Study enough information theory, and it will start seeming obvious on a gut level that a library containing all possible books has less information in it than a library containing one book.
I'm not well-versed in information theory, so this may be a misinterpretation. But this strikes me as a beautiful truth that rewards careful pondering.
A library containing all possible books may contain all the gems of knowledge in the world, but it also contains far more nonsense: Books full of random letters that mean nothing, as well as real books with changes made to them, both unnoticeable and noticeable. Such a library, though containing all information, would not contain any information about which books contain real information and which do not. A single real book, though, while it may contain factual errors, can at least be guaranteed to contain the beliefs and assumptions of its author, which is more than you can say about the infinite library.
Look at it this way: an infinite list of strings containing all possible strings (or even a finite list of strings containing all possible strings shorter than a given length) contains less information than a program or essay that I wrote, because the infinite list gives no indication about which strings are "important", "informational" or "useful", while my one essay or program contains concise, easily-extractable information intended for a particular purpose.
One more example, just for fun: Imagine I'm a stupid bank robber who keeps all the plans for my heists and the names of my partners in one Word document on my hard drive. If the police arrest me, they have enough information to send me and my partners to jail. However, imagine that I've written a program that makes a million copies of my secret plan and changes the names and locations inside all of the copies to random values. Now, if I get caught, without knowing which document is real and which are fake, the authorities don't have any useful information.
And it would also contain an infinite number of books which claim to list which books contain real information, but actually contain some titles that contain nonsense & lies and/or are missing some titles that contain real information. So the chance of you finding a fake information-listings book is far greater than you finding the real one.
Hmm, sorry. I thought you were referring to the book being difficult to find because it would be mixed in with an infinite number of other books, not specifically that there would be an infinite number of fake versions of that book.
It requires only a very short program to print out all possible books, and a much longer program to print out one particular book.
To look at it another way, by the trivial isomorphism between any class and its complement, the library containing all possible books has just as much information as the library containing no books.
A better way of stating this would be to look at how much information you can extract from the library with one book in a given time, as compared to the library that contains "all" books: nearly all such books would have gibberish in them.
Eliezer's statement would be a lot less surprising with a proper definition of "all" :)
Think about how you would specify a book in the "all possible books" library - you would need a number at least as large as the book itself, in the general case! This might be easier to see if you imagine the "library" as the set of all possible numbers (in binary) up to a certain reasonable maximum. So you specify the book...by giving the binary representation of the book, which is simply its index if you order them by their numerical value. So you can see how having all books in your library is the same as having no books.
I think he's relating to the "book of five rings" / "art of war" way of writing books.
Theres many big long books that are merely churned out, or are written before an author has reached a point of completeness about the subject he's writing on. Worse, they could be written as "wines of fear" where they are written not to directly offend anyone.
On the other hand, there are a few beautiful books that are short, yet establish mental frameworks that cover all the other books. The book of five rings is touted as one.
Personal example: There's volumes of business books that all harken back to people being some mixture of evangelist, maven, or relater. Books like "the human fabric" or religious books like the "bhagavad gita" establish the same framework.
So.. we should just think about doing our work and not actually do it. Because imagining, and not reaching the finished product, is the most fun? Uh.. right.
I like the implementation in his house, but it doesn't apply to business or creating things. In some ways, what he's done is created something by creating nothing - but very rarely does that occur elsewhere.
Awesome, awesome post. The great Ze Frank has a similar video [1] where he calls this "brain crack" (because your mind is addicted to the idea more than what you actually accomplish...or don't accomplish).
Reminds me of John Cage's 4'33", a composition where no notes are played and instead the audience is expected to observe the sounds of the environment.
My wife and I visited the Guggenheim a number of years ago, and came across Robert Rauschenberg's "White Paintings", which even as I am a fan of minimalism, seemed kind of silly. All the while, we decided that we would one day duplicate the paintings for our own house. If it's good enough for the Guggenheim...
A year or so later, we bought four canvases, some gesso and some white paint, and made four white paintings, and upon hanging them up, found that it actually looked really nice. We did some research on the original "White Paintings" series, and discovered that John Cage and Robert Rauschenberg were friends, and that 4'33" was "a direct response" to the White Paintings.
It was doubly cool because while I've worked in the visual arts, my wife is a composer. These seemingly silly paintings have come with us to our new house, and hang in the living room, and I'm quite fond of them, even though they required fairly minimal effort for my wife and I to create. Hooray for minimalism!
Frank Stella said (during a 1964 radio interview): "I knew a wise-guy who used to make fun of my painting, but he didn't like the Abstract Expressionists either. He said they would be good painters if they could only keep the paint as good as it is in the can. And that's what I tried to do. I tried to keep the paint as good as it was in the can."
