In most things there is a tiny group of productive paths and entropy rules the infinite remainder.
The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type a given text, such as the complete works of William Shakespeare.
At first this seems true if not obviously so. Would not a piano player eventually reproduce any song? After all there are only so many notes. Yet we mustn't be confused by a challenge where every move excludes a possibility... increasing the odds of success with each key stroke...with a challenge containing unknowable amounts of unknowables where each blind step increases the odds of failure while excluding nothing. Entropy compounds error through interference patterns inevitably forming a Production Horizon beyond which all significant information is trapped. Such is the nature of low order compared to high.
Production Horizon is the point of no return where entropy overwhelms production in the absence of capitalization by intelligent agency. It separates the low order and the high order. Randomness cannot penetrate its mysteries.
This classic theorem is an exercise in vanity and wishful assumption. As with so many scientific fallacies it fails to recognize the limitations of automation and only suggestively implies a tautology thwarted by practical limits of time. Yet it arrogantly assumes that given an impossibly long time, all combinations of letters would appear several times. That may not be true. Billions of years of entropic evidence out in the universe would imply that until an intelligent agent(eg. lifeforms) arrives we won't get anything more than primitive objects like planets and stars and raw forces.
Wind chimes won't randomly play Smoke On The Water and though a monkey might write its own poems, Shakespeare was and will forever be the only Shakespeare.
I just don't agree that information works the way the theorem suggests. There's a difference between low order data and high order information and there are an infinite number of configurations between them. After a very primitive point, a brute force text string fails to achieve coherency in ever greater degrees. Entropy dictates that there are always more failing outcomes than succeeding. Only capitalization can overcome the production horizon. Therefore it's clearly not possible for a high order product like Shakespeare to be produced through any automation process even one with all the time in the world.
Infinity is a concept describing something without any bound or larger than any natural number.
My claim is that NOT all combinations of numbers are possible through automation of ANY kind including brute force random number generation. While others claim that it can because it feels mathematically symmetrical I assert that some combinations of numbers can only be achieved through actual causality because a Production Horizon separates low order from high order information. Such is the case with any original work such as Shakespeare.
It doesn't scale that if a problem can be solved in 1 minute by one processor that it can be solved in 1/10 the time if 10 processors are used. There's an order of operations that must occur and this will have opportunity costs in the real world and will require intelligent capitalism to overcome to any degree at all. Works like Shakespeare are produced only through real-time capitalization over long periods of unique circumstances.
Experience cannot be reproduced any other way, not even by accident given infinite tries. One thing is not another thing. It's always something less or more. Some things simply can't follow other things and some things can ONLY follow other things. High order requires something more than chaos even given infinity which at best would generate something that appears very similar (isolated from time and place) without ever achieving a true match.
Time/causality exists for a reason and that reason is to generate the numbers that chaos can't.
Capitalization, informed by ever growing scientific knowledge, is the method through which we over come the "Production Horizon" to create original things. A conversation that produces the anthropic elements of value which fuels further conversation. The basis of all life. Capacity plus opportunity equals production. I say that some numbers (all information can be represented as a number of some size) can only be generated through capital production. That time/causality is the mechanism through which life capitalizes on opportunity to generates the numbers that randomness fails to produce given any amount of time.
As such only "possible" combinations can appear and what's possible depends intimately on opportunities that once missed may never come around again. Just as accelerating space eventually draws currently visible galaxies beyond a causality horizon...spending time on one thing might preclude another forever denying our monkey access to what it needs to complete the task. Any patterns of a higher order require very specific circumstances that simply won't exist for a monkey or anyone else not part of the flow of actual events. There can be only one bard in his time and place.
Capitalism is the art of capitalizing on opportunity by levering capacity to produce more than one consumes.
No, even in an infinite universe it wouldn't be possible for a monkey to type Shakespeare. "Can happen" doesn't mean anything will happen. A monkey is not Shakespeare and won't randomly type out Shakespeare's thoughts which were formed the only way they can be...through actual experience of real world events in a given time and place which will never happen again.
Original works are NOT inevitable. A monkey will loop through endless repetition of diverse combinations of gibberish accomplishing nothing more than the occasional appearance of coherence. A word or even a sentence emerging here and there almost suggesting a proof of sorts for the theorem...but some things just don't scale and when each character becomes an infinite loop there's just no end in sight.
We mustn't mistake sound for music. Intelligence for happenstance. Sound can happen by accident but music cannot. Even given infinity. The difference is the element so critical to the word "possible" in "all possible outcomes". Order of operations. Cultivation of value through capacity and opportunity. In other words... capital production.
The assumption that even infinite random attempts could produce Shakespeare is analogous to a Big Bang skipping all of cosmic evolution to produce a 10 hour commentary tape for a baseball game that never took place. Impossible. Not the same as brute forcing a password at all. There's no way to check for validity. Each character typed can't even be compared to match the original work without violating the randomness of it all. Then infinite attempts eventually go on infinitely, too. So add infinite moneys working simultaneously...still doesn't help.
Infinity times zero is still zero. It reminds me of those who presume AI is inevitable. This kind of faith based thinking can trap us in a fallacy of dilution and confusion. When someone claims that a monkey has something greater than zero chance to write Shakespeare on its first try because there's no physical law that would prevent it from typing a key... they ignore a much larger part of the equation than the typing. Namely the authoring. It's NOT within known physics for a monkey or even another human to "accidentally" produce Shakespeare, even if given infinite time.
Even considering the character count is cheating, but it won't help the monkey at all, just the scientists who groom results to fit the desired outcome. One can not just "invent" information from nothingness. It must be developed incrementally in a feedback system which gradually refines data into something original and valuable. If you think you have infinity to make something happen through pure chance then infinity plus one is also there working to prevent the effort. Entropy reins supreme in a universe without creative production. Only the most simple and repetitive patterns will emerge without original thought to overcome the drive for chaos. Over and over again.
A tale told by an idiot, full of sound and fury, signifying nothing. - Macbeth
Every keystroke, absent the authentic cultivation of value(the living and authoring), faces an impossible task. As impossible as predicting the future without any measurements. As impossible as knowing something without any causal connection to a source for the information. Only an authentic process of capital production can ever overcome a "Production Horizon". Supernatural powers don't exist. Magic isn't real. Once again let us remind ourselves as we look out into the vast universe and see no high order beyond that which we ourselves have made through productive capitalization. Time alone cannot produce anything high order without the procession of experience to constrain it.