Borges' Proof

(From Borgs to Borges - what an unfortunate coincidence.) I just finished reading the Penguin translation of Jorge Luis Borges' The Aleph, including prose works from The Maker. This is from the latter:

Argumentum Ornithologicum

I close my eyes and see a flock of birds. The vision lasts a second, or perhaps less; I am not sure how many birds I saw. Was the number of birds definite or indefinite? The problem involves the existence of God. If God exists, the number is definite, because God knows how many birds I saw. If God does not exist, the number is indefinite, because no one can have counted. In this case I saw fewer than ten birds (let us say) and more than one, but did not see nine, eight, seven, six, five, four, three, or two birds. I saw a number between ten and one, which was not nine, eight, seven, six, five, etc. That integer--not-nine, not-eight, not-seven, not-six, not-five, etc.--is inconceivable. Ergo, God exists.

For some reason, Borges' proof reminded me of the story of Hippasus of Metapontum, the disciple of Pythagoras who applied his master's famous theorem to a right triangle with legs each one unit long. The length of the resulting hypotenuse is the square root of two, a number that cannot be calculated, like pi. Pythagoras' cult sought to create a perfect philosophical system based on a geometry of whole numbers; so Hippasus' calculation was a dangerous heresy. Legend has it he made the discovery on board ship and was immediately drowned by his colleagues when they realised its implications - the real story may be a little more involved.

Hippasus' hypotenuse exists; pi exists. Are they inconceivable, in the same sense as the number of Borges' birds, for being incalculable? And in order for them to be perceived by the mind of God, would it be necessary for Him to calculate them to the last decimal? Even an infinite being would take an infinite time to calculate an infinite series of decimals, I would assume, unless the universe began with the exact value of pi existing ab initio in the mind of God. In any case, if we can conceive of a line with a length that cannot be precisely calculated, we can surely think of a flock without it containing a precise number of birds.

And it's certainly a good thing I didn't take Borges' little joke as seriously as these guys.

Three Borges stories:

The Aleph (original translation)
The Library of Babel
The Zahir and I

More on Borges