killyourbuddha @ 2008-03-07T20: 53:00
The arrival of spring, moves in me mixed feelings. To save me from madness, fortunately, there is still the mental masturbation. The latest adventure concerns a Borges, a scholar of my greatest systems has enjoyed to criticize during the past week.
failing to desire to make things flowing, I copy-paste the important pieces of the debate, which took place by email and sms.
The topic is "The Library of Borges is finite or infinite?"
"With a finite, albeit large, of information, you can not describe an infinite number of concepts.
Taking the simplest example, many combinations can give infinite concepts, think of such binary alphabet: 1 and 0. No matter how many combinations you can make sure you can not express any number if you do not have endless places to put the figures. If you say it enough Tell me what is the limit ... There will be a finite number of places where your reasoning allows to express infinite numbers, and infinite words, infinite-machine codes, etc. ... "
" But then he thinks the Peano axioms. Five finite strings of information can generate a "cycle, the successor function" all natural numbers (and therefore there any countable set). It does not matter if you have not mentioned, the information you have created the same, in a way perhaps more elegant than with the mere "recitation" of the sequence of numbers. If desired, every piece of information can be made as close you want the continuous, though now I can cut her arms talking about qubits and the like. "
" Each element has a successor. So there is no limit. Try to define the library with something similar to the five Peano axioms. It does not hold, Peano does not limit the length of "figures" (wlog since we can speak of numbers, but the discussion can be extended with little effort), which instead Borges ago, I think even elegant ... "
To give you an example, take a circle and its radius. Detach from a corner radius that it is an irrational multiple of 2 \\ pi: for sure, sooner or later will touch all points of the compass, but does not exit. You have to go that route for an infinite time, just because an irrational number is aperiodic. Gambling: somewhere in the decimal expansion of \\ pi there is a string in binary, in easdecimale or any other alphabet contains letters of decoding the entire Library. And advancing space for as many libraries similar or different from countless others for a single letter, or different at every point, or resulting from disruption to the order of the letters ... "
"In my opinion nothing to do with you ... You are true examples of the continuous, but remember that the Library is, by definition, consists of a set of discrete elements (mind you, discreet even finite cardinality ). If anything to be infinite (and periodic) is the sequence of Libraries are the same, I guess I heavily heavily stoic but also elegant. I mean, I might try using Cantor's diagonal argument show that there is at least one book outside of what I thought were "all" books. But it does not work because you did not book-length infinite .. suppose you have listed all the books. You'll then have the opportunity to "godelizzare" all B (the Library) by assigning each book a certain natural number, no matter that the second rule. At that point, however, realize that numbers are finite. If I choose the number (ie the book, given that I built Fichissima bijection) which differs from the first book in the first letter, the second book in the second letter, the third in the third and so on, I get a book that is not in ... no, wait, we're sure! Just who is in all those books that have an index greater than K, where K = c * r * p (c are the characters, the lines r, p pages ... should be around 900,000 if I remember bad), books that have every right to exist and which, when assembled, I'll bet my copy of De Ludo Alea, the book contains different from all K. So no nonsense, you can actually enumerate all the books.
See, the problem I think is more logical and analytical, combinatorial or algebraic. You could give an interpretation "bibliophile" of Godel's theorem? And most importantly, how to define objectively a "concept"? As I told you the natural numbers are constructible by enumeration (brute method, but rich in information) or axiomatization (using information-poor but also concise and elegant) ... the two formulations are equivalent or not?
[Pause, weariness of the contestants]
"My criticism is not on the finiteness of the alphabet, but on the books ... If the definition of the concept is not well placed, then we are metaphysics and the value given the truth of words is entirely arbitrary. But if we in the realm of spsotiamo concretoe associate with the term "concept" to mean "information" then gives me all right. Except, perhaps, but I've just thought, if the finiteness of the universe imply that the information and concepts can be expressed. "
" Exactly what is the thesis of Borges. Finiteness of the concepts back to the eternal condemnation = = sadness because we are players in a game rules unknown to warp minds inconceivable to us that are scrambling to find the Book and the keeper but his prey of their mortality. But go ahead, I'm interested. "
" The question now becomes physical. The universe seen expanding at the speed of light, then a nap is finished considerrarlo hazardous. In addition, space is not quantized. If it were it might spin, but ONLY say that as long as the library itself is not a new concept of its own. "
" On the quantization of space I do not comment. But to say that the universe expands admit that is an implicit "the universe is finite, giving the word universe the value of" being organized and responsive to laws of general relativity. " We are organized in a magma bubbles chaotic, is a very elegant way to see the universe, to think in terms of random foam. Instead it is the last interesting concept. There is a book about the Library and describes it? There is a book about the book that describes the library? Certainly yes, but I do not think that this alone is enough to make it down the finiteness. The problem is the end: generalize. If A is an alphabet made from k symbols, the possible information strings of infinite length are obviously infinite (countable). But if we want only the combinations p symbols long, the question is more intricate! Will be, like, combinations with repetition of symbols taken k groups of p strings ... But one aspect is that Borges does not elaborate on the interdependence between the symbols: of course there are books that refer to other books, but what about books about topics that can not drain completely in a string of p symbols? They are obviously allowed paraphrases, summaries or notations that shortening the exposure, but to what extent? Is there a "quantum of information" so small or so primitive that it can not be attributed to any other entity? There is a book about the subject in x p characters, its existence implies that of a book about in x p -1 characters and replaces the w ith a letter to the case (indeed, there are 21, one for each letter as possible). The same reasoning leads to understand how there can be a book about x in p -2 characters, and 21 * 21 books that occupy the last two posts for more. And so on, but I wonder, can legally reverse the induction far as to say that there is a book about of x in A character and uses p -1 characters to say more? The next step would be an absurd dialectic, a book which exhausts a subject in 0 characters! Not to mention the fact that if this were possible, every book is about any subject, would be enough to build some kind of sequence (and you know this is always possible) that starts with a good book and get the book arbitrary j -th, and justifies the fact that a book that consists only of a series of methodical mjhkdl speak, in our view, the face of God. is loses any kind of logic convention, everything about everything and nothing. Surprising. "
" I think I've reached a topic that exhausts the question. The library expands, because it must include the idea of a book in its entirety without that book. It must contain the following in turn the concept of wholeness of the library before him. And so on, ad infinitum, the books are endless, as each book must be different from others and that for every book that speaks of the totality of s books there is one who speaks for all of s +1 books. "
here ... did not know what to say and I retired to think, but the question remains. The question is really low? What is the final hypothesis in the last note of the story, that of the "silky handbook" endless pages of infinitely thin? applications are based on the finiteness? There is a unique way to define "concepts" and "information" that could help the discussion? What is also of ; supposed periodic structure of the library?
willing to turn the question, the friend who supported me has been "spent" enough but I still want to deepen.
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