Tuesday, January 25, 2011

Planck Apes

L'horrible masse des livres révèle et cache la rivière et ses origines : j'aime à dire que les sources attirent les savants parce qu'elles sont libres de savants ! - Michel Serres, Le Tiers-Instruit

Like all men of the Library, I have traveled in my youth; I have wandered in search of a book, perhaps the catalogue of catalogues . . . - Jorge Luis Borges, The Library of Babel

Suppose that every public library has to compile a catalog of all its books. The catalog is itself one of the library's books, but while some librarians include it in the catalog for completeness, others leave it out, as being self-evident. Now imagine that all these catalogs are sent to the national library. Some of them include themselves in their listings, others do not. The national librarian compiles two master catalogs - one of all the catalogs that list themselves, and one of all those which don't. The question is now, should these catalogs list themselves? The 'Catalog of all catalogs that list themselves' is no problem. If the librarian doesn't include it in its own listing, it is still a true catalog of those catalogs that do include themselves. If he does include it, it remains a true catalog of those that list themselves. 

However, just as the librarian cannot go wrong with the first master catalog, he is doomed to fail with the second. When it comes to the 'Catalog of all catalogs that don't list themselves', the librarian cannot include it in its own listing, because then it would belong in the other catalog, that of catalogs that do include themselves. However, if the librarian leaves it out, the catalog is incomplete. Either way, it can never be a true catalog of catalogs that do not list themselves. This catalogue problem is a good illustration of the Russell paradox, discovered by Bertrand Russell in 1901, which showed that the set theory of Frege leads to a contradiction, thus causing a deep crisis in the foundation of Mathematics. 

 The Library of Babel (David R Godine edition, 2000). Front cover by Erik Desmazières

Borges's total library concept was the main theme of his widely-read 1941 short story "The Library of Babel", which describes an unimaginably vast library consisting of interlocking hexagonal chambers, together containing every possible volume that could be composed from the letters of the alphabet and some punctuation characters:
The orthographical symbols are twenty-five in number. This finding made it possible, three hundred years ago, to formulate a general theory of the Library and solve satisfactorily the problem which no conjecture had deciphered: the formless and chaotic nature of almost all the books. One which my father saw in a hexagon on circuit fifteen ninety-four was made up of the letters MCV, perversely repeated from the first line to the last. Another (very much consulted in this area) is a mere labyrinth of letters, but the next-to-last page says Oh time thy pyramids. This much is already known: for every sensible line of straightforward statement, there are leagues of senseless cacophonies, verbal jumbles and incoherences.”

Giuseppe Arcimboldo, The Librarian, 1566

In a 1939 essay entitled "The Total Library", Borges traced the infinite-monkey concept back to Aristotle's Metaphysics. Explaining the views of Leucippus, who held that the world arose through the random combination of atoms, Aristotle notes that the atoms themselves are homogeneous and their possible arrangements only differ in shape, position and ordering. In On Generation and Corruption, the Greek philosopher compares this to the way that a tragedy and a comedy consist of the same "atoms", i.e., alphabetic characters. Three centuries later, Cicero's De natura deorum (On the Nature of the Gods) argued against the atomist worldview:  
“He who believes this may as well believe that if a great quantity of the one-and-twenty letters, composed either of gold or any other matter, were thrown upon the ground, they would fall into such order as legibly to form the Annals of Ennius. I doubt whether fortune could make a single verse of them.”
A great quantity of letters . . .

Borges follows the history of this argument through Blaise Pascal and Jonathan Swift, then observes that in his own time, the vocabulary had changed. By 1939, the idiom was "that a half-dozen monkeys provided with typewriters would, in a few eternities, produce all the books in the British Museum." (To which Borges adds, "Strictly speaking, one immortal monkey would suffice.")  Borges then imagines the contents of the Total Library which this enterprise would produce if carried to its fullest extreme: Everything would be in its blind volumes.

Given enough time, would a hypothetical chimpanzee typing at random, as part of its output, produce one of Shakespeare's plays?

Everything: the detailed history of the future, Aeschylus' The Egyptians, the exact number of times that the waters of the Ganges have reflected the flight of a falcon, the secret and true nature of Rome, the encyclopedia Novalis would have constructed, my dreams and half-dreams at dawn on August 14, 1934, the proof of Pierre Fermat's theorem, the complete catalog of the Library, the proof of the inaccuracy of that catalog. Everything: but for every sensible line or accurate fact there would be millions of meaningless cacophonies, verbal farragoes, and babblings. Everything: but all the generations of mankind could pass before the dizzying shelves,shelves that obliterate the day and on which chaos lies, ever reward them with a tolerable page.

Recently, a team of british mathematicians (above) explained the monkey theorem as follows:

“I’m going to divide the universe into Planck-sized regions, and put a monkey in each one. You will ask what the monkey is made of, when nothing can be smaller than the Planck scale, and I will say that it is not made of anything – it is a single, fundamental monkey particle. One in every Planck sized region of space. These regions are very small - there will be nearly as many monkeys inside the space occupied by a single atom as there are atoms in the universe. And there will be monkeys in the spaces not occupied by atoms too. And they will type faster. How fast can a thing happen? Just as there is a shortest possible distance, there is a shortest possible time, and it’s called the Planck time.

Planck Ape

The Planck time is how long it would take you to cover one Planck length if you travelled at the speed of light. My monkeys will type at a rate of one keystroke per Planck time (approx. 5.39 * 10^-44 s). They will type so fast because the energy required to confine a monkey to such a small region will make the monkey extraordinarily hot. You will ask what the typewriter is made of, and I will say it is not separate from the monkey: typing is what a monkey particle does. (I don’t know what happens to the letters that the monkeys type. There is no room for them or anything else, as the cosmos is jampacked with hot monkey particles. But I’m not going to let this stop me.) So, from the Big Bang, with a monkey in every last tiniest unit of space possible, typing at the fastest speed there is, for the entire history of the growing Universe, and do we have a deal?

A Planck Ape contemplating Shakespeare

Yes! The first four lines of the sonnet “SHALL I COMPARE THEE TO A SUMMERS DAY THOU ART MORE LOVELY AND MORE TEMPERATE ROUGH WINDS DO SHAKE THE DARLING BUDS OF MAY AND SUMMERS LEASE HATH ALL TOO SHORT A DATE” will be knocked out somewhere in the cosmos several times a second! This is good! In fact, every few dozen thousand years, it’ll come together with the next word – SOMETIME - to boot. Will we ever get the next two words (SOMETIME TOO)? We might be lucky – there’s something like a one in three chance in the age of the universe.

A couple of Sulawesi Macaque monkeys hard at work not writing Shakespeare


I don't know about you, but I think that's rather impressive. Howevwer, if you want the Complete Works, as the theorem says, you'll have to wait so much longer than a cat's life. The full text of The Library of Babel is here.


  1. I thought the link to "the full text of the Library of Babel" was to a website which _actually contained the Library of Babel_ as envisioned by Borges. Sadly, I was disappointed to see one more copy of the short story.

    I shall now begin a short story in which I shall replace every instance of "text" in Borges' short story with "The Famous Short Story by a slightly more-famous Argentinian author".

  2. We are not amused, though our librarian friend may just wee herself. Anyone seen a big monolith around here?

  3. This essay should be supplemented with a discussion of Kolmogorov complexity. The monkeys need not type the compleat works of Shakespeare -- they need simply type the most compact computer program that generates these works. That should hasten their completion by many quadrillions of years.

  4. Hi Weimar! This is very nice, but I am not "a team of british mathematicians" :-)
    I'd prefer it if you credited the author, or at least gave a link to the source of the quoted text. Would you please edit the post to include this. Thank you.