Let the peal of a gong be heard in the last half of a minute, a second peal in the preceding 1/4 minute, a third peal in the 1/8 minute before that, etc. ad infinitum. … Of particular interest is the following puzzling case. Let us assume that each peal is so very loud that, upon hearing it, anyone is struck deaf — totally and permanently. At the end of the minute we shall be completely deaf (any one peal being sufficient), but we shall not have heard a single peal! For at most we could have heard only one of the peals (any single peal striking one deaf instantly), and which peal could we have heard? There simply was no first peal.

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

Various writers throughout the 19th century confidently reported that they’d found the true and exact value of π. Unfortunately, they all gave different answers. In 1977 DePauw University mathematician Underwood Dudley tried to make sense of this by compiling 50 of their pronouncements:

He concluded that π is decreasing. The best fit is π_t = 4.59183 – 0.000773t, where t is the year A.D. — it turns out we passed 3.1415926535 back in 1876 and have been heading downward ever since.

And that means trouble: “When π_t is 1, the circumference of a circle will coincide with its diameter,” Dudley writes, “and thus all circles will collapse, as will all spheres (since they have circular cross-sections), in particular the earth and the sun. It will be, in fact, the end of the world, and … it will occur in 4646 A.D., on August 9, at 4 minutes and 27 seconds before 9 p.m.”

There is some good news, though: “Circumferences of circles will be particularly easy to calculate in 2059, when π_t = 3.”

(Underwood Dudley, “π_t,” Journal of Recreational Mathematics 9:3, March 1977, p. 178)

Andy fires a shot at the goal, but it’s deflected by his opponent Bill. If Bill had not reached the ball, it would have struck Charlie, Andy’s teammate. Roberto Casati asks, “Should Bill get credit for the save?”

He: Not quite. After all, the ball was not going to score anyway; it would have hit Charlie’s body.

She: But neither would it be right to say that anything happened thanks to Charlie. After all, Charlie did nothing.

He: But then who is responsible for spoiling Andy’s shot?

“Cases like this one are indicative of a deep conceptual tension,” Casati writes. “I am walking in the rain. My umbrella is open and I am wearing a hat, so my head is not getting wet. But why is that so? It’s not because of the umbrella, because I’m wearing my hat. And it’s not because of my hat, for I have an umbrella.”

From Casati’s excellent book Insurmountable Simplicities. See also In the Dark.

614,656 = 28⁴
6 + 1 + 4 + 6 + 5 + 6 = 28

1,679,616 = 36⁴
1 + 6 + 7 + 9 + 6 + 1 + 6 = 36

8,303,765,625 = 45⁶
8 + 3 + 0 + 3 + 7 + 6 + 5 + 6 + 2 + 5 = 45

52,523,350,144 = 34⁷
5 + 2 + 5 + 2 + 3 + 3 + 5 + 0 + 1 + 4 + 4 = 34

20,047,612,231,936 = 46⁸
2 + 0 + 0 + 4 + 7 + 6 + 1 + 2 + 2 + 3 + 1 + 9 + 3 + 6 = 46

From the Soviet magazine KVANT, 1986:

On an n × n checkerboard, a square becomes “infected” if at least two of its orthogonal neighbors are infected. For example, if the main diagonal is infected (above), then the infection will spread to the adjoining diagonals and on to the whole board. Prove that the whole board cannot become infected unless there are at least n sick squares at the start.

The key is to notice that when a square is infected, at least two of its edges are absorbed into the infected area, while at most two of its edges are added to the boundary of the infection. Thus the perimeter of the infected area can’t increase; in order for the full board (with perimeter 4n) to become infected, there must be at least n infected squares to begin with.

At any given instant, an arrow in flight is where it is, occupying a space equal to itself. It cannot move during the instant, for that would require the instant to have parts.

This seems to mean that motion is impossible. Aristotle writes, “If everything, when it is behaving in a uniform manner, is continually either moving or at rest, but what is moving is always in the now, then the moving arrow is motionless.”

Bertrand Russell adds, “It is never moving, but in some miraculous way the change of position has to occur between the instants, that is to say, not at any time whatever. … The more the difficulty is meditated, the more real it becomes.”

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.

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