Someone in whose power I am tells me that I am going to be tortured tomorrow. I am frightened, and look forward to tomorrow in great apprehension. He adds that when the time comes, I shall not remember being told that this was going to happen to me, since shortly before the torture something else will be done to me which will make me forget the announcement. This certainly will not cheer me up, since I know perfectly well that I can forget things, and that there is such a thing as indeed being tortured unexpectedly because I had forgotten or been made to forget a prediction of the torture: that will still be a torture which, so long as I do know about the prediction, I look forward to in fear. He then adds that my forgetting the announcement will be only part of a larger process: when the moment of torture comes, I shall not remember any of the things I am now in a position to remember. This does not cheer me up, either, since I can readily conceive of being involved in an accident, for instance, as a result of which I wake up in a completely amnesiac state and also in great pain; that could certainly happen to me, I should not like it happen to me, nor to know that it was going to happen to me. He know further adds that at the moment of torture I shall not only not remember the things I am now in a position to remember, but will have a different set of impressions of my past, quite different from the memories I now have. I do not think that this would cheer me up, either. For I can at least conceive the possibility, if not the concrete reality, of going completely mad, and thinking perhaps that I am George IV or somebody; and being told that something like that was going to happen to me would have no tendency to reduce the terror of being told authoritatively that I was going to be tortured, but would merely compound the horror. Nor do I see why I should be put into any better frame of mind by the person in charge adding lastly that the impressions of my past with which I shall be equipped on the eve of torture will exactly fit the past of another person now living, and that indeed I shall acquire these impressions by (for instance) information now in his brain being copied into mine. Fear, surely, would still be the proper reaction: and not because one did not know what was going to happen, but because in one vital respect at least one did know what was going to happen — torture, which one can indeed expect to happen to oneself, and to be preceded by certain mental derangements as well. If this is right, the whole question seems now to be totally mysterious.
– Bernard Williams, Problems of the Self, 1973
Well, it’s our old friend the mysterious pouch. Today the pouch contains a random quantity of marbles, and we’re going to withdraw a handful. But first, consider:
- If the bag contains an even number of marbles, then we are equally likely to withdraw an even or an odd number. For instance, if it contains 4 marbles, then we are equally likely to withdraw 2 or 4 as 1 or 3.
- But if the pouch contains an odd number of marbles, then we’re more likely to withdraw an odd number, as there’s one more way of choosing an odd number than an even number. For example, if the pouch contains 5 marbles then we’re more likely to draw 1, 3, or 5 than 2 or 4.
This is troubling. Without even opening the pouch we seem to have decided that, on balance, we’re more likely to withdraw an odd number of marbles than an even. Indeed, this seems to mean that handfuls in general are more commonly odd than even. How can this be?
In 1688, John Locke received a letter from scientist William Molyneaux posing a curious philosophical riddle: Suppose a blind man learned to identify a cube and a sphere by touch. If the shapes were then laid before him and his vision restored, could he identify them by sight alone?
Locke responded, “Your ingenious problem will deserve to be published to the world,” and he included a formulation of the problem in the second edition of the Essay Concerning Human Understanding.
Three hundred years later, it’s still an open question. (Locke agreed with Molyneaux that the answer is probably no: “The blind man, at first sight, would not be able with certainty to say which was the globe, which the cube, whilst he only saw them; though he could unerringly name them by his touch, and certainly distinguish them by the difference of their figures felt.”)
When I think of a unicorn, what I am thinking of is certainly not nothing; if it were nothing, then, when I think of a griffin, I should also be thinking of nothing, and there would be no difference between thinking of a griffin and thinking of a unicorn. But there certainly is a difference; and what can the difference be except that in the one case what I am thinking of is a unicorn, and in the other a griffin? And if the unicorn is what I am thinking of, then there certainly must be a unicorn, in spite of the fact that unicorns are unreal. In other words, though in one sense of the words there certainly are no unicorns–that sense, namely, in which to assert that there are would be equivalent to asserting that unicorns are real–yet there must be some other sense in which there are such things; since, if there were not, we could not think of them.
– G.E. Moore, Philosophical Studies, 1922
It deals with a game that [Theodore] Roosevelt and I used to play at Sagamore Hill. After an evening of talk, perhaps about the fringes of knowledge, or some new possibility of climbing inside the minds and senses of animals, we would go out on the lawn, where we took turns at an amusing little astronomical rite. We searched until we found, with or without glasses, the faint, heavenly spot of light-mist beyond the lower left-hand corner of the Great Square of Pegasus, when one or the other of us would then recite:
That is the Spiral Galaxy of Andromeda.
It is as large as our Milky Way.
It is one of a hundred million galaxies.
It is 750,000 light-years away.
