## The Library of Babel

As films are to tv series, short stories are to novels and I invest heavily in the former for both categories. Everything subsidiary to one central motif is cut back and you get to have a single story told in a pure form, with no padding, or dragging side plots. Science fiction and mystery novels are particularly keen on this format, mainly because they used to be sold to monthly magazines.

Outside the world of science fiction there is one short story that I keep going back to called The Library of Babel by Jorge Luis Borges, an Argentinian author. The English Translation is only seven pages long and so if you have quarter of an hour it is well worth the read. But I'll give a brief outline here as well.

The book describes a world which is entirely made up of a gigantic library. Each hexagonal room is identical in layout and has bookshelves lining four of the walls. There are little recesses for sleeping and using the toilet and in the other walls there are doorways to travel to other rooms. Each book on the shelves is identical in format with exactly 410 pages made up of 40 lines of around 80 characters each. Each character is either a space, a comma, a full stop or one of 22 letters. However the pages seem to be written completely randomly with complete nonsense filling them.

Occasionally the inhabitants of this world will find a book which has a few words which make sense to them, or a long sequence of repetitions. In true random we seek patterns and will eventually find them; we are creatures that see links where they do not exist.

The narrator mentions two rules that they have been able to observe:

1. Each character is one of 25 distinct choices

2. No book is repeated

From this we are able to work out that there must be a finite number of books. You see, this library appears not to be infinite, simply it is massive. Every human thought capable of being expressed in 410*40*80=1312000 characters is contained in the world somewhere once. Similarly ever thought possible, but with a single misplaced error, will appear 1312000 times and so.

We can also think that every single book is simply a cipher for another thought. To have every such perfect idea already existing in the world (an inconceivably many times) drives many of the inhabitants mad. Much of the story discusses how different groups of people react to this set up, with many driven to superstition or cult like behaviour.

I want to talk about what would happen if we had a single book of infinite length, with the characters appearing at random. It would be tempting to think that the same sort of logic would apply: that every thought would appear at some point. For instance at some point we would have the word alaric 100 times in a row and then later we would have the entirety of Hamlet. Surely infinity contains all of these finite strings eventually? Well, actually the answer is that for any particular infinite book we're not sure, but almost certainly not.

Take for instance the decimal expansion of Pi. We could encode that onto the alphabet in loads of different ways, but let's just consider it in its native home of mathematics. Does Pi contain every finite string of numbers within its expansion? For instance if I pick the number 123456789, does it appear in Pi? The answer is yes, starting at the 523,551,502nd decimal place. However, there was no guarantee that it had to appear anywhere, even though Pi goes on to infinity. I find this a weird thing to get my head around.

However there are numbers which do contain every finite string. Would you like to see this arcane, beautiful number which magically contains an encoding of every name you have ever loved, every dream you have had and every secret you have buried?

...

0.1234567891011121314...

And another infinitely many boring variations on it. Not mystical in the slightest.