Wasn’t the saying an infinite number of monkeys on an infinite number of typewriters? If so then they’d write Hamlet and indeed every other book written or ever will be written in however long it would conceivably take to type them out if you were copying them.
I don’t really know how this myth? paradox is supposed to work? I know infinity isn’t a number but a concept and in theory I understand what it’s trying to say, but if I have an infinite amount of scrap yards and infinite amount of tornadoes, they can go on forever, but they’ll never assemble a Boing 747.
An infinite number of monkeys typing randomly on an infinite number of typewriters, so long as the writing is truly random, will eventually write every novel. Once you factor in the infinite number of monkeys, every novel in existence will not only be written, it will be written an infinite number of times.
It’s like saying if you had a random number generator and gave it an infinite amount of time generating 16 numbers at a time, it would eventually generate every bank card number ever an infinite number of times. Give that task to an infinite number of random number generators and they will generate every bank card number an infinite number of times instantaneously.
Come to think of it, if the tornado throws around junk completely randomly, and provided there’s enough material in every junkyard to assemble a plane, the tornado will eventually assemble it. That’s the power of infinity and randomness.
Once you factor in the infinite number of monkeys, every novel in existence will not only be written, it will be written an infinite number of times.
You don’t need an infinite number of monkeys to ensure that. The cardinality of an infinite collection of 2-tuples (monkey, char) is the same as the cardinality of an infinite sequence of characters, just as the cardinality of the rational numbers is the same as the cardinality of the integers.
And in a countably infinite sequence of uniformly random characters, there is no assurance that any particular finite sequence will occur only a finite number of times.
Not the same the monkeys have all the capabilities and tools to cohesively combine letters words and white space. A tornado cannot weld and program controllers and solder. But a monkey can type randomly even wacking randomly. The idea is that given an infinite truly random output of text by the nature of infinity the text of Shakespeare will be outputted in its entirety eventually
The idea is that given an infinite truly random output of text by the nature of infinity the text of Shakespeare will be outputted in its entirety eventually
Only for a certain kind of randomness. For example, it’s possible to construct a random process that at each step emits a uniformly distributed character, but which also includes a filter that blocks the emission of the string “Falstaff” if it occurs. Such a process cannot ever produce the complete works of Shakespeare, since the complete works include that string, though it will still contain (for example) every lost work of Aristotle, as well as an infinite number of false and corrupted versions of those works.
But yeah, an unconstrained uniform-random-distributed countably infinite sequence of printable English characters and whitespace cannot be proven to not contain the complete works of Shakespeare, or any other finite sequence. I believe it’s also impossible to exclude any countably infinite sequence, but I might be wrong on that part, since my mathematics education happened a very long time ago.
I guess that was kinda what I was trying to convey in the truly random part. Truly random in which you have no idea what character will be next, no filter. In that case yes which I believe is what most people think of when they think of random
The problem here is the implied entropic outcomes of each system.
If the monkeys were replaced with true-random number generators, then you’d eventually get shakespear. But they aren’t RNG engines, and they aren’t quantumly random.
Instead, the monkeys have a large-ish but very finite number of logical branches that they can take in their decision-making processes, with slight variations within a fixed genetic scope. They will slam away at the typewriters in a very specific monkey kind of way.
At the end of the thought experiment, you will end up with an infinite amount of monkey-gibberish and a slightly smaller infinite amount of soiled or destroyed typewriters.
If the monkeys’ probability distribution function can be transformed to a uniform distribution by a continuous function, the outcomes are equivalent enough for this exercise. (There are probably some discontinous functions that’d also work). So, unless there’s some genetic weirdness in monkeys that prevents their ever hitting certain keys, they’re adequate RNG engines. But at that point, you’re really tweaking the assumptions based on how realistically you think monkeys are portrayed in the thought experiment.
And I don’t believe “quantumly random” is a necessary condition here.
