I have two problems with the concept (even distribution of random letters producing a work of Shakespeare): one mathematical, one philosophical.
Mathematical
IANAM, but…
If the distribution of letters in the random generator is vastly different from that of the work, then the chance of success on any given try is going to be virtually indistinguishable from zero. Whether this is expressed in terms of probablility (i.e., 1 x 10[sup]-200[/sup] chance of success on a try) or in time required to get a success if a try takes x amount of time (i.e., 1 x 10[sup]200[/sup] years on average to success), I would say we are no longer dealing with numbers and units that are truly related to those we use on a daily basis. In other words, 1 x 10[sup]200[/sup] is indistinguishable from infinity. There are not that many of anything in the universe.
I also reject the notion of “infinite computers.” This is obviously a contradiction. There could never be an infinite number of a physical object. You can say it’s just speculation, but I could also conceive of a griffin swooping from the sky with a scroll of Hamlet in it’s beak. I don’t see how this is any different from conceiving of an infinite number of computers. Hence, the claim that an infinite number of computers really would produce a text instantly is nothing more than a fantasy.
Philosophical
I have a more interesting (I hope) objection to the concept under discussion, as well as the idea that any possible course of events would occur granted an infinite number of universes (i.e., in a different universe a cat could scratch a Shakespearean sonnet on the sidewalk with its claws, inasmuch as we can see no logical contradiction in this).
To wit, things like texts are products of intent that cannot come into being without a suitable cause. They are only superficially conceiveable as products of random generation, and a random number generator is incapable of reproducing them.
Shakespeare’s works are the product of a particular brain at a particular time and place. They could not have been written in the year 1350, nor could they be written today. In fact, I’ve read Hamlet and, even if I had a billion years I would not be able correctly to transcribe it. Yet my chances of being even slightly correct are much better than that of a machine.
Just cause something can’t exist physically doesn’t mean we can’t talk sensibly about it. But, since you do have that objection, let’s consider one computer operating for an unbounded (not infinite) amount of time. At any given trial, there’s a non-zero probability that it will produce the script of Hamlet. This implies that, given enough trials, the probability that the computer never produces Hamlet is as close to zero as you’d like.
And from a theoretical standpoint, 10[sup]200[/sup] is so small that confusing it with infinity is laughable.
In addition to my points above, it’s certainly reasonable to expect a machine to generate Hamlet. I can easily write a program that will cycle through all texts the same length as Hamlet, and it will eventually output the script.
Mathematical: Mathematicians don’t worry about insignificant stuff like whether a real physical object could be designed to do this. 1^200 may be ‘indistinguishable’ from infinity in the practical sense that you couldn’t count by ones to reach that number, even with a fast computer doing the counting for the entire history of the universe. But they’re very different in the theoretical mathematical sense.
Philosophical: Really, your objections work together as a philosophical objection. While it’s ‘possible’ in the theoretical mathematical sense for the random characters to spell out the text of Hamlet, it’s not ‘physically’ possible in any reasonable sense of the word. If a computer, or a monkey, or an upright ape, spits out the text of Hamlet; or even, for that matter, “To be, or not to be; that is the question,” that is conclusive evidence that something non-random is going on.
The mathematician can say, “Yes, but if we really had an infinite number of monkeys …” To which we can respond, “Any sentence that starts that way is true, because a conditional with a false antecedent is true, and it’s not actually possible to have an infinite number of monkeys.” Which is a good thing, unless you like the idea of having an infinite amount of monkey poop thrown at you.
I have no doubt that the “concept” is valid in a logical sense. I am simply claiming that it has no relevence whatsoever–not even a superficial relevence–in a physical system.
And of course I wasn’t claiming that 10[sup]200[/sup] really is infinity, or is “close” to infinity. I’m saying that a number this large can have no relevence within our physical system.
Well, I think it’s like this. A writer will write something for a popular magazine or whatever, and say something like, “If the number of universes is infinite, then right now there is a version of you doing [interesting or absurd or violent act].”
The million monkeys typing out Shakespeare is one of these.
The examples are intended to compel an imaginary image and to code that with the “real” marker, thereby instilling in one a particular emotional reaction. It really would happen!
I say “BS” to that. The concept is logically consistent but completely incompatible with the physical universe. If writers want to talk about systems that are logically consistent, that’s fine; but I will oppose their encoding an inappropriate emotional context into the presentation.
So your objection is not a mathematical one. Calling it mathematical is what’s caused people to jump on you. When you start talking about what’s physically possible, or “relevance to a physical system,” you leave the mathematical part of the question behind.
