Journal: Percutio

Cover: S. Bianciardi

An annual published in hard copy in New Zealand and France paying special attention to the creative process and issues relating to translation.


Percutio is dedicated primarily to reflections upon the creative process, particularly in relation to work that bridges cultures.

Cover: N. Bunn

Percutio may, therefore, feature poetry, essays, extracts from novels, choreography, approaches to composition and journal entries in English and the language of creation.

Cover:A. Loeffler

The deadline each year is April 10th.

Submissions are welcome.

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The Random Infinite Part One of an essay on the mathematical puzzle of the random infinite (added 14.06.2010). By Nick Ascroft


The Random Infinite

Nick Ascroft


Two statements on infinity are below, the second seems to be demonstrably untrue, but how does the first differ?

(1) Any infinite random string of units will contain every possible combination of those units.

(2) Any infinite non-random but unrestricted string of units will contain every possible combination of those units.

The first claim is the (necessarily) modified version of the assertion that a monkey (or an infinite number of monkeys) jabbing randomly at a typewriter would eventually type the complete works of Shakespeare. I bracket out 'necessarily' as I find the hidden but wilful abstraction of this idea irritating. The nature and finiteness of monkeys is ignored. Point one, no monkey will produce a random string. Point one clarified, no monkey, even no chimp, will hit the space bar enough, nor hold the shift key down enough for capitals, let alone hit all of the keys. Point two, no monkey has an infinite life, the word monkey defines a member of a set of very real animals with specific lifespans. This characteristic is essential to their monkeyhood. A godlike immortal monkey is quite a fictitious and fantastical place to start. Nor is it possible for there to be infinite monkeys, whose finite populations will over the course of the universe eventually become extinct.

This last point however is the easiest to let slide into hypothetical abstraction. And such pooh-poohing stickler pragmatics may seem absurd and unbefitting of an essentially mathematical problem simply being colourfully illustrated. (I find a good counter-illustration to express why the monkey-typewriter scenario is a dud: imagine an immortal chimpanzee (who never learns) faced with the typesetting program InDesign: if placed in front of an immortal computer running this software forever, would the monkey eventually typeset an exact replica of the Oxford University Press's latest edition of The Oxford Shakespeare: The Complete Works -- including page numbers, the running heads, the font style variation, the italicisation? Clearly not. It is an ape. The typewriter likewise is too difficult a technology.)

But if the assertion is to be truly considered, such obvious stumbling blocks should be removed. Nevertheless the idea behind the conundrum is easily modified and it's best to get to its heart, that these imagined monkeys, not understanding the meanings of the typewriter keys would produce a random string of letters, numbers, symbols and punctuation.

Must then a truly random infinite string contain all possible variations of that string? One could quibble over the definition of 'random' . It has been argued that this is a fictitious notion, that all random generators to date are based on some algorithm or other. Is it possible for instance for any existing random number generators to produce a nine repeatedly, and only a nine for four billion years? This, like any slice, is a small slice of the infinite and just as random an occurrence as any other four-billion-year string.

However, let's imagine that randomness is real, and that a truly random generator exists. In fact let's make it even easier. Imagine this generator only generates words and only those which are in Shakespeare's vocabulary. Ignoring punctuation and spaces, over an infinite period must this generator write Shakespeare's complete works in perfect order? A better and more probing question is this: is it possible for there to be an infinite string created by this generator which does not contain Shakespeare's complete works? Just as it is possible for there to be an infinite string of numbers which does not contain the number three (all even numbers for instance), is it conceivable that the generator could go forever creating combinations of Shakespearean words which every time failed to be the complete works? There are after all infinite strings of Shakespearean words which do not contain the complete works. For instance, an infinite string of Shakespearean words in which the word 'but' never occurs. Could the random generator not generate one of these?

To limit the generator further, imagine if it generated a random string of Shakespearean words the exact length of the complete works. There are only a finite number of combinations of this string and one of them clearly is the complete works. But if left to generate these forever would it necessarily go through all combinations, or could it perhaps just endlessly repeat combinations that weren't the complete works?

This is like asking: would a random number generator set to endlessly pick numbers between zero and nine ever fail to pick the number five for eternity? Or, could a coin flipped endlessly always land on heads? Instinct yells No! but hold on. Some might rush to the defence in the name of randomness, saying that an inherent aspect of randomness is that any combination is capable of appearing and therefore over enough time all must. This is a definition of randomness based around this very problem and so is self-answering. If randomness is defined at the level of each single instance, however, the prospect seems more believable. With each instance just as likely to be any of a variety of options, on each occasion it is just as likely to be any one of these, and if one of these is heads over tails, it is just as likely to be heads each time even if it has been heads for a massively large number of times on preceding flips.

To go further, imagine a series of infinite binary strings, and instead of one and zero you have heads and tails. The first might start HTTTHHTHTHHTTT ... and a second might start TTTHTHHTTTHHT ... How many of these possible infinite binary strings exist? Clearly this too is infinite. Numbered among these infinite possible strings however, must there not be one combination which begins HHHHHHHHHHHHH and continues like that forever? And if a truly random generator was to select one of these possible infinite binary strings would it not be just as likely to select the one that was always heads as any other? That is to say, doesn't this mean an infinite random series might not contain every possible combination of the units it is choosing from?

Let's return to the random complete-works-length generator running forever. Among all the possible random infinite strings of complete-works-length sets of random Shakespearean words, is there not a string which, for instance, starts with the complete works exactly but for the last two words being in reverse order and then, like endless heads, repeats this same combination forever (or any other of the many possible infinite iterations of the generator which do not include spitting out the complete works)? And of all possible infinite strings of complete-works-length sets of random Shakespearean words (of which there are an infinite number), is this one not just as likely to be selected randomly as any other?

I believe so, but perhaps there is a hole in my logic as infinity is clearly a slippery body. But if so, the first proposition above is false. Not every random infinite string need contain every possible iteration of the units of that string. That is, the highly hypothetical monkeys even over infinity can fail. Which leads to the second proposition above, widely implied as true and from which many dubious entailments follow. This I will discuss shortly ...