Earth is Probably Unique

[pErvinalia]

It does require infinite time as that is its specific formulation.

[JimC]

It does require infinite time as that is its specific formulation.

Incorrect. Someone may say that infinite time will guarantee the production of all of Shakespeare's works by some random process, but that does not mean that it would not occur given a sufficiently vast but finite time.

Truly requiring infinite time is another way of saying it would never be completed...

[pErvinalia]

https://en.m.wikipedia.org/wiki/Infinite_monkey_theorem

[JimC]

However, an infinite length of time implies no end whatsoever. If, after the entire age of the Universe, multiplied by a googleplex, we eventually have Shakespeare's works, we haven't reached infinite time.

Still no complete works?

Repeat the previous process a googleplex more times...

Rinse and repeat. When the works do emerge, it will still be after a finite length of time...

[pErvinalia]

But MM's point is that the formulation of the thought experiment is based on infinite time, not finite time.

Addressing your specific point, there is a non-zero chance that the monkey may never output Shakespeare. So there is no guarantee that it will happen in finite time.

[JimC]

But MM's point is that the formulation of the thought experiment is based on infinite time, not finite time.

Addressing your specific point, there is a non-zero chance that the monkey may never output Shakespeare. So there is no guarantee that it will happen in finite time.

I read your link, but I have a philosophical disagreement with it.

After some vast amount of time, the monkeys still haven't cut the mustard.

Fine.

Multiply that time by any vast number, and check again.

Repeat as necessary.

You can never decide that "these fucking monkeys can't do it", because you can always give them x amount more of time.

The moment they have accomplished the task, it, by necessity, has occurred in finite time. So, if they actually achieve it, it has not taken infinite time...

[pErvinalia]

But if their output is truly random, then we have to allow that it is possible that every combination of letters could occur. If every possible combination of letters could occur, that set would have to contain at least one combination that doesn't equal the works of Shakespeare. And it's easy to imagine one such combination of letters: the same letter pressed repeatedly (or thereabouts). The probability of this is probably minutely small, but it's possible nonetheless.

[JimC]

But if their output is truly random, then we have to allow that it is possible that every combination of letters could occur. If every possible combination of letters could occur, that set would have to contain at least one combination that doesn't equal the works of Shakespeare. And it's easy to imagine one such combination of letters: the same letter pressed repeatedly (or thereabouts). The probability of this is probably minutely small, but it's possible nonetheless.

Sure. That string which is not the works of Shakespeare occurs.

In finite time.

Then, the tireless monkeys start again...

There will not be a single moment in time where you can say "OK, that's it, it can't be done". You keep going, of course; when it finally happens, it will be in finite time.

[rainbow]

But if their output is truly random, then we have to allow that it is possible that every combination of letters could occur. If every possible combination of letters could occur, that set would have to contain at least one combination that doesn't equal the works of Shakespeare. And it's easy to imagine one such combination of letters: the same letter pressed repeatedly (or thereabouts). The probability of this is probably minutely small, but it's possible nonetheless.

Sure. That string which is not the works of Shakespeare occurs.

In finite time.

Then, the tireless monkeys start again...

There will not be a single moment in time where you can say "OK, that's it, it can't be done". You keep going, of course; when it finally happens, it will be in finite time.

Indeed. Now we can use virtual monkeys to speed up the process. We can in finite time with a large number of processors working at speeds not yet possible, on all the habitable planets in the universe, produce a sonnet, or two.
...but that might mean that we'd need life on other planets - which brings us back to topic on hand.

[pErvinalia]

But if their output is truly random, then we have to allow that it is possible that every combination of letters could occur. If every possible combination of letters could occur, that set would have to contain at least one combination that doesn't equal the works of Shakespeare. And it's easy to imagine one such combination of letters: the same letter pressed repeatedly (or thereabouts). The probability of this is probably minutely small, but it's possible nonetheless.

Sure. That string which is not the works of Shakespeare occurs.

In finite time.

Then, the tireless monkeys start again...

There will not be a single moment in time where you can say "OK, that's it, it can't be done". You keep going, of course; when it finally happens, it will be in finite time.

You actually can say that, as the monkey could press the same key, or thereabouts, infinity times in a row.

[JimC]

But if their output is truly random, then we have to allow that it is possible that every combination of letters could occur. If every possible combination of letters could occur, that set would have to contain at least one combination that doesn't equal the works of Shakespeare. And it's easy to imagine one such combination of letters: the same letter pressed repeatedly (or thereabouts). The probability of this is probably minutely small, but it's possible nonetheless.

Sure. That string which is not the works of Shakespeare occurs.

In finite time.

Then, the tireless monkeys start again...

There will not be a single moment in time where you can say "OK, that's it, it can't be done". You keep going, of course; when it finally happens, it will be in finite time.

You actually can say that, as the monkey could press the same key, or thereabouts, infinity times in a row.

And exactly when would you decide that he had?

"You" saying that is an event in time. Once you've said it, there is still an infinity of years for the poor bloody monkey to keep typing...

Deep down, this is the conundrum at the heart of how we regard infinity. As an abstract mathematical proposition, it can perhaps be "actualised"

When applied to a measurement in the actual universe, i.e. time, it cannot.

[pErvinalia]

But if their output is truly random, then we have to allow that it is possible that every combination of letters could occur. If every possible combination of letters could occur, that set would have to contain at least one combination that doesn't equal the works of Shakespeare. And it's easy to imagine one such combination of letters: the same letter pressed repeatedly (or thereabouts). The probability of this is probably minutely small, but it's possible nonetheless.

Sure. That string which is not the works of Shakespeare occurs.

In finite time.

Then, the tireless monkeys start again...

There will not be a single moment in time where you can say "OK, that's it, it can't be done". You keep going, of course; when it finally happens, it will be in finite time.

You actually can say that, as the monkey could press the same key, or thereabouts, infinity times in a row.

And exactly when would you decide that he had?

At every point in time until infinity (i.e. forever).

"You" saying that is an event in time. Once you've said it, there is still an infinity of years for the poor bloody monkey to keep typing...

And the same odds that he will repeat the same letter over and over will still apply. Absolutely nothing has changed.

[pErvinalia]

Deep down, this is the conundrum at the heart of how we regard infinity. As an abstract mathematical proposition, it can perhaps be "actualised"

When applied to a measurement in the actual universe, i.e. time, it cannot.

I can't really comment on that but it is probably transferable to the probability issue. The probability of [non-Shakespeare] will asymptote towards zero. What that actually means for the problem I can't really answer.

In any case, it's not relevant to the thread subject, as MM said, because the universe isn't infinitely old.

[JimC]

Deep down, this is the conundrum at the heart of how we regard infinity. As an abstract mathematical proposition, it can perhaps be "actualised"

When applied to a measurement in the actual universe, i.e. time, it cannot.

I can't really comment on that but it is probably transferable to the probability issue. The probability of [non-Shakespeare] will asymptote towards zero. What that actually means for the problem I can't really answer.

In any case, it's not relevant to the thread subject, as MM said, because the universe isn't infinitely old.

Your last point is very true in the real world, which does not, and cannot, contain actual infinities of any sort...

And the "asymptote to zero" idea is vital; by definition, it never actually reaches zero, so you cannot ever say that it is impossible (probability = 0) for the monkeys to do the job...

Whether that means it is guaranteed to be possible is another matter, perhaps...

[pErvinalia]

No, you've got that wrong. You should say that it is incorrect to say that the monkeys can't do the problem (i.e. probability = 0). And the corollary is that you can't guarantee that it is possible they will.