## Infinite monkey theorem meets infinite sentence theorem

You’ve no doubt heard the theory that if you gave a monkey a typewriter and infinite time he would eventually compose, in order, the complete works of Shakespeare. This may take him a very, very long time, but it possibly, if this other theory is correct, would not require infinite time.

The number of possible sentences in the English language is apparently finite. Indeed, the Macquarie University has calculated that there are 10570 possible sentences in English. They used these suppositions:

• that English has about 500,000 words (there are about 450,000 in the 20 volume Oxford English Dictionary, but this excludes many colloquial forms – although it does include many obsolete forms),
• that English sentences can be up to 100 words in length (a fairly reasonable working assumption)
• that any individual word can occur 0 to 100 times in a single sentence (an unrealistic assumption)
• that words can be combined in any order (a false assumption)

Now, they even acknowledge that some of their assumptions are unrealistic and false – but using these assumptions they’ve given us a pretty reasonable guess at the number of permutations available in English. I guess you could go a step further and introduce “texts” to the equation – if each text is assumed to be 100,000 sentences long, or something unreasonably high, we’re still in the realm of calculable numbers.

Then we could figure out how fast a monkey produces a sentence, and get an upper limit for how long a monkey would take to produce every possible combination of words within those parameters to figure out the maximum amount of time required for a monkey to produce the works of Shakespeare. Simple.

The Macquarie University used the calculations above to demonstrate that English is an open language – full of unique sentences that have never been written before. My aim when I write, is to produce as many unique sentences as possible. To claim my place in the lexicon of life. I am pretty sure that at the point of writing, those two sentences are unique and have not been read by imaginary dinosaurs before. This process of trying to achieve the maximum number of unique sentences may lead me to introducing an odd adjective, and a MacGuffin, in every tremendously Jurassic sentence. Then I could rewrite the same sentence over and over again in order to claim maximum mileage from the one creative work (ie the writing of a unique sentence).

Lets try.

• As I wrote this, sitting next to a coffee pot plant, with my hot wife, I produced a uniquely blue sentence.
• As I wrote this, sitting next to a ceramic pot plant, with my hot wife, I produced a uniquely green sentence.
• As I wrote this, sitting next to a yellow dinosaur, with my hot wife, I produced a uniquely green sentence…

Producing unique sentences in an open language is relatively easy. Here’s what the Macquarie researcher had to say:

Grammatical rules would greatly reduce this number of sentences, as would the requirement that all sentences be meaningful, but the resulting number of possibilities would still be extremely large (more than could ever be spoken in the entire history of human languages let alone during the much shorter life span of an individual language). So for all practical, non-mathematical, purposes we can say that the English language, or any other living language (1), is an open system. It’s actually quite easy to come up with a unique, never before produced, sentence. To do so, for example, combine an unlikely (or impossible, or meaningless) event with a particular named person on a particular date. For example: “On 31st October 1999, whilst writing a lecture on animal communication, Robert had a colourless green idea.” (2) Once this sentence has been written or spoken, subsequent productions of this same sentence are not unique, but unique sentences may potentially be generated from it by making slight changes to it (eg. change “green” to “red”).

Got any unique sentences? Care to claim your place in history?

And perhaps a mathematically minded commenter might like to resolve this infinite monkey conundrum for me – if I have an infinite number of monkeys with typewriters will one monkey come up with the works of Shakespeare on his first go?