Challenge 560 Response
Mining the Imageverse
with Jo Wharton Heath
How does the essay convey tongueincheek humor?
One intriguing and amusing idea in Michael O’Farrell’s wonderful “Imageverse”, is the thought that the universe of ideas, obviously infinite (ask the gypsy!), can be corralled into the finite, since each particular idea can be described via pixels on finitely many 8.5 by 11inch pages with margin allowance. He shows with excruciating precision that the number of pages of this type is a large but finite number.
Challenge title [“Indefnite Infinity”]: if n ÷ 0 = ∞, and 0 ÷ n is an indefinite number, what is 0 ÷ 0?
A reader’s response:
Answer: 0 ÷ 0 = (0×∞)/n. Here’s why:
Start with the given: Multiply both sides by 0: Divide both sides by n: 
n ÷ 0 = ∞ (0×n)/0 = 0×∞ 0 ÷ 0 = (0×∞)/n Q.E.D. 
You’ll get lots of variations on this. One can prove almost anything if one ignores the fact that numerous properties hold for all nonzero numbers only, and that ∞ isn’t a number at all and satisfies very few ordinary number properties.
Copyright © 2014 by Jo Wharton Heath
Thank you, Jo! As far as I can make out, the idea — can we really call it a formula? — 0 ÷ 0 gives us a famous song by George Gershwin.
Michael O’Farrell’s “Imageverse” shows that the possibilities in all the pixels on a standardsize sheet of paper are finite in number, although that number is impossibly large. On a sheet of paper the size of the universe, with or without margins, the number of possibilities will be larger still and nonetheless finite.
Could even a standardsize sheet of paper be of any possible use? The bureaucrats in Steven C. Levi’s “A Behemoth in the Barn” seem to think so and will, I’m sure, report their results in due course. In the process they will once again revalidate Sturgeon’s Law, that 90 percent of everything can be found on the floor of farmer Brown’s chicken coop.
Dividing anything by zero gives an unsettling but mathematically proportional view of ourselves in the universe, according to William Blake:
To see a world in a grain of sand,
And a heaven in a wild flower,
Hold infinity in the palm of your hand,
And eternity in an hour.— “Auguries of Innocence” (1803?)
Blaise Pascal also had it right:
Car enfin qu’estce que l’homme dans la nature? Un néant à l’égard de l’infini, un tout à l’égard du néant, un milieu entre rien et tout [...] également incapable de voir le néant d’où il est tiré, et l’infini où il est englouti.
— Pensées (bef. 1662)

In the end, what is man in nature? Nothing in infinity, everything in nothing, a middle ground between nothing and everything [...] equally unable to see the nothingness from which he is taken and the infinity in which he is immersed. 
Any summary will be prosaic, but we can take comfort in it: the musician, the poet and the scientist reassure us that there are infinities everywhere we look. And when we set pencil to paper, there’s no end to them!
Copyright © 2014 by Don Webb