1.) At infinity, probability disappears, since things are only probable if they can be otherwise at some time.

2.) One gets not just Hamlet or all the books in the British museum, but all of them backwards. Any signal will be buried in infinite noise. All possible information would not be usable or accessible. Sure, we would account for complexity, but how would would be able to locate it in anything outside of us?

3.) One doesn’t get all outcomes if they are tied to a time constraint: to get *Hamlet *fifteen minutes into one monkey typing is possible, but it does not become possible at infinity.

4.) It is a mistake to think that because we cannot be astonished to the point of incredulity at complexity arising from chance *at infinity* we therefore have a reason not to be so astonished at the claim that it arises in a finite time, or even in any finite time.

5.) We slip into a cosmological gambler’s fallacy when we think that a longer finite time (“billions and billions” or whatever) is more *relevantly like* the infinite than, say, twelve minutes. Infinite time does not make some possible outcome necessary because it is a really big number of trials but because it does away with any time for a possible outcome to exist, and so requires that it be actual. But this is exactly what can never be removed from a finite time.

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## David Lentini said,

April 23, 2016 at 12:25 pm

I’m not sure what you mean by “at infinity”. AFAIK, there is no place called “infinity” (other than a car brand). The question, it seems to me, is that given an arbitrarily long time period (i.e., one than can continue without end), will a group of monkeys produce the desired product?

I think your point about detection is interesting. The exercise becomes like reading telemetry, as was common during the Cold War when both the US and USSR would embed key data in long continuous streams of nonsensical signals. (Of course, by design they had the means to extract the wheat from the chaff.)

Your last point seems to reflect this: Since infinity isn’t a some fixed state, it can’t be treated like an endpoint at which all probabilities, however infinitesimal, will come true. As Keynes said, “In the long run, we’re all dead.” Requiring an arbitrarily long time to prove a point is, in the end, pointless.

## David Lentini said,

April 23, 2016 at 12:26 pm

Oh! And of course, nothing will enable an event of zero probability to happen. Like, I suspect, monkeys producing Shakespeare.

## James Chastek said,

April 23, 2016 at 1:23 pm

The question, it seems to me, is that given an arbitrarily long time period (i.e., one than can continue without end), will a group of monkeys produce the desired productIf that’s the question, the answer is no. A possible outcome of a trial becomes necessary only if you posit infinite trials. This is true of coin flips too, FWIW.

I’m not sure what you mean by “at infinity”. AFAIK, there is no place called “infinity”

Agreed. But do you say this to all limit arguments or physical theories that posit an infinite past time? At any rate, your last paragraph seems to get the gist of the argument.

In a universe with an infinite past time there is nothing surprising about complexity arising from chance, since it is

necessaryfor every possible complexity to not only have arisen, but to have done so an indefinite number of times. Things become more interesting if we lose this postulate of infinity – perhaps some finite time is enough to give us a reasonable expectation of complexity from chance alone or perhaps it isn’t. We’d be just as reasonably shocked if a billion fair flips didn’t give us a heads as if ten minutes was enough time for the monkey to tap out Shakespeare.## David Lentini said,

April 23, 2016 at 3:39 pm

Agreed. But do you say this to all limit arguments or physical theories that posit an infinite past time? At any rate, your last paragraph seems to get the gist of the argument.I guess this gets to a point that I think Hilbert or Poincaré made regarding the physicality of arguments that rely on infinity. IIRC, it’s one thing to allow for an indefinite period over which something may occur at some finite probability at a point along the infinite process. It’s another thing for an event that will happen a point that converges to infinity. This is a huge flaw in the standard DSGE model of economics, as I also recall.

So, a cosmology that posits a “big bang” at some point of infinity in the past is not really physical.

## theofloinn said,

April 23, 2016 at 6:40 pm

I don’t think the big bang is supposed to have occurred an infinite amount of time in the past, but only about 14 billion years ago.

## James Chastek said,

April 23, 2016 at 8:14 pm

Sean Carroll also gives a Big Bang model with an infinite past T value. He needs it, inter alia, to give a Naturalist explanation of the low entropy of the initial state.

## David Lentini said,

April 23, 2016 at 8:25 pm

Sounds like he need God. 😉

## David Lentini said,

April 23, 2016 at 7:58 pm

I don’t think the big bang is supposed to have occurred an infinite amount of time in the past, but only about 14 billion years ago.Yes. I wasn’t referring to The Big Bang Theory, but a hypothetical one which is assumed to have occurred in some “infinitely” distant past. The oscillating universe theory might fit a description like that.

## dwightk said,

April 24, 2016 at 6:49 am

This seems related to Borges’ Library of Babel. Which, if I understand it, is actually finite. Every possible configuration of 400 page book is in the library (with a limited set of characters). So you could go in there and find the complete works of Shakespere (if one play needs more than 400 pages you’d find it in multiple volumes). The signal is buried in a great amount of noise but it is a finite amount of noise. The works would be scattered all over the library. Also you may have the first 99% of a play with a page of nonsense at the end, or at each page in the middle.