In math, there’s a huge difference between zero and “virtually indistinguishable from zero”; it’s what Calculus is based on.
It’s going to be “virtually” indistinguishable, in practice, even if the letter distribution matches perfectly. But not mathematically indistinguishable.
Your argument is not really a mathematical one at all. Mathematically speaking, 10[sup]200[/sup] is not indistinguishable from infinity. It is as different from infinity as ten is. The fact that extremely improbable events are effectively impossible — on human time-scales, or even the universe’s time-scales — doesn’t make them truly, strictly impossible. Mathematics clearly distinguishes these two cases. It just means that the universe, vast as it is, is still too small and short-lived to churn out every conceivable physical event before it goes through its Heat Death.
(… At which point the typing monkey experiment must sadly end, since there’ll be no more starlight for growing the bananas.)
It is capable of reproducing them, it’s just extremely extremely unlikely to produce Hamlet, Prince of Denmark under any circumstances we’ll ever find ourselves in. Humans don’t live long enough. Stars and planets don’t live long enough. The universe as a whole probably won’t live long enough. There’s no limit to how improbable events can become and yet still remain possible, because there’s no limit to how tiny a positive number can get and still be non-zero.
I can certainly agree that human authors are far more efficient at writing literature — at least enjoyable literature — than random character generators are.
Would you say the same of, say, portraits? If you saw a portrait of Lenin, say, surely that would be evidence of some artist’s intent to create that portrait? Not neccessarily. The Bad Astronomer, for instance, once saw the face of Lenin in a water splotch on his shower curtain. Was it unlikely for the water to form such a vivid image? You betcha. Did it actually happen? Yup. Does this mean that the water had any intent to form a protait? Nope.
There is once sense in which one might say that it’s impossible for a team of monkeys or a systematic computer program to create pieces of literature: It would take at least as much work to find the meaningful pieces of literature in amongst the noise, as it would take to create the literature in the first place. If you didn’t know Hamlet to begin with, and I handed you a big stack (a very big stack) of all of the monkeys’ output, you wouldn’t be able to find the Hamlet in it.
Really? Obviously, 10[sup]200[/sup] is a number. Infinity is not a number. So they are different. But my point is that 10[sup]200[/sup] is only being used as a number superficially in talking about the “concept” (supposing it were the result of probability calculations).
Perhaps this is, indeed, not strictly a mathematical argument. I suppose what I really wanted to say is that mathmatics was being applied improperly.
My response to this is that there is no way to test our mathematical models for highly improbable events, so we don’t know if they really correspond to the physical universe. Further, if we were actually to get a “hit” with a random number generator, we would assume fraud before we actually believed that Hamlet had been produced.
And some events are qualitatively impossible. It is hard to know which is which, but events of infintessimal probability are as good as impossible, I would say, for the reasons I’ve already stated.
This is interesting, but it’s a completely different thing. First, it’s just a fact that all kinds of face-like patterns can appear in clouds, the grain of wood, etc. Er, in photographs of Mars, even. Nor is this statistically unlikely, since the elements of a face are none too complex.
Second, since there are a nigh-infinite number of events going on at any given time, it is outright probable that something odd will happen in a given period of time. I once bit off half a piece of candy and chucked at a metal garbage can. Later on, when I came back into the room, I was surprised to find it sitting on the thin rim of the can! I bet if you pitched the candy at the rim trying to make it land there on purpose, you couldn’t do so in a million tries.
The candy on the rim and the Lenin water splotch are interesting and unusual after the fact–but no, they are not signs of intent.
Third, in the case of the water splotch, the area in which order must be preserved is rather flexible: the splotch could have been larger or smaller, and it only had to come together in this one smallish area. In contrast, the complete Hamlet text would have to be orderly throughout. Whereas, in any large chunk of randomly generated letters, I’m sure you can pick out a few small words.
Yes, this is also a good point. And there would have to be the intent to find it. I suppose another argument is that in a theoretical space in which all possible permutations of specified elements are realized (pretty much the Platonic forms, eh?), that intent to find that permutation is equivalent to creation.
The Yahoo! link in that thread (the first thread I ever initiated here at the SD, so it’s very dear to my heart) isn’t active anymore, but the original AP story can still be accessed here.
1^200 may be ‘indistinguishable’ from infinity in the practical sense that you couldn’t count by ones to reach that number, even with a fast computer doing the counting for the entire history of the universe. But they’re very different in the theoretical mathematical sense.