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

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

Henry is driving in the countryside with his son. For the boy’s edification, Henry identifies various objects on the landscape as they come into view. ‘That’s a cow,’ says Henry. ‘That’s a tractor,’ ‘That’s a silo,’ ‘That’s a barn,’ etc. Henry has no doubt about the identity of these objects; in particular, he has no doubt that the last-mentioned object is a barn, which indeed it is. Each of the identified objects has features characteristic of its type. Moreover, each object is fully in view, Henry has excellent eyesight, and he has enough time to look at them reasonably carefully, since there is little traffic to distract him.

Given this information, would we say that Henry knows that the object is a barn? Most of us would have little hesitation in saying this, so long as we were not in a certain philosophical frame of mind. Contrast our inclination here with the inclination we would have if we were given some additional information. Suppose we are told that, unknown to Henry, the district he has just entered is full of papier-mâché facsimiles of barns. These facsimiles look from the road exactly like barns, but are really just façades, without back walls or interiors, quite incapable of being used as barns. They are so cleverly constructed that travelers invariably mistake them for barns. Having just entered the district, Henry has not encountered any facsimiles; the object he sees is a genuine barn. But if the object on that site were a facsimile, Henry would mistake it for a barn. Given this new information, we would be strongly inclined to withdraw the claim that Henry knows the object is a barn. How is this change in our assessment to be explained?

— Alvin I. Goldman, “Discrimination and Perceptual Knowledge,” Journal of Philosophy, November 1976

“I am fond of the businessman’s paradox due to Lisa Collier: The president of a certain company offered a reward of $100 to any employee who could offer a suggestion which would save the company money. One employee suggested: ‘Eliminate the reward.'” — Raymond Smullyan

Let’s play a game. You’ll flip a coin, and if it comes up heads I’ll give you $1. If you flip heads again I’ll give you $2, then $4, then $8, and so on. When the coin comes up tails, the game is over and you can keep your winnings.

Because I’m taking a risk, I ought to charge you an entrance fee. What’s a fair fee? Surprisingly, it seems I should charge you an infinite amount of money. With each new flip your chance of success is 1/2 but your prospective earnings double, so your total expected earnings — the earnings times their chance of being realized — is infinite:

E = (1/2 × 1) + (1/4 × 2) + (1/8 × 4) + … = ∞

Nicholas Bernoulli first described this problem in 1713. One proposed resolution is that it ignores psychology — we’re considering the monetary value of the prize rather than its value to us. Gold shines more brightly for a beggar than for a billionaire; once we’ve amassed a certain sum, the appeal of greater riches begins to diminish. “The mathematicians estimate money in proportion to its quantity,” wrote Gabriel Cramer, “and men of good sense in proportion to the usage that they may make of it.”

(Thanks, Ross.)

Take a whole stick and cut it in half. Half a minute later, cut each half in half. A quarter of a minute after that, cut each quarter in half, and so on ad infinitum.

What will remain at the end of a minute? An infinite number of infinitely thin pieces? Writes Oxford philosopher A.W. Moore, “Do we so much as understand this?”

Does each piece have any width? If so, couldn’t we reassemble them to form an infinitely long stick? If not, how can we assemble them to form anything at all?