Infinitely many monkeys?
Warning: what follows may have implications with regard to how man evolved from monkeys...creationists are particularly invited to comment...
A common phrase goes something like "infinitely many monkeys, typing away at infinitely many typewriters, must eventually produce the works of Shakespeare." Now of course this has no rigorous meaning; rather it is intended to express the notion that "some successes are purely by accident," or perhaps "put enough heads together, and something interesting is bound to happen."
But let's take it rigorously, in that we assume we have the set of all monkeys M, that it is infinite, and that they all do nothing but type on their typewriters. Further assume that for any infinite subset S of M, there exists a monkey m in S such that m types exactly the works of Shakespeare.
Is the set G (of "genius" monkeys) that finish the works of Shakespeare finite or infinite? Suppose that it were finite. Then the complement of this set must be an infinite set of monkeys, none of whom have amounted to anything as playwrights. Contradiction. So G is infinite.
Is the complement of G, call it D (consisting only of "dumb" monkeys), finite or infinite? Suppose that it were infinite...oops, again a contradiction. So D is finite.
So M consists of an infinitude of genius monkeys, and finitely many dumb monkeys. Seems reasonable to me, in that it agrees with experimental data thus far.
Or, you could go with the "quantum computing" philosophy: the probability of hitting the exact works of Shakepeare is on the order of (1/26)^(a lot), so it would take at least every atom in the known universe, making a bajillion calculations per second, from the dawn of time, to achieve even a tiny likelihood that this would happen.
In any event, I like to tell this story when people utter this phrase, because "infinite" has a precise mathematical meaning, and its misuse can lead to nonsense. The same is true of the notion of "probability," which only applies to a model within a model, and not a model within reality. I think these distinctions are relevant to many discussions on this forum, and I wish more philosophy folk were aware of this.
Anyway, I apologize for all the typing. I'm starting to get in to this whole "internet forum" thing. This probably signifies the beginning of the end of its popularity.
(PS...As far as I know, I made this crap up, but maybe its one of those ghost memories that you don't remember is a memory when you remember it. I'm sure one of those Barnes & Noble's books on the "mysteries of infinity" has something analogous, although I can't remember reading any of them, so please don't accuse me of (intentional) plagiarism [img]/images/graemlins/smile.gif[/img])
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