mesh cage at the bottom, which serves as a kind of self-serve food dispenser for neighborhood cats.

“The cat feeder by its design is self-cleaning since the cat quickly learns to remove the sparrow from the cage.”

Rest Stop

rest stop puzzle

A gold miner lives at point A, 3 miles north of the river and 5 miles upstream from the gold mine at point B, which is 2 miles north of the river. On the way to work he must stop at the river to give his burro a drink. At what point on the river should he stop in order to minimize the length of the trip?

Click for Answer

“Money and a Friend”

– I.J. Reeve, The Wild Garland, 1865