It consists of one hundred billion suns, each larger than our sun.
After an interval Colonel Roosevelt would grin at me and say: ‘Now I think we are small enough! Let’s go to bed.’
– William Beebe, The Book of Naturalists, 1944
To divide 8101265822784 by 8, move the 8 from the number’s head to its tail.
To multiply 1034482758620689655172413793 by 3, move the 3 from the tail to the head.
During the Russian revolution, the mathematical physicist Igor Tamm was seized by anti-communist vigilantes at a village near Odessa where he had gone to barter for food. They suspected he was an anti-Ukrainian communist agitator and dragged him off to their leader.
Asked what he did for a living, he said he was a mathematician. The sceptical gang leader began to finger the bullets and grenades slung round his neck. ‘All right,’ he said, ‘calculate the error when the Taylor series approximation to a function is truncated after n terms. Do this and you will go free. Fail and you will be shot.’ Tamm slowly calculated the answer in the dust with his quivering finger. When he had finished, the bandit cast his eye over the answer and waved him on his way.
Tamm won the 1958 Nobel prize for physics but he never did discover the identity of the unusual bandit leader.
– John Barrow, “It’s All Platonic Pi in the Sky,” The Times Educational Supplement, May 11, 1993
Suppose we illustrate. You put a ball on a billiard-table, and, holding the cue lengthwise from side to side of the table, push the ball across the cloth. Here, in a rough way, the ball represents the ship, the cue the wind, only, as there is no waste of energy, the ball travels at the same rate as the cue; evidently it cannot go any faster. Now, let us suppose that a groove is cut diagonally across the table, from one corner-pocket to the other, and that the ball rolls in the groove. Propelled in the same way as before, the ball will now travel along the groove (and along the cue) in the same time as the cue takes to move across the table. The groove is much longer than the width of the table, double as long, in fact. The ball, therefore, travels much faster than the cue which impels it, since it covers double the distance in the same time. Just so does the tacking ship sail faster than the wind.
– “Some Famous Paradoxes,” The Illustrated American, Nov. 1, 1890
- The sum of the numbers on a roulette wheel is 666.
- ANTITRINITARIANIST contains all 24 arrangements of the letters I, N, R, and T.
- The Empire State Building has its own zip code.
- 63945 = 63 × (-9 + 45)
- “Isn’t it strange that we talk least about the things we think about most!” — Charles Lindbergh
In 1938, Samuel Isaac Krieger of Chicago claimed he had disproved Fermat’s last theorem. He said he’d found a positive integer greater than 2 for which 1324n + 731n = 1961n was true — but he refused to disclose it.
A New York Times reporter quickly showed that Krieger must be mistaken. How?
A pleasant anecdote is told of Partridge, the celebrated almanac maker. In traveling on horseback into the country he stopped for his dinner at an inn, and afterward called for his horse that he might reach the next town, where he intended to sleep. ‘If you would take my advice, sir,’ said the ostler, as he was about to mount his horse, ‘you will stay where you are for the night, as you will surely be overtaken by a pelting rain.’ ‘Nonsense, nonsense,’ said the almanac maker, ‘there is sixpence for you, my honest fellow, and good afternoon to you.’ He proceeded on his journey, and sure enough he was well drenched in a heavy shower. Partridge was struck with the man’s prediction, and being always intent on the interest of his almanac, he rode back on the instant, and was received by the ostler with a broad grin. ‘Well, sir, you see I was right after all.’ ‘Yes, my lad, you have been so, and here is a crown for you, but I give it you on condition that you tell me how you knew of this rain.’ ‘To be sure, sir,’ replied the man; ‘why the truth is we have an almanac in our house called Partridge’s Almanac, and the fellow is such a notorious liar, that whenever he promises us a fine day we always know that it will be the direct contrary.’
– The Golden Rule, and Odd-Fellows’ Family Companion, Oct. 16, 1847
- Can God sin?
- The Thinker’s right elbow is on his left knee.
- 48625 = 45 + 82 + 66 + 28 + 54
- MARASCHINO is an anagram of HARMONICAS.
- “Genius is nothing but continued attention.” — Helvetius
Here is a curious old story that is something like a puzzle: A crocodile stole a baby, ‘in the days when animals could talk,’ and was about to make a dinner of it. The poor mother begged piteously for her child. ‘Tell me one truth,’ said the crocodile, ‘and you shall have your baby again.’ The mother thought it over, and at last said: ‘You will not give it back.’ ‘Is that the truth you mean to tell?’ asked the crocodile. ‘Yes,’ replied the mother. ‘Then by our agreement I keep him,’ added the crocodile; ‘for if you told the truth I am not going to give him back, and if it is a falsehood, then I have also won.’ Said she: ‘No, you are wrong. If I told the truth you are bound by your promise; and, if a falsehood, it is not a falsehood, until after you have given me my child.’ Now, the question is, who won?
