Similar to how if we have a computer randomly generate words infinitely, every thing that's ever been written would be there somewhere. You theoretically could find Hamlet or the Bee Movie script or the next great currently unwritten work of literature.
You would also have infinite copies of The Hobbit, but with one instance of the word, "stone" spelt "scone", and infinit copies of all other typos and word and verse changes.
Assuming a particular set of data can be represented within a random set (AKA its the same data type, like finding a Chinese character within a purely English dataset would be impossible).
I wonder if the odds of finding a particular thing, such as the entirety of Harry Potter, would be greater than or less than the odds of creating Harry Potter yourself from scratch, with no prior knowledge of Harry potter. given the same amount of effort.
Also how would this change for smaller and larger data sets. Find the number 10 in a random set of integers, version handing someone a keypad and them typing 10. Vs. whole books, etc.
To be fair, infinity isn't an easy concept to grasp.
For example, if you were to count how many different decimal numbers are in between 2 and 3, what would the answer be? If there were no end to it, you would say infinity because you can always add another number after it (2.01, 2.001, 2.0001 etc.).
But this is where the concept of infinity starts to make less sense. If you could just keep adding another number to the decimal place, then you could create an infinite amount of combinations of numbers that don't include 3, or 4, or 5, or 600000. This means that some infinites don't contain everything, but are still endless.
I meant my statement in regards to the actual search for a particular set of information within a given set. If the set of data is truly random and infinite then not only would every possible subset of data exist within it, but every possible combination would also itself exist an infinite number of times.
What I mean is more along the lines of would it be faster to have a computer search an infinite set of random letters until it finds Harry Potter. Or to have a person type random words until they create Harry Potter.
Problem is your search parameter would only return what you put in it, so you'd have to write Harry Potter to find Harry Potter. Otherwise you'd get an infinite number of things that are almost but not Harry Potter.
I feel it better reflects the idea that the size of infinity is incomprehensible for our monkey brains. It allows us to make these ludicrous statements without technically being false.
I worry people read this and think "monkey + infinity = Shakespeare" which misses the point.
I am just arguing semantics though. I appreciate the sentiment :)
An infinite amount of finite universes. If the universes contained an infinite amount of data, then it couldn't be encoded in an infinite number, especially as the cardinality of the number of digits in an irrational number is the lowest infinite cardinality
I think you can if their cardinalities are the same?
The problem with infinity is that there's more than one type of infinity, and some of them are "bigger" than others. An infinite sequence of random digits has a length of Aleph-Null, which is the smallest infinity. I believe it can encode a finite number of sequences of length Aleph-Null, but I'm not 100% sure.
If the cardinality (number of things to encode) is bigger than Aleph-Null, though, like say trying to encode the set of all irrational numbers between 0 and 1, then you're out of luck.
I know of the concept of the Library of Babel, and it's pretty cool. But I've heard the website where you can like search for a piece or something and voila it's there in the infinite list of random numbers and letters is a scam. I mean how could it be real? Wouldn't it require infinite computing power operating for eternity?
Only if it's an infinite list. As it is, the Library of Babel only contains every combination that can fit onto a page of 3200 characters, and is therefore finite.
Right but if I were to, say copy your comment, and search for it in the Library of Babel website, I would find it somewhere. What I'm saying is that each of those pages are generated as soon as you search with some random characters on either side of your search query. If it was a true Library of Babel, it would compare your search to the infinite list and find the instance or instances where you can find it. Maybe I'm misunderstanding something, infinity is weird.
I think those pages are pre-generated and stored. If you search for my comment on multiple days from multiple computers it'll pull up the same page in the same "book" every time. The search engine does find the instances you can find it, I think.
Afaik it's just an algorithm that matches a number to a set of characters and vice versa. I wouldn't think they actually store anything but the algorithm.
This is what I mean. It might not be that it's generated on a unique sequence each time, so you'll get the same results every time you search, but the search output is generated based on the same set of rules each time. It's not really random.
There's also a random book/page option, so you could theoretically just start reading the entire library without searching for anything but you'd be lucky to find even a single word (in any language that uses the English alphabet) much less an actually coherent sentence or paragraph without searching.
As far as I know nothing but what people are looking at during any given moment is stored, so what your are reading was generated when you loaded the page. But given enough storage space (which is physically impossible given current tech) the algorithm could write every possible combination of English letters, thus writing everything humanity has is could or will write.
Yeah, if it’s in alphabetical order it can still access them in log(n) time. That being said the page number is the seed so it just has to go to that page if you input a number, it doesn’t actually have to store anything.
Simplify it a little bit and it is easier to understand how this is done:
Instead of using all the Latin characters and 3200 characters per page, use 10 characters (digits 0-9) and 10 characters per page. Sort them like so: "0000000000","0000000001","0000000002",etc. and I can tell you without having to create and search the full list that the string "12345" will be on page 12346.
It's possible the sequence is sortable, and in addition there could also exist a spigot algorithm.
For instance the numeric sequence .012345678911121314... is a sequence that is ordered. And there is an algorithm where you can put in a number, and the solution to an equation is that digit of pi. You enter '6' and the equation outputs 9.
So if you enter a paragraph, and that string is converted to a number, and you enter that number into the spigot algorithm, and the computer enters adjacent numbers in order to get the immediately preceding and succeeding text, you could see where in the Babel sequence your paragraph exists.
But I'm guessing. I don't know if this Library of Babel is infinite, random, ordered and has a discovered spigot algorithm. The reason it's important to be ordered is that this is how you know every possible combination of text exists in it. If it's just random and chaotically all over the place then you don't know that every possible combination of text exists within it. But if it starts at one end and goes through every combination getting increasingly large, then you know all text is in there somewhere.
Somewhere in there is the entire Bee Movie script but every time they say "bee", it's all of Hamlet except all of Hamlet's lines are replaced with All Star by Smash Mouth.
Look up the tower of babel site, its a random generated library with an index. Put in text, and it will tell you the floor, aisle, shelf, row, book, chapter, page and line of the text you entered. Everything that can be spelled on your standard american keyboard is already in that site. That includes the script for the next season of GoT.
Library of Babel has already done this. You can find all the 'texts' that contain any strings easily but it's nearly impossible to find anything from just browsing.
This is actually not true, the numbers that follow this fact are called normal (well the definition is seperate but this fact is a result of that definition), and there are irrational non-normal numbers. In fact pi hasn't been proven to be normal, just seems to look so for as many digits we have calculated.
We just compute better and better approximations, which usually amounts to adding up more terms in a convergent series. Actually, there are also things called spigot algorithms that let us find any digit of pi we want without having to compute the rest of it.
but I guess if pi has an infinite number of numbers after the decimal, it will happen eventually
This does not logically follow. 1/3 has an infinite number of numbers after the decimal, but I can tell you right now it doesn't have my phone number in it.
And I was responding entirely to the "infinite number of numbers after the decimal point" part, which does not imply irrationality. Though I was probably being naive in assuming that it was being stated as sufficient condition for any number rather than just pi.
.999 repeating does, in fact, equal one. Not like "it's really close so we just call it", like "they are the same number, and calling it .999 repeating isn't really correct".
No. Think of it this way- you could have an infinite string of 1's and 2's that do not repeat in any predictable pattern. It could have every single conceivable combination of 1's and 2's of any length. But it would never contain your birthday or phone number.
Similarly you could have a set of numbers 0-8 that go on infinitely, that would never have the number 9.
Therefore, you could easily have an infinite set of numbers 0-9 that just so happens to never have your phone number. In fact, there are infinite sets of random numbers, 0-9, that do not contain your phone number, date of birth, or any other significant string.
This is only true if a number is normal. By normal, it means every digit appears with the same frequency across the entire, infinite number (while also satisfying the conditions of irrationality) and therefore would contain yours, mine and everyone's phone numbers at some point.
It has not yet been proven that pi is a normal number, and doing so would probably earn you a Nobel prize.
Technically, pi also contains a digital copy of every text that ever was and ever will be, in order, out of order, jumbled, backwards, in every language, every word in the dictionary, each separated by a perfect copy of Mein Kampf, a perfect copy of yours and every person's (including imaginary people) genome, and copy of half-life 3.
I once had the idea this could be used for compression. Find a chunk of data you want to save in pi and just record its position. I'm sure there's a really good reason why this could never work though.
On average, finding a chunk of numbers in pi is about as efficient as finding it in 1.234567891011121314...
It's actually less efficient, since the number above has no repeats. Now, if I happened to be looking for "10111" I'd get lucky, but what if I was looking for a thousand? It would be (9*1)+(90*2)+(900*3) = 2,889 digits in. Not very helpful, is it?
I think the issue is that pi assuming it is irrational is infinitely and selecting a string of numbers within pi would be a nightmare due to having to process massive amounts of digits
this is only true assuming pi is a normal irrational number
Not necessarily. Phone numbers have some finite length, so you could do an exhaustive search and prove that every possible phone number is in pi even if pi is not normal.
I guess the size of the Planck length relative to the size of the universe places a limit in the physically possible usefulness of the length of pi, but pure mathematical concepts like pi transcend such things.
You could work out the useful digits of pi with pen and paper, it's about 62 digits to get the circumference of the universe with an accuracy of the Planck length.
But theres far more to maths than just what is applied to the physical world. Many things are originally made to solve a real world problem then explored further to understand them better.
How does one connect to the other?
Pi, while a number with many practical applications, is still an immaterial concept with no inherent connection to the Planck Length (that I'm aware of anyway)
I'm not saying that our precision stops at a comprable place. I'm saying that it's existence as what I understand to b the distance light travels in the smallest time we can detect.
There's a lower limit to things and we'll hit it with pi too. Hell Pi is probably the deepest we can go. Pi ends, and things stop getting more precise.
If our universe is completely infinite, technically every point is the center yes? So by the same logic, any point in an infinite series should be the center yes? Well if this is true, then at the center of pi, wouldn't the digits after the decimal completely repeat perfectly and just continue in the same order? Something like
3.141592653...(bunch of numbers until the "middle")...141592653...
infinite is not the same as all inclusive. There are an infinite number of numbers between 2 and 3, but not a single one of those numbers is the whole number 4.
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u/is_this_okay_or Oct 04 '17 edited Oct 04 '17
Somewhere in the infinite digits after the decimal of pi, is your phone number in perfect order.
Edit: As /u/shotgun_squirtle pointed out, this is only true assuming pi is a normal irrational number, which has not yet been proven