Wasn’t the saying an infinite number of monkeys on an infinite number of typewriters? If so then they’d write Hamlet and indeed every other book written or ever will be written in however long it would conceivably take to type them out if you were copying them.
I don’t really know how this myth? paradox is supposed to work? I know infinity isn’t a number but a concept and in theory I understand what it’s trying to say, but if I have an infinite amount of scrap yards and infinite amount of tornadoes, they can go on forever, but they’ll never assemble a Boing 747.
An infinite number of monkeys typing randomly on an infinite number of typewriters, so long as the writing is truly random, will eventually write every novel. Once you factor in the infinite number of monkeys, every novel in existence will not only be written, it will be written an infinite number of times.
It’s like saying if you had a random number generator and gave it an infinite amount of time generating 16 numbers at a time, it would eventually generate every bank card number ever an infinite number of times. Give that task to an infinite number of random number generators and they will generate every bank card number an infinite number of times instantaneously.
Come to think of it, if the tornado throws around junk completely randomly, and provided there’s enough material in every junkyard to assemble a plane, the tornado will eventually assemble it. That’s the power of infinity and randomness.
You don’t need an infinite number of monkeys to ensure that. The cardinality of an infinite collection of 2-tuples (monkey, char) is the same as the cardinality of an infinite sequence of characters, just as the cardinality of the rational numbers is the same as the cardinality of the integers.
And in a countably infinite sequence of uniformly random characters, there is no assurance that any particular finite sequence will occur only a finite number of times.
🤓
Not the same the monkeys have all the capabilities and tools to cohesively combine letters words and white space. A tornado cannot weld and program controllers and solder. But a monkey can type randomly even wacking randomly. The idea is that given an infinite truly random output of text by the nature of infinity the text of Shakespeare will be outputted in its entirety eventually
Only for a certain kind of randomness. For example, it’s possible to construct a random process that at each step emits a uniformly distributed character, but which also includes a filter that blocks the emission of the string “Falstaff” if it occurs. Such a process cannot ever produce the complete works of Shakespeare, since the complete works include that string, though it will still contain (for example) every lost work of Aristotle, as well as an infinite number of false and corrupted versions of those works.
But yeah, an unconstrained uniform-random-distributed countably infinite sequence of printable English characters and whitespace cannot be proven to not contain the complete works of Shakespeare, or any other finite sequence. I believe it’s also impossible to exclude any countably infinite sequence, but I might be wrong on that part, since my mathematics education happened a very long time ago.
I guess that was kinda what I was trying to convey in the truly random part. Truly random in which you have no idea what character will be next, no filter. In that case yes which I believe is what most people think of when they think of random
You know it? That’s nice. A lot of people think they know a lot of things that aren’t really true.
Now prove it.
Same way in the infinite random non repeating numbers of Pi is the binary of a 4k resolution photo of Betty White nude holding a snake on a tiger.
It’s random forever and eventually the 1s and 0s fall into place. The problem is the monkeys repeating themselves.
The problem here is the implied entropic outcomes of each system.
If the monkeys were replaced with true-random number generators, then you’d eventually get shakespear. But they aren’t RNG engines, and they aren’t quantumly random.
Instead, the monkeys have a large-ish but very finite number of logical branches that they can take in their decision-making processes, with slight variations within a fixed genetic scope. They will slam away at the typewriters in a very specific monkey kind of way.
At the end of the thought experiment, you will end up with an infinite amount of monkey-gibberish and a slightly smaller infinite amount of soiled or destroyed typewriters.
If the monkeys’ probability distribution function can be transformed to a uniform distribution by a continuous function, the outcomes are equivalent enough for this exercise. (There are probably some discontinous functions that’d also work). So, unless there’s some genetic weirdness in monkeys that prevents their ever hitting certain keys, they’re adequate RNG engines. But at that point, you’re really tweaking the assumptions based on how realistically you think monkeys are portrayed in the thought experiment.
And I don’t believe “quantumly random” is a necessary condition here.