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