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Hopelessly Egocentric

Conditions

  • Recall that a bird is egocentric when it responds to itself with itself: x(x) => x. A bird is "hopelessly egocentric" if, no matter what you call it with, it always responds with itself: x(<anything>) => x.
  • A bird is "fixated" on another if, given any input, it responds with that input. (That is, a bird that is hopelessly egocentric is one that is fixated on itself.)
  • The Kestrel is a bird where, for any values of x and y, K(x)(y) => x. (That is, K(x) is fixated on x.)

Problem

As with problem 1, composition holds, and the the forest contains a Mockingbird. It also contains a Kestrel. Show that at least one bird is hopelessly egocentric.

Solution

The individual conditions of problem 1 are a red herring—you don't need to (as I first tried) somehow compose the Mockingbird and the Kestrel. Instead, it's a consequence of problem 1 that's important: the fact that, under its conditions, every bird is fond of at least one other.

That means the Kestrel is fond of another bird—call it A:

K(A) === A;

Since those two are equivalent, so are the results of calling each with another arbitrary bird, x:

K(A)(x) === A(x);

But since the Kestrel is fixated on its first argument, A, we know that K(A)(x) always returns A. Given that K(A)(x) is equivalent to both A(x) and A, we can state that A is hopelessly egocentric: it always returns itself.

// Both true:
K(A)(x) === A;
K(A)(x) === A(x);

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