“It has been asserted (by C.S. Lewis, for instance) that no determinist rationally can believe in determinism, for if determinism is true, his beliefs were caused, including his belief in determinism. The idea seems to be that the causes of belief, perhaps chemical happenings in the brain, might be unconnected with any reasons for thinking determinism true. They might be, but they need not be. The causes might ‘go through’ reasons and be effective only to the extent that they are good reasons.”

— Robert Nozick, “Reflections on Newcomb’s Paradox,” 1974

“If … [determinism] is true, then the intellectual or cognitive operations of its upholders, including their choice or decision to maintain the thesis, … are themselves only the effects of inexorable forces. But if this is so, why should the thesis … be accepted as valid or true?”

— Alan Gewirth, Reason and Morality, 1978

The digits 1-9 can work some impressive tricks:

The first formula, found by B. Ziv in 2004, produces the first 10 digits of pi.

The second, astonishingly, reproduces e to 18,457,734,525,360,901,453,873,570 decimal places. It was discovered by Richard Sabey, also in 2004.

(Thanks, Robin.)

“A man wrote to say that he accepted nothing but Solipsism, and added that he had often wondered it was not a more common philosophy. Now Solipsism simply means that a man believes in his own existence, but not in anybody or anything else. And it never struck this simple sophist, that if his philosophy was true, there obviously were no other philosophers to profess it.”

— G.K. Chesterton, St. Thomas Aquinas, 1933

214358976 = (3 + 6)² + (4 + 7)⁸ + (5 + 9)¹

1000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000000000000000000000000
000002569 is prime.

In “Partial Magic in the Quixote,” Borges quotes philosopher Josiah Royce:

Let us imagine that a portion of the soil of England has been levelled off perfectly and that on it a cartographer traces a map of England. The job is perfect; there is no detail of the soil of England, no matter how minute, that is not registered on the map; everything has there its correspondence. This map, in such a case, should contain a map of the map, which should contain a map of the map of the map, and so on to infinity.

This sequence tends to a single point, the point on the map that corresponds directly to the point it represents in the territory.

Cover England entirely with a 1:1 map of itself, then crumple the map into a ball. So long as it remains in England, the balled map will always contain at least one point that lies directly above the corresponding point in England.

Related (sort of): In The Humor and Drama of Early Texas (2003), George U. Hubbard notes that one day in 1865, Thomas Jefferson Chambers was standing in this house in a room containing his portrait when someone fired a shot through the second-story window. “The bullet passed through Chambers’ body and lodged in his portrait on the wall. The citizens of Anahuac thought it very singular that the bullet that killed him struck the portrait in exactly the same place it had passed through his body.” The crime was never solved.

See Garganta and Papered Over.

Draw a semicircle and surmount it with two smaller semicircles.

A line drawn through A, at any angle, will divide the perimeter precisely in half.

This probably has some romantic symbolism, but I’m not very good at that stuff.

06/19/2024 Very belated update: This is called the cardioid of Boscovich, after Roger Boscovich, 1711-1787. There’s a proof of the theorem in Alsina and Nelsen, Icons of Mathematics.