A King of Infinite Space in a Nutshell

The argument that “given infinite time and space, all possibilities becomes actualities” is based on an ambiguity of speech. The word “possible” is used in three ways, first, to say something is practical, second, to say something is imaginable, third, to say something is nonsense.

Sober men, within living memory, once dismissed the moonshot as impossible, by which they meant that the engineering obstacles involved were so great, that no foreseeable efforts could surmount them. After the moonshot became history, those who cheered that America had done the impossible were using the word to means that by an unforeseen level of effort, the engineering obstacles were overcome. An impractical thing had been done for reasons of glory going beyond the practical. This is the way the word is used to describe climbing mountains, or like feats.

The word is also used to refer to things one can imagine as hypothesis one cannot reduce to theory. In this case, the ballistics of how to shoot a shell to the moon have been known since the time of Jules Verne, if not since the time of Isaac Newton, even if the mechanics of rocketry were not well articulated enough, in those days, to reduce the hypothesis to theory. Whether or not Impey Barbicane can fire a man inside a shell from his Columbiad gun at such a velocity as will reach lunar orbit without killing the passenger, is a separate question; in that sense of the word, a thing can be possible without being possible, that is, the word is used for those things where no objection exists to the hypothesis, but no feasible means are seen. However, so long as the hypothesis does not violate any known facts, no matter how unlikely the event, it is hypothetically possible.

Finally, the word is used to refer to those things not logically impossible, which are merely fantastic images in the mind, or phrases in speech combined without reference to reality. It is unimaginable that a hypercube the size of Texas should collide with the earth at twice the speed of light, because, first, no one can imagine a hypercube, and, second, according to the standard model of physics, the speed of light is absolute, such that adding more acceleration only decreases travel time within the moving frame of reference of the thing accelerated. However it is not, in this final sense of the term, impossible, if it is possible for the standard model to be wrong, or if it is possible for a three dimensional cross section of a hypercube to protrude into our dimension and cross paths with the earth.

On the other hand, inventing a new moral code, inventing a new primary color, or filling the gas tank of your motorcar with that primary color to allow you to run over next Tuesday at the intersection at the speed of the ultimate prime number while fleeing from a Snark who enjoys puns, only to collide with the rock God made to be too heavy for Himself to lift, is actually impossible. Because a snark, so the Bellman told me, always looks grave at a pun.

And what he tells me three times is true.

If the possibilities in infinite time and space involve some mechanical process, such as having an countless number of immortal monkeys type out every combination of every key on a Roman alphabet typewriter, and if no accident interferes with the process, why, then, yes, eventually one will have every Shakespeare sonnet ever written typed out correctly. One will also have a large set of sonnets with one letter, space, or punctuation mark wrong; and a larger set with two wrong; an even larger with three; each set being geometrically larger than the last. Eventually, after every letter has been changed and changed back in every combination, one will also have a complete set of all the poems of Byron, Tennison, Keats, as well as everything else written in any Latinate language the letters on the keys can form. One will have every possible English or French or Italian translation of all the poems of Sappho while one is at it, but, of course, one will not have the poems as written in Greek, because there are no Greek keys on the typewriter, nor will there ever be, no matter how long the monkeys keeps typing.

This is not to say our monkeys could not type out an English language description of the Greek letters, via words like “alpha” and “beta” and “gamma” and so on, so in that sense, given enough time, even Sappho’s poems in a form of Greek will be available.

But good luck find those poems lost in the infinite reaches of the Library of Babel, which, in honor of Borges, is the name of the repository where all the infinite monkeys’ infinite manuscripts have been stored.

Yes, in theory, there is a book somewhere in the stacks which describes correctly the position of all the other books in the stacks, but the book which gives instructions on finding this book has myriads of copies which are missing a space, punctuation mark, or letter making the instructions unreliable.

Finding the correct version is impossible.

But having the infinite monkey typing out the Unendurable Word which is that word which, if typed out by monkeys, destroys the world, is impossible, because there is no such thing as the Unendurable Word.

It is not imaginable, that is, it is not one of the outcomes one can reach by any combination of keyboard strokes of a Roman alphabet keyboard.

Nor can the monkeys type out a new moral code, nor describe a new primary color, nor type out a new word so long that it is longer than itself. These things are impossible because they are merely nonsense phrases.

So when we say that given infinite space and time, all possibilities become actualities, we are making one assumption and reaching one trivial conclusion.

The false assumption is that a sufficient cause to actualize every potentiality in the universe exists. It may not. If our hypothetical monkeys, for example, suffer a staffing shortage, and are eaten by an infinite number of jaguars before every possible keyboard combination of every possible length is typed out, many are the Shakespeare sonnets which may never be typed.

The trivial conclusion is to say that every possible combination of letters is, in fact, a possible combination of letters.

So, yes, hypothetically, given enough time, and a crew of infinite monkeys, a repeated mechanical process could type out all of Shakespeare’s sonnets. But the process will produce infinitely more random word and letter combinations, and the writings of James Joyce.

The act of finding the volumes in the Library of Babel which contain the same works men could have created with creative effort would itself require more effort than the act of writing it to begin with.

So, yes, in theory, the Library of Babel contains all the books ever written or will be using any combination of Roman letters. This means not a single four-color comic book is present, or a sheet of music, even if volumes able to describe these things by means of typed letters does exist.

But what the library does not contain is every possible thing I can imagine, if I am imagining a thing not to be written in Roman letters. Nor can it contain those things no one can imagine because the words used to depict them are nonsense words.

Nor is there a snark, nor any living creature, in the library, even if there are many books, a countless number, describing every aspect of a snark, and there are even fewer four-sided triangles.

So let us check on the progress of our thought experiment!

At the moment, one quadrillion kalpas of trillion-eon periods into the experiment, my monkeys have only been typing the smallcase “a” over and over again, as “aa” then “aaa” then “aaaa” and so on.

The head monkey tells me he will have the boys start working on all the possible combinations of words, sentences, manuscripts and volumes starting with “ab” only once the team has reached an infinitely long string of smallcase letter “a” and exhausted those possibilities. He will add one more “a” to each string of “a” until he runs out.

You see, the monkey believes that, given enough time and space, and given infinite monkeys to type, every possibility can one day be actualized.

I may have infinite monkeys, but they are not infinitely intelligent, it seems.