"Space," it says, "is big. Really big. You just won't believe how vastly, hugely, mindbogglingly big it is. I mean, you may think it's a long way down the road to the chemist's, but that's just peanuts to space, listen..."You might think that matter and information are incommensurate, and I think you might be right, but bear with me and perhaps you'll see what I mean.
Douglas Adams, The Hitchhikers Guide to the Galaxy
These thoughts were prompted by reading 'The Library of Babel' in Jorge Luis Borges' "Labyrinths". Borges imagines a library consisting of every possible book of a certain size (letters per page, pages per book). Most of them are full of sets of nonsensical sequences of letters, but among them are also all possible meaningful books in every language that uses that script. (It is in the same vein as the 100 monkeys typing Shakespeare.) The script in the story has 22 letters (only one case), commas, full-stops and spaces, so there are 25 different characters to consider. (Obviously a much simplified script.)
The number of books of course is, I was going to say astronomical, but astronomical is way to small, if by that we mean of the order of, say, the number of atoms in the universe.
It may sound a bit daft, but I start out thinking about how close could I approach to generating a mini Library of Babel. That is I thought that I could write a programme to generate all strings of text of length n, so I wondered how long realistically could I make n.
For just one character, n=1, there are 25 books, for n=2, 25^2 = 625 and so on with the number of books being 25^n = 10^(1.398n) in general. I'd hoped I might be able to get to big enough n so as to generate word or two. I thought it would be nice to see some meaningful text emerge within the possibilities. Of course you will already seen this was a hopeless idea - just n = 4 requires more than 1/3 million books.
In the Library of Babel we are told:
"Each book contains 410 pages; each page, 40 lines; each line, about 80 black letters."That makes n = 410 x 40 x 80 = 1,312,000.
Although I knew from the start that the Library of Babel was big, I don't think I'd quite thought through how big.
According to Wolfram Alpha, there are about 10^80 atoms in the known universe which requires n = 80/1.398 = 58. That is to say, if the book had only one page with one line of text, there'd be more books in the library than there are atoms in the universe.
Much of the universe is empty, of course, so now lets imagine that it was instead solid with matter. Again from Wolfram Alpha the diameter of the observable universe is 8.8 x 10^26 metres. An atom is about an angstrom in diameter so you could fit about (8.8 x 10^26/10^-10)^3 = 6.81 x 10^110 atoms in the universe. That would require about n = 111/1.398 = 80 - still only one line of text! Suddenly space doesn't sound quite so big...
I could go on, but you get the picture. 'All' I'm doing, of course, is pointing out how fast exponential growth is, but I also - and I know I need to be careful here - think it suggests something about the significance of information within the universe.
Knowing 'how much stuff (material)' we have, is much less significant than knowing how that stuff is arranged.