A Pinkie Parable of Hilbert’s Hotel · 1:12am Nov 22nd, 2017
New story out today: Infinitely Too Many Pinkie Pies
I have a vague recollection that someone suggested I should write a Pinkie Pie story about Hilbert’s Infinite Hotel. Unfortunately I can’t remember who it was, but thank you.
I was a bit cautious about doing this as it’s Not My Field. I know I teach math, so I should understand it, but I do the applied end of the mathematical spectrum, dealing with the numerical tricks you need to do physics and engineering. The nature of infinity is a bit too abstract for my taste.
More interesting for me, was to look at Hilbert’s Hotel as a Science Communication case study. It is one of those rare parables that explains a particular academic idea so well that it has come to define it, as well as providing inspiration for countless works of fiction. The only comparable example I can think of off-hand is Schrödinger’s Cat (see: Quantum Mechanics, Schrödinger's Cat and the My Little Pony Movie Trailer).
The original story was thought up by German mathematician David Hilbert and described in a lecture in 1924. I looked up the original text. Although my skill at reading German is rather limited, I got the gist. Hilbert’s original story was quite basic, but it has been greatly embellished by the many writers who have retold it.
Hilbert wanted to illustrate the concept of a countable infinity (∞), which can be indexed by the set of natural numbers (1,2,3…), or pictured as a hotel with an infinite number of rooms and an infinite number of guests.
This is then used for various mathematical proofs to show that ∞+1=∞ (the hotel can accommodate an extra guest); ∞+n=∞ (the hotel can accommodate a finite number of extra guests); ∞+∞=∞ (the hotel can accommodate an infinite number of extra guests).
This is as far as I went in my story, but you can continue to show the hotel can accommodate an infinite number of infinitely large groups of new guests. All this is done by showing that you can map the elements of these sets to the infinite set of natural numbers (hence you can find rooms for all the new guests). The next chapter is to show that there are some infinities bigger than this. The number of real numbers between 0 and 1 is infinite, but this is an uncountable infinity—the set can’t be mapped one-to-one with the natural numbers—there’s no room for them all at the hotel. There’s quite a nice blog post explaining this by Steven Strogatz.
I wrote my first draft of this to kill time on a transatlantic flight back in August. I never intended to do anything more than explain the concept of Hilbert’s infinite hotel through a pony fic. Unfortunately writing a fic with an infinite number of characters turned out to be more problematic than expected and it took a lot of redrafting to get it to a point where I felt I could submit it. I kept having to put this to one side to finish more urgent tasks, so it’s good to have finally got it done.
So there’s going to be another chapter?!
Now which pony do you use to explain complex numbers...?
Pinkie plus Hilbert's Hotel is a match made in… well Hilbert's Hotel, I suppose. Awesome!
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Discord, of course
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Not anytime soon. Much as I love Pinkie Pie, after writing this story, I've had quite enough of her for a while.
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Hmm... which pony has both a real and imaginary component?
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They all do, come to think of it....
¡Bravo! ¡You replaced genocide with mathematics! ¡I love it!