A visitor to Niels Bohr’s cottage noticed a horseshoe nailed over the door.
“Surely you don’t expect that a horseshoe will bring good luck?” asked the visitor.
“No, I don’t,” Bohr said. “But they say it works even if you don’t believe in it.”
See The Misfortune Field.
What’s the key to this curious sequence of numbers?
1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, …
When read aloud, each term describes the one that precedes it. The first term consists of “one 1,” the second of “two 1s,” the third of “one 2, then one 1,” and so on.
That seems pretty arbitrary, but it opens a door into an unsuspected mathematical universe. Start with any number (except 22, an obvious dead end) and it will produce a string of digits that lengthens by about 30 percent with each generation — indeed, the percentage approaches a predictable constant (30.3577269 …) as the length approaches infinity.
More amazingly, the growing string will organize itself into a series of recognizable finite substrings that evolve predictably with each generation. John Horton Conway, who discovered all this, identified 92 such substrings, which he named after the chemical elements. Thus “uranium” (3) decays into “protactinium” (13), which becomes “thorium” (1113), and so on.
Thus an infinitely complex universe can arise from simply reading the number 1 aloud.
For any convex polyhedron,
vertices – edges + faces = 2
In 1988, readers of the Mathematical Intelligencer judged this the world’s second most beautiful theorem — behind Euler’s identity.
Trisect the angles of any triangle and you’ll find an equilateral triangle at its heart.
This theorem has a curious cousin: If you trisect the sides of any triangle and erect an equilateral triangle outwardly on the middle third of each leg, then the outermost vertices of these equilateral triangles will themselves form an equilateral triangle.
When we look at another person’s face, her eyes and mouth convey the most information about her mood.
Indeed, when a face is inverted we can have trouble recognizing it because we can’t read its expression.
So in 1980 University of York psychologist Peter Thompson tried inverting everything but the eyes and mouth.
Most people can recognize the face at left and assign a mood to it, but they’re often surprised to see it right side up.
“Further research into this illusion might help determine whether face recognition is a serial or parallel process,” Thompson wrote in Perception that summer. “It might even tell us something about Margaret Thatcher.”
O Cleinias, are those who learn the wise or the ignorant?
He answered that those who learned were the wise.
Euthydemus proceeded: There are some whom you would call teachers, are there not?
The boy assented.
And they are the teachers of those who learn — the grammar-master and the lyre master used to teach you and other boys; and you were the learners?
Yes.
And when you were learners you did not as yet know the things which you were learning?
No, he said.
And were you wise then?
No, indeed, he said.
But if you were not wise you were unlearned?
Certainly.
You then, learning what you did not know, were unlearned when you were learning?
The youth nodded assent.
Then the unlearned learn, and not the wise, Cleinias, as you imagine.
— Plato, Euthydemus
“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
