Why The Infinite Monkey Theorem Might Really Be False

I realise this blog is inactive, and it is entirely possible that the current AS class never subscribed to it, but nonetheless, as last year’s ‘Nature of Numbers and Equality’ debate demonstrated, I am not capable of letting a discussion go, especially when it involves both maths and philosophy. Therefore, I’m going to show why the infinite monkey theorem is false, again given David did so in 2013 here, only unlike Churchland, this probably won’t cover any grounds for divorce.

1. Givens

The theorem as stated is: “If infinite monkeys with infinite typewriters randomly press letters for infinity, they must eventually produce the entire works of Shakespeare.”

We must of course assume that “monkey” is a substitute for “entirely random letter generator.”

The statement when stated mathematically is: “every process that produces a random string of characters according to proper probability distribution will eventually produce any finite string of characters with probability 1.”

Similarly, we must assume that the random generation is truly random and totally uninfluenced by previous events (i.e conforms to proper probability distribution), and ignore the practical issue that there is nothing really random, for the sake of the case.

Finally, we assume that only letters are characters that can be produced and omit capitalisations and punctuations, so that a production of Hamlet would begin something like this: entertwosentinelsfirstfranciscowhopacesupanddownathispostthenbernardowhoapproacheshim
and this would be a valid outcome as the entire works of Shakespeare.

2. Almost surely, or, why p=0 does not mean “can never happen”

To begin with, we need to assert that p=1 does not actually mean that it is impossible for anything else to happen, in this case at least, despite having always been taught that p=1 means an absolute certainty and p=0 the opposite. Of course, our maths teachers were not wrong in saying so – when dealing with almost all circumstances, probability 1 does indeed mean that X must occur, however, the most notable exception to this is when dealing with infinities.

Formally, for the fellow mathematicians: Let (Ω, ƒ, P) be a probability space. An event E∈F happens almost surely if P[E]=1. Equivalently, E happens almost surely if the probability of E not occurring is zero: P[Ec]=0. (Jacod, Jean; Protter, (2004). Probability Essentials. Springer. p. 37)

In words: something happens almost surely if the probability is 1 or if the probability of it not occurring is 0.

So far so good, however, almost surely does not mean surely, in the same way that p=1 does not mean always, and this example demonstrates why:

Consider a sample space – the possible outcomes of any probabilistic experiment – that is not countable, and denoted by Ω

We assign to each sub interval a probability equivalent to its length (proof two three):

[a,b] ⊆ [0,1] ⇒ P([a,b]) = b – a

if the set [a,b] is a subset of [0,1], then the probability of [a,b] is b – a.

The direct consequence of this is that every single possibility is a 0-probability outcome:

∀∞ ∈ Ω, P({∞}) = P([∞,∞]) = ∞ – ∞ = 0

for all sets ∞ that are a member of the sample space, the probability is ∞, which equals the probability of the set [∞,∞], which equals ∞ – ∞, which equals 0

What this example shows is that a 0-probability event is not an event that never happens, because in the case where the sample space cannot be defined it is possible for a 0-probability event to occur, because all events are p=0 and yet something must occur. Forgive the crude translations of notation to words, please.

This means that in cases where infinities are involved, p=1 does not in fact mean “surely,” but rather, “almost surely.”

3. 0-probability monkeys

Caitlin and Alec were, of course, right in asserting that as you reach infinity, probability becomes closer and closer to 0, meaning that once you reach infinity, any event has a probability of 0. In reference to the monkeys, they applied this to the case of the letter A being pressed every single time, and were right in saying this has a probability of 0. However, so does the case where the monkeys eventually type the entire works of Shakespeare, because, as shown above, every event has a probability of 0 where the sample space is infinity, which it is in this case because we have infinite monkeys on infinite type writers typing for infinity, yielding an infinite possible combination of typed letters (despite the fact that there are only 26 letters).

Yet, the monkeys must type something, the generator must produce some letters, despite the fact that every single possible combination has a probability of 0.

What this means, is that it is entirely possible that the monkeys do at some point produce the entire works of Shakespeare, or Plath, or Marlowe, or every volume in the Bod. It is entirely possible that within the infinite string of letters produced by the monkeys, all these finite strings are included. But that does not mean they will be. There must be the possibility that only the letter G is typed, or S, or that they type and type real words but never the right ones in the right order.

Therefore, the fact that there must be other possibilities other than somewhere including all of Shakespeare demonstrates that one cannot assert that the monkeys must type all of Shakespeare’s works.

Josephine’s motor

An illustrative story

One fine day in October, Josephine Muppet is on her way to work at Dagenham Community College, where she teaches a joint course in Celebrity Studies with Global Ecology. On this particular morning, her Renault Clio (which she chose, in preference to several other similarly-priced but more reliable cars, because she liked the colour) seems to be lacking its usual zippy acceleration, and there is an unfamiliar and disconcerting vibration.

She takes the car to the garage, where a misfire is instantly diagnosed. The mechanic’s first thought is that it will almost certainly be the inherently dodgy electrics that are to blame: probably a coil, or the ignition leads. But then again (thinks the mechanic) this is a Clio, so in fact it might alternatively be prematurely-worn piston rings, a broken valve, a failed injector or, for that matter, almost anything else. Ah well… let’s see, now… Closer investigation duly reveals a frayed ignition lead, which (along with its fellows, just in case) is replaced. The engine runs smoothly again. Josephine hands over £100, and everyone goes home happy.

Four weeks later, Josephine is back with, as she says, the same fault. The car’s acceleration is once again feeble, and there is the same strange vibration as before. This time, she knows that what she has is called a ‘misfire’ and, needless to say, she is not happy. Clearly, in her mind, the garage must have bodged the repair last time, and as a result the same fault has reappeared. At all events, the mechanic gets to work and the car is once again made to run normally. Josephine (reluctantly) hands over a second wodge of cash (she is not taken in by what she sees as the transparently dishonest insistence that it was a different fault this time), and everyone goes home.

Another five weeks pass, and now Josephine is really miffed: the same fault has occurred yet again. One thing is for sure: she’s not going anywhere near that garage full of crooks, which has so far lightened her purse to the tune of £200 while failing on two occasions to fix the problem. Christmas is coming, and she needs to control her finances more tightly. Instead, she invites Danny, a student with whom she is conducting an ill-advised affair, to have a look at the car for her.

(The affair is ill-advised not merely because, given Danny’s age and Josephine’s position, it involves criminality on her part, as well as a serious breach of the terms of her contract of employment. In addition, Danny, unknown to Josephine, is secretly in it only with a view to threatening, at some later stage, to post compromising pictures of Josephine on the internet. He hopes that, in this way, he will be able to blackmail Josephine into recommending for him a Pass Certificate for his course, despite the array of facts that might countermand such an award, namely: he is rarely in class; he is invariably under the influence of drugs on the few occasions when he is present; and he has yet to begin – let alone proofread – a single piece of his required coursework.)

Although Danny’s extravagant appetite for controlled substances is doing no favours, in Josephine’s estimation, to either his academic achievement or his sexual performance, Danny does have a bit of know-how when it comes to motors. This is thanks mainly to his father, who steals them for a living and who is a moderately significant player in the vibrant local banger-racing scene. When Danny looks under the bonnet of the Renault, he therefore immediately notices the following:

1. the new ignition leads correctly fitted by the garage in October;

2. the new set of fuel injectors correctly fitted by the garage in November; and

3. a large crack in the cylinder head adjacent to cylinder number 3.

Danny turns the key in the ignition, and the engine grinds, reluctantly and lumpily, into life.

‘So… it’s, like, a misfire, Miss, like… innit?’ he suggests.

‘I know that!’ snaps Josephine. ‘But it’s been fixed twice! Why does it keep coming back?’

‘So… like, … it’s not, like, an it, izzit? It’s, like, a them, innit?’ replies Danny, warring simultaneously against unfairly-redoubled enemies – the sheer conceptual complexity of the wisdom that he must convey, and the bafflingly uncompliant syntax of his native tongue. ‘I mean, like, they’re all, like, misfires…? But they’re all like, different, like, kinds… know what I mean? So… anyway… reckon my Dad’ll give you, like, fifty pound for the motor…? So… are you, like, gettin’ your kit off or what?’

Poor Josephine! She has a terrible job, an appalling boyfriend, and a Renault Clio! Why oh why did she never take the opportunity to sit in on the Level 1 Motor Mechanics course at the College? She still doesn’t really know what is going on with her car, nor whom to trust.

In desperation, she rings the Philosophy Helpline. And her life improves.

It is patiently explained to her that she is quite right: her car has had, repeatedly, what it makes sense to call the same fault: it has been uncharacteristically underpowered. A car engine (she learns) does not work by magic, nor via the diligent ministrations of a team of fairies resident under the bonnet. Accordingly, the fault was, on every occasion, nothing other than a physical state of the engine.

What physical state? Well, as she is told, her car has been, successively, in a series of physical states, all of which fall under the heading ‘misfiring’. These states can all be said to be of the same physical type, in that they all consist in a failure, for well-understood physical reasons, by one cylinder to ignite fuel and produce power; hence the common label ‘misfire’. So, in a way, it was the same physical problem each time. But the various instances of this single type of fault – misfiring – were also all, at a different and deeper level of description, of different types: one was an electrical failure, one was an injector failure, and the third was a leaking cylinder.

‘So was it the same kind of fault or different kinds of fault?’ she asks.

‘Well, there’s no single answer to that,’ comes the reply. ‘It all depends on how you individuate kinds of physical state.’

Following additional advice from the Philosophy Helpline, Josephine confiscates Danny’s mobile telephone the next day, when he pulls it out in class in order to update his Facebook status from ‘stoned’ to ‘stonked’. She ‘forgets’ to return it, and on her way home from work throws it, complete with its collection of photographs, into the River Beam. She trusts, correctly, that Danny will be too ignorant to realise that copies of the files on his mobile will have been automatically archived on his provider’s server. She dumps Danny, and for good measure has him expelled from the College for gross idleness; she also calls the police and alerts them to the shady dealings of Danny’s father. She sells the Clio to the unfairly maligned garage for £120 and buys, from the same source, a much older Volvo. She quits her job, takes a course in motor mechanics, gains an apprenticeship, and fifteen years later inherits a successful independent Volvo specialist’s garage from her kindly and childless employer. She sells the business, moves to France, falls in love with a handsome and romantic writer-cum-architect named Pascal, and lives happily ever after, gaining on the way a worldwide reputation of her own as a portraitist and essayist, and still driving and servicing the indefatigable Volvo. The beauty and happiness of her life are rivalled only by its length and the extent to which she is loved, respected, and admired by all.

‘All’ traffic

Next exhibit in my mini-campaign against absurd road signage.

sign

This is the sign that you see if you are travelling in the opposite direction on the very same road as the sign in the last similar post.

Where to begin?

Well, start with the obvious: ‘All’ traffic? Really? What weird idiosyncratic meaning of the word ‘all’ has the authority dreamed up here? The sign says that you can go left. So it isn’t true that all traffic has to go straight ahead – not unless ‘all’ has a special new meaning. Or is there a new and technical sense of ‘traffic’ that they have secretly invented? Is ‘traffic’ now to denote only vehicles travelling on roads that are not at some reasonably proximate distance subject to restrictions affecting motor vehicles?

Or is it – just possibly – that they cannot think, or would not think? (I find it secretly pleasing that this sign seems to confirm my suspicion about the ‘other traffic’ sign that what is really at work is an officious and slightly spooky desire on the part of the council to control free citizens’ behaviour more tightly than do the legal constraints that are in fact applicable to them.)

The added joy here is that, together with the large sign’s unnecessary complexity and inaccuracy/dishonesty, we have one of the appallingly numerous and completely unnecessary ‘traffic enforcement cameras’ warning signs that litter the city. Is anyone unaware that there are cameras all over the place? Where would you have to have been living for the last ten years to be unaware of this? Jupiter? The signs do nothing to affect driving behaviour; they bear little relation to the exact presence of cameras; and motorists don’t need, or deserve, to warned of the presence of cameras. It’s just another instance of the ugly thinking that seems to underlie so much of the hideousness:  ‘Why have just one sign on a post if you can fit another on there?’

(I can forgive the ‘Ring Road’ sign as the sort of thing that is potentially useful for visitors to the city. I am not against all signs: indeed it is a major part of my opposition to excessive, confusing, misleading, and excessively complex signage that it distracts attention from the useful ones.)