– Pennsylvania School Journal, March 1887
The first few powers of 5 share a curious property — their digits can be rearranged to express their value:
25 = 52
125 = 51 + 2
625 = 56 – 2
3125 = (3 + (1 × 2))5
15625 = 56 × 125
78125 = 57 × 182
It’s conjectured that all powers of 5 have this property. But no one’s proved it yet.
Suppose you borrowed $10 from Tom and $10 from Bob. On your way to repaying them you are robbed of everything but the $10 you had hidden in your shirt pocket. By no fault of your own, you now face the following paradoxical dilemma:
(1) You are obligated to repay Tom and Bob.
(2) If you pay Tom you cannot repay Bob.
(3) If you repay Bob you cannot repay Tom.
(4) You cannot honor all your obligations: in the circumstances this is impossible for you. (By (1)-(3).)
(5) You are (morally) required to honor all your obligations.
(6) You are not (morally) required to do something you cannot possibly do (ultra posse nemo obligatur).
– Nicholas Rescher, Paradoxes, 2001
String together the numbers 1 through 19 in reverse order:
Fittingly, the resulting number is evenly divisible by 19.
- 34425 = 34 × 425
- A running joke is a standing joke.
- RESTAURATEURS balances two identical sets of letters on either side of the central R.
- Does an artwork have value if no one sees it?
- “Marriage is a covered dish.” — Swiss proverb
6205 = 382 + 692
3869 = 622 + 052
5965 = 772 + 062
7706 = 592 + 652
Suppose that in one night all the dimensions of the universe became a thousand times larger. The world will remain similar to itself, if we give the word similitude the meaning it has in the third book of Euclid. Only, what was formerly a meter long will now measure a kilometer, and what was a millimeter long will become a meter. The bed in which I went to sleep and my body itself will have grown in the same proportion. When I wake in the morning what will be my feeling in face of such an astonishing transformation? Well, I shall not notice anything at all. The most exact measures will be incapable of revealing anything of this tremendous change, since the yard-measures I shall use will have varied in exactly the same proportions as the objects I shall attempt to measure.
– Henri Poincaré, Science and Method, 1908
If we take a cube and label one side top, another bottom, a third front, and a fourth back, there remains no form of words by which we can describe to another person which of the remaining sides is right and which left. We can only point and say here is right and there is left, just as we should say this is red and that blue.
– William James, The Principles of Psychology, 1890
Sir John Cutler had pair of silk stockings, which his housekeeper, Dolly, darned for a long term of years with worsted; at the end of which time, the last gleam of silk had vanished, and Sir John’s silk stockings were found to have degenerated into worsted. Now, upon this, a question arose amongst the metaphysicians, whether Sir John’s stockings retained (or, if not, at what precise period they lost) their personal identity. The moralists again were anxious to know, whether Sir John’s stockings could be considered the same ‘accountable’ stockings from first to last. The lawyers put the same question in another shape, by demanding whether any felony which Sir John’s stockings could be supposed to have committed in youth, might legally be the subject of indictment against the same stockings when superannuated; whether a legacy left to the stockings in their first year, could be claimed by them in their last; and whether the worsted stockings could be sued for the debts of the silk stockings.
– Thomas de Quincey, “Autobiography of an English Opium-Eater,” from Tait’s Edinburgh Magazine, September 1838
J.J. Sylvester was a brilliant mathematician but, by all accounts, a lousy poet. The Dictionary of American Biography opines delicately that “Most of Sylvester’s original verse showed more ingenuity than poetic feeling.”
What it lacked, really, was variety. His privately printed book Spring’s Debut: A Town Idyll contains 113 lines, every one of which rhymes with in.
Even worse is “Rosalind,” a poem of 400 lines all of which rhyme with the title character’s name. In his History of Mathematics, Florian Cajori reports that Sylvester once recited “Rosalind” at Baltimore’s Peabody Institute. He began by reading all the explanatory footnotes, so as not to interrupt the poem, and realized too late that this had taken an hour and a half.
“Then he read the poem itself to the remnant of his audience.”
See Poetry in Motion.
“Light crosses space with the prodigious velocity of 6,000 leagues per second.”
– La Science Populaire, April 28, 1881
“A typographical error slipped into our last issue that it is important to correct: the speed of light is 76,000 leagues per hour — and not 6,000.”
– La Science Populaire, May 19, 1881
“A note correcting a first error appeared in our issue number 68, indicating that the speed of light is 76,000 leagues per hour. Our readers have corrected this new error. The speed of light is approximately 76,000 leagues per second.”
– La Science Populaire, June 16